Shaping and Control with Nonlinear Optics

does not discriminate between polar and axial order. The only way to get such a discrimination is by using NLO techniques, sensitive to the polar order. The build up of the order parameter with poling time t may be well described by a monoexponential function (cf. Refs. 42, 43) (25) where ∞ is the saturation value and the time constant of the orientation process τ depends on the poling temperature and poled polymer itself (viscosity of polymer, mobility of chromophores, etc).

Fig. 12. Build up of the order parameter

for a side chain liquid crystal polymer at different poling temperatures:
- T = T g + 5°C, - T = T g,  - T = T g- 5°C,  - T = T g- 10°C,  - T = T g - 15°C (after Ref. 43).

In order to link the order parameter to the poling field it is important to know the orientation distribution function G(θ ) giving the probability of finding the molecular axis in the direction defined by angle θ (for poled polymers the azimuthal symmetry is obvious). The average value of the function f(θ ) is given then by the following expression (26) where N A is the normalization constant 112

(27) At higher temperatures (above or slightly below the room temperature), when molecules may rotate (free gas model) one uses for G( θ ) the Gibbs - Boltzmann distribution function given by

(28) where E p is the poling field. For dipolar molecules the ordering energy U is given by (29) where µ is the dipole moment of molecule. First expression on RHS of Eq. (29) describes the dipolar interaction energy with poling field and the second one that due to the induced dipole moment by the applied field. Neglecting the induced dipole moment interaction energy with the poling field (isotropic model) one can develop the orientation distribution function into the Legendre polynomials obtaining (30) where the expansion coefficients can be expressed in terms of the modified spherical Bessel 34 functions (31) with the parameter (32) expressing the ratio of the dipolar interaction ordering energy to the thermal randomization energy. The following recurrent relations obey for the modified spherical Bessel functions (33) where

(34) Similarly for Legendre polynomials (35)

113

with (36) and (37) (38)

(39) and (40) Introducing the development (Eq. (30) into Eqs. (15)-(16)) one can calculate the induced birefringence by the poling field which is depicted in Fig. 13. It is clearly seen that the poling process, with poling field directed perpendicular to the thin film surface, increases the extraordinary index of refraction and decreases the ordinary one. The variation depends on the x parameter. The second observation seen in the variation of the linear absorption spectrum (cf. Fig. 9)) is the shift of the maximum absorption wavelength towards larger (red shift) or smaller (blue shift) wavelengths (see. Fig. 9 and Refs. 31, 32). This shift is due to the DC Stark effect. The polar orientation induced by the poling field leads to establishment of a high DC field experienced by molecules. The corresponding shift of the maximum absorption wavelength may be related to the difference ∆µ = µ 11 – µ 00 between molecule 32 dipole moments in excited (1) and fundamental (0) states (41) where is the Planck constant divided by 2π and c is the light velocity. Thus depending on the sign of ∆µ one observes a red or blue shift in the optical absorption spectrum. POLAR ORDER As already mentioned, the measure of the variation of the linear absorption spectrum gives information on axial order, although, in some cases, especially in DC poling, this is confounded with the polar order. One can get the true information on polar order by using NLO techniques, such as SHG or linear electro-optic effect, sensitive to the noncentrosymmetry. For poled polymers with point symmetry ∞mm, functionalized with 1D charge transfer active molecules, characterized by an enhanced first hyperpolarizability β zzz in the CT direction z and provided that the Kleinman conditions are satisfied (usually this is done far from the absorption band) there exist two non zero χ (2) (- ω3 ;ω1 , ω 2 ) tensor components (cf. Eq. (7)) (42) 114

and

Fig. 13. Calculated variation of the induced birefringence of optical dielectric constant with x = µE p / k T

(43) Similarly as before the capital letters (X,Y,Z) refer to the laboratory reference frame whereas lower case to the molecular system.

Fig. 14. Calculated variation of the diagonal (χ µE p /kT.

(2)

zzz) and off diagonal (χ (2) zzz) tensor components with x =

115

Again by using the orientation distribution function (cf. Eq. (30)) one can express the corresponding tensor components in terms of the averages . As consequence one obtains (44) and

(45) Both diagonal and off diagonal χ(2) tensor components increase at small poling field E p (cf. Fig. 14). At higher values of the ratio x (cf. Eq. (32) the diagonal tensor components χ(2) zzz is still increasing while the off diagonal χ(2) xxz starts to decrease. The efficiency of the polar ordering is given by the ratio (46) which varies between 1 and ∞ . The last value is reached for perfectly ordered structures (all dipole moments pointing in the same direction). The parameter is a = 3 for a free electron gas (isotropic model). For side chain liquid crystalline polymers a values as high as 18 have been obtained 35. Case of liquid crystals The formalism described above for polar order applies to polymers with no initial order. However, in some cases, like in liquid crystals (LC’s) and side chain liquid crystalline polymers (SCLCP’s) such initial axial order is present. In that case the model proposed by Singer, Kuzyk and Sohn (SKS model)36 or by Vand der Vorst and Picken37-38 , the last based on Maier - Saupe theory 39,40 applies (MSVP model). Orientation mechanism Singer, Kuzyk and Sohn (SKS) model36 . In order to take into account the preexisting axial order such as found in LC’s and described by the order parameters and 36

Singer et al neglected the interaction energy with induced dipolar moment and added another term to the molecular energy in Eq. (29) describing the tendency of rod-like molecules to have their long axes more or less parallel to a preferred direction. They did not introduce any explicit analytical expression for this term. Instead, they assumed that the axial order of the LC « host » is transferred to the NLO « guest » molecules. In this model, the dipolar energy is first expanded into Taylor series up to first order in the poling field Ep . Then, the Gibbs-Boltzmann distribution is expanded in terms of Legendre polynomials (cf. Eq. (30)) with coefficients given by Eq. (31). In this case, the configurational average values of and <sin² θ cos θ>/2 (cf. Ref. 35) are linear functions of the poling field strength through the term x (cf. Eq. 32) and depend on the intrinsic axial order of the LC through the microscopic order parameters and

.

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MSVP model 37,38. In the SKS model, the transfer of axial order from the mesogenic rods to the NLO molecules assumes implicitly the existence of latent mesogenic properties in the NLO molecules. In the MSVP model, the intrinsic mesogenic properties of the NLO molecules themselves are taken explicitly into account. The natural tendency of rod-like molecules to align mutually parallel can be described by the effective single particle energy U0 ( Θ ), introduced originally by Maier and Saupe 39-40 (47) where ξ is a parameter describing the strength of the anisotropic interactions. It takes a constant value for a given liquid crystal. In the Maier - Saupe theory this parameter is proportional to the clearing temperature Tc (at zero field strength) through the following relation T c) (48) The order parameter has to be determined selfconsistently by using numerical methods. This value, injected into Eq. (29) (with neglect of U2 ( θ), cf. Eqs. (47) and (29)) allows to determine the orientational averages and <sin² θ cos θ>/2. Van der Vorst and Picken37-38,41 extended this model to the case of poling of single rod systems. In addition to U0( θ) and U 1(θ) they introduced the dipolar moment interaction energy term, induced by poling field, similar to that in Eq. (29) (49) where ∆α is the anisotropy of molecular linear polarizability of the rod-like NLO molecules. The resulting orientational averages and <sin² θ cos θ>/2 are given in Table 2. Table 2. rientational averages of and <sin² θ cos θ>/2 in different statistical orientational models (after Ref. 35, for details see text).

a) u = µ0E p/kT b) The order parameters

and

are assumed to be independent of the electric field strength and are to be determined before the poling field has been applied. c) The order parameter

has to be determined selfconsistently by using numerical methods. This value, injected into Eq. (47) allows to determine the orientational averages and <sin² θ cosθ>/2.

Indeed, the SHG experiments performed on a series of isotropic and side chain liquid crystalline polymers with different axial order parameters show a correlation between the second order NLO susceptibility χ(2) as well as the a parameter (cf. Eq. (46)) on one side and the order parameter on the second, respectively. As the degree of orientation depends on the product µE, the DC poling technique limits the kind of molecules to which it can be applied. As already mentioned it applies to

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dipolar molecules only, which always show a tendency to aggregate, thus increasing the propagation losses. This effect limits the amount of active molecules in polymer matrix, which on other side, is so important for macroscopic nonlinear optical response (see Eqs. (42)-(43)). Moreover heating to glass transition temperature may lead to unwanted, thermally induced chemical reactions. For practical applications thermostable polymers are required, with high glass transition temperature (200-250°C) ensuring a better temporal stability of the induced polar order. In situ second harmonic generation measurements The induced polar order has been studied by in situ SHG measurements. Figure 15 36-37 shows a typical temporal growth of SHG signal for an isotropic polymer . This growth can be described by a triexponential function

(50) where l is the thin film thickness, τ ‘s are the time constants and P’s are the maximum contributions of the different mechanisms taking part in molecular orientation process, respectively. The sum of the poling saturation limits: P = P 1 + P 2 + P 3 represents the overall poling efficiency under the experimental conditions (temperature, geometry, poling field, atmosphere). The superscripts p and r in Eq. (24) refer to polymer thin film and reference harmonic intensities, respectively. Both the time constants and the overall poling efficiency

Fig. 15. Temporal growth of the SHG signal during poling. Points depict experimental data whereas dashed, broken and solid lines show least square fit with a mono - , two- and three - exponential quations, respectively (after Ref. 35).

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PHOTOASSISTED POLING Dumont and coworkers 15,16,44,45 have observed that shining doped (or functionalized) polymer thin films, with non-centrosymmetric dipolar chromophores, in the chromophore absorption band induces a significant increase of electro-optic coefficient, corresponding to a better , polar orientation of chromophores. The measurements have been done using the attenuated total reflection technique, and the optical field polarization was perpendicular to the applied low frequency external electric field to the thin film (cf. Fig. 16). A better stability of induced orientation was observed in the case of functionalized polymers than in guest - host system, as it is usually the case with the static field poled polymers. The chromophores orient with dipolar moments perpendicular to the optical field (and parallel to the applied static (or low frequency) field. As it will be discussed later, the chromophore orientation is going through trans - cis izomerization process (cf. Fig. 17).

Fig. 16. Attenuated total reflection set - up (a) and temporal growth of the electro-optic coefficient as function of laser illumination at 514.5 nm: (b) - experiment, (c) - theory (courtesy of M. Dumont).

Figure 16 shows the experimental set-up used by Dumont and coworkers together with the observed and calculated dependence of the electro-optic coefficient on the external 119

light illumination (green Argon laser line). The studied, poled film is placed between two electrodes, one of them being a silver electrode deposited on the side of a prism. The condition to excite surface polaritons in thin film depends on its refractive index and thickness as well as on the refractive indices of electrode material and prism. By varying incidence angle it is possible to satisfy this condition, what is observed as a dip in the intensity of reflected beam. By varying refractive index of thin film with applied voltage one changes the resonance conditions (or in other words the coupling angle). The technique is very sensitive to thin film thickness and refractive index variation. As the variation of refractive index depends on thin film electro-optic coefficient the technique serves to determine it with a high precision. The variation of refractive indices is measured with a lock-in amplifier. By shining the studied film with HeNe laser emitting at 632.8 nm or with an Argon laser at 514.5 nm parallel to the poling field (optical electric field perpendicular to it) Dumont et al observed a noticeable increase of the measured electro-optic coefficient (cf. Fig. 16 ). The effect was significantly larger in the case of grafted polymer than with a guest - host system. Switching off the light source leads to a decrease of electro-optic coefficient, which is connected with the decrease of the polar order. The relaxation of polar order is faster in guest - host system than in side chain polymer, as it is usually observed.

Fig. 17. Light induced reorientation of DR #1 molecule in the presence of the static field and photoizomerization process. From almost parallel orientation to the exciting optical field the molecule dipole moments reorient to an almost perpendicular direction.

A possible explanation of this effect is shown schematically in Fig. 17. The active molecule, which is disperse red #1 (DR #1) (cf. Fig. 18) undergoes the trans - cis izomerization (cf. Fig. 19) under light illumination. The double N=N bond is flexible at excited state and through rotation or translation, molecule may change configuration from 46, 47 and is significanly slower in solid trans to cis form. This process is very fast in liquids 48 state (Kajzar et a1 estimated it for a grafted polymer as taking about 150 ns). The inverse transformation (although sometimes the process is irreversible) from cis to truns form may go only through nonradiative channels and is very slow (of the order of few seconds in solids). Molecules can return to the previous configuration, but will be again excited by incoming light. Thus the only stable position will be obtained if molecule orients with the dipole moment perpendicular to the incident light polarization, thus parallel to the applied electric field. As consequence, the light induced molecular re-orientation will lead to an increase of the electro-optic coefficient as it was observed experimentally15,16,44-45 ( s e e a l s o Fig. 16). Using a simple rate equation for the trans - cis izomerization process Dumont et al 45 describe well the observed temporal behavior of electro-optic coefficient during attenuated total reflection measurements (cf. Fig. 16). 120

Fig. 18. Chemical structure of side-chain copolymer of DR1 in PMMA with 35% molar content and its optical absorption spectrum.

Fig. 19. Light (or heat) induced photoizomerization process (a) and electron transitions between fundamental and excited singlet states diagram (b) with ωC >ω T . The trans to cis transition time is much faster than the reverse one and depends on molecular environment (usually much faster in liquids than in solids). In cis form, which often is unstable, the volume of the molecule is smaller, thus easing its rotation.

ALL OPTICAL POLING The first observation of a polar orientation of chromophores in a grafted PMMA DR1 polymer has been done by Charra et al 50 in a four wave mixing geometry with two picosecond pump beams at 1.064 µm and a probe beam at doubled frequency (cf. Fig. 20). The observed signal at 0.532 µm rose slowly with time, up to a saturation value. A spontaneous SHG was observed after switching off the probe beam, with a fast relaxation 121

component at the beginning 48 . The maximum second order nonlinear optical susceptibility 50. value obtained in these experiments was of 3 pm/V This experiment has shown that using purely optical fields one can obtain a polar orientation of chromophores in a functionalized or a doped polymer film.

Fig. 20. Four wave mixing geometry leading to the observation of all optical poling in a side chain polymer PMMA - DR1. Pump beams I 1 and I 2 are at 1.064 µm while probe (I 3 ) and signal (I 4 ) beams at 0.532 µm (after Ref. 50).

Fig. 21: Experimental set-up for collinear seeding geometry. A fast photodiode synchronizes the sampler. SHG intensity is measured with the photo-multiplier tube (PMT). P: polarizers; F: interferential filter at 532 nm; SFS : spatial filtering system; R: dielectric mirror for fundamental rejection; S: shutter synchronised with insertion of the green blocking RG670 Schott filter. BK7 glass plate is fixed on a rotating stage for phase adjustment between ω and 2ω beams.

Significantly larger χ(2) value was obtained in seeding geometry using two collinear picosecond beams at 1.064 µm and 0.532 µm. The experimental set-up used for optical preparation of polymers is shown in Fig. 21. It consists in a pulsed picosecond Nd:YAG laser delivering both fundamental (1064 nm) and harmonic (532 nm) wavelengths at 10 Hz repetition rate. Energies are 500 µJ and 2.5 µJ, respectively at 1064 nm and 532 nm, and beam diameter is 2 mm at sample location. With the same polymer as that used in the four wave mixing geometry the best obtained value for susceptibility was of 76 pm/V, close 51-52 . This geometry has been already used for optical to that obtained in corona poling poling of glass fibers, where also an efficient SHG has been obtained after a seeding procedure 53-54 .

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Fig 22. Schematic view of the orientation mechanism in all optical poling with azo-dye molecules (a) and electron transition diagram between fundamental and excited singulet states (b). In the case of azo dyes, the trans -cis izomerization is achieved through both one and 2 photon excitations.

Poling field <E3>t ≠ 0 The mechanism of creation of χ(2) grating in glass fibers has been described by Baranova and Zeldovich 5 5 in terms of a polychromatic interference of input fields at ω and 2 ω frequencies, leading to a non-zero temporal average field (51) where ∆φ is the relative phase difference between E ω and E 2 ω fields. This interference leads to a non-zero poling field, as it is shown schematically in Fig. 23. However the microscopic mechanism in polymers is different than that in glass fibers, where color centers and defects are at origin of the created polarization. Similarly as in photoassisted poling and in the case of azo dyes, the polar orientation of dipole moments can be explained by the trans-cis izomerization. Contrariwise to the photoassisted poling, where this izomerisation is induced by a one photon transitions (cf. Fig. 17) in all optical poling this is excited simultaneously by 2 photons with frequency ω and one photon with frequency 2 ω (cf. Fig. 22). As already mentioned, the used azo dyes absorb strongly light if the exciting optical field is parallel to the dipole transition moment. The double N=N bond in excited state is mobile and molecule changes conformation to cis-form, with a smaller volume. Subsequently the molecule relaxes slowly to the trans form through the non-radiative channels. In photoassisted poling a stable orientation of chromophore is achieved when the dipole transition moment is perpendicular to the optical, exciting field. A given polar orientation is imposed by the applied DC field. In all optical poling, due the existence of a non-zero temporal average “poling” field (cf. Eq. (51)) a stable chromophore orientation is obtained when the dipole moment change is directed oppositely to the exciting field). There is a fundamental difference between photoassisted poling and all optical poling in the obtained chromophore orientation. In the first case chromophores orient perpendicularly to the optical field, whereas in the 3 second - parallel (to the resulting E poling field). Dependence on phase mismatch between ω an 2ω beams As it is seen from Eq. (51) the sign of the temporal average of cubic interference is determined by the relative phase between ω and 2ω beams. As in condensed materials, fundamental and harmonic frequency propagation velocities are different, the optical polarity alternates between positive and negative extremes. This can be easily verified by changing the phase between ω and 2 ω fields. It has been done by introducing a BK7 slab in optical path of fundamental and harmonic beams (cf. Fig.21) and rotating it along an axis 123

perpendicular to the beam propagation direction. Because of BK7 dispersion of refractive index the relative phase between ω and 2 ω fields is varied continuously. Figure 24 shows an experimental verification of this interference. A good agreement is observed between the observed and calculated dependence of SHG signal on phase mismatch between ω and 2 ω fields.

Fig. 23. Polar cubic interference between optical electric fields at fundamental and harmonic frequencies. Double arrows are the excited molecules.

Fig. 24. Experimental dependence of the SHG intensity induced after 20-minutes preparation-time, on the relative phase ∆Φ of the ω and 2ω beams. Solid line corresponds to a theoretical dependence with ∆ n = n(2ω ) – n (ω ) = 0.3. Sample was 0.1 µm thick with 0.3-optical density at 532 nm.

124

Fig. 25 : Experimental real-time growth and decay dynamics of typical second harmonic signal amplitudes ( α |χ (2) |) in a side chain polymer DR #1 - PMMA with 35% molar dye content seeded at 1064+532 nm. (2) Negative times correspond to the seeding preparation process, positive times correspond to χ decay. Insert gives the different seeding-beam polarizations checked (after 62).

Growth of polar order. The growth and decay kinetics of the induced polar order was studied by Fiorini et al for a guest host system and side chain polymer using the apparatus shown in Fig. 21. The seeding procedure was alternated with reading procedure by using a mobile Schott filter RG670-glass. The filter was alternatively placed or removed from the input beams path. When it was placed in, the green beam was blocked and fundamental ω beam generatedω2 beam from earlier created noncentrosymmetry. In the other case seeding procedure took place. The filter removal (or placing) was controlled by computer and consequently the lecture of induced SHG signal (corresponding to the moments when filter was in place). The temporal growth of SHG intensity (and consequently of the induced non-centrosymmetry) is shown in Fig. 25 for a grafted polymer PMMA-DR #1 with 35 mol% of chromophore content. The d 33 (-2 ω;ω,ω) susceptibility reaches a value of 45 pm/V after 2 hours of preparation. 52 It is as high as what is currently obtained using corona electric-field 58, 59 poling . After seeding-type preparation, the decay dynamics of the induced d 33 is the same as with the same polymer prepared using corona poling. This corresponds to an infrared electrooptic coefficient r 33 close to 10 pm/V in the plane of the film.60 Spatial profile of the induced polar order. The spatial profile of the induced second-harmonic efficiency has been studied by using a small diameter probe beam at fundamental frequency and scanning the poled area by translating thin film. The result is shown in Fig. 26. Unlike what is currently observed in seeded bulk glasses, 53-54 the spatial profile along the polymer-film plane is uniform. This confirms that optical poling of polymers results in a local effect. Our preparation technique thus permits patterning of micro structures using the laser spot. In this respect, an interesting question concerns the effect of the accumulated spacecharge on the nonlinear optical coefficient. Space-charge fields are indeed known to dominate SHG in seeded optical glass-fibers 53-57 . Considering that we get the same nonlinearity as using corona-poling, we can expect internal fields as large as 1 MV/cm in the 125

optically poled region. We thus prepared a polymer film using various angles of incidence. It results that SHG signal is independent on the angle of incidence, proving that space-charge field effects are inefficient with this polymer.

Fig. 26. Spatial profile of the second harmonic generation intensity along the film plane.

Symmetry of the induced χ(2) susceptibility. The symmetry of the induced χ (2) susceptibility was studied on a seeded film by SHG measurements placing it between two polarizers, as shown in Fig. 27. The SHG intensity was collected when rotating the analyzer. The result, displayed in Fig. 28 shows that the induced χ(2) susceptibility has similar symmetry to that observed in corona poled polymers with where X is the poling field direction (polarization direction of ω and 2 ω beams) as for moderately poled polymers (isotropic model). The main difference consists in the direction of the induced net polarization. It is perpendicular to the thin film plane in corona poling and in plane (parallel to the electric field

Fig. 27. Schematic representation of experimental arrangement for the study of the symmetry of induced χ(2) susceptibilty.

126

polarization of collinear ω and 2 ω beams).

Fig. 28: Polarisation dependence of seeded SHG coefficient. Angle θ is measured with respect to polarisation axis around the beam-propagation direction.

Seeding power dependence. The optically induced χ (2) is proportional to

Fig. 29. SHG-amplitude growth at different seed intensity-ratios (I 2 ω /I ω ×2000).

An optimized poling efficiency also requires a good interference between one and twophoton absorption effects in order to maximize the polar E3 -term. It requires equalization of . Obviously, χ (2) -initial

one and two-photon absorptions; that is growth-rate is proportional to E

3

which is also I ω

× I

1/2 (2) 2 ω .χ

saturates as 127

when poling becomes as efficient as loss of polar orientation resulting from axial excitation processes. Such behavior is observed in Figure 29 in which I 2 ω / I ω -ratio was varied while keeping sample and phase-∆Φ constant. It is clear that optimization of optical-preparation conditions can lead to orders of magnitude gain on poling efficiency. Of course, optical-poling conditions must be optimized as concerns stationary parameters such as phase and relative intensities of the beams,61 but also as concerns dynamic parameters such as excitation and orientation diffusion rates.62 RELAXATION One of the important parameters determining practical applicability of poled polymers is the stability of the induced polar orientation. This is studied usually through the temporal behavior of the χ (2) susceptibility or of the electro-optic coefficient r at elevated temperatures. Relaxation studies have been almost done for static field poled polymers. The temporal decay with t of the χ (2) susceptibility is usually described by the Kohlrausch Williams-Watt (KWW) stretched exponential function (52) where τ is the relaxation time constant, depending on temperature and β (0< β <1) describes the width of relaxation (departure from a monoexponential behavior).63 The relaxation of poled polymers (isotropic or liquid crystalline may be also well described by a biexponential function

(53)

Fig. 30. Temporal decay of SHG intensity for a polyacrylate functionalized with cyanobiphenyl chromophore. Dash-dotted and solid lines have been computed using one and biexponential functions, respectively (after Ref. 35).

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where 1 is the thin film thickness, is the second harmonic intensity of the studied thin film (p) and of reference r, respectively. The constant C in Eq. 53 characterizing the residual orientation at the experimental time scale which is important for practical applications, R’s are relaxation rates and τ 1 and τ 2 are relaxation times of different processes contributing to the molecular disorientation, respectively. All these parameters depend on the measurement temperature. Closer the glass transition temperature is to the measurement temperature, smaller are time constants τ and larger are relaxation rates. This behavior is true for both isotropic and liquid crystalline polymers, as it was observed by Dantas de Morais et al.28 . Fig. 30 shows an exemple of such a biexponential fit of χ(2) relaxation curve. CONCLUSIONS As already mentioned, poled polymers have known a very important development in last few years, not only from the point of view of chemical synthesis and characterization but also from device applications. They exhibit the most important progress, concerning the obtained values of second order NLO susceptibilities, thermal and orientational stability as well as light propagation properties. High rate, 100 Ghz signal modulation has been demonstrated with close commercialization of electro-optic modulators64 . Stacked electrooptic modulators have been also fabricated for parallel signal processing. 65 , which cannot be anticipated with single crystals like the standard material LiNbO3 . A successful integration of poled polymer based electro-optic modulators with semiconductor technology has been also demonstrated. 66 . One can reasonably expect their forthcoming, commercial application in high rate electro-optic modulation.. Among the herein presented oriented poled polymer film preparation methods a better adapted poling efficiency is expected with photo-assisted and all optical poling. In particular, the all optical poling technique represents several important advantages, with respect to the classical static field poling method. Among them, we quote : (i) the absence of electrodes and consequently no charge injection, which usually leads to dielectric breakdown (ii) a higher damage threshold observed with optical fields E, allowing consequently higher poling efficiencies, with no “ point effect ”, observed sometimes with electrodes (iii) automatic phase matching, assured by the seeding procedure and useful for the frequency conversion (iv) possibility of auto-regeneration, as frequency conversion is done simultaneously with seeding. It allows an automatic χ (2) grating adjustment for harmonic conversion to e.g. refractive index variation due to e.g. heating (v) poling can be done at any temperature, especially at room (“ cold poling ”) or at any operation temperature (vi) possibility of poling other than dipolar molecules as it was shown with octupolar molecules. 67 Although the poling mechanism is still not as clear as with azo dyes, it is the only technique which allows to orient such interesting molecules. ACKNOWLEDGMENTS This paper describes results of numerous collaborations and discussions. The authors would like to thank warmly all people who contributed to it, and in particular Fabrice Charra, Pierre-Alain Chollet, Michel Dumont, Céline Fiorini, Maryanne Large, Claudine Noël and Paul Raimond.

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REFERENCES 1. J. R. Heflin, Y. M. Cai and A. F. Garito, J. Opt. Soc. Am. B, 8, 132( 1991). 2. A. Willets, J. E. Rice, D. M. Burland and D. P. Shelton, J. Chem. Phys., 97, 7590(1992). 3. F. Kajzar, Electric Field Induced Second Harmonic Generation, in « Characterization Technique »s, R. A. Lessard and E. Franke Eds., Proceed. SPIE, vol CR69, Bellingham 1997, p. 480. 4. J. Zyss and J. L. Oudar, Phys. Rev. A, 26, 2028(1982). 5. F. Kajzar, K. Horn, A. Nahata and J. T. Yardley, Nonl. Optics, 8, 205( 1994). 6.2. E. L. Havinga and P. Van Pelt, Ber. Bunsenges. Phys. Chem., 83, 816(1979). 7. S. Myata, N. Yoshikawa, S. Tasaka and M. Ko, Polymer J., 12, 857(1980). 8. O. A. Aktsipetrov, N. N. Akhmediev, N. N. Mishina and V. R. Novak, Soviet Phys. - JE Lett., 37, 207( 1983). 9. C. Bosshard, Oriented Molecular Systems, in "Organic Thin Films for Waveguiding Nonlinear Optics: Science and Technology ", F. Kajzar and J. Swalen Eds., Gordon & Breach Sc. Publ;,p. 163. 10. Poled Polymers and Their Application to SHG and EO Devices, S. Myata and H. Sasabe Eds., Gordon & Breach Sc.Publ, Amsterdam 1997. 11. S. Bauer, Appl. Phys. Rev., 80, 5531(1996). 12. J. Le Moigne, in « Organic Thin Films for Waveguiding Nonlinear Optics », F. Kajzar and J. Swalen Eds., Gordon & Breach Sc. Publ; p. 289. 13. P. A. Chollet, F. Kajzar and J.Le Moigne, Mol. Eng. 1,5(1991). 14. F. Kajzar, Y. Okada - Shudo, C. Meritt and Z. Kafafi, Fullerenes and Photonics IV, Proceed. SPIE, vol. 3429, in print. 15.Z. Sekkat and M. Dumont, Nonl. Optics, 2, 359( 1992). 16. M. Dumont, G. Froc and S. Hosotte, Nonl. Optics, 9, 327(1995). 17. F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond and E. Idiart, Opt. Lett., 18, 941( 1993). 18. J. M. Nunzi, C. Fiorini, F. Charra, F. Kajzar and P. Raimond, All-Optical Poling of Polymers for Phase-Matched Frequency Doubling, in « Polymers for Second-Order Nonlinear Optics », G. A. Lindsay and K. D. Singer Eds, ACS Symposium Series, vol. 601, p. 240, ACS, Washington 1995. 19. M. Eich, B. Reck, C. G. Wilson, D. Y. Yoon and G. C. Bjorklund, J. Appl. Phys., 66, 2559( 1989). 20. B.K. Mandal, J. Kumar, J. C. Huang and S. K. Tripathy, Makromol. Rapid Commun., 12, 63(1989). 21. B. Comizzoli, J. Electrochem. Soc. : Solid Science and Technology, 134 : 424( 1987). 22. K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, B. Comizzoli,, H. E. Katz and M. L. Schilling, Appl. Phys. Lett., 53, 1800( 1988). 23. M. Date, T. Furukawa, T. Yamaguchi, A. Kojima and I. Shibata, IEEE Trans. Electr. Electr. Insul., 24, 537( 1989). 24. S. Yilmaz, S. Bauer and R. Gerhard-Multhaupt, Appl. Phys. Lett., 64, 2770( 1994) 25. B. Gross, R. Gerhard-Multhaupt, A. Berraisoul and G. M. Sessler, J. Appl. Phys., 62, 1429( 1987). 26. M. Houe and P. D. Townsend, J. Phys. D. 28, 1747(1995). 27.G. M. Yang, S. Bauer-Gorgonea, G. M. Sessler, S. Bauer, W. Ren, W. Wirges and R. Gerhard-Multhaupt, J. Appl. Phys., 64, 22( 1994). 28. D. Gonin, PhD Thesis, University Paris 6, September 1995. 29. G. Gadret, F. Kajzar and P. Raimond, Nonlinear optical properties of poled polymers, in "Nonlinear Optical Properties of Organic Materials IV", K. D. Singer Ed., Proc. SPIE, vol. 1560, pp. 226-237, Bellingham 1991. 30. R. A. Norwood and G. Khanarian, Electron. Lett., 26 , 2105(1990).

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31. M. A. Mortazavi, A. Knoesen, S. T. Kowel, B. G. Higgins and A. Dienes, J. Opt . Soc . Am. B, 6 , 733(1989). 32. R. H . Page, M. C. Jurich, B. Reck, A. Sen, R. J. Twieg, J. D. Swalen, G. C. Bjorklund and C. G. Wilson, J. Opt. S o c. Am . B , 7: 1239(1990. 33. D. Gonin, B. Guichard, C. Noël and F. Kajzar, Macromol. Symp ., 96, 185(1995). 34. J. W. Wu, J. Opt. Soc. Am. B, 8, 142(1991). 35. C. Noël and F. Kajzar, Proceed. of 4th International Conference on Frontiers of Polymers and Advanced Materials, P. Prasad, Ed., Plenum, in press. 36. K. D. Singer, M. G. Kuzyk and J. E. Sohn, J. Opt. Soc. Am. B, 4, 968(1987). 37. C. P. J. M. Van der Vorst and S. J. Picken, Proc. SPIE, 866, 99(1987). 38. C. P. J. M. Van der Vorst and S. J. Picken, J. Opt. Soc. Am. B, 7, 320(1990). 39. W. Maier and A. Saupe, Z . Naturforsch ., 14a, 882( 1959). 40. W. Maier and A. Saupe, Z. Naturforsch., 15a, 287( 1960). 41. G. R. Möhlmann and C. P. J. M. Van der Vorst, « Side Chain Liquid Crystal Polymers as Optically Nonlinear Media », in « Side Chain Liquid Crystal Polymers », C. B. McArdle Ed., Blackie, Glasgow 1989, pp. 330-356. 42. T. Dantas de Morais, C. Noël and F. Kajzar, Nonl. Optics , 15, 315(1996). 43. D. Gonin, B. Guichard, M. Large, T. Dantas de Morais, C. Noël and F. Kajzar, Journal of Nonlinear Optical Physics and Materials, 5 , 735(1996). 44. M. Dumont, Z. Sekkat, R. Loucif-Saibi, K. Nakatani and J. Delaire, Nonl . O p t., 5 , 229(1993). 45. M. Dumont, Mol. Cryst . Liq. Cryst ., 282, 437( 1996). 46. J. Wachtveil, T. Nägele, B. Puell, W. Zinnh, M. Krüger, S. Rudolph-Böhner, D. Osterhelt and L. Moroder, J. of Photochemistry and Photobiology A : Chemistry, 105, 283(1997). 47. T. Nägele, R. Hoche, W. Zinnh and J. Wachtveil, Chem. Phys. Lett., 272, 489(1997). 48. F. Kajzar, F. Charra, J. M. Nunzi, P. Raimond, E. Idiart and M. Zagorska, i n Proceedings of International Conference on Frontier of Polymers and Advance d Materials, Jakarta, January 1993, P. N. Prasad Ed., Plenum 1995. 49. R. Loucif-Saibi, C. Dhenaut, K. Nakatani, J. Delaire, Z. Sekkat and M. Dumont, Mol. Cryst. Liq. Cryst., 235, 251(1993). 50. F. Charra, F. Kajzar, J. M. Nunzi, P. Raimond and E. Idiart, Opt. Lett., 18, 941(1993). 51. C. Fiorini F. Charra, J. M. Nunzi and P. Raimond, J. Opt. Soc. Am . B, 11, 2347(1994). 52. J. M. Nunzi, C. Fiorini, F. Charra, F. Kajzar and P. Raimond, All-Optical Poling of Polymers for Phase-Matched Frequency Doubling, in Polymers for Second-Order Nonlinear Optics, G. A. Lindsay and K. D. Singer Eds, ACS Symposium Series, vol. 601, p. 240, ACS, Washington 1995. 53. U. Osterberg and W. Margulis, W.; Opt. Lett., 11 , 516(1986). 54. R. H.; Stolen and H.W.K Tom,; Opt. Lett., 12, 585(1987). 55. N. B. Baranova and B. Ya. Zeldovich, JETP Lett., 45, 717(1987). 56. F. Ouellette, Photoinduced Self-organization Effects in Optical Fibers , SPIE Procs. ; vol.1516, 1991. 57. F. Ouellette, Photosensitivity and Self-Organization in Optical Fibers and Waveguides; SPIE Procs., vol.2044, 1993. 58. P. A. Chollet, G. Gadret; F. Kajzar, P. Raimond and M. Zagorska, Nonlinear Optical Properties of Organic Molecules, G. Möhlmann Ed., Proc. SPIE , vol. 1775 , Bellingham 1992, p. 121. 59. D. Broussoux, P. Le Barny, P. Robin, Rev. Tech. Thomson-CSF , 20-21, 151(1989). 60. C. Fiorini, P.-A. Chollet, F. Charra, F. Kajzar, J.-M. Nunzi, Nonlinear Optical Properties of Organic Materials VII, G.R. Mohlmann Ed., SPIE Vol. 2285, Bellingham 1994, p. 92. 131

61. C. Fiorini, F. Charra, J.-M. Nunzi, P. Raimond, J. Opt. Soc. Am. B, 108, 1984 (1997). 62. J.-M. Nunzi, C. Fiorini, A.-C. Etilé, Nonlinear Optical Properties of Organic Materials X, M.G. Kuzyk Ed., SPIE Vol. 3147, 1997, p. 142. 63. F. Kohlrausch, Pogg. A n n. Physk ., 119 , 352 (1863), G. Williams and D. C. Watts, Trans. Faraday Soc., 66, 80 (1970). 64. L. R. Dalton and A. W. Harper, Photoactive organic materials for for electro-optic modulator and high optical memory applications, in « Photoactive orgnic materials: science and application », F. Kajzar, V. M. Agranovich and C. Y.-C. Lee Eds, NATO ASI Series, High Technology, vol. 9, Kluwer Academic Publ., Dordrecht 1996, p. 183. 65. P.R. Ashley, Component integration and applications with organic polymers, in « Photoactive orgnic materials: science and application », F. Kajzar, V. M. Agranovich and C. Y.-C. Lee Eds, NATO ASI Series, High Technology, vol. 9, Kluwer Academic Publ., Dordrecht 1996, p. 199. 66. Van Tomme, P. Van Daele, R. Baets and P. E. Lagasse, IEEE J. Quantum Electron, ED-27, 778 (1991). 67. C. Fiorini, F. Charra, J.M. Nunzi, I.D.W. Samuel and J. Zyss, Opt. Lett. 20, 2469 (1995).

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NONLINEAR PULSE PROPAGATION ALONG QUANTUM WELL IN A SEMICONDUCTOR MICROCAVITY

V.M. Agranovich 1 , A.M. Kamchatnov 1 , H. Benisty 2 , and C. Weisbuch 2 1 2

Institute of Spectroscopy of Russian Academy of Sciences, Troitsk, Moscow obl. 142092 Russia Ecole Polytechnique, Laboratoire P.M.C., 91128 Palaiseau cedex France

1 INTRODUCTION The possibility of propagating stable nonlinear optical pulses near resonances stemming from Wannier-Mott as well as organic Frenkel excitons has been the subject of intense theoretical effort for more than two decades. Attractive stabilizing mechanisms are (i) self-induced-transparency (SIT) and (ii) Kerr nonlinearity which allows solitontype solutions. The former require very short pulses ( p< T 2 ) of given “area” (2π− pulse) whereas the latter applies to longer pulses ( p > T 2 ). Such pulses were considered to propagate either in bulk materials [1]–[13] or along surfaces and waveguides [14]–[17]. Renewed interest recently arose from hopes to use such pulses in the field of high-bit rate communication and ultrafast optical devices. Such hopes have been recently substantiated by first experimental observations: Ref. [18] demonstrated the distortion-free propagation of polariton pulses near (below) excitonic resonance in CuCl due to giant Kerr nonlinearity. These pulses are stabilized by the anomalous nonlinear dispersion associated with the biexciton two-photon transition and are restricted to narrow frequency domains. They thus differ from polariton solitons considered in the framework of Ref. [12], based on a dispersionless Kerr nonlinear constant. In a second example [19] propagation of pulses of self-induced transparency was achieved taking advantage of local impurity states (donor-bound exciton) in a CdS platelet. There is no doubt that these pioneering experiments will trigger an increased interest in this field in the next few years and applications to a variety of systems. In this paper, we propose to lay a theoretical basis for the understanding of nonlinear pulses propagating along a quantum well in semiconductor microcavities, which seems good candidates not only for observing such phenomena, as explained below but also for tailoring them through cavity parameters. Imbedding quantum wells (QWs) in semiconductor microcavity (MC) allows interaction of confined excitons and cavity photons with specific dispersion law. In a planar microcavity, upon increasing mirrors reflectivity, the continuum of photon modes condensates into Fabry-Perot and waveg-

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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uide quasimodes yielding, for small in-plane wavevectors, an increase of light electric field by more than two orders of magnitude. In this situation, we can expect optical processes such as three or four-wave mixing and stable nonlinear pulse formation may be realized for smaller incident light intensity. The same applies of course to light scattering processes like Brillouin and especially Raman scattering, whose investigation has begun with some success [20]. Saturation effects have also been observed for QW in MC [21]. It seems therefore timely to investigate the formation and propagation of nonlinear soliton pulses along a planar microcavity due to nonlinear interaction of cavity modes with QW excitonic transition. Excitons in QWs are more stable than excitons in bulk materials raising reasonable hopes to experimentally observe the phenomena which we describe here. We chose a configuration where the same quantum well supports both the (linear) polaritons mode and the nonlinear process for simplicity. But the degrees of freedom of state-of-the-art molecular deposition techniques (MBE, etc.) allow a variety of related configurations where for example the host cavity material or a second, different QW would sustain the nonlinearity whereas the light-coupled polaritons would stem from the first QW. Combining these and others functionalities in a microcavity opens the road to engineering of photonic properties for applications. For such configurations, the formalism presented here can be easily extended as long as active layer thicknesses are much smaller than the working wavelength. In the next section we introduce a semiclassical formalism to obtain the field response to QW polarization in a microcavity. We explicit both TM and TE field polarization for completeness, and show that only the latter lends itself to efficient coupling schemes. In the following section, we apply the above results to very short pulses of SIT propagating along a quantum well in a microcavity and point out the specificities due to photon confinement. In the fourth section, we turn to third–order nonlinearity, at longer time scale. For this case, we first take into account the linear polarization leading to the split of cavity polariton branches. On this basis, we include nonlinearity and show that it leads, in some approximation, to nonlinear Schrödinger equations and soliton-type solutions. We also make estimates of the pulse energy required to form such solitons for realistic cavity parameters. The last section is devoted to a conclusion. 2 GENERAL EXPRESSIONS FOR ELECTROMAGNETIC FIELD AND POLARIZATION IN QUANTUM WELL We suppose that a single QW is located in the middle (z = 0) of the cavity (–L /2 ≤ z ≤ L/2) with dielectric constant ε, surrounded by a cladding medium (|z| > L /2) with dielectric constant We assume for simplicity that all interfaces are plane (see figure 1); generalization on cylindrical geometry is straightforward. The QW is treated as a thin film (its thickness l is much smaller than L and the medium wavelength) having polarization P, which can be taken into account by means of modification of the boundary conditions at z = 0 for fields in cavity (see, e.g., [14, 15, 16]). For a given k = (k, 0, 0), the three directions of P give rise to L, T or Z quantum well exciton branches with respectively P||x, P ||y, P ||z. The first two cases respectively couple to TM and TE modes. The Z case can be treated by combining the TM approach with the appropriate boundary conditions given in [16]. Close to normal incidence, however, only L and T QW branches couple well to outside waves making the Z branch less suited to nonlinear effects at low intensities. Hence next sections will focus only on the L and T cases. Following sections will exemplify the TE case for simplicity.

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2.1 TM MODES Let us consider at first the TM mode propagating along x axis: H = (0, H y , 0), E = ( Ex , 0, E z ). We assume that these field components do not depend on y coordinate. Then the solution of the wave equation leads to the following dependence of Hy component on z after Fourier transform from space–time coordinates (x , t ) to the wave vector k and frequency ω :

(1)

where q = and similar solutions for E x and Ey . We have taken into account that only outgoing fields components must be present at z → ±∞. Let H >, E > denote fields above some boundary and H <, E < below it. Then at z = ± L/2 we have usual boundary conditions of continuity of tangential electric and magnetic fields: (2) However at z = 0 occurs a discontinuity of H ≡ H y because of the surface currents along the QW: (3) With the use of Maxwell equations we find immediately from (1) and (2) that

(4) where

(5) is the value which evidently is related to reflection coefficient of surfaces z = ± L /2. Then the equations (3) can be written as

(6) i.e. antiFor the cavity normal modes (when Px = 0) we find symmetric (r = –1) H ~ cos β z, E ~ sin β z, or symmetric (r = 1) H ~ sin β z, E ~ cos β z modes. Elimination of A3 from (6) gives

and the equation us to connect E x and P x in the following way:

permits

(7)

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where (8) and r is given by eq. (5). In space–time representation we have the equation (9) Using the material nonlinear relation between P x and E x , we are able to obtain the equation for E x which describes the propagation of pulses along our system. 2.2 TE MODES Now let us consider the TE modes with E = (0, E y , 0), H = (H x , 0, H z ). Again the geometry of the system is shown in figure 1. The solution of the wave equation has the same form as in eq. (1), but now it is more convenient to write it for the E y component:

(10)

where q and β have the same meaning as before. At z = ±L/2 the electric and magnetic fields must satisfy usual boundary conditions (2) which again yield (4) but now with (11)

Figure 1. Scheme of the cavity model with a quantum well inside.

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At z = 0 the boundary conditions have the form (12) and, on inserting Eq. (10), we obtain (13) 1 i.e. we In this case the normal modes (at Py = 0) correspond to A 2 = ± A 3 , r = have symmetrical E ~ cos β z ( r = –1) and antisymmetric E ~ sin β z ( r = 1) modes. In the same way as for TM modes we obtain the equation (14) where now (15) and r is given by eq. (11). In space–time representation we have (16) In the following, we consider only the TE configuration for which we shall proceed to some applications of the derived equations. 3 SELF–INDUCED TRANSPARENCY PULSES , where is the relaxWe consider here intense short pulses with duration ation time of donor–bound QW excitons. We assume exact resonance of cavity mode frequency and donor–bound exciton transition, so that the polarizability of the QW, the same as in [19] can be modelled as a two–level medium [1,2]. Considering bound QW excitons in this way we neglect exciton–exciton screening effects which usually can be taken into account in the frame of optical Maxwell-Bloch equations. But for QW thicknesses small in comparison not only with wavelength but also with bound exciton radius the effect of screening is small and can be safely neglected. To have a non–vanishing electric field at z = 0, we must take the symmetric TM ( r = 1) and T E ( r = –1) modes when E ~ cos β z. In both cases the frequency and the wave vector of the plane wave solution (17) satisfy the dispersion relation for normal modes: F( Q, Ω ) = 0.

(18)

We can look for the solution of equations (9) or (16) in the form of a pulse: (19) where the envelope amplitude changes slowly enough,

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(20) and Q and Ω satisfy the above dispersion relation (18). Then we have (21)

Assuming that the function F(k, w) has first-order changes in the spectral width of the pulse, we substitute its expansion (22) into equations (9) and (16). For example, in the case of TE mode (16), taking into account (18) and (21), we obtain the equation (23) where

is the group velocity of the pulse and the amplitude of polarization Py (x, t) has been introduced according to (24) In the case under consideration of exact resonance between the wave and QW exciton transition frequencies we have [1, 2] (25) where µ is the dipole moment of the transition, n is the concentration of donors (two– level “atoms”) per unit area of the QW, and the argument in the sine is (26) so that equation (23) reduces to the well–known sine–Gordon equation for the function Ψ ( x, t). For Ψ (x, t) depending only on = t – x/u, u being the pulse velocity, we obtain the equation (27) where (28) For real solution

138

p

— what we assume here and will check below — equation (27) has the

corresponding to the self–induced transparency (SIT) soliton described by the electric field (29) that is has a meaning of the pulse duration. A similar result can be obtained for TM modes. Let us calculate for TE cavity modes of a cavity confined by perfectly reflecting walls. This corresponds to and according to equation (11), we get (30) For a symmetric mode with r = – 1, the wavevector component β is quantized according to (31) β L = π N, N = 1, 3,... Easy calculation give the required derivatives

and from Eq. (28) we have, (32) Thus, we evidence the possibility of propagating SIT pulses along the QW at the resonance condition wl a s e r = w b o u n d e x c i t o n . It is important to note that the pulse velocity is necessarily smaller than the group velocity of the bare wave. Let us provide some quantitative estimates using typical parameters of the CdS system where SIT pulses have already been observed in bulk CdS. To calculate the quantity (33) we must know Ω, , ε, L , and the concentration of donors per unit area n (n = n o l where n o is a bulk concentration of donors and l the QW thickness). Going to estimates, 5 ⋅ 10 –29 C ⋅ m, a frequency Ω = 4 ⋅ 10 15 s –1 , and we find that for we take from [19], µ a CdS QW of thickness l = 100 with a bulk donor concentration n o = 10 16 cm –3 in a L = 1µm= 10 –4 cm cavity of dielectric constant ε = 10 and a pulse duration = 5 ps we obtain a pulse velocity µ2

(34) evidencing that the soliton moves much slower than the group velocity. The amplitude of the soliton electric field, given by

139

(35)

has the following order of magnitude ε0

26 CGSE = 8.103 V / cm,

(36)

Finally, the pulse energy for a cavity cross section of S picojoule range according to

100 µ m 2 is found in the

(37) If the pulse energy falls below this critical value, then it fails to satisfy the well–known “area theorem”. It is unstable and transforms during propagation into linear waves. It is interesting to compare the properties of SIT pulses in bulk and in cavity. Comparing Eq. (33) to its counterpart for bulk material, we see that only the relation between and u changes: in the bulk case, the difference between v g and u can be much larger due to n o l being replaced by n o L. Note that the cavity may be made still narrower and still more the wave (guide) cross sections, which would allow to lower the soliton critical energy and would shorten pulses. To get more flexible design, we could go to the case where the cavity polaritons and optical nonlinearity arise from different QWs, allowing separate choice of e.g. v g and soliton frequency w 0 More generally, the degrees of freedom provided by present day growth techniques allow full tailoring of cavity and QW parameters, and thus obtaining pulses of given energy and duration as long as the first criterion is satisfied, that is for rather short pulses. Still, the possibility to generate in this range calibrated solitons together with the simple and selective excitation scheme provided by microcavity quasi-modes at reasonable angles of incidence is attractive. For longer pulses we have to exploit the possibly slower third-order Kerr nonlinearity, as explained in the following section, focusing again on the TE case. 4 NONLINEAR SCHÖDINGER EQUATION FOR POLARITON SOLITONS IN WELL Let us consider long pulses sustained by TE waves with duration is the relaxation time of QW excitons), when the QW polarizability can be expressed as a sum of linear and nonlinear terms, here supported by the same well (see comments in the introduction): (38) where the linear part (39)

140

is responsible for the so-called vacuum Rabi splitting of cavity-polaritons. Before we can take the nonlinear part into account, our first task is to include these linear effects, as made in the next few paragraphs. We thus modify the general formalism by including the linear polarizability in the bare cavity response. We will thus give the bare cavity photon dispersion relation and deduce the modified relation associated to the modified response function Using this, we will be able below to take properly into account the nonlinear term. It is straightforward to obtain from Eq. (16) the new form: (40) if we just define (41) The modified dispersion relation giving rise to linear cavity polaritons is given by the zeroes of this new function: (42) Let us consider again, as for SIT, the symmetric TE mode in the case of cavity confined by the ideally reflecting walls, so that the bare function (43) vanishes for modes with r –1. The value β = of the N = 1 mode and the definition of β lead to the dispersion law for the bare cavity (44) where

We assume that the dielectric susceptibility

has a resonant behavior (45)

where A is proportional to the oscillator strength per unit area. Now we can find the dispersion law for cavity polaritons near the resonance ω ≈ Ω (k) ≈ ω0 , using the expansion of Eq. (43). Then Eq. (41) leads to the dispersion relation (46)

141

where the first term of the expansion vanishes for bare cavity photon ω = Ω (k). In the limit >> ε we have from (30) (47) Thus, from Eq. (46) we obtain the expected quadratic equation with respect to ω :

which yields two (upper and lower) cavity polariton modes, (48) separated by the vacuum Rabi splitting (49) In dimensionless units

the dispersion relation (48) takes the form (50)

where δ = (ω 0 – Ω c ) / Ω c is a relative detuning and

is a dimensionless Rabi splitting. This “modified” dispersion law (48) will be considered from now on as known and we shall write it in a general form (51) Now we are ready to include nonlinear effects. As is well-known [22]-[24], for long enough pulses the nonlinear part of susceptibility of the QW is determined mainly by the phase–space filling effect when the excitons and the electron–hole pairs created by the pulse prevent further excitation due to the exclusion principle. The lifetime of the excitons at room temperature is determined by their ionization due to collisions with thermal LO phonons and is about 0.1–0.4 ps. The resulting electron–hole pairs live for very long time ~ 20ns and produce approximately the same blocking effect as the excitons. Therefore, for rough estimation, we suppose that the electromagnetic pulse creates only one long–living species (electron-hole pairs in bound or free state) which number N is determined by the rate equation (52) where is the decay time of pairs. In the experiment [22] it depends on recombination and diffusion time and, as was mentioned above, has the order of magnitude 2 0ns. For very long pulses with duration greater than we have the stationary state with saturated plasma density

142

According to [22]–[24], the nonlinear part of the QW polarizability can be written as

that is the steady–state nonlinear susceptibility is given by (53) and for AsGa QW has the experimental (balk) value about (54) For short pulses with duration of time and is proportional to

the nonlinear susceptibility becomes the function

For rough estimate, we approximate this integral by 2 Im duration, and obtain for the nonlinear susceptibility the value

being the pulse

(55) that is we replace the variable nonlinear susceptibility by its mean value and take into account the quantum well width l. For p = 1 ps and l = 10 –6 cm we find numerical value (56) So we can proceed as if the QW were characterized by the “Kerr type” nonlinearity: (57) Then, as is well known, such a nonlinearity can be compensated by the dispersion of group velocity, i.e., we must take into account the quadratic terms in the expansion of (k, w) : (58) where all derivatives are taken at k = Q, ω = Ω. As . above, we shall look for the solution of equation (40) in the form of a pulse: (59) where the envelope amplitude changes slowly enough as stated by Eq. (20), and Q and Ω satisfy the above dispersion relation (47). Then we have eqs. (21) and analogous formulas for higher derivatives. On inserting of (57) and (58) into (40), we obtain with the use of (21) the following equation for the envelope function ε (x, t) : (60)

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Introducing the group velocity and its dispersion

and neglecting the terms which correspond to the cubical corrections to the expansion (58), we arrive at the well known nonlinear Schrödinger (NLS) equation [25] (61) in a moving coordinate system defined by ξ = x, η = t – x /υ g , where (62) Now we have to calculate It follows from eq.(20),(26) that (see e.g. eq. (24) and (22)) and to first order in ∆/ω0 , we have = –2i k L c 2 /ω 2. It means that (63) The sign of k" plays a crucial role in the solution of the NLS equation. We have plotted in figure 2 the inverse dispersion relation k (ω) for a typical splitting of two percent. It shows that both signs of the second derivative can be obtained and a large range of magnitude of k" is possible. The near-resonance trends would however be smeared out within the system’s absorption linewidth. Let us discuss solutions for both signs of k". If k" < 0, (as it takes place for ω above the anticrossing region), then this equation, in the case of positive χ (3) , takes the form of the “focusing” NLS equation (64) and has the soliton solution [25]. As in Ref. [12], we shall write here the solution propagating in the laboratory frame with the group velocity of linear approximation which is enough for qualitative estimations. Thus, the soliton solution of Eq. (64) has the form (65) where the amplitude ε 0 depends on the pulse duration

p

according to (66)

If k" > 0, then the equation (61) can be transformed to the form

144

Figure 2. Dispersion relation in the form k (ω): reduced wave number

= k L/π is a function

of the reduced frequency

and again has the soliton solution for

< 0, where now

If k" < 0, < 0 or k" > 0, > 0, we come to the “defocusing” NLS equation which does not have soliton solutions (but permits the so called dark soliton solutions). 4 Supposing that χ > 0, we shall estimate the field strength at k = 5 ⋅ 1 0 cm–1 for –0.9 ⋅ 10 –23 CGS. Then we obtain upper branch of the dispersion curve, where k"

The energy of the pulse is given by the formula (67) where S is the cross-section of the cavity.

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5 CONCLUSION We proposed a formalism enables us to describe accurately the interaction between the electromagnetic field and linear as well as nonlinear polarizations from a quantum well in a microcavity, considering on the same footing either true guided modes or Fabry-Perot quasi-modes. We showed that stable pulses propagating along quantum wells in such a microcavity can be sustained by cavity modes for two kinds of nonlinear mechanisms at different time scales. The first mechanism is ultrafast two-level unharmonicity leading to the self-induced transparency for 2π-pulses. In this case, we could derive soliton velocity as well as other pulse characteristics and compare them to bulk ones. We found pulse energies in the picojoule range. We gave some hint of the advantages of quantum wells and possible ways to flexibly design a system in view of given pulse requirements. The second mechanism is the slower third-order nonlinearity. In this case the linear polarizability of the quantum well giving rise to the so-called vacuum Rabi splitting has to be included and new split modes have to be obtained before nonlinearity is taken into account. The obtainment of the soliton solution is found to depend critically on the sign and magnitude of the curvature of the new dispersion relation for which suggestive examples are given. In view of experiments, laser pulses for self-induced transparency seem available. For Kerr nonlinearity, the detailed effect of inhomogeneity on splitting and linewidth has to be studied in more detail to evaluate the potential of this mechanisms in novel systems for soliton generation. ACKNOWLEDGEMENTS V.M.A. is thankful to Laboratoire PMC URA 1254 du CNRS for hospitality and support as well as to DRET, Grant 93 811-62/A000. He is also thankful for partial supports through INTAS Grant 93-461, Grant 96-0334049 of Russian Foundation of Basic Research and Grant 1-044 from Russian Ministry of Science and Technology

References [l] S.L. Mc Call and E.L. Hahn, Phys. Rev. 183:457 (1969). [2] A. Allen and J.H. Eberly, Optical Resonance and Two–Level atoms, Wiley, New York (1975). [3] A. Schenzle and H. Haken, Opt. Commun., 6:96 (1972). [4] H. Haken and A. Schenzle, Z. Phys., 258:231 (1973). [5] J. Goll and H. Haken, Phys. Rev. A18:2241 (1978). [6] V.M. Agranovich and V.I. Rupasov, Sov. Phys. Solid State 18:459 (1976). [7] O. Akimoto and K. Ikeda, J. Phys A: Math. Gen 10:425 (1977). [8] K. Ikeda and O. Akimoto, J. Phys A: Math. Gen 12:1105 (1979). [9] K. Watanabe, H. Nakamo, A. Honold and Y. Yamamoto, Phys. Rev. Lett., 62:2257 (1989). [10] S. Koch, A. Knorr, R. Binder and M. Lindberg, Phys. Stat. Sol. (b) 173:177 (1992).

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[11] A. Knorr, R. Binder, M. Lindberg and S. Koch, Phys. Rev., 46:7179 (1992). [12] I.B. Talanina, M.A. Collins and V.M. Agranovich, Solid State Comm., 88:541 (1993); Phys. Rev.. B49:1517 (1994). [13] I.B. Talanina, Solid State Comm., 97:273 (1996); J. Opt. Soc. Am., B13:116 (1996). [14] V.M. Agranovich, V.I. Rupasov, and V.Y. Chernyak, JETP Lett., 33:185 (1981). [15] V.M. Agranovich, V.Y. Chernyak, and V.I. Rupasov, Opt. Commun., 37:363 (1981). [16] V.M. Agranovich, V.I. Rupasov, and V.Y. Chernyak, Sov. Phys. Solid State, 24:1693 (1983). [17] A. Boardman, G.C. Cooper, A.A. Maradudin, and T.P. Shen, Phys. Rev., B34:8273(1986). [18] K. Ema and M. Kuwata-Gonokami, Phys. Rev. Lett., 75:224 (1995). [19] M. Jütte, H. Stolz, and W. van der Osten, Phys. Stat. Sol., (b) 188:327 (1995); J. Opt. Soc. Am., B13:1205 (1996); J. Luminescence, 67:45 (1996) [20] A. Fainstein, B. Jusserand, V. Thierry-Mieg, and R. Planel; in Microcavities and Photonic Bandgaps: Physics and Applications, J. Rarity and C. Weisbuch, eds., Kluwer, Dordrecht (1996), p. 105. [21] R. Houdre, J.L. Gibernon, P. Pellandini, R.P. Stanley, U. Oesterle, C. Weisbuch, M. Ilegems, J. O’Gorman, and B. Roycroft, Phys. Rev., B (in press); T. B. Norris, J.K. Rhee, C.Y. Sung, Y. Arakawa, M. Nishioka, and C. Weisbuch, Phys. Rev., B50:14663 (1994). [22] D.S. Chemla, D.A.B. Miller, P.W.Smith, A.C. Gossard, and W. Wiegmann, IEEE J. Quant. Electr., QE-20:265 (1984). [23] S. Schmitt–Rink, D.S. Chemla, and D.A.B. Miller, Phys. Rev., B32:6601 (1985). [24] S. Schmitt–Rink, D.S. Chemla, and D.A.B. Miller, Adv. Phys., 38:89 (1989). [25] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations, Academic, London (1982).

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SOME ASPECTS OF THE THEORY OF LIGHT–INDUCED KINETIC EFFECTS IN GASES

Stanislaw Kryszewski University of Gda sk Institute of Theoretical Physics and Astrophysics ul. Wita Stwosza 57, 80-952 Gda sk, POLAND E-mail: [email protected]

LIGHT AS A THERMODYNAMIC FORCE One of the most frequently used experimental setups of quantum optics consists of a container with gaseous mixture irradiated by external laser radiation. The number of phenomena occurring in such a system is enormous. It is impossible to cover all of them even in a large monograph. Vast amount of work is devoted to such studies, it is therefore, quite impossible even to list all relevant literature, except a few books reviewing the subject [ 1 ] – [ 5 ] . It is necessary to restrict attention to some more specific cases. First of all, we specify the gas in the container to be a mixture of two species. One kind of atoms (molecules) is called active (A), these atoms are coupled to the incident radiation field. The second species (P) is inert, not coupled to the light field. P atoms serve as perturbers, or a thermal bath for the system since they interact collisionally with active ones and constitute a reservoir of energy and momentum. Let us observe, quite generally, that two basic kinds of physical phenomena, namely the collisional and radiative ones, occur in the discussed case. Therefore, we may expect that the physical effects and properties of active atoms will depend mainly on the interplay between collisional and radiative processes. The aim of this lecture is to present one of the possible approaches to the description of such effects. Light-induced kinetic effects (LIKE) in gases are one of the examples of the phenomena in which such an interplay plays an essential role. The applications of the proposed theoretical methods to LIKE will thus serve as an illustration of the usefulness of our model. Firstly, we shall briefly discuss the LIKE introducing basic notions necessary to describe these effects. They occur due to the light-induced modifications of the velocity distributions of active atoms (or molecules) interacting collisionally with perturbers and with incoming light. For example, light-induced drift (LID) is possible when the active particles immersed in the much denser buffer gas are excited in a velocity-selective manner. This induces oppositely directed fluxes of ground and excited state atoms. The velocity selectivity, however, is

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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usually not sufficient to observe macroscopic fluxes. When the atoms in either of the states suffer different diffusive friction, the two fluxes do not cancel and the macroscopic drift is observed. LID is just one of the manifold of LIKE in atomic or molecular gases. Such effects are reviewed by Hermans [ 6 ] , de Lignie and Eliel [7,8] . The examples given further in this paper do not exhaust the variety of situations in which kinetic effects, connected with the translational degrees of freedom, are coupled with or influenced by external irradiation. Much more complete reviews can be found in the above given references (see also the materials in Ref. [9]). Secondly, we will introduce some theoretical methods necessary to describe the interconnections between collisional and radiative processes. Generally, collisions influence the internal state of active atoms and their translational degrees of freedom, or, in other words, velocities. The first type of collisions is a typical subject of investigations within the classic lineshape theory. We shall call these collisions dephasing ones. They can be dealt with within a simple impact approximation in which the homogeneous linewidth may be accounted for by a phenomenologically adopted linewidth and lineshift. The other type of collisions are frequently called velocity-changing collisions (VCC). They are responsible, for instance, for thermalization of the atomic velocity distributions, that is due to these collisions the system attains and preserves thermal equilibrium. When the system is light irradiated, then active atoms undergo simultaneously radiative and collisional processes and the incident light modifies the velocity distributions of the active atoms both in the ground and in the excited state. The considered system finds itself in the steady state which often differs from thermal equilibrium. The aim of this work is to give a theoretical method for modelling the VCC occurring in the described irradiated mixture of gases and to show how it can be applied to describe some of the physical effects due to VCC. We will present a theoretical model which incorporates VCC explicitly. The necessary assumptions essential for the applicability of the model will be introduced and discussed. Let us note that a fully quantum-mechanical treatment is possible. However, the classical kinetic theory of gases seems to provide tools sufficient for our aims. Thus, discussing collisional effects due to VCC we will confine ourselves to the modified classical methods. On the other hand, consistent description of radiative processes obviously requires a quantum approach. This is easily obtained within the model of optical Bloch equations. Combining the two kinds of physical processes we arrive at the Bloch-Boltzmann equations which are the fundamental theoretical framework for our main aim – study of the interconnections between radiative and collisional processes affecting the active atoms. It should be, however, stressed that the presented approach is just one of the alternatives. We will try to show that this alternative is an attractive one, attractive in the sense that it seems to allow a satisfactory description of a wide scope of physical phenomena occurring in the discussed physical system. Finally, we will present some specific applications of our theoretical method based upon the Bloch-Boltzmann equations. Since the presentation of the method itself is central to this work, we will briefly give only some results, mainly concerning LIKE. We shall, however, also indicate other possible fields in which our method seems to be useful. We will discuss some possible generalizations as well as limitations. The prospective areas of future investigations will be indicated and discussed. We will try to avoid many of the technical or mathematical details which can be found in the original papers. In the Appendix we will briefly summarize some main results concerning the formal solutions to the Boch-Boltzmann equations. They seem to be relevant for the presentation of the applications for which such solutions are essential. Let us also observe that the process of photon absorption results in the recoil effect, atomic momentum changes by amount with being the wave vector of incident radiation. Similarly, the emission process changes the atomic momentum by – , where in the

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stimulated process, but not in the spontaneous one. Recoil effects play an important role in many physical phenomena such as radiation pressure or radiative cooling [10] . The changes of atomic momentum due to interatomic collisions are at least by an order of magnitude greater. Therefore, in the case when VCC play an important role the effects connected with photon recoil can usually be neglected. In our discussion of the collisional processes in the gaseous mixture we will leave the photon recoil out of the picture. Such an approximation seems to be well justified and frequently used in literature.

Light–induced kinetic effects in gases Light-induced kinetic effects (LIKE) in gases started to receive considerable attention in the early eighties, with the first theoretical predictions by Gel’mukhanov and Shalagin [11] and then experimental demonstration of light-induced drift (LID) of active atoms by Antsigin et al [12]. LID is reviewed in great detail by Eliel [7,8] . Many various aspects of LIKE in gases were then investigated both theoretically [13], and experimentally [9]. To understand the essential aspects of LIKE let us consider active atoms modelled by simple two-level ones, with |1 〉 and |2 〉 denoting the lower and upper states, respectively. When such atoms are irradiated slightly off-resonance, the detuning ∆ = ω L – ω21 ≠ 0 (ω L is the frequency of the laser radiation, while ω 21 is the atomic frequency). Then, only those atoms which are Doppler-shifted into resonance may be excited. This implies that only the atoms possessing velocity determined by the relation kvr = ω L – ω21 (where k is the magnitude of the wave vector of the radiation) can appreciably absorb. This results in creation of the Bennett hole in the velocity distribution of the ground-state atoms (see e.g., Ref.[3]). The number of the ground-state atoms with velocities close to vr is depleted, while there is a certain number of excited atoms with the same velocities. We may say that the radiation selects atoms, the velocity of which coincides with resonance one vr . The scheme for such a velocity selection is given in Fig.1 which presents the corresponding distributions.

Figure 1. The scheme of one-dimensional velocity distributions for two-level atoms irradiated along the z –axis. The light field is tuned in the blue wing of the Doppler-broadened absorption profile, i.e., ∆ > 0, or equivalently ω L > ω 21 . ρ ii (v z ) are the velocity dependent populations for ground-state atoms ( i = 1) and for excited-state ones (i = 2). W (vz ) is the Maxwellian distribution. vr i s t h e Doppler-selected velocity, as in the text above.

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From Fig. 1 it is evident that radiation introduces asymmetries into the velocity distributions and may, therefore, result in nonvanishing macroscopic fluxes and flows. The situation is especially interesting when the sum of two velocity distributions ρ11 (v) + ρ 22 (v) = f ( v ) does not equal the Maxwellian W (v). Such a case occurs when the collisional cross section for collisions is state dependent, that is when it is different for ground- and excited-state atoms. Let us also note, that for exact resonance (ωL = ω21) the curves similar to those in Fig.1 are symmetric with respect to the vertical axis, and the selected velocity vr = 0. Off-resonance velocity selectivity results in asymmetric distributions and, in effect, we have nonvanishing fluxes since partial average velocities do not vanish. This is, however, usually insufficient to observe macroscopic flows. The two opposite fluxes cancel each other. The situation changes when diffusive friction is state dependent. It is reasonable to expect that excited atoms have larger collisional cross section. Hence, they suffer larger diffusive friction, or in other words, have smaller mobility and smaller diffusion coefficient. This results in the fluxes which do not cancel. Net macroscopic flow is then observed. It may be said that the LIKE in gases are due to radiation-induced nonequilibrium velocity distributions combined with state-dependent collisions. In the forthcoming considerations we will briefly describe some of kinetic effects occurring in such a situation. Further on we will present a theoretical model which allows the consistent description of these phenomena. It is well known from the kinetic theory of gases [14] that the collisions produce entropy. Only in thermal equilibrium the entropy is constant and the velocity distributions of the considered particles are Maxwellian. Light irradiating a sample, which was initially in equilibrium, changes the situation. Steady irradiation forces the considered system into a new state which is stationary but nonequilibrium. Since the ordered macroscopic fluxes are predicted, we may expect some decrease of entropy. The problem of entropy production was considered in great detail by van Enk and Nienhuis [15] . They have shown that the entropy of the matter (i.e., of active atoms and/or perturbers) may decrease. This confirms the notion of radiation introducing some order in the system. On the other hand, stimulated and spontaneous radiative processes are also sources of entropy. Entropy of photons increases by several orders of magnitude more, so that the total entropy increases, as it should. Radiation is the source of these effects. Hence, following the usual convention[14, 15] we may call light to be a thermodynamic force. To show that light indeed acts as a thermodynamic force let us discuss one [16] of the approaches to LIKE. These effects are determined both by the radiative and collisional phenomena, the interplay of which results in macroscopic flows. Obviously, the former effects are microscopic and, as such, occur on the time scale much faster than the flows. This allows one to separate the usual Liouville–von Neumann equation for the density operator of active atoms into two parts. The resulting equation is formally written as with

(1)

Operator L 1 gives the free (macroscopic) flow term and operates on the slower time scale. It gives rise to slow diffusive flows between macroscopically separated positions. Operator L 0 governs the evolution of active atoms due to radiative processes and collisions. The specific form of operator L0 is unimportant for the present purposes, it will be discussed in much detail in further sections. Let us, however, note that L0 depends on the intensity of incident light and, therefore, may depend implicitly on the position within the sample due to possible spatial variations of intensity, but we assume that it does not contain derivatives with respect t o Moreover, L 0 contains radiative effects, so it must depend on the Doppler shift Hence, we expect this operator to be anisotropic in the velocity space. Since L 0 describes the processes which occur on the rapid time scale (of the order of several atomic lifetimes) we expect that L0 drives the density operator at any position to its

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local steady state before the diffusive flow, expressed by L1 , has had any chance to produce appreciable changes of the local atomic density n( , t ) which is defined as (2) It is therefore, reasonable to seek a solution to Eq.(1) by elimination of rapid variables. This may be done by assuming the density operator in the form where may be viewed as the local steady state solution to the equation (3) Local stationary density operator is thus specified by fast processes, radiative and collisional. depends on position only parametrically (implicitly) via the position-dependent light intensity. Moreover, for any position we impose the normalization condition (4) which is consistent with (2). Employing the well-known technique of elimination of rapid variables we have obtained [16] the general diffusion equation for light-induced effects in the following form (5) with the atomic flux given by three terms (6) Technical details of the derivation of the equation (5) are given in Ref.[16]. Therefore we proceed to physical discussion of the terms appearing in the atomic flux (6). The first (0) term contains velocity which is related to local density operator by (7) It is just the average velocity of active atoms in the state which is stationary with respect to L 0 . Velocity (0) may be called drift velocity and be associated with light-induced drift. In the forthcoming sections we will show that this is indeed the case and we will discuss its properties in more detail. Now, we only note that when the density operator is a symmetric function of velocity (as is the case on resonance) then the drift velocity is expected to vanish. In absence of light certainly corresponds to the Maxwellian because all atoms are in the ground state and the system is in thermal equilibrium, so there is no drift. The next quantity in the atomic flux (6) is the light-modified diffusion tensor It is given by the fairly complicated formal expression (8) Operator L 0 possesses a stationary solution, hence it has an eigenvalue zero. It can be shown that the second term in the brackets ensures that the integrand in the time integral approaches zero on the rapid time scaleand the integral converges. Light-modified diffusion tensor depends on local light intensity (via and it is thus an implicit function of Since operator L 0 is anisotropic, due to its dependence on the light wave vector , the diffusion tensor must also be anisotropic and it must have the cylindrical symmetry. It is, therefore,

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determined by two constants, a transverse diffusion constant Dt , and the parallel one D p , so that it has matrix elements (9) This clearly illustrates that light indeed modifies the kinetic properties of the considered system [17] . It is important to distinguish between the light-modified diffusion tensor (8) and diffusion coefficients D1 and D 2 describing active atoms in either of the atomic states. The constants D1 and D 2 depend on collisional cross sections (or mobility) of active atoms in the perturber gas, while they do not depend on the intensity of the light. We expect that light-modified diffusion tensor is a complicated combination of constants D1 and D 2 . Obviously, for zero light intensity the diffusion tensor must reduce to the isotropic one and has the strength of the diffusion constant for the ground state atoms in the buffer gas. The last term appearing in (6) contains velocity (1) which we call gradient velocity. It is a correction to the effective drift velocity, and it arises from the variation of the local with position. The formal expression for gradient velocity is density operator (10) The integrand in the time integral again vanishes on the rapid time scale. Since and (0) depend on the position exclusively through the dependent intensity, we may express (1) in the form , with the cylindrically symmetric second-rank tensor (11) Gradient velocity defined formally in Eq.(10) plays an essential role in the effect of lightinduced diffusive pulling. The general diffusion equation (5) with the flux (6) thus describes some of LIKE. Each of the components of the atomic flux is driven by the light intensity or its gradient. We conclude that light field is indeed a thermodynamic force. The presented results, although quite formal, give the essential insights into the LIKE in gases. Obviously, any concrete calculations require one to specify the local evolution operator L0 . We shall do that in the next section where we will present the theoretical methods allowing consistent modelling of LIKE. Before doing so, we will briefly discuss the physical effects connected with the formal results. The full discussion of the methods to compute the above given quantities is given in Ref.[16] which also gives the relevant references. We also note that the obtained lightmodified diffusion tensor and gradient velocity are closely related to Green–Kubo relations for Navier–Stokes transport coefficients [14] . It seems necessary to stress that the presented approach is one of the several possibilities. Another theoretical framework to describe gas kinetics in a light field is due to van Enk and Nienhuis [18]–[21] .

Light–induced drift Let us assume that a sample of active and perturber atoms is irradiated by monochromatic radiation detuned into the blue side of resonance, that is ∆ = ω L – ω 21 > 0. Let us also assume that radiation wave vector determines positive direction of the z axis of the laboratory coordinate frame. Resonance velocity vr is then positive, as it follows from the Doppler shifted resonance condition = ∆ , (this is the case depicted in Fig.1). Then, as evident from Fig.1, the average partial velocity of ground-state atoms is directed in the negative direction of z axis, i.e., towards the light source. At the same time the partial average velocity of the excited-state atoms has positive direction. We have two oppositely directed

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fluxes of atoms in either of the states. When the incident radiation is tuned to the red wing of the resonance ( ωL < ω 21) then directions of the fluxes are reversed, while irradiation exactly on resonance gives rise to no fluxes, since then the corresponding velocity distributions are symmetric and the average partial velocities are zero. If the collisions do not discriminate between the atomic states, then the fluxes of groundand excited-state atoms will be equal, as it follows from the particle number conservation. Then, there will be no macroscopic flows. On the other hand, when collisional cross sections are state dependent, we may expect that due to differences in the diffusive friction the net flow will occur. Excited atoms have larger collisional cross section, so they diffuse more slowly. This situation is schematically presented in Fig.2.

Figure 2. Light-induced drift. Active atoms are irradiated at the blue side of resonance. Excited atoms are pushed by light, while the ground-state ones are pulled towards the light source. Since the excited atoms suffer greater diffusive friction the drift with velocity (0) arises. The net effect is that active atoms are pulled by light.

The essence of the LID, may be summarized by a simple proportionality (12) Each term in this relation has a clear physical meaning. The resonance velocity is positive for irradiation in the blue side of the resonance, negative for the red wing case and zero for exact resonance. The next fraction in (12) is the ratio of the density of excited-state atoms to the total density of the active atoms. Below saturation level this fraction is proportional to the intensity of the incoming light, so it vanishes in absence of light. In the saturation regime the situation is more complicated and must be considered in more detail. The last term in Eq.(12) may be viewed as the relative change of the collision cross section upon excitation. Usually, we expect σ 2 > σ 1 , as discussed above. When the interatomic interaction does not distinguish between atomic states (i.e., when σ1 = σ2) then LID vanishes as expected. The general expression (7) for drift velocity must incorporate the features discussed here in an intuitive manner. We will show that this is indeed the case. It seems worth noting that an interesting phenomenon occurs when the sample of active atoms and perturbers is optically thick. Then, irradiation of the sample in the red wing of the Doppler absorption profile results in pushing the atoms from the entrance window. The active atoms in the part of the container close to the entrance window are pushed by light relatively strongly. When the penetration depths increases, the light intensity decreases due to absorption and the drift becomes weaker. Hence, a cloud of active atoms gathers in the vicinity of the entrance. The atomic density is locally higher than in the other parts of the capillary. As a result of the drift effect, a cloud of active atoms travels from the

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entrance window towards the other end. This is the effect of optical piston demonstrated in an experiment by Werij et al [22] and described theoretically by Nienhuis [23] . The behavior of the atomic density n( , t) may be given a soliton-like interpretation. The LID effect is still studied and used in many practical applications, see for example Refs.[24, 25, 26]. The implications or applications of LID in other fields as isotope separation, investigations of molecular properties, solid-state physics or astrophysics are very interesting. However, it is not our purpose to review these questions, so we refer to the paper by Eliel[8] for a review of experiments and theory concerning LID effect.

Light–induced diffusive pulling Light irradiating the sample of active and perturber atoms usually does not fill the whole cross section of the cell, or its intensity is not constant across the beam. Then, in the region of higher intensity there are more atoms in the excited state than in the nearby regions of lower intensity. The excited atoms have larger cross section for collisions, hence, they diffuse at a rate slower than the ground state ones. Therefore, the diffusive flows of active atoms out of the region of higher intensity is, on average, smaller than the flux of ground-state atoms into the considered region. In other words, the higher mobility of ground state atoms induces a diffusive flux of ground state atoms towards high-intensity regions that more than counterbalances the opposite diffusive flux of excited atoms. As a result, the concentration of active atoms inside the high intensity region is higher than outside. This is the effect of light-induced diffusive pulling (LIDP) [27, 28, 29] .

F i g u r e 3 . Light-induced diffusive pulling. The light beam (indicated by gray) does not fill the container’s cross section and its intensity varies across the beam (white curve). Active atoms in ground state (small circles) and in excited state (larger, filled circle) diffuse out of the beam, the latter ones at a slower rate. Ground-state atoms from outside diffuse into the beam. The fluxes do not balance, more atoms enter the beam than flow out. Concentration inside the beam is higher than outside.

It is evident that the LIDP effect depends on two factors: firstly, the diffusion coefficients of the ground-state and excited atoms must be different; secondly, there must be some gradients of light intensity. The intuitive considerations can be easily put into more formal, though approximate interpretation [16] . The light-modified diffusion tensor (8) may be viewed as a weighted average of the diffusion constants of atoms in the ground and excited states. To illustrate that, let us assume that velocity autocorrelation functions for atoms in

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either of the states depend negligibly on the shape of the stationary velocity distributions. We, however, assume these correlation functions to be state dependent. This approximate assumption allows the conclusion that diffusion tensor becomes isotropic, because anisotropies of the velocity distributions are no longer important (are neglected). Hence, the diffusion tensor can be expressed as a weighted average of diffusion constants D1 and D 2 corresponding to atoms in two states. Then we have (13) where (14) are the fractions of atoms in each state. The same approximation applied to gradient velocity (10) yields (15) The gradients of the populations result from intensity variation with respect to position. Since an obvious condition p1 + p 2 = 1 must hold, we get As it was argued, we expect that D 1 > D 2 . So we conclude that gradient velocity gives rise to a diffusive-like flow of atoms towards the region of larger p2 , i.e., towards region of higher light intensity where the number of excited-state atoms is greater. This also explains the term gradient velocity. Within the discussed approximation the atomic flux (6) becomes (16) This relation includes the basic features of LID and of diffusive pulling. To see that more clearly, let us assume that atoms are irradiated at resonance, so that the drift velocity vanishes and only the diffusive part remains in Eq.(16). In the closed cell, the steady state flux must vanish and then Eq.(16) implies = const., or, equivalently (17) Since D 1 > D 2 we see that in the higher intensity region (where p2 is appreciable) the atomic density n must be larger than in the darker regions. This is the essence of the light-induced diffusive pulling effect. Consequently, the term is essential in physical description of LIDP. Previous derivations [13, 23] of the diffusion equation for light-induced effects did not include the LIDP. This was due to another approximation method concentrated rather on the difference between collisional cross sections for two atomic states. This did not permit to account for the light-modifications of the diffusion tensor. Hence, the diffusive flow connected with gradient velocity was also absent. The approach given here includes the effect of collisions to all orders and generalizes earlier studies. This qualitative discusion of the light-induced diffusive pulling gives some basic explanation of the phenomena which are due to the joint effects of radiation and collisions. The other light-induced effects are also possible, but their presentation goes beyond the scope of this work, we only refer to already mentioned literature. To proceed further we need an operator L 0 necessary to find the coefficients of the diffusion equation, drift velocity (7), light-modified diffusion tensor (8) and the gradient velocity (10) which are expressed by integrals over time dependent correlation functions, time evolution of which is governed by the fast operator L0 .

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THEORETICAL FRAMEWORK Discussing LIKE in gases we have associated an operator L0 with radiative and collisional effects which occur on the rapid time scale, and the free-flow operator L1 with the slow macroscopic fluxes. We have introduced the local stationary density operator which parametrically depends on position within the sample. We eliminated the fast variables and obtained the diffusion equation (5). This section is, therefore, devoted to establishing the explicit form and the properties of operator L 0 . To construct operator L0 we must account for radiative and collisional processes. Firstly we briefly recall the theoretical description of radiative ones.

Optical Bloch equations Realistic atoms used in the experiments on LIKE usually possess complicated internal structure. For theoretical purposes, giving all essential insights it is, however, frequently sufficient to consider a simple two-level model with ground state denoted by | 1 〉 and the excited state by | 2 〉 with being the energy difference. The density operator for this atom is a 2 × 2 matrix and in general, its matrix elements are functions of: position within the cell, velocity and time t. Hence, we write for a, b = 1,2. Time evolution of the density operator is governed by the standard optical Bloch equations. Since these equations are very well documented in literature [ 5 , 3 0 , 3 1 , 3 2 ] , it seems that they require no comments or justification. Bloch equations allow many generalizations and applicable to a variety of physical problems. The Bloch equations include the atom-light-field coupling and the effects of spontaneous emission. The standard optical Bloch equations in the rotatingwave approximation read (18a) (18b) (18c) The notation and the physical meaning of the terms appearing in Eqs.(18) with is as follows. where is denotes Rabi frequency, defined as the local intensity of the electric field of the incoming light, is the electric dipole moment of the two-level atom, which without loss of generality, may be assumed to be real. Rabi frequency may depend on position within the sample, because of the possible variations of light intensity, for example due to absorption. A is the Einstein coefficient for spontaneous emission from upper to the lower level. The velocity-dependent detuning includes the Doppler shift and is given as with ω L and being the frequency and the wave vector of the incident monochromatic radiation. When the duration of the collision is short as compared to any other characteristic time scale then it can be shown [5, 31] , that in many practical cases, the effect of collisions on the internal atomic state, is accounted for by the following contribution (19) with γ ph being the homogeneous collisional linewidth and δ is the corresponding collisioninduced lineshift[5] . Both of these quantities may be considered as position and velocity

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independent. The meaning of Eq.(19) is that the effect of collisions on atomic coherences consists in the dephasing of the oscillations of the atomic dipole. This is typical to the classic theory of pressure broadening of spectral lines and it is the essence of the impact approximation, which we assume to hold. Such a situation arises, for example, when the lower atomic state is much less polarizable than the upper one [33] . The presence of dephasing leaves the populations unchanged [31] . We assume that collisional lineshift δ is already included in the atomic frequency ω 21 , while the homogeneous linewidth ΓC = A /2 + γ ph . The solutions to optical Bloch equations depend on velocity only implicitly, via the Doppler shift. Since we are interested in the study of the interplay between radiative and collisional effects we look for velocity distributions. Therefore, we must augment equations (18) by suitable terms describing VCC occurring in the system. In order to do so, we will adopt a series of assumptions which facilitate the theoretical descriptions, but which are not very restrictive, and are applicable to a large class of experimental situations.

Description of velocity-changing collisions In the active-atom (A) and perturber (P) mixture three kinds of collisions occur, namely the A–A, A–P and P–P ones, Henceforward, we will assume that the density NA of the active atoms is much lower than the density of the perturbers NP , i.e.,: N A << N P . P–P collisions are much more frequent than A–P ones. A–A collisions are still much more rare than the former ones, and as being extremely rare, can be neglected and dropped out of consideration. P–P collisions occur frequently enough so that the thermalization of the perturber’s velocity distribution is much more rapid than that for active atoms. Thus, we approximate the P-atom velocity distribution by the equilibrium one, that is, by the spatially uniform Maxwellian (20) with = 2k BT / mP being the square of the most probable velocity of perturbers at temperature T. We conclude that only A–P collisions are considered relevant and need to be discussed. This may seem quite restrictive but it is not really the case, since it corresponds to a fairly common experimental situation (see, for example Refs.[1]-[5] and [8, 9, 11, 12]). Furthermore, we assume that the average thermal energy of the active atoms 3k B T /2 is much smaller than the energy separation . Moreover, the energy of the first excited level of the perturber is taken to be much larger than Therefore, the energy transfer from the excited atom A to the perturber is impossible. There is no non-radiative deexcitation (quenching) of atom A during the collision. Finally, we take the density of perturbers as not too high so that the A–P collisions can be treated within the binary collision regime. Stated conditions are well satisfied in great majority of typical experiments, so they may be considered reasonable and justified. When these assumptions are met, the collisions are accounted for by means of the quantum-mechanical Boltzmann equation. Such an approach was proposed by Snider [34] . , Berman [33, 35] elaborated further on the subject in the context similar to one discussed here. Rautian and Shalagin [3] also give an extensive and modern discussion of the quantummechanical Boltzmann equation. The rate of change of the matrix elements of the active-atom density operator due to collisions is formally written as (21) Fully quantum-mechanical expressions for collision kernels and rates are quite complicated since they involve scattering amplitudes for active atoms in both states. On the other hand, as it is generally accepted, classical kinetic theory is quite appropriate for determination of

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the velocity distributions of not too dense gases. The investigated gas can be considered as a mixture of three species: perturbers, active atoms in the ground and in the excited state. Due to our assumptions, when applying the classical kinetic theory [36, 37, 38] we take the velocity distribution of the perturbers to be a Maxwellian (20), and we arrive at the linear Boltzmann equations for the populations ρaa which are formally the same as Eq.(21). The collision kernel for populations is then expressed in a simple, intuitive form, namely

(22) The delta functions ensure momentum andenergy conservation. The corresponding collisional is then given as rate γaa (23) (or are the velocities of an active atom, and (or of the perturber after where (or before) collision. (or are the relative velocities. m A , m P and µ denote the masses of the active atom, perturber and the reduced mass, respectively. Expressions (22) and (23) instead of quantum-mechanical scattering amplitudes now contain differential cross section for the active-atom-perturber scattering in the center-of-mass frame, with χ s being the scattering angle. It should be stressed that the cross sections for various atomic levels are in general different, due to differences in the interaction potentials. It seems necessary to clarify terminological distinction between linearized and linear Boltzmann equations. When we assume that the velocity distribution only slightly differs from the Maxwellian, we can take it as Then, in the Boltzmann equation we can retain only the terms linear in the correction φ This leads to the linearized Boltzmann equation. On the other hand, when we consider a mixture of gases with one component much denser, then we can assume that denser component thermalizes rapidly and its velocity distribution is Maxwellian. Due to this assumption the Boltzmann equation becomes linear in the velocity distribution of the dilute component and attains the form of Eq.(21). Thus, our model corresponds to the linear Boltzmann equation. The approach via classical kinetic theory to the influence of VCC on coherences is questionable, if at all possible. First of all, we note that the kernel and rate defined in Eqs.(22-23) are real. This is consistent with the probabilistic interpretation of populations. The coherences are complex, and so are the corresponding quantum-mechanical kernels and rates. Thus, they do not have classical analogues. Nevertheless, classical approach gives useful insights in describing the effect of VCC, especially when the influence on coherences may be reduced to dephasing only. We shall return to the problems raised by the coherence collision kernels and rates later, after discussing quantities corresponding to populations. First of all, we note that the number of particles must be conserved during the collisions. This requirement implies that the integral over d of both sides of Eq.(21) must yield zero and entails the general requirement (24) The integral term in Eq.(21) is the gain one and it gives the number of particles which change velocity from before, to after the collision. Hence, the collision kernel K aa is a measure of transition probability between and velocity groups. The term with the rate γaa is the loss one, and it gives the number of particles escaping from velocity interval to any other one. γ aa can be also viewed as the collision frequency, and its

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inverse can be interpreted as the average time between collisions. Hence, the names: collision rate and frequency, can be used interchangeably. The given probabilistic interpretation of the kernel and frequency is fully consistent with requirement (24). Since the kernel gives the transition probability between various velocity groups, it satisfies the detailed balance condition in equilibrium [36] : (25) where WA

is the Maxwellian distribution for active atoms (26)

with u 0² = 2kB T / m A being the square of the most probable velocity. Integrating requirement (25) over and using Eq.(24) we obtain (27) So, the Maxwellian WA must be the stationary solution to the kinetic equation. It is equivalent to say that WA is the eigenfunction of the right-hand side of Eq.(21) with the eigenvalue zero. This result is also evident from the notion of the equilibrium distribution (which is known to be Maxwellian) as the stationary solution to the kinetic equation (21). With the aid of the particle number conservation requirement it can be shown [36] that WA is a unique steady-state solution. Hence the discussed zero eigenvalue is nondegenerate. It is perhaps of interest to note that the linearized Boltzmann equation has five scalar invariants (eigenfunctions) corresponding to the zero eigenvalue of the collisional integral. They are 1, and v² and correspond to mass (or equivalently, particle number), momentum and kinetic energy conservation, respectively. In our case, we consider the linear Boltzmann equation and relation (24) corresponds to mass (particle number) conservation while the perturber gas serves as the reservoir of momentum and kinetic energy. Furthermore, it can be shown [36] that the collision kernel Kaa and the corresponding collision rate γ aa satisfy the relation (28) This inequality does not have any simple interpretation. It is, however, important for the forthcoming considerations. The above given relations summarize the most important properties of the collision kernels and rates. Although these properties are simple and physically understandable, they do not facilitate the computation of the kernel and rate. The presented expressions are very general and difficult to deal with in practice, either in quantum or semiclassical case. Finding the collision kernels and rates requires the knowledge of the scattering amplitudes or, equivalently, the corresponding T-matrix elements [3, 39] in the quantum case. Moreover, real atom possesses spatially degenerate levels, which is still another complication, because then the T-matrix will have off-diagonal elements between various m-states. In the classical case, we need the collisional cross sections. They can be found, at least in principle, when the interatomic potentials of the A–P interaction are known, which is rarely the case. The introduction of realistic potentials usually leads to intractable analytic expressions for the kernels and requires extensive numerical calculations [40, 41] . Thus, their determination, even for a simple two-level atom, is a formidable task. Hence, in practical investigations one has to resort either to approximate methods, or to some models. The latter approach is adopted by many authors and seems to be quite

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effective. When an analytical model kernel is adopted, the rate follows by Eq.(24). The simplest model, called the strong collision (SC) one, is based on the assumption that even a single collision fully thermalizes the velocity distribution of active atoms. In such a case, Eq.(21) reduces to (29) where γ ( sc) is a constant, while ƒA is an arbitrary velocity distribution for active atoms. The other useful model was introduced in 1952 by Keilson and Storer [42]. Although it is not derivable from any physically reasonable interatomic potential, it seems to be the most frequently used model for VCC, see for example [43, 44] . The kernel for this model is (30) where γ (k s ) is a constant, while and parameter αks ∈ (0,1). It is straightforward to check that both given models satisfy all the necessary requirements. The KeilsonStorer model possesses one interesting property, which allows relating the average velocity after the collision to the average velocity before the collision

(31) Therefore, αks is interpreted as the persistence ratio for an atom having velocity and is related to the masses of the active atoms and perturbers 1 – αks = 2 m A /( m A + mP ). Nevertheless, in practical applications the parameter αks is often treated as adjustable. For example, when αks → 0, the Keilson-Storer model reduces to the SC one. Another interesting feature of the KS model is, that it has a set of eigenvalues and eigenfunctions [43] . These eigenfunctions coincide with the ones of the kernel found from (22) for Maxwell molecules, i.e., for the potential V (r) ∝ r –4 . These eigenfunctions play important role in the present considerations, so they will be discussed in more detail later. Apart from these two simple models it is also possible to construct other models. For example, a kernel for hard spheres or the difference kernel are also sometimes used. When a model analytical kernel is employed in practical computations, then, there immediately arises the question whether the adopted model can be found from physically justified cross section. That is, whether a given model can be derived from relation (22) which relates the kernel and the interatomic potential (via a corresponding cross section). The answer to such a question is usually either negative, or very difficult to be given. Berman et al [45] addressed this question by relating the suitable moments of the collision kernel to the collisional integrals Ω (l,s) known from kinetic theory of gases and closely connected with transport coefficients [37] . These authors analyzed the KS kernel, the hard spheres one and the difference kernel. They have found that the KS kernel gives the results closest to the expectations following from the kinetic theory of gases. Thus, they have concluded that the KS kernel may be treated as the model yielding the results more reliable physically than the other models. This is so, regardless of the fact that it is not derivable from interatomic potentials. The introduced collision kernels and rates (with suitable generalizations for coherences) are ideal candidates for theoretical description of VCC. Simply, the microscopic equation of motion for the atomic density operator elements is written as (32)

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where the radiative part corresponds to the right-hand sides of the optical Bloch equations (18) and collisional one to the expressions of the type of the right-hand side of Eq.(21). Then one arrives at the set of equations which constitute the operator L0 describing the radiative and collisional processes and it is possible to proceed to investigate LIKE in gases. Concluding this section, we note that any modelling method is in fact arbitrary, because any adopted analytical model for description of VCC is, to say the least, loosely if at all connected with realistic interatomic potentials. Moreover, the coherence collision kernels are a problem in the analysis via linear Boltzmann equation. Therefore, we propose a different approach which will allow a consistent theoretical method of modelling VCC. The proposed method will be shown to avoid at least some of the questions raised in this section.

Collision operators and their properties To facilitate the presentation and discussion of our modelling technique it is convenient to introduce the concept of the collision operator [46, 47] which stems directly from the linear is an arbitrary velocity distribution, the collision operator Boltzmann equation. If on is defined as (33) where, for sake of clarity, we have temporarily neglected the indexation. The collision kernel and rate are combined into a single entity and the corresponding equation of motion (34) together with the given initial distribution is the kinetic equation which gives the time evolution of the velocity distribution of active atoms due to the influence of the VCC with perturber particles. It is worth stressing that the introduced collision operator is, by definition, neither time nor position dependent. The properties of collision operators were discussed previously [47] . It seems however, useful to review briefly the main concepts. It is convenient to introduce a vector space F of velocity distributions (35) For ƒ, g ∈ F, we define the scalar product (36) The particle number conservation given by Eq.(24) when reexpressed in terms of a collision operator reads (37) Furthermore, as a consequence of the detailed balance condition (25) for collision operator we find that for any two distribution functions ƒ, g ∈ F we have (38) which means that the collision operator is Hermitian in space F. Moreover, the rate γ and kernel satisfy relation (28) which yields [47] the inequality: (39)

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which holds for arbitrary velocity distribution ƒ . This relation may be called the nonpositivity property of collision operator. Since the collision operator is Hermitian in space F, there exist left and right eigenvectors of which are equal and the eigenvalues are real. Denoting the eigenfunctions by and the real eigenvalues by λ α we write (40) The eigenfunctions

form a complete set of orthonormal vectors (41)

The general property (39) of the collision operator implies that the eigenvalues are nonMoreover, as it was discussed after Eq.(27), the Maxwellian positive W is a unique stationary solution to the kinetic equation and it corresponds to the nondegenerate zero eigenvalue. So, there must exist an index α, henceforward denoted by 0, such that λ 0 = 0 and Since it is a unique eigenvalue, we have λ α < 0 for all α ≠ 0. Expanding the solution of the kinetic equation (34) we see that all initially present deviations from Maxwellian (i.e., equilibrium) distributions behave as exp[– λ α ( t–t 0 )]. Hence, requirement λ 0 = 0 ensures that the Maxwellian is the stationary solution corresponding to thermal equilibrium, as it should be. Within the presented formalism the kinetic equation (34) can be written as (42) with the kernel expanded in terms of the eigenfunctions as (43) It is the basis of the modelling method introduced and discussed in [47]. Namely, if we know a certain set of functions to be the eigenfunctions, then any collision operator can be constructed according to Eq.(43), provided the eigenvalues are known. Thus, we can model a whole class of collision operators by a suitable choice of the eigenvalues. In the paper [47] we treated the eigenvalues as free parameters, and we have also shown how to reconstruct the Keilson–Storer and strong collision models by the proper choice of the eigenvalues.

On coherence collision operators. Introducing the concept of collision operators we focused attention on the velocity distribution functions which have a simple probabilistic interpretation. Hence, all discussion given above is certainly valid for populations. Coherences, however, are in general complex and as such, have no probabilistic significance. Corresponding collision operators do not need to satisfy the dicussed requirements. We will argue, however, that it is possible to generalize the introduced concepts to coherences. As it was discussed in connection with optical Bloch equations the atom suffers not only velocity changing collisions but also the dephasing ones. We propose following procedure for coherence collision operators. The dephasing collisions are accounted for as it was done in the Bloch equations (18) via the homogeneous linewidth ΓC . In other words, dephasing is always included in the corresponding equations of motion. Since the coherences are conjugates of the coherence collision operators must also have this property. Hence, each other effects due to VCC, but other than dephasing, are assumed to be incorporated in a complex ) with zeroth eigenvalue λ (c0 = 0 (as for populations). Other collision operators 164

(c)

(c)

eigenvalues λ α are complex, with Re{ λ α } < 0 which ensures the correct time dependence of coherences. In such a manner we can easily extend our modelling technique to the case of coherences. However, operators do not have to satisfy number conservation, or detailed balance conditions. On the other hand, when collisions are isotropic it can be argued [3] that only the diagonal (i.e., population) operators play significant role and those for coherences are negligible. Then, only dephasing contributes to evolution of coherences. This statement is a kind of a postulate and its validity (i.e., the necessity of including coherence collision operators other than simple dephasing) has not been investigated, although it is frequently employed in practical calculations. Nevertheless, in the view of the above given remarks, our modelling method can be easily adjusted to the case when the coherence operators must be included.

Eigenfunctions and eigenvalues of collision operators The collision operator was shown to be Hermitian and therefore it possesses a complete and orthonormal set of eigenfunctions. Solving the eigenproblem and finding exact eigenfunctions and eigenvalues for the collision operator requires the knowledge of a collision kernel and rate from interatomic potential. So it is still more difficult than computation of the kernel and rate directly from the potentials. In order to avoid this difficulty we propose a different approach which was introduced earlier [46, 47]. Namely, we adopt a set of orthonormal functions as the eigenfunctions of the collision operator and define them as follows (44) where φ α

might be taken in the Cartesian coordinates, as (45)

where vi are Cartesian components of the velocity and H n (.) are the Hermite polynomials (all special functions in our work are taken according to Ref.[48]). Equivalently, we may choose become spherical coordinates and then the eigenfunctions φ α (46) ( +1/2)

with L nl being the associated Laguerre polynomials and Ylm the spherical harmonics of the angles determined by the spatial orientation of the vector The two sets of eigenfunctions for either of the coordinate systems are distinguished by different subscripts and the context should make it clear which eigenfunctions are considered. It is worth noting that the slightly modified functions (47) are the usual eigenfunctions of the standard quantum-mechanical harmonic oscillator [49] of is replaced by u 0² . mass m osc and frequency ω osc , such that the factor The connection between the adopted eigenfunctions of collision operator and oscillator ones allows to see that all the requirements imposed on are indeed satisfied. The adopted eigenfunctions of the collision operator are the eigenfunctions of the Keilson– Storer kernel, as demonstrated by Snider [43] . The eigenfunctions of the KS model correspond to eigenvalues (superscripts (c) and (s) distinguish Cartesian and spherical cases) (48)

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We should, however, stress that adopting the given functions, in either of the coordinate frames, as eigenfunctions of the collision operator is a postulate. This is a postulate, because our choice is in fact arbitrary, although well-justified. We assume that the chosen eigenfunctions are good approximations to the true eigenfunctions of any physically reasonable collision operator. The fact that the selected eigenfunctions are the ones corresponding to the KS model is one of the arguments supporting our choice. In our earlier paper [47] we have given a detailed discussion and justification that such an approach is reasonable and that the functions can be considered as good approximations to the true eigenfunctions. On the other hand, in we have assumed [47] that the eigenvalues are unknown, and thereby treated as free parameters of the theory. In particular, the eigenvalues can be different from the ones corresponding to the KS kernel. In this manner, using expansion (43) and allowing the eigenvalues to be free parameters we are able to model a whole class of collision operators. These assumptions allowed us to propose a method for modelling various physical phenomena in gaseous mixtures subjected to electromagnetic irradiation. The examples of applications of our modelling approach are given in our previous papers [47, 50] . We can use either Cartesian form (45), or the spherical one (46) of the eigenfunctions of the collision operator. The latter form of the eigenfunctions is sometimes more convenient and it is usually used in the literature [43, 45] . However, in practical calculations the choice between the forms (45) and (46) is, in fact, a matter of convenience. In the papers devoted to modelling the physical phenomena eigenfunctions as given in (45) were mainly used. One of the advantages of the eigenfunctions taken in the Cartesian form (45) is that they are factorized in a way which facilitates calculations of various integrals. Moreover, the case with axial symmetry (the symmetry axis being determined by the incident radiation) is essentially a one-dimensional one. Then, the eigenfunctions (45) are especially convenient since all the physically interesting quantities are expressed by integrals over a single component of velocity. The choice of either of the possible forms of the adopted eigenfunctions determines the numbering of the eigenvalues. The situation is similar as in the case of a harmonic oscillator. It is perhaps worth noting that eigenvalues (or energies) of the harmonic oscillator are strongly degenerate. In the spherical case the energy corresponds to the principal quantum number N ( s) = 2n + l , while in the Cartesian case N (c) = n 1 + n 2+ n 3 . The energy levels are g ( N ) = ( N + 1) ( N + 2) times degenerate (with N being the principal quantum number for either of the cases). However, in this work we consider the collision operator which differs considerably from the oscillator Hamiltonian. Therefore, it is reasonable to expect that the degeneracy specific to the harmonic oscillator will be, at least partially, lifted. The eigenvalues of the collision operator can be found from the relation Substituting the collision operator according to its definition (33) into scalar product (36), then using the property (24) of the collision rate, we arrive at the expression (49) and either Cartesian eigenfunctions (45) or spherical ones (46) can be used. In a modelling approach, eigenvalues can be considered as free parameters, and as such can be used to fit theoretical results to the experimental data. The closed theory, however, should not contain free parameters, but only the quantities with well-defined physical significance. Computation of the eigenvalues according to Eq.(49) requires the knowledge of the collision kernel. For populations, K aa is specified in Eq.(22) via the cross section. The cross section is usually unknown since the potentials for A–P interaction are also unknown. It is, however, possible to circumvent this difficulty [51] . The idea is to express the eigenvalues of the collision operators via the transport coefficients which are directly measurable physical quantities. This is done in several steps. Transport coefficients can be computed in the kinetic theory of gases within Chapman-Enskog approach which gives a

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method to find successive approximations to various physically significant quantities. Within the first order (for the case when N A << N P , as assumed in this work) this method gives transport coefficients via collision integrals Ω( l,s ) [37] . For example, for active-atom molecules in the perturber gas, diffusion coefficient, viscosity and heat conductivity are given as (50) The collision integrals Ω(l,s ) for integers l and s are defined as (51) where W R is a Maxwellian with u 2R = 2kB T/µ = u 2P + u 02 being the square of the most probable speed of the relative motion of A and P particles. Hence, W R corresponds to the distribution of the velocities of relative motion of the collision partners. The quantity Q(l) (v) which appears in (51) is expressed by the cross section via the following integral (52) relative velocities before and after the collision. χ s is the scattering angle between and On the other hand, the eigenvalues (49) of collision operators are given as integrals, over velocities of an active atom before and after the collisions, of the integrand which includes the cross section via the collision kernel [see Eq.(22)]. This suggests that eigenvalues are closely related to the collision integrals (51). The derivation of this relation consists in constructing the generating function which, by differentiation, allows one to express the eigenvalues directly via a combination of the collision integrals. The employed mathematical methods are technical and fairly complicated [51] . Therefore, we present here only the final results. Although it is possible to obtain the discussed relationship both for the spherical case (for eigenfunctions (46)) and for Cartesian ones (45), we concentrate on the Cartesian case, since it is more useful for the system with axial symmetry. The eigenvalues associated with the Cartesian eigenfunctions (45) are labeled by three nonnegative integers. As the general property of collision operators requires, we have λ000 = 0, while for the next several eigenvalues we have (53a) (53b) (53c) Eigenvalues differing by permutation of indices are degenerate, because no direction is privileged. This is a general property of the Cartesian case [51] . Calculation of next Cartesian eigenvalues is certainly possible, but results are more and more complicated. Eigenfunctions of the harmonic oscillator in spherical or Cartesian coordinates are connected by a unitary transformation [49] . Since eigenfunctions of the collision operator are proportional to the oscillator ones, the mentioned transformation can be employed to obtain a connection between the eigenvalues corresponding to two coordinate systems. One easily obtains that the eigenvalues for Cartesian (c) and spherical (s) cases are related as (54)

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Transport coefficients (50) can easily be expressed by the eigenvalues given in Eqs.(53). Conversely, eigenvalues can be rewritten in terms of combinations of transport coefficients. This fact puts the eigenvalues on the firm physical ground. They are no longer free parameters, but are fully expressed by physically understandable transport coefficients. For example, for the first eigenvalues (53a) from (50) we obtain (55) Expressing some of the first eigenvalues by the collision integrals or transport coefficients we can construct an approximate (because of finite number of eigenvalues used) collision operator (43). Thus, it is written in terms of the quantities which are directly measureable. In this way we are able to construct the collision operator without invoking any particular analytical model. Therefore, we avoid all the difficult questions arising when adopting any specific model. Interatomic potentials determine differential cross sections which appear in the collision integrals, and thus determine the eigenvalues. Thereby the right potential is automatically accounted for in our (although approximate) collision operator constructed according to Eq.(43). Circumventing the problems connected with any particular model of collision kernel we provide a method to construct a whole class of collision operators. The obtained relationship between the eigenvalues of the collision operators and collision integrals Ω (l,s ) or transport coefficients clarifies the previously employed modelling method. We have shown that only several eigenvalues of the collision operator usually suffice to describe physical phenomena occurring in the gaseous mixtures [47, 50] . Hence, it would be of interest to reexamine the results of modelling in the view of the present ones. It would be interesting to use explicit expressions for the eigenvalues in the equations for the quantities modelled by eigenvalues – previously taken as free parameters. This may provide some new information on the light–induced kinetic effects in gases. Moreover, it will be easier to compare the theory with potential experiments, since the physically significant quantities will be expressed by other ones, which are measurable and have well known physical meaning. This may also be an interesting subject for further investigations.

Bloch–Boltzmann equations and modelling method In previous sections we have presented a theoretical method allowing the detailed description of the velocity-changing collisions. As we have discussed it earlier, we are interested in the interplay between collisions and radiative effects which is incorporated in operator L0 . We follow the lines established in Eq.(32). The radiative part of L 0 is thus given by the optical Bloch equations, while the collisional operators introduced above provide the necessary collisional terms. In this manner we arrive at the set of equations for local, microscopic evolution of the density matrix elements of active atoms. Such an approach may be called phenomenological, but it seems quite successful and is used by virtually all authors dealing with similar problems. The obtained equations of motion corresponding to operator L 0 c a n be called Bloch-Boltzmann equations and are as follows (56a) (56b) (56c) with ρ 12 = ρ *21 . The notation in Eqs.(56) is the same as in (18), but for simplicity, we have omitted the arguments indicating the dependence of ρab on velocity, position and time. We

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recall that Rabi frequency may be implicitly depend on the position. Thereby, the matrix elements of the density operator are also parametrically dependent on position We also note that, in general, these equations are subject to initial conditions ρab t = 0) = ρ 0ab The stationary solution , however, is expected to be independent of the initial conditions. and kernels Kaa (a = 1, 2), incorporate the rates γaa Collision operators and represent the influence of VCC on the atomic populations. The coherence collision operator are assumed to have the properties introduced earlier. Dephasing contribution is included in the homogeneous linewidth ΓC . The operator in (56c) describes collisional effects other than dephasing. The term containing can frequently be neglected and influence of collisions on coherences reduces to dephasing [3, 47] . Bloch-Boltzmann equations together with the definitions of the quantities appearing in their right-hand sides constitute the main theoretical framework sufficient to investigate the interplay between the collisional and radiative effects. When using this model we will assume that: 1 . All collision operators in the above equations correspond to the same set of eigenfunctions {ϕ α } given by Eq.(45) or (46). This assumption is thoroughly discussed and justified in Ref. [47]. 2. Each population collision operator is parameterized by a different set of eigenvalues. (a) for a = 1,2. The choice of two different sets That is, a set {λα } corresponds to of eigenvalues for ground- and excited-state atoms reflects the fact that, in general, the A–P interaction depends on the internal state of the active atom. Different collisional cross sections imply different transport properties and hence two different sets of eigenvalues. On the other hand, these eigenvalues can be expressed via the collision integrals or transport coefficients characterizing active atoms in either of the atomic states. Therefore, λα(1) and λα(2) are not free parameters, but have specific physical significance, as it is implied by above given considerations. 3. If the effect of collisions other than dephasing must be included, then the eigenvalues of collision operators are treated as complex parameters. Otherwise, these collision operators can be neglected. Since collision operators satisfy the requirement of particle number conservation (37) the density operator is normalized (57) as required by Eq.(4). This requirement must be also satisfied by any initial condition ρ0 Before proceeding further, we add some remarks on the calculational procedure. The computation of the light-modified diffusion tensor and the gradient velocity is most easily performed when the time integrals in Eqs.(8) and (10) are calculated with a limiting procedure. The integrand is multiplied by the factor e –st and, simultaneously, the limit s → 0 + is taken. Performing the integration over time, after simple manipulations we obtain for light-modified diffusion tensor (58) and for the gradient velocity (59)

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The operator L 0 has a zero eigenvalue (since it possesses a stationary solution, as in Eq.(3)), but in these equations its inverse operates on quantities which yield zero after taking the trace over the atomic states and integrating over velocity. This ensures the existence of the limit in the above equations. In order to evaluate the integrands in (58) and (59) we need to calculate the quantities of the type of (60) where σ0 is a velocity dependent matrix satisfying the requirement (61) The right-hand side of Eq.(60) may be viewed as the Laplace transform of the time-dependent matrix the time evolution of which is governed by the Bloch-Boltzmann equations (56), i.e. by the operator L 0 . Then, the quantity (60) can be evaluated by solving the Bloch-Boltzmann equations in the Laplace domain with σ0 being the initial condition. Hence, we need (62) to obtain (58) and (59), respectively. This discussion clearly illustrates a twofold meaning of the Bloch-Boltzmann equations. The solution with initial conditions satisfying the requirement (57) would yield the time evolution of the local density operator determined by collisional and radiative processes. Stationary solution is then the limit for t → ∞. On the other hand, solutions with the special initial conditions (62) specify the integrands in expression (58) for light-modified diffusion tensor and in (59) for gradient velocity. Therefore, the explicit solution to the problem of LIKE may be obtained via the detailed analysis of Bloch-Boltzmann equations with initial conditions chosen according to the current needs. The general solutions to the Bloch-Boltzmann equations are found in the Laplace domain [47] . Since such solutions are rather complicated, they are briefly summarized in the Appendix. We note that the obtained expressions contain collision operators or their inverses acting on the initial functions of velocity. This difficulty can be avoided, because the physical quantities of interest such as drift velocity (7), light-modified diffusion tensor (8) or gradient velocity (10) are expressed as integrals over velocity of simple functions of velocity, multiplied by the solutions of Bloch-Boltzmann equations, which via the initial conditions are also functions of velocity. This suggests a simple procedure. The functions of velocity are expanded in terms of the eigenfunctions of the collision operators. The solutions to Bloch-Boltzmann equations depend on the combinations of the eigenvalues and eigenfunctions. The arising integrals over velocity can then be easily computed. It appears, that due to orthogonality of the eigenfunctions only a very few expansion terms are of importance. We will illustrate this procedure in more detail in the next section taking drift velocity as an example.

Absorption equation In our discussion we have noted that the local steady-state density operator may parametrically depend on the position within the sample due to the spatial variations of light intensity. Therefore, to close the system of equations we need an equation governing the dependence of the light intensity within the sample. Analysis of Maxwell equations for a light wave propagating in the gaseous medium [30] yields the equation (63)

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where is the direction of light propagation. α denotes the saturated absorption rate which, in general, depends on the velocity distributions of active atoms in both states. Hence, it is a functional of the local light intensity I It can be shown [30] that (64) In the optically thick systems, due to strong absorption, the light intensity varies in space and Rabi frequency χ changes accordingly. This is so, because Rabi frequency and the light intensity are connected as (65) with B being the Einstein’s absorption coefficient and c the speed of light. As a result, elements become dependent. Since atomic populations depend on intensity via Rabi frequency the resulting set of equations is strongly nonlinear. In absence of saturation the absorption rate is directly proportional to the intensity I which considerably simplifies the problem. In the case of optically thin systems further simplification may be obtained, if one assumes that the intensity variations across the sample are negligible. In any case, Eq.(63) closes our physical model, because without it we would be unable to determine the variation of light intensity within the sample.

APPLICATIONS AND DISCUSSION The derived theoretical model seems to be quite elastic to enable the description of a variety of physical situations. We will, however, restrict our attention and describe only some applications directly connected with LIKE in gases, and focus especially on the drift velocity. Our purpose is to illustrate how the developed formalism may be used to analyze physical phenomena. We will not dwell upon mathematical details but discuss the method, the basic results and the possibility of approximations useful in some particular cases. We will also discuss the generalized correlations of the functions of velocity which seem to be quite important, since they may be viewed as Green-Kubo relations for Navier-Stokes transport coefficients for light irradiated systems. These correlation functions allow us to put lightmodified diffusion tensor (8) and gradient velocity (10) into a single theoretical scheme. Other applications are certainly also feasible. We have employed Bloch-Boltzmann equations (56) to the analysis of radiation distribution (i.e., of the spectra of resonance fluorescence) [50] in presence of VCC. The narrowing of spectral lines, similar as in the Dicke effect, was then predicted. Also, using a decorrelation approximation, Bloch-Boltzmann equations can be reduced to the rate equations which are especially useful for the discussion of the case with broadband irradiation. The drift velocity and light-modified diffusion tensor were investigated for such a case [17] .

Drift velocity Physical concepts underlying LID effect were already discussed above. We can, theredefined by Eq.(7) which we fore, concentrate on theoretical investigation of drift velocity rewrite as (66) where the trace of the local steady-state density operator is replaced by a total velocity found in (A.14). The components of velocity are proportional to distribution function

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the eigenfunctions (45) with one index being a unity and two other ones zeroes. Since is expressed by the upper state population (see Eq.(A.13)) we expand into the same set of eigenfunctions, and we write (67) When using (A.13) in (66) we see that the first term is odd in velocity, so it does not contribute. Using the orthonormality of the eigenfunctions we obtain (68) When the incoming light propagates along the z-axis of the system then Bloch-Boltzmann equations (and thereby the solutions) depend only the z component of velocity. This is the case with axial symmetry and since it corresponds to a typical experimental situation, in the forthcoming, we will concentrate on this case only. Generalizations to three dimensions do not pose any problem. Thus, all velocity distributions are axially symmetric and the expansion coefficients as in (67) with first two indices equal to zero are the only nonvanishing ones. In such a case, drift velocity given by Eq.(68) reduces to a single term, which is (69) for consists in finding the expansion coefficient The final step in calculating the local, stationary upper state population. These coefficients are given in the Appendix (Eq.(A.17)). For axial symmetry, when first two indices are zeroes, the drift velocity becomes (70) and with operators following from Eqs.(A.6) and (A.8) in the Appendix. Deriving (70) we have accounted for the fact that collision operators acting on the Maxwellian WA give zeroes. Note, that the operators depend solely on the z component of velocity and, therefore, all integrals are one-dimensional. The integrand in Eq.(70) is not an eigenfunction of collision operator, so further calculations and results are fairly complicated [47] . Thus, we restrict ourselves to discussing the general result. Due to relation (55) the eigenvalues appearing above can be identified with the diffusion constants for excited and ground state atoms. This conclusion coincides with the experimenequivalent to requirement D 1 ≠ D 2 , is tally confirmed results [8] . Condition necessary to observe the drift. This condition summarizes the physical requirement that the collisions must be state dependent and reflect the dependence of the A–P interaction on the internal state of the active atoms. The same conclusion stems directly from Eq.(66) when (A.16) is taken into account. Moreover, (as discussed previously) we expect that D 1 > D 2 , which implies that eigenvalues satisfy the inequality (71) All eigenvalues must be nonpositive, so we see that the first factor in (70) is positive. For axial symmetry ∆ = (ωL – ω 21 ) – kv z . therefore when light field is tuned to the blue side of resonance (i.e., when ω L > ω 21) the contribution of v z > 0 dominates in the integrand of (70) and the integral is positive. Drift velocity becomes positive – atoms are pulled by the light. Conversely, for irradiation at the red wing of the resonance the contribution of v z < 0

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dominates and becomes positive. Atoms are pushed by light. It is straightforward to see that drift velocity vanishes on resonance, because on resonance the integrand in (70) is an odd function of velocity (everything apart from the factor vz is even in velocity). Frequent collisions quench the drift effect since appreciable thermalization occurs during atomic lifetime. As a result we expect the drift velocity to decrease when the buffer gas pressure increases. The eigenvalues of collision operators are proportional to perturber density as it follows from Eqs.(53) and can be proved in a general case [51] . Thus, drift velocity decreases when N P increases. This is evident in Eq.(70). The dependence of drift velocity on Rabi frequency is also evident. It vanishes when χ → 0, because, then, there are no excited atoms. Conversely, when χ is very large, the operator factor in (70) tends to and yields zero when acting on the Maxwellian W A so that → 0 for χ → ∞ . This conclusion corresponds to the fact that power broadening smoothens out the differences in velocity distributions for atoms in both states. The decrease decreases. of velocity selectivity entails, and We conclude the general discussion of drift velocity by noting that all the features of LID are properly accounted for in the presented formal approach. Our method being independent of any particular collision models is, therefore, quite effective in the description of light-induced drift. In some physical situations it is possible to obtain simpler expressions. In order to present such a case, let us assume that coherence collision operators may be neglected (as is frequently done by many authors). Then, operator becomes an ordinary function proportional to the Lorentzian profile with width determined by the homogeneous linewidth ΓC . The inhomogeneously broadened Doppler absorption profile is usually much broader. Then the Doppler approximation is possible, and it consists in approximating the relatively much narrower homogeneous profile by suitably normalized delta function. If this approximation is applicable, the integrals in (70) can be computed explicitly and we obtain (72) where we have denoted (73)

(74) The derived expression (72) for drift velocity within Doppler approximation is still independent of any collision model. The eigenvalues can be expressed via the transport coefficients, giving drift velocity in terms of measureable quantities. The described approach illustrates the possibilities inherent in our modelling technique. Drift velocity given by relation (72) has the properties similar to ones discussed above. The dependence of on Rabi frequency is the only difference. In the present, approximate case, does not vanish when χ gets very large. This is so, because Doppler limit is not valid when there is an appreciable power broadening. This restricts the Doppler limit to weak or modest light intensities. The presented Doppler limit corresponds to a restricted, but well-defined class of experiments and is mathematically much simpler than the general one. The Doppler-limit expressions, exhibiting the major features of LID may be, therefore, quite useful in many practical applications also ones different from LIKE. For example, calculation of fluorescence spectra [50] can be done within this approximation.

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Generalized correlation functions The correlations of various functions of velocity play an important role in the transport theory. We briefly show how these correlations can be dealt with in our approach. For illustration purposes, we first give a simple one-dimensional example. We consider an active-atom-perturber mixture in equilibrium, without any radiation field. For integers p and q, we define a time dependent correlation function of powers of velocity [47] as (75) where the double angular brackets denote averaging over the equilibrium velocity distribution, which in this simple case, is just a Maxwellian WA (v). Thus, we can write (76) Collision operator (for A–P collisions) appears here, since it is the one which governs the evolution of A atoms in equilibrium. Using the one-dimensional eigenfunctions ϕ n (v) of collision operator to expand the above expression and the scalar product (36) we transform the correlation function into (77) where we also employed the completeness and orthonormality of eigenfunctions. Taking ϕn (v) in the Cartesian coordinates (45), we can easily compute the scalar products (integrals). Using the generating function of Hermite polynomials [48] we obtain (78) with the integrals Jn (p) given as (79) Due to the properties of Hermite polynomials integrals Jn (p) vanish when p and n are of different parity. What is more important, integrals J n(p) vanish for n > p. Therefore, for given integers p and q, the sum in (78) contains only a finite number of terms. We conclude that the considered one-dimensional correlation function is expressed by a finite number of the eigenvalues of collision operator. The quantities of physical interest are then expressed as the time integrals over the function G eq , that is, via (80) and its a trivial matter to find such integrals from (78). Instead of considering a general case, let us focus on a particular example. The one-dimensional diffusion coefficient for A atoms in the perturber bath may be defined [14] as (81)

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Substituting expression (78) we note that it contains only one term (since p, q = 1), and we get (82) which is positive since the eigenvalue λ1 must be negative. This is a one-dimensional equivalent of relation (55), thus it proves the self-consistency of our approach. The generalization to three dimensions poses no difficulties. Other functions of velocity can also be easily analysed. Problems similar to our example are, however, not very interesting because they concern the equilibrium, when there are no excited-state atoms and the velocity distribution is Maxwellian. The situation changes when the A–P mixture is light irradiated. The system attains the stationary, but nonequilibrium state. Its time evolution is much more complicated, since it is governed by the operator L0 . For example, instead of a simple diffusion coefficient (82) we need to deal with the light-modified diffusion tensor (8). We will briefly discuss how the method outlined in the above simple case can be generalized for systems excited by incident light beam. Let and be two velocity dependent quantities. They may also have the operator character (as it is in case of gradient velocity), hence, their sequence may be of importance. By analogy to Eqs.(80) and (76) we define [52] the generalized correlation function as a double integral (83) The generalization consists in three points. First, we use evolution operator L0 corresponding to the Bloch-Boltzmann equations. Next, the averaging is now performed over the local stationary density operator . Finally, there is an additional term 〈b〉 ∞ denoting the average (84) where is the stationary state total velocity distribution of the active atoms as in (A.14). This average is necessary in Eq.(83) in order to ensure the convergence of the time integral, and it is due to the nonequilibrium features of the light-induced stationary state. Calculating g(a, b) we first apply the limiting procedure, as that leading to Eqs.(58) and (59). This yields the expression (85) The last terms have an obvious property (86) which follows directly from Eq.(84) and from normalization (4) of the density operator. Due to requirement (86) the limit in Eq. (85) is finite. Expression (85) can be considered as equivalent definition of the generalized correlation functions. Comparison of the definition (85) with previous results allows us to put the light-modified diffusion tensor and the gradient velocity into a single general scheme, so that they can be expressed as (87)

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Returning to generalized correlation function (85) we note that the expression in curly brackets can again be viewed as a Laplace domain solution to Bloch-Boltzmann equations with being the initial condition. Thus we are back to the already discussed computational procedure. The essential role is played again by Bloch-Boltzmann equations. The formal solutions (A.10) given in the Appendix with suitably chosen initial conditions can be used in (85). Then, expanding the functions of velocity in the eigenfunctions of collision operators we express the correlation function g(a, b) entirely via the physical parameters characterizing the irradiated system and via the eigenvalues. Since the latter are related to transport coefficients we obtain closed expressions for quantities od interest. Finally, let us discuss behavior of generalized velocity correlation functions in the case when the light field is absent. The limiting procedure is rather lengthy, so we state the final result, best expressed in a form expanded into the eigenfunctions (88) which reduces to a finite sum, when functions and have polynomial character. It also seems interesting that when the collisions do not distinguish between atomic internal (or, equivalently when for all multiindices α), the states, that is when general expression (85) reduces to (89) where the right–hand side is given in (88). This means that the A–P interaction ”does not see” excited atoms which collide with perturbers exactly as the ground–state ones do. Although due to irradiation we have for collisional processes all atoms behave as those in the ground state. For , the A–P collisional interaction is simply insensitive to the presence of excitation. Hence, in the stationary state the total velocity distribution must correspond to the equilibrium one, that is to the Maxwellian as in (A.16). The specific calculations based on Eqs.(87) for light-modified diffusion tensor (8) or gradient velocity (10) are fairly complicated, thus we give only a discussion of the results. The light-modified diffusion tensor consists of two contributions, one being the coefficient D 1 corresponding to the no-light-field case, and the second one displaying the interplay of collisional and radiative processes. When the irradiation specifies an axis of symmetry, the elements of diffusion matrix exhibit axial symmetry D 11 = D 22 = D t,

D33 = D p .

(90)

with D t and D p each being of the form D 1 term plus complicated functions of system parameters and eigenvalues corresponding to both atomic levels. The axial symmetry is preserved = 0). In absence of irradiation, or for excitation insenon resonance (when drift velocity sitive collisions the second term vanishes (as in (89)) and the elements of the diffusion tensor reduce to D 1 , as expected. This confirms the notion of light as a thermodynamic force. Very similar comments apply also to gradient velocity, for which we obtain This is an expected result, which fully agrees with the discussed idea of diffusive pulling of atoms towards the regions with higher light intensity. The components perpendicular to the incident light are different from which is parallel to the light beam, as it is required by the axial symmetry. Gradient velocity also does not vanish on resonance, but it tends to zero when χ → 0. The obtained formulas [52] for gradient velocity are quite involved. Their mathematical form does not bring anything new to the understanding of the physical effects, thus we do not present them here.

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Final remarks We have constructed a closed theoretical model allowing the description of physical phenomena occurring in the system consisting of a dilute active-atom vapor immersed in the much denser perturber gas and submitted to the external radiation field. The fact that radiative and collisional processes are much faster than the macroscopic light-induced flows allows the separation of the evolution of the considered system into two distinct parts. The first part, corresponding to macroscopic flows, is the slow one. The flows enter the generalized diffusion equation (5) and (6) which depends on the light-modified coefficients. These coefficients, in turn, are determined by the interplay between radiative and collisional processes occurring on a rapid time scale. Radiative processes are accounted for by means of the optical Bloch equations for a simple two-level atom. The employed model allows us to include all essential effects. It poses no problems to generalize Bloch equations either to an atom with more levels, or to a model with spatially degenerate levels specific to realistic atoms. More complex models, however, entail much more involved mathematics or can be dealt with only by numerical calculations. Comparison of theoretical results with experiments usually requires more elaborate models, but it seems that a two-level one is sufficient to give insight into the physics of the problem. Optical Bloch equations must be augmented by terms describing the collisions. We treat the dephasing collisions (influencing the internal atomic state) within the impact approximation by introduction of a homogeneous linewidth ΓC to the equations of motion for coherences, while velocity changing collisions (VCC) are treated by means of the classical linear Boltzmann equation which is valid when the perturber gas is much denser than the active-atom one. The arising terms are combined into a single concept – the collision operator. Collision operators are shown to be Hermitian in the suitably chosen space of velocity distributions. As such, they possess real eigenvalues and a complete, orthonormal set of eigenfunctions. We have discussed the theoretical approach towards determination of collision operators via the eigenvalues which were expressed by the transport coefficients. On the other hand, some experimental measurements were also done to determine the collision operators (or kernels) (see Refs.[53, 54], and the references given therein). The basic idea of these experiments is to excite a resonance transition in a velocity selective manner, and then to probe it by a second (usually weak) laser field by excitation to a higher third level. The spectra of fluorescence from this third level provide information about velocity redistribution in the first excited level. Haverkort et al [53] applied such a method to obtain the best fit to the Keilson–Storer kernels. Gibble and Gallagher obtained the necessary spectra and then, by means of an appropriate deconvolution procedure found the corresponding (one-dimensional) collision kernels. It is very interesting to relate the experimental fluorescence spectra directly to collision operators. An attempt in this direction, however with only one exciting- laser field, was presented in [50] . It may be expected that a general and possibly rigorous treatment of the fluorescence spectra corresponding to the experimental cases will be quite complex and probably will require a suitable approximation scheme. It is an interesting subject for further studies, and it seems, at least for the weak probe case, that it will be possible to express the spectra via the eigenvalues of the collision operators. Then, comparison of experimental spectra to the theoretical ones should directly yield the eigenvalues and hence the (approximate) collision operators according to Eq.(43). The collision operators corresponding to each element of the density operator of active atoms are then added to optical Bloch equations. The obtained set of equations of motion is called Bloch-Boltzmann equations (56) and it constitutes the essential theoretical framework

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of our approach since it gives the sought joint description of rapid radiative and collisional processes. They attain a local stationary, but nonequilibrium state, which is found as the steady-state solution to Eqs.(56). This solution, denoted by depends parametrically on local light intensity (or, equivalently, on local Rabi frequency). Hence, it is necessary to close the system of equations by the absorption equation (63) and (64). In such a way the description of the system is complete. Let us note, that even for simple two-level atom this system of equations is strongly nonlinear, mainly due to complicated behavior of light intensity within the sample. The formal solutions determining the quantities of physical interest are expanded in terms of the eigenvalues and eigenfunctions of collision operators. This results in relatively simple expressions depending only on a few eigenvalues of collisions operators. The eigenvalues, in turn, are connected with the transport coefficients following from the kinetic theory of gases. Therefore, the obtained results contain no free parameters, perhaps except the complex eigenvalues of the coherence collision operators. The latter, however, can frequently be neglected and left out of consideration. In such a case our approach gives a slightly simpler description of the considered physical system. Introducing our model we have discussed several underlying assumptions. They mostly concern the validity of the linear Boltzmann equation and the postulated form of the adopted eigenfunctions of the collision operators. On the other hand, our approach is not restricted by the field strength, detuning or other parameters. This allows easy and well controlled approximations, such as the Doppler one employed for the analysis of the drift velocity. Moreover, we are not restricted by any specific choice of an analytical model for VCC. We are able to avoid difficult questions, because our results involve the collision-operator eigenvalues which are directly expressible by the transport coefficients. The latter automatically include the interatomic potentials. This seems to be an advantage of our method since it provides a bridge between quantum optics and kinetic theory of gases. The usefulness of our method is illustrated by two simple examples. The number of effects studied within our formalism is much larger and it may be successfully used for investigations of other effects or different physical situations. We have described a method allowing investigation of various parameters and phenomena due to the interplay of radiative and collisional effects. We have argued that light may be viewed as a thermodynamic force driving macroscopic flows. We also stress the essential role played by VCC. Most of the effects disappear when collisions do not diuscriminate between atomic states. Within our model we have shown that local stationary total velocity distribution function reduces to the Maxwellian when the collisions do not depend on atomic internal state, that is when collision operators are the same for an atom in either of its states. One of the still unresolved problems concerns the coherence collision operators. They are frequently left out of the picture, that is, simply neglected. There is no known criterion indicating precise conditions which would determine the situation when omission of these operators is indeed justified. It seems, that finding such a criterion will require a complex fully quantum-mechanical investigations. Classical linear Boltzmann equation is insufficient because coherences are complex and do not classical analogues. Coherences are responsible for phase relationships between quantum states and thus must be treated quantum-mechanically. Our method allows modelling VCC for coherences via complex eigenvalues with negative real parts, but does not provide any tools for resolving the discussed problem. These are still open questions. Recently, some anomalous features of light-induced drift were reported [55, 56] . The simple analysis in previous sections shows that drift-velocity as the function of detuning ∆ is expected to have just one zero (see Eq.(72)). Anomalies consist in observation of more than one zero. The experiment [55] is performed in a perturber gas which is a mixture of two

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noble gases, and as the authors state, still requires some enhanced resolution, but indicates the strong dependence of anomalies on the composition of the buffer gas. The calculations of Gel’mukhanov [56] connect anomalies with the high temperature effects. It seems that our formalism may be extremely useful for investigations of such anomalies. The connection with Navier-Stokes formalism via the transport coefficients can be especially useful in studies of temperature effects. The detailed study of the generalized correlation functions as GreenKubo relations may also be useful in better understanding of light-induced macroscopic flows. Our approach seems to be helpful in establishing connections between quantum-optical and statistical points of view on light-induced phenomena. One more interesting problem is connected with nonlinear effects. The Bloch-Boltzmann equations determine the local stationary state due to radiative and collisional effects but description of the long-range, macroscopic effects requires both the diffusion equation (5) and the absorption one (63). When no simplifying assumptions are made, the equations are strongly nonlinear. It is of interest how the light intensity variations are combined with other phenomena. To our knowledge, the only one work devoted to this subject is due to Gel’mukhanov et al [57]. These authors consider the evolution of light intensity inside an infinitely long sample, while describing the VCC in the simplest strong-collision model (see Eq.(29)). They study both low and high field intensity. For the high field case they obtain and discuss the nonlinear Burgers which leads to soliton-like behavior of light-induced drift. The study of LIKE in gases in the complete manner is thus a separate and quite difficult problem which, to our knowledge, is almost unexplored. Hence, we conclude that the full (i.e., including absorption equation) description of LIKE is largely an open problem. Application of our approach due to its strong relations to kinetic theory may also prove quite useful in further, more detailed analysis of nonlinear effects occurring in the propagation of light across a gaseous mixture. This is, however, a possible subject for further studies. One more factor deserves attention. All the considerations presented here are restricted to bulk effects. That is, we have left out all surface, or boundary effects. Phenomena occuring at the walls of the container may strongly influence LIKE in gases. For example, accomodation coefficients different for an atom in either of its states may affect the lightinduced drift, or even lead to a surface LID [8] . However, all kinds of wall or boundary effects are entirely beyond the scope of this work. Concluding, we feel justified to say that despite the discussed limitations, the approach via Bloch-Boltzmann equations, eigenfunctions and eigenvalues of the collision operators seems to be quite elastic and applicable to the description of a large collection of experimental situations. Results presented in this review together with previously published ones seem to have considerable potential and a consistent, well justified physical basis.

APPENDIX Formal solutions to Bloch–Boltzmann equations. The Bloch Boltzmann equations (56) play an essential role in our approach. Their solutions determine varoius quantities of interest. Therefore, we briefly discuss the formal solutions and their properties. We consider here a general case in which a 2 × 2 matrix with a, b = 1,2, is assumed to satisfy Eqs.(56) with general initial conditions given by . We do not specify the meaning of the matrix R ab, it can be adjusted according to current needs. First, we transform the system of equations to Laplace domain by (A.1)

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where the tilde denotes Laplace transforms. After the transformation, the set of equations becomes algebraic (but includes initial conditions explicitly) and may be solved formally. We omit lengthy, but otherwise straightforward calculations in the Laplace domain, we only present the final results. The element is (A.2) The velocity distribution

is expressed via the already known element

as (A.3)

Similarly, the off-diagonal matrix element is (A.4) The auxiliary quantities depending on initial conditions are defined as (A.5) We have also introduced several auxiliary operators. They are defined as follows (A.6) (A.7) (A.8) (A.9) Selecting appropriate initial conditions, we can assign specific meaning to the obtained formal solutions and we can use them to construct the needed physical quantities such as, for example, the generalized correlation functions. The corresponding expression is (A.10) where [b] denotes the dependence of the matrix, which is a solutions of Bloch-Boltzmann equations, on initial conditions as discussed in the main text. The stationary solutions to Bloch-Boltzmann equations are obtained by multiplying Eqs.(A.2-A.4) by a factor s and taking the limit s → 0+ . It is easy to notice that (A.11) so that the term does not contribute to stationary solutions. The term requires some care. When the initial condition F 0 is expanded into the eigenfunctions, the zeroth So expansion term contains ϕ 0 = W A which belongs to eigenvalue zero of the operator the zeroth term in is of the form When the stationary limit is taken, this term is the only one which survives, and we have (A.12) and it gives the nonzero contribution to the stationary solutions.

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With the above remarks it is a simple matter to construct the velocity distributions They are given as corresponding to the local steady-state density operator (A.13) (A.14) (A.15) where the operators with subscript zero follow from (A.6–A.9) for s = 0. When the A–P collisions do not discriminate between the states of active atoms, then As a consequence, equation (A.14) yields (A.16) although there is nonzero field, and therefore Finally, we note that expansion coefficients as defined in Eq.(67) can be found from (A.13) and using the scalar product (36) we obtain (A.17) The eigenfunction taken in the Cartesian frame, according to (45). We can certainly also use spherical eigenfunctions, but for the case with axial symmetry the Cartesian ones seems to be more convenient due to their factorization.

Acknowledgment Partial support by Gda sk University through grant BW/5400-5-0305-7 is gratefully acknowledged.

References [1] A.B. Demtroder, Laser Spectroscopy – Basic Concepts and Instrumentation, Springer, New York (1982). [2] H. Haken, H.C. Wolf, The Physics of Atoms and Quanta. Introduction to Experiments and Theory, Springer, Heidelberg (1994). [3] S.G. Rautian, A.M. Shalagin, Kinetic Problems of Non–Linear Spectroscopy, North-Holland, Amsterdam (1991). [4] I.I. Sobelman, Atomic Spectra and Radiative Transitions, Springer, Heidelberg (1979). [5] S. Stenholm, Foundations of Laser Spectroscopy, Wiley, New York (1984). [6] L.J.F. Hermans, Light-induced kinetic effects in molecular systems, in: Quantum Optics and Spectroscopy, J. Fiutak, J. Mizerski, M. Zukowski, eds., Nova Science Publ., (1993). [7] M.C. de Lignie, E.R. Eliel, Coherent and incoherent light-induced drift, in: Quantum Optics and Spectroscopy, J. Fiutak, J. Mizerski, M. Zukowski, eds., Nova Science Publ., (1993). [8] E.R. Eliel, Adv.At.Mol.Opt.Phys.30, 199 (1993). [9] Light Induced Kinetic Effects on Atoms, Ions and Molecules, Proceedings of the Workshop, Marciana Marina, Elba, Italy, May 2-5, 1990, L. Moi, S. Gozzini, C. Gabbanini, E. Arimondo, F. Strumia, eds., ETS Editrice, Pisa (1991). [10] C. Cohen–Tannoudji, Atomic motion in laser light, in: Fundamental Systems in Quantum Optics, Les Houches Session LIII, J. Dalibard, J.M. Raimond, J. Zinn–Justin, eds., Elsevier Science Publ. (1991). [11] F.Kh. Gel’mukhanov, A.M. Shalagin, Pis’ma Zh.Eksp.Teor.Fiz.29, 773 (1979). [12] A.D. Antsigin, S.N. Atutov, F.Kh. Gel’mukhanov, A.M. Shalagin, G.G. Telegin, Opt.Comm. 32, 237 (1980).

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[13] G. Nienhuis, Phys.Rep.138, 153 (1986). [14] D.J. Evans, G.P. Morris, Statistical Mechanics of Nonequilibrium Liquids, Academic Press, London (1990). [15] S.J. van Enk, G. Nienhuis, Phys.Rev.A 46, 1438 (1992). [16] G. Nienhuis, S. Kryszewski, Phys.Rev.A 36, 1305 (1987). [17] S. Kryszewski, G. Nienhuis, Phys.Rev.A 36, 3819 (1987). [18] G. Nienhuis, Phys.Rev.A 40, 269 (1989). [19] S.J. van Enk, G. Nienhuis, Phys.Rev.A 41, 3757 (1990). [20] S.J. van Enk, G. Nienhuis, Phys.Rev.A 42, 3079 (1990). [21] S.J. van Enk, G. Nienhuis, Phys.Rev.A 44, 7615 (1991). [22] H.G.C. Werij, J.E.M. Haverkort, J.P. Woerdman, Phys.Rev.A 33, 3270 (1986). [23] G. Nienhuis, Phys.Rev.A 31, 1636 (1985). [24] F.Kh. Gel’mukhanov, G. Nienhuis, T.I. Privalov, Phys.Rev.A 50, 2445 (1994). [25] B. Nagels, M. Schuurman, P.L. Chapovsky, L.J.F. Hermans, Phys.Rev.A 54, 2050-2055 (1996). [26] E. Bu , Hrade y, J. Slovák, T. Têthal, I.M. Yermolayev, Phys.Rev.A 54, 3250–3253 (1996). [27] S.N. Atutov, S.P, Pod’yachev, A.M. Shalagin, Opt.Comm.57, 236 (1986). [28] S.N. Atutov, S.P. Pod’yachev, A.M. Shalagin, Zh.Eksp.Teor.Fiz.91, 416 (1986). [29] F. Wittgrefe, J.L.C. Saarloos, S.N. Atutov, E.R. Eliel, J.Phys.B 24, 145 (1991). [30] L. Allen, J.H. Eberly, Optical Resonance and Two–Level Atoms, Wiley, New York (1975). [31] C. Cohen–Tannoudji, J. Dupont–Roc, G. Grynberg, Atom-Photon Interactions, Wiley, New York (1992). [32] G. Compagno, R. Passante, F. Persico, Atom–Field Interactions and Dressed Atoms, Cambridge Univ. Press, Cambridge (1995). [33] P.R. Berman, Phys.Rep.43, 101 (1978). [34] R.F. Snider, J.Ch em.Phys.32, 1051 (1960). [35] P.R. Berman, Adv.At.Mol.Phys.13, 57 (1977). [36] C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York (1988). [37] J.H. Ferziger, H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland, Amsterdam (1972). [38] R.L. Liboff, Introduction to the Theory of Kinetic Equations, Wiley, New York (1969). [39] J.R. Taylor, Scattering Theory. The Quantum Theory of Non-relativistic Collisions, Wiley, New York (1972). [40] Tak–San Ho, Shih–I Chu, Phys.Rev.A 33, 3067. [41] P.R. Berman, T.W. Mossberg, S.R. Hartmann, Phys.Rev.A 25, 2550-2571 (1982). [42] J. Keilson, K.E. Storer, J.Appl.Math.10, 243 (1952). [43] R.F. Snider, Phys.Rev.A 33, 178 (1986). [44] S. Kryszewski, G. Nienhuis, J.Phys.B 20, 3027 (1987). [45] P.R. Berman, J.E.M. Haverkort, J.P. Woerdman, Phys.Rev.A 34, 4647 (1986). [46] S. Kryszewski, Eigenexpansion of collision operators for description of light-induced effects in gases, in: Lasers – Physics and Applications, A.Y. Spasov, ed., World Scientific, Singapore, p. 104 (1989). [47] S. Kryszewski, G. Nienhuis, J.Phys.B 22, 3435 (1989). [48] H. Bateman, A. Erdelyi, Higher Transcendental Functions, vol.2, McGraw–Hill Book Co., New York, (1953). [49] C. Cohen–Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Wiley, New York (1977). [50] S. Kryszewski, G. Nienhuis, J.Phys.B 24, 3959 (1991). [51] S. Kryszewski, J. Gondek, to appear in Phys.Rev.A (1997) . [52] S. Kryszewski, unpublished (1997). [53] J.E.M. Haverkort, J.P. Woerdman, P.R. Berman P.R., Phys.Rev.A 36, 5251 (1987). [54] K.E. Gibble, A. Gallagher, Phys.Rev.A 43, 1366 (1991). [55] F. Yahyaei–Moayyed, A.D. Streater, Phys.Rev.A 53, 4331 (1996). [56] F. Kh. Gel’mukhanov, A.I. Parkhomenko, J.Phys.B 28, 33 (1995). [57] F.Kh. Gel’mukhanov, J.E.M. Haverkort, S.W.M. Borst, J.P. Woerdman, Phys.Rev.A 36, 164 (1981).

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TEMPORAL AND SPATIAL SOLITONS: AN OVERVIEW

A D BOARDMAN, P BONTEMPS, T KOUTOUPES and K XIE Photonics and Nonlinear Science Group Joule Laboratory Department of Physics University of Salford Salford, M5 4WT United Kingdom

Abstract This chapter contains a fundamental review of envelope temporal and spatial solitons. A substantial effort has been made to give an account of both the historical background and the physical concepts. Mathematical detail is given to justify the generic nonlinear equations and guide to the inverse scattering method is presented. 1. INTRODUCTION Long ago, in August 1834 [1] John Scott Russell, a naval architect, was working for the Scottish Canal companies to establish the possibility of rapid steamboat transit on canals. As part of this investigation, he was observing a boat being pulled along, rapidly, by a pair of horses. For some reason, the horses must have stopped the boat rather suddenly. What happened next was to change science in the most dramatic way. The stopping of the boat caused a very strong wave to be generated. This wave, in fact, a significant hump of water stretching across the rather narrow canal, rose up at the front of the boat and proceeded to travel, quite rapidly, down the canal. Russell, immediately, realised that the wave was something very special. It was 'alone', in the sense that it sat on the canal with no disturbance to the front or the rear; nor did it die away until he had followed it for quite a long way. The word 'alone' is synonymous with 'solitary' and, Russell soon referred to his observation as the Great Solitary Wave. The word solitary is now routinely used, indeed even the word 'solitary' tends to be replaced by the more generic word 'soliton'. Once the physics behind Russell's wave is understood, however, solitons, of one kind or another, appear to be everywhere but it is interesting that the underlying causes of soliton generation were not understood by Russell, and only partially by his contemporaries.

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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Russell, after graduating in 1824, at the age of sixteen, was a brilliant observer, a beautiful writer and had a great gift for lecturing. It is not surprising, therefore, that he also carried out very careful laboratory experiments on shallow water containers, long enough to see the generated waves evolve. Incidentally, it is incredibly simple to do this; all that is necessary is to strike the water at one end of the water tank with moderate force. What Russell saw was that the solitary waves have a speed proportional to their amplitude and that they pass through each other [2] without destruction, or change. These results did not agree with the work of Airy [3] who asserted that large amplitude waves would self-steepen and break up. Solitary waves were not predicted so something was missing from the theory. In the meantime, in 1847, Stokes [4] worked out that, for deep water, CW (periodic) waves (not localised) with finite amplitude can exist without breaking up i.e. in mathematical language they have permanent form. What then was missing from the earlier by Airy with respect to localised waves? The answer is strikingly simple, once the point is realised. The explanation of the observations of Russell needed a study of the delicate balance between dispersion and nonlinearity. This point was appreciated by both Boussinesq (1871) [5] and Rayleigh (1876) [6] but even they never arrived at what we now term a nonlinear partial differential equation, for which Russell’s wave is a solution. It took until 1895 for two Dutchmen, Korteweg and de Vries [7] to produce this differential equation. One of the solutions of the KdV equation is the solitary wave observed by Russell. Normally, a hump of water like Russell’s solitary wave is thought of as a packet of waves, all travelling with different speeds. This way of looking at things comes from Fourier, but it is a linear viewpoint with the end result being the destruction of the hump due to dispersion. Large amplitudes mean lots of power, so the waves in the packet are now forced to interact with one another. The forces trying to restore equilibrium are no longer just proportional to the height of the wave and the speed now depends on displacement: this is a nonlinear system. In a balanced situation, the power of the wave acts against dispersion. The wave then remains intact and a soliton is born! As is sometimes the way of science, this nonlinear work was then ignored as being of marginal interest. Indeed, the natural philosophers of the time seemed blissfully unaware that our world is really rather a nonlinear one. Russell, on the other hand, was convinced of the importance of his wave, and its curiosity, right up to his death in 1882. But it took much later research on waves in crystal lattices, by Zabusky and Kruskal in 1965, to make plain [8] the elegant generic nature of Russll’s observation to a wider public, and to emphasise that nonlinearity is important to the propagation of such waves; Zabusky and Kruskal realized that the form of their equations was exactly like that of Kortweg and de Vries. A really striking feature of the 1965 work, however, was in demonstrating that solitary waves retain their shape (this time in a crystal), even after colliding with each other. It is fascinating that Russell had quietly observed this 130 years earlier. The 1965 authors invented the name soliton for these waves, just to emphasize that although a soliton is a solitary wave it retains its identity, even after a collision. The change of ‘ary’ to ‘on’ is not surprising, incidentally - physicists call things ‘ons’ at almost every opportunity (electron, meson and photon, for instance) because the Greek word ‘on’ means solitary, and in each instance the ‘on’ suffix signals the principle that the entity concerned retains a characteristic particle nature. The fundamental characteristic, for us, is that a soliton can exist that retains its shape. The fundamental shape of Russell’s soliton happens to be a sech2 function but that it is not important, at the moment. The real point is that Zabusky and Kruskal ended up, also, with the Korteweg-de Vries equation, even though they were working on discrete crystal lattices. In other words, they got what Russell saw on shallow water. They got what is now known as a Korteweg-de Vries soliton.

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In broad, physical terms, nonlinear water waves can easily be appreciated, by glancing at figure 1, which is a sketch of what can happen to water waves approaching a shore. Under certain conditions, to be discussed, properly, later on, dispersion and nonlinearity will balance and then solitons on the water surface will appear. Otherwise, a variety of unbalanced conditions occur. SOLITONS IN WATER • Waves approaching a shore (beach) • Wave speed has small ‘nonlinear’ dependence on height so crest travel faster • Water waves also exhibit dispersion - as the water behind it disperses ‘rounding’ the edge • In the right conditions dispersion & nonlinearity balance Figure 1. Sketch to illustrate some water wave behaviour. Note that solitons are readily generated on the sea.

Achieving this balance can be thought of in a crude sort of way but it is really quite a sophisticated step by Nature, as will be shown later. In fact, it is a chirp balance that is operating. Chirp, although it is a concept that arouses interest, immediately, will not be defined, at this stage, however. It is better to introduce it later on. Earlier on, in this chapter, the Russell observation was referred to as a solitary wave and the word soliton was, confidently, introduced. Some sort of qualification of this language is now necessary, before passing on to greater detail. Basically, it is quite simple. If solitary waves pass through each other, experiencing nothing other than a phase shift, they are elevated to the status of solitons. Of course, linear beams and pulses pass through each other without any change but this is to be expected because linear physics permits simple superposition. The big surprise comes when nonlinear beams and pulses do not destroy each other when they interact. In the interaction region, nonlinear forces operate forbidding the process of simple superposition. Self-destruction is expected but solitons survive this. Solitary waves, or solitons, also come in various forms, some of which are shown in figure 2. Some of the most famous members of this large, extended, family are • Korteweg-de Vries:first (and easily) observed on shallow water by Russell and has a velocity that is proportional to its amplitude easily observed on deep water and is a solution of the nonlinear • Envelope: Schrödinger equation and has a velocity independent of amplitude. The most famous application is in optical fibres but they are the ones that will be the centre of attention here for both pulses and beams dislocations in solids are described by these solitons and are kinks or • Sine-Gordon: anti-kinks with velocities that are independent of amplitude.

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Solitary Waves or Solitons?

Figure 2. Illustrations of solitary waves and some useful comments.

Nonlinear equations are classed as integrable or non-integrable [2] and the integrable ones have soliton solutions that are preserved during collisions. Non-integrable equations can have, for certain parameters, solitary wave solutions but these are not always preserved during collisions. The point about solitons is that they are among Nature’s generic entities, i.e. although their first observation was on shallow water they also turn up in optical fibres, in the same form as those on deep water, and should be looked for wherever dispersive excitations, or waves, can be created. In order to be selective, this chapter concentrates on bright envelope solitons, which come in the form of temporal pulses or spatially distributed beams. These are not Russell’s solitons but another member of the family. Dark solitons (black holes!) are also possible but they will be introduced only briefly here. A typical, undamped, evolution of what is called a findamental bright envelope soliton is shown in figure 3 in which a pulse with arbitrary (normalised) intensity is selected. It varies with time across its intensity distribution and the plot shows the shape of the pulse with distance due to propagation. Figure 3, deliberately, for the benefit of this discussion, has no units attached to it. If a specific system is in mind, however, and that system is in the form, for example, of a thin film, then the units, i.e. the scales will, of course, matter. This plot will apply, equally, to a beam. All that is necessary to make it applicable to a beam is to replace time with distance across the beam.

Figure 3. The fundamental [lowest-order] bright envelope soliton, in the absence of damping.

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2. DISPERSION AND AND TEMPORAL SOLITONS A temporal pulse is thought of, in a linear regime, as a superposition of waves with different frequencies so that a spectrum of frequencies (its bandwidth) exists around ω0 . If this bandwidth is ∆ω, then the condition ∆ω << ω 0 is necessary, if the concept of carrier frequency is to mean anything. In other words, although, in principle, a wave packet [pulse] can be decomposed into Fourier components, their amplitudes are small [negligible] outside ∆ω . Hence, as shown in figure 4, ω(k) is represented rather well by a Taylor expansion about ω0 . In fact, figure 4 shows that the linear dispersion for a vacuum contrasts rather nicely with the typical (ω,k) variation that can occur for a material. It is the presence of a finite value of

in the Taylor expansion that causes pulse

dispersion, i.e. spreading with local time as the evolution progresses. It is important to observe the following points [10] •

the group velocity, Vg ( ω ), depends upon the frequency

•

the arrival times of the frequency components spread around ω 0 , at a particular point z, are

•

for a bandwidth ∆ω, the spread in arrival times of all the ω values is equal to the pulse width ∆τ, where

Figure 4. Comparison of pulse behaviour in a dispersionless (vacuum) medium and a medium with material dispersion.

In figure 4 local time is measured across the pulse in a frame of reference moving with a speed equal to the material group velocity, vg , hence, local time is

where t is

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real time. We will, in fact, make the transformation

→ t and use this new t as

a definition of local time, unless specified otherwise. It is always important to check in any problem whether the local time or the laboratory frame time is being used. This local time model is a standard and convenient representation. If a laboratory frame is used then all that will happen to figure 4 is that the pulse cross-sections will be centred over a line drawn at an angle to the z-axis. This is, quite simply, inconvenient! Finally, the local time is plotted from left to right, taking t = 0 at the pulse centre.

Figure 5. Pulse spreading in the time-domain due to material dispersion in a linear medium. There is not a corresponding spreading in the frequency-domain.

Figure 5 proves, rather dramatically, that it is the existence of a finite value of

and hence

that causes dispersion. In optics,

is called the group-velocity

dispersion. This is so-called because Vg( ω) is a function of ω and this alone causes arrival time spreading of signals. Dispersion is a material property so a pulse in a vacuum [11] can also be said to be a solitary wave also! While this is strictly correct we can choose any pulse shape to travel in a vacuum. Hence, the vacuum ‘solitary wave’ is not unique, in the sense that the stationary solution of the KdV equation, or the nonlinear Schrödinger equation, is unique. Also pulses propagating in a vacuum can pass through each other but no nonlinearity is involved so it is pointless to consider them as solitons. To take this discussion further, Figure 6 gives a few examples of what can happen to a temporal pulse, or spatial beam, because of the presence, or absence, of dispersion, or nonlinearity.

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F’igure 6. Solitary waves: to be or not to be! That is the question! [with apologies to William Shakespeare]

3 . CHIRPING AND BRIGHT PULSE (TEMPORAL) SOLITON FORMATION At this stage of the discussion, it is useful to return to the concept of chirp. The discussion begins with figure 7, which summarises the idea of phase.

Figure 7.

Summary of the concepts of phase and chirp parameter.

Suppose that the time (local, or otherwise) is t and that p = ωt. The definition that phase is p = ω t, is a very familiar one. The definition that

seems, at first sight, to be

unnecessary, for such a simple relationship. Why not just use p/t? Such a step leads, immediately, into error, if p is not as simple as ωt. The problem is that it is perfectly possible for the frequency to deviate from the carrier frequency, as time is swept across a pulse i.e. the instantaneous frequency can be lower (higher) than the centre (carrier) frequency as the front, or back, end of the pulse is reached. It is perfectly possible for a smooth change to be established through the existence of a so-called chirp parameter C that can be positive or negative. For interest, the historical background of the concept of chirp is summarised in figure 8. In a Bell Laboratory report in 1951, B M Oliver [12],

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referring to work on frequency-modulated radar, introduced the use of chirping. The aim was to produce compressed radar signals and a famous remark, attributed to him, is "not with a bang but a chirp". The word chirp is, perhaps, more obviously attributable to birds, because they emit chirped (frequency-modulated) pulses as a matter of course. To recognise that what birds, and bats [13] do, naturally, lies at the heart of our particular soliton work can be said to be an example [12] of Pasteur’s view that " fortune favours the prepared mind". The original idea in the B M Oliver report is, first, to generate a square pulse envelope, in contrast to a continuous wave, which is uninterrupted, and has a frequency ω 0 , for example. A rectangular or square pulse envelope simply ‘chops’ off the wave, front and rear, encapsulating a carrier, which has a frequency ω 0 , all the way across the pulse. In other words, in the time-domain ω = ω 0 is not a function of time across the pulse i.e. it is unchirped. This situation can be changed by making the frequency vary with time across the pulse, A linear 'frequency ramp' is shown in the second segment of figure 8 and the frequency variation (chirping) that is established across the pulse is shown in the last segment of figure 8.

Figure 8.

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Basic idea on chirped radar put forward, long ago, by Oliver.

Unchirped input pulse: same colour through the pulse same proportion of frequency everywhere

Anomalous-Dispersion Regime: higher frequency travels faster than lower frequency leading edge moves towards the blue trailing edge moves towards the red

Figure 9. Illustration of pulse broadening in a dispersive medium. Time increases from left to right; the envelope is the bell-shape that encloses the pulse oscillations; it is the locus of the extreme values of the amplitude.

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Nonlinearity changes the refractive Index results in a phase profile across the pulse

results in a frequency shift across the pulse

In presence of Anomalous-Dispersion leading edge has concentration of lower frequencies (red): slows down tail has concentration of higher frequencies (blue): speeds up

Figure 10. Illustration of nonlinear pulse propagation in a dispersive medium.

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Figure 11. Diagram designed to illustrate the competing roles of dispersion and nonlinearity in preserving the shape of a fundamental (lowest-order) envelope temporal soliton. Time is measured left to right.

Since, in this section, the emphasis is on dispersion as a material characteristic, the discussion of chirp is conveniently centred upon temporal pulses. If dispersion was unimportant, however, the emphasis would switch to beams which invoke diffraction. It is diffraction as a beam characteristic that leads, through the introduction of a spatial frequency (a wavenumber), to an identical argument concerning spatial chirp. A simple analysis shows that spatial frequency chirps are introduced into the spatial beams as they are transmitted. Having made this point clear about the spatial analogue, the arguments will now return to the temporal case. The optical fiber (fibre) is the usual transmission vehicle [14] for temporal envelope solitons, but other waveguides can perform this task as well. It all depends on the state of the dispersion and the magnitude and type of the nonlinearity. More details will be given later but the ability to transmit a temporal soliton (usually at optical frequencies) depends upon the sign of

[the group velocity dispersion] and γ [the nonlinear coefficient].

γ has not been mentioned before but, as the pulse power increases, nonlinearity affects the wavenumber (or frequency)by shifting it in value by an amount proportional to the power. γ is simply the constant of proportionality. For the time being, all materials to be

considered here have γ > 0, so it is the sign of Lighthill who discovered that, for a carrier frequency condition for soliton existence. If dispersion and if

that counts, Indeed, it was is a necessary the material is said to have normal

the material is said to have anomalous dispersion. Figures

9, 10 and 11 show the effect of dispersion and nonlinearity on pulse propagation, together

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with a diagram that shows how a balance is obtained. Figure 9 shows that in an unchirped pulse, in a linear medium, the ‘proportion of frequency’ is the same everywhere in the pulse. In an anomalous dispersion regime, the higher frequencies move faster than the others. Hence, considering the bandwidth of the pulse, arising from the envelope shape, higher frequencies will arrive first, causing a blue → red distribution i.e. chirping. In a purely nonlinear medium the opposite occurs. If there is nonlinearity and dispersion, chirping, because of the nonlinearity, causes a red → blue distribution, at the beginning of the passage of a pulse, so the tail of the pulse speeds up. In an unbalanced state, therefore, a nonlinear pulse in a dispersive medium could compress if the power is high enough. Figure 11 shows how a balance can be interpreted by reading ‘frequency effects’ upwards/downwards and ‘time effects’ forwards/backwards. The figure can be used in the following way. Chirping arises from both nonlinearity and dispersion and the balance achieved in the fundamental soliton keeps the frequency distribution (proportion of frequency) evenly distributed across the pulse. Suppose that anomalous dispersion disturbs the frequency distribution. In an anomalous dispersive medium, high frequencies (blue) travel faster [14] and so arrive earlier. This is shown in the sketch as an arrow pointing to earlier times. Nonlinearity, on the other hand, lowers lead frequencies and this is shown by the downward frequency arrow at the lead end of the pulse. Once lowered, these frequency components will travel slower and arrive later, as required by an anomalous dispersive medium, Upon being redistributed to the tail of the pulse the nonlinearity then proceeds to increase the frequencies, once again. The bottom line is that going around this loop results in the balanced state that is the envelope soliton. Both material dispersion and power level (nonlinearity) add a chirp to an evolving pulse without external intervention, In the original radar example, the technique was used with a delay line to compress pulses i.e. because different frequencies have different propagation times through the delay line, delaying the first arrivals causes ‘a traffic jam’ and, hence, a compression. As an illustration of the behaviour of pulses in a material, figures 12-14 contain numerical simulations that show in detail what happens as a pulse evolves.

Dispersion of a Sech Input in a Linear Medium

Figure 12. Dispersion of a pulse in a linear medium, The figures refer to beams if time is replaced with a spatial coordinate.

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Figure 12 shows that, in the absence of any nonlinearity, i.e. no power dependence of the frequencies, a pulse will disperse. As can be seen the phase p, which is a constant across the pulse at the beginning of the propagation develops a local time dependence as the evolution proceeds. The derivative

where t is the local time, is the chirp and across

the centre of the pulse a very clear linear chirp develops. The slope is negative so the type of dispersion selected is called anomalous and the chirp parameter is said to be negative. If the medium could be purely nonlinear then a significant power level is possible, without dispersion! Of course, this cannot ever be completely true [dispersion is usually present] but it is, nevertheless, more or less possible, especially at the start of a propagation. It is illustrated in figure 13 to make an important point.

Behaviour of Sech Input In A Purely Nonlinear Medium

Figure 13. Behaviour of a pulse, in the time-domain, in a purely nonlinear medium. Replacing time with space makes the figures appropriate to beams.

This point is that the action of nonlinearity alone does not compress the pulse in the timedomain. Indeed, in the time-domain the pulse progresses without change of shape. The phase behaviour, on the other hand, is very interesting. A positive chirp develops as the evolution progresses, What then happens in a nonlinear dispersive medium? The answer is that it is possible for the chirps from dispersion and nonlinearity to cancel, precisely. When this happens, a sech-shaped eigensolution of the system is formed and it this which we will refer to as an envelope soliton. The solitons are nonlinear states of the system and can exist in various forms, dependent upon the energy available and it is the lowest-order one that is shown in figure 14. Clearly the chirps cannot cancel if hardly any energy enters the material so it is to be expected that some power threshold, for a sech(t) pulse, should be reached before a chirp cancellation, and hence a soliton, will appear. That this is, indeed, the case is shown in figure 15, where a 0.2 sech(t) input disperses, completely, and a 0.8 sech (t) input does not. The 0.8 sech(t) input sheds (disperses) energy it does not require but, because it is above the threshold energy for soliton formation, then goes on to form a 195

stable, lowest-order soliton. Another important point is that, if the threshold energy level is exceeded, the precise input shape does not matter. A dramatic example is given in figure 16 in which a rectangular input pulse, above the threshold energy, "gives birth to a soliton".

Figure 14. How the lowest-order, envelope, soliton arises.

Figure 15. A numerical check on the power needed to create the lowest-order, envelope soliton. Note that local time need only be replaced by distance for these figures to relate to beams.

It is interesting that the emerging bright envelope soliton has the sech shape it is required to have, as a solution of what we have been calling the nonlinear Schrödinger equation. 196

This equation will be shown below, to govern the behaviour of pulses and beams. It is an equation that has rather special properties because an Nsech(t) input generates a fundamental soliton if N = 1 and higher-order solitons if N > 1. Technically, fundamental solitons are of great interest because they do not change their shape as they propagate i.e. they remain sech-shaped. Higher-order solitons N = 2, N = 3,… change shape [they breathe but, fortunately, so does the chirp!] as they propagate but keep returning to Nsech(t), periodically. The period is known as the soliton period and is a useful length scale of the system. Apart from that, higher-order solitons are not expected to be useful in a switching device or communication system.

Figure 16. A Square input pulse gives birth to an envelope soliton . The transverse direction is time (pulse) or distance (beam).

This is because a higher-order, bright, envelope soliton is a bound state of N =1, sechshaped, fundamental solitons, with zero binding energy, so it is extremely vulnerable to perturbations. Hence an N = 2 soliton, for example, could degenerate quickly to two N = 1 solitons. This means that N is the soliton content or soliton number for a given input pulse. Figure 17 contains a diagramatic explanation of what happens when either a Bsech(t) input pulse is used for a nonlinear system described by the nonlinear Schrödinger equation, or a rectangular input pulse B rect(t). The soliton "content" is shown as a function of B. For example, if a pulse Bsech(t) is entered into a dispersive material then, provided 0.5 < B < 1.5, a fundamental soliton is created. On the other hand, if B rect(t) is entered then a fundamental envelope soliton pulse is created only if 0.5π < B < 1.5 π . These conclusions come from an exact mathematical treatment of the nonlinear Schrödinger equation, using inverse-scattering theory [15] [IST] that is to be discussed later on. The conclusions are easily confirmed out by numerical simulations.

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Figure 17(a). How solitons are created from Bsech(t) input pulses or Brect(t) input pulses.

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Figure 18 Simple idea of diffraction.

4. DIFFRACTION, SELF-FOCUSING AND BRIGHT SPATIAL SOLITON FORMATION We are well aware that a beam of light, even if it is emerging from the now familiar laser pointer used by lecturers, will spread out and change shape as it propagates (see 199

figure 18). In air, this is all that will happen. But what about light propagation in nonlinear media? In this case light always heads for the region of highest refractive index (think of how a lens works), and a nonlinear material, such as a quantum well AlGaAs structure, forces the index up, precisely, and only, where the beam is travelling. This is just what is needed to stop light rays from inside the beam from bending outwards and causing a change in beam shape. When beam spreading is exactly stopped by the nonlinearity, a spatial soliton [16] is formed and the beam can now be used as a stable optical guide, just like the familiar optical fibre. The spatial soliton is, however, better that the fixed, ‘hard’ waveguide such a fibre constitutes. The reason is that it is a ‘soft’ guide that is easily tunable by changing the beam size. Many applications can be imagined for these novel waveguides, such as using them for optical wiring, to steer one beam with another in addressable arrays, and to act as switching devices and as logic units for a whole range of information processing and computing. Spatial solitons [16,17,18-30] are a significant outcome of modern soliton and materials science because of the ease with which they can be manipulated. Although there are still worries about how fast many materials can react to light, or changes of beam direction, work on spatial solitons should herald an era of new, all-optical processing devices that are cheap and easy to implement. Russell’s observation did not change physics or technology in his own century nor most of our own. But with the real possibility of optical soliton communication systems and all-optical processors now on the horizon, it may well transform them next century. It is many years now since the basic physics, concerning the self-trapping [30-32], of a powerful beam of light in a nonlinear medium was thoroughly discussed. Some of the imagery evoked in those early, pioneering, discussions will be produced here, through the consideration of a beam of light with a rectangular distribution of energy, a radius r and a wavelength λ , propagating through a dielectric, non-magnetic, medium that has a linear refractive index n 0 . First, as stated above, the beam will want to spread out, due to diffraction. Indeed, at a given radius r the light (laser) beam can be imagined to act as its own aperture and rays of light will diffract through it, with an approximate diffraction angle θ d = λ/2nr. If the material becomes nonlinear, in a Kerr-like manner, [i.e. third-order only, with nonlinearity proportional to intensity], with a nonlinear coefficient α , then the refractive index of the linear medium will change, by the small amount α|E|2 , where E is the electric field carried by the beam. If the medium has a positive α then the refractive index increases to n + (α|E| 2) and, as shown in figures 19 and 20, it is possible to balance the tendency of the beam to self-diffract with a nonlinear tendency to self-focus. In these figures can be seen a simplistic, use of diffraction through a circular aperture of radius r, but the example is designed to illustrate what nonlinearity can do [30,31] to achieve diffraction-free beams. In figure 19, rays setting out from the beam axis will either escape, or be totally internally reflected. This approximate model uses an aperture to create a beam of radius r. Since diffraction does not permit a geometrical transmission, rays making an angle θ c with the axis are totally internally reflected at the beam boundary but other rays can escape the geometrically defined boundary. In principle, therefore, beam broadening occurs. Because the electric field E of the beam causes a nonlinear change in refractive index, the critical angle depends upon E and, hence, upon power. As figure 20 shows, most of the beam power is enclosed [30-32] within ray angles making an

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Figure 19 Showing that the critical angle depends upon electric field or power.

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Figure 20 Illustration of circular hole/single slit diffraction plus introduction of the concept of critical power.

angle

to the beam axis. Figure 19 is not completely true (it lacks rigour) [33]

but we ought to be able to learn quite a lot from it! The main idea being introduced is that if the aperture is circular then 2θd is the direction of the first minimum in the (far field) Fraunhofer diffraction pattern. This means that the bulk of the energy in a diffracting beam is, more or less, associated with rays making an angle of θd, or less, to the horizontal axis. For a single slit the first minimum in the pattern should be at θd but, of course, the intensity will not be zero on the scale of figure 20 because the distance between the slit and the screen is not infinite, as it should be for (Fraunhöfer diffraction: far field), when the source is at infinity. For figure 20, the observation plane is rather close to the circular hole, or slit, so the actual geometrical shadow if it would also be wider than that which is shown. The greater the distance between hole/slit and screen the more the Fresnel (: near field) pattern goes over to a Fraunhöfer pattern. These points need not worry us in this kind of what we call a ‘hand waving’ or ‘ball park’ exercise, so we will set

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(approximately) and use it to get an estimate of some distance [30] LD at which a beam can double in width, due to diffraction. Since this is the case, then θc = θ d is the critical condition for the beam to have its diffraction killed by the nonlinearity. This leads to a critical power Pc , which, rather importantly, does not depend upon r. Another way to look at this problem is to introducc the concept of nonlinear length LNL, as demonstrated in figure 21. At balance, the diffraction balances the self-focusing and diffraction-free beams are the result. Such diffraction-free beams are called self-trapped and are spatial solitons. Troublesome questions now arise about Pc. Can it be controlled to produce stable spatial solitons?

Figure 21. Introducing diffraction length (LD ) and nonlinear length (LN ) .

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The important feature is that, if P > P cr , rays are forced towards the axis of the beam, in a self-focusing action. If P < Pc r, then diffraction takes over. These properties of the medium occur because Pcr is fixed, i.e. ε0, n, c, α and λ are fixed parameters of the system and it is necessary for any launched input power P to be precisely equal to P cr for stability to occur [21,23,30,32]. Inevitably, in any real experiment the power launched into a waveguide will be either P < Pcr or P > P cr, even if the difference is only infinitesimal. In either case, such a fluctuation will lead to instability taking the form of a 'runaway' to diffraction or to self-focusing. This, then, is the problem with self-trapped beams in which diffraction and self-focusing is finely balanced in a bulk (infinite with no boundaries) medium. Mathematically, it means that such beams are, by their nature, unstable in a multi-dimensional system and this has been established, for a long time, in the literature. The situation described above, although interesting, in principle, at first sight appear to have very little experimental promise because of its instability. Some forms of stabilisation have been proposed, however, that are proving to be rather successful. The first stabilisation method involves controlling the perturbations that can lead to instability by a method based upon a modulation due to interference fringes [34]. This is, relatively, complicated so it will not be discussed here because the aim of this chapter is to study straightforward spatial solitons, which can be created within [16] a nonlinear planar waveguide. This procedure reduces the propagation to a one-dimensional diffraction. Given the propagation distance as the other dimension this is then called (1+1) propagation. The instability, referred to above, that arises in a three-dimensional medium has a P c r that is fixed, in the sense that self-adjustment of the beam such as its radius, while focusing, cannot change it. On the other hand, if a beam is launched into a planar waveguide that has interfaces parallel to the x-z plane, then guiding confinement will occur in the y-direction. Suppose then that the guide is weakly nonlinear, so that the modal field intensity is unaffected by nonlinearity, to first order. In this case, the power needed for diffraction and self-focusing to balance becomes [21]

where 2d is the thickness of a guiding layer with linear refractive index n0 . r is now the largest beam dimension and πrd is, approximately, the area of a beam with most of its energy confined within an elliptic cross-section, and within the guide. This is shown in Figure 22, which also contains a schematic explanation of the stability. This new value of P cr does not now arise from the cancellation of r2 but has, left within it, a factor

A s

r changes, due to attempts to self-diffract or self-focus, the system is always forced back to stability. This is the elegant feature of a waveguide being able to induce (1+1) propagation and created some excitement in the field [16] when it was announced. Naturally, since open waveguides, made of transparent dielectric materials, are used, real beams carry some part of their energy outside the waveguide. This is because tangential field components are continuous at

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Figure 22 Stable beam propagation in a planar waveguide .Launched powers P < Pcr or P > Pc r return towards P = P c r. Diffraction/self-focusing balance in the x-direction; linear guiding in y-direction; propagation in z-direction.

the boundaries but, as shown in figure 23, most of the energy is carried, inside the waveguide, as sketched in figure 22. Hence the 'balancing' arguments are correct, even for realistic cases.

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Figure 23 Contour plot of the intensity distribution of a stabilised beam in a planar waveguide. Note that the contours are further apart as the field weakens into the cladding regions.

In figures 24 and 25 an outline of what self-focusing means, physically, is given. Basically, if an unbounded crystal is used and beam of power P is entered into it with P > P c , or P >> Pc then phenomenon of self-focusing occurs. This means that because the power of the beam is more than is needed to achieve the balance power (Pc ) needed to kill diffraction, the beam will continue to focus. In figure 24 pure self-focusing is conveniently illustrated in terms of an equivalent focusing [30,32] lens of focal length LN . In fact, while L N is (equivalently) only associated with a focusing lens, diffraction is associated with a defocusing lens of focal length LD . The combined effect is obtained in this, geometrical, ray picture by obtaining a combined focal length, in the usual way. Of course, self-trapping (spatial soliton creation) is always going to be possible so the lens model is only useful upto a point. According to figure 24 as P → Pc , LND → ∞, yet a self-trapped filament can be formed. A thorough investigation of focusing region is needed, therefore, but this is far beyond the scope of this article. It has been addressed, however, and some answers can be found in the literature that abounds on this topic [32]. Figure 25, sketches out the case when beams go supercritical and hints that several filaments (self-trapped) beams can be created, each with P = Pc. The figure finishes with a final sketch of a spatial soliton labelled as a self-trapped beam.

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Figure 24 An illustration of self-focusing. Only the equivalent positive (focusing) lens is shown, to demonstrate the dramatic consequences of self-focusing. The focusing region is not accounted for by this simple theory. Self-trapped filaments can form, for example, rather than the beam becoming smaller and smaller.

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Figure 25 Paraxial geometrical optics will not accurately describe the focusing region. Many filaments (spatial solitons) could be created, for example.

5. GENERATING THE NONLINEAR SCHRÖDINGER EQUATION [NLS] The nonlinear Schrödinger equation [NLS] is a familiar sight in the literature and one of the preferred generic forms is (5.1) [the literature often shows ½ multiplying the second term but a simple re-scaling of Z places the 2, as in (5. l)], where U is an envelope function that is slowly varying, with respect to Z. This envelope function has been seen pictorially in the previous sections and is literally the “envelope” that “holds”, or “encloses” the rapidly oscillating wave. In the form (5.1) the equation has been stripped of dimensions and any particular physical application. As shown in figure 25, it is entirely generic in character.

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Figure 26 pulse/beam solitons are solutions of the same equation.

Even though it is generic in form, equation (5.1) has preferred forms, as outlined in figure 27, so that applications to beams of pulses can be emphasised.

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Figure 27. Preffered form of the NLS.

Having stated what the NLS looks like, how does the equation arise? There are several answers to this question to be found in the literature, all leading to the generic equation. At one extreme, a complicated first principles calculation can be undertaken. At the other, a simple, straightforward, examination of the dispersive nature of the system also reveals the nonlinear Schrödinger equation. It is this latter approach that will be adopted here. Suppose that the dispersion equation is written in terms of kz , the z-component of the wavenumber in the manner shown in figure 28.

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Figure 28 Summary of expansion of dispersion equation about an operating point ( ω 0 , k0 ). This procedure is like Taylor expanding the characteristics of an electronic device about an operating (bias) point.

Dispersion in a system simply means that the wavenumber (k) has a frequency (ω ) dependence that is not a straight line. This means that phase velocity vp = ω/k and group velocity

varies, according to which point on the dispersion curve is selected.

Temporal pulses and spatial beams are modulations of carrier waves [9,14]. If only temporal pulses are being considered, then it is clear that the act of producing an envelope introduces a frequency spread ∆ ω about the carrier frequency ω 0. Provided that we consider only a small bandwidth around ω0 , all frequency components in a linear system lie fairly close to the centre (carrier) frequency ω0 . If, in addition, there is a small spread of wavenumbers about k0 then the wavenumber of the complete beam-pulse-like disturbance can be Taylor expanded [9,15,35] around (ω0,k0) and the series can be safely truncated at the second-order or third-order terms. The number of terms to be kept will depend upon the application we have in mind. Figure 28 shows propagation down the z-axis of such a 211

pulse/beam envelope, associated with some envelope function u(x,z,t) that varies much more slowly than the rapidly varying exp i[ ω0 t-k 0 z], as the propagation proceeds. Note also that y is ‘frozen’ so no variation with y is allowed. Because of the slowly varying assumption only

will eventually appear in the NLS, amounting to a neglect of

In the expansion given in figure 28, since the system is confined in the y-direction, i.e. guiding along z occurs and the x- and z-directions are each allowed to reach ± ∞ . The guiding confinement means that the wavenumber being used here is already the guided wavenumber. The expansion looks at the deviation, from the guided wavenumber k z0, because of the introduction of wavenumber and frequency bandwidths [ ∆ kx , ∆ k z ] and ∆ω, respectively. Since the whole point of the exercise is to look at beam and pulse propagation in a nonlinear medium, the deviation, away from k z0 , originating from the power is also added in, intuitively, as a term

|u|² term, where u is the

(complex) envelope amplitude. Because both spatial and temporal effects are included, the final envelope equation accounts, simultaneously, for diffraction and dispersion. Some details will now be given that may help with the derivation. These are the relationships •

k x , ω, |u| ² are to be treated as independent variables (5.2a)

• kz is a dependent variable; k depends upon ω and |u|² (5.2b)

(5.2c) [k x , ω are independent variables in this theory] (5.2d)

(5.2e)

[k x |u|² are independent variables] (5.2f)

At the operating point k z = k z0 = k, k x = 0, ω = ω0 so that

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(5.3a) (5.3b)

Figures 29, 30 and 31 show further developments that lead to the familiar generic equation. Figure 29 contains the fundamental step that ∆ ω , ∆ k x and ∆ k z are in the space-time domain,

and

exist as operators on u(x,z,t). In other words, the NLS is

the application of the inversion to the time-space domain of the operator

to u(x,z,t).

Figure 29(a) Using the operator derived in figure 27 upon u to get the NLS. The result of leaving out diffraction is given.

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Figure 29(b) NLS appropriate to diffraction cases.

Figure 30 Illustration of length scales in dispersion and diffraction limited propagation in a nonlinear medium.

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Figure 31 How to reach the generic form of the NLS

Bright (B) or dark (D) solitons can be generated from the temporal, or spatial, equations by considering the sign of γ and/or the sign of β 2. This is the application of the famous Lighthill criterion, which tests whether β2γ < 0, or β 2γ > 0. Diffraction always has the same sign and so is always there, even if the material is not; hence a laser beam in a vacuum will always spread out because only material-induced self-focusing can stop it. Nevertheless, in a self-defocusing material dark (D) soliton beams are possible. The situation is summarised in figure 32. Dark solitons will not be discussed any more here, except to point out that they are, strictly speaking, ‘holes’, or ‘absence of light’ in a continuous infinitely extending background. In practice, finite backgrounds can be used such as a ps hole ‘dug out’ from an ns pulse. These ‘holes’ propagate like the bright pulses or beams. A dark soliton has a tanh cross-section. For spatial solitons the preferred notation is (5.4) and the general (fundamental) solution for this equation is displayed and demonstrated in figure 33.

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Figure 32 Summary of bright/dark conditions.

Figure 33(a) First steps in checking the general, fundmental solution of the NLS. 216

Check

Figure 33(b) Final check on the fundamental (first-order) solution of the NLS, formulated for a spatial soliton beam.

In this beam case, ξ is interpreted as the angle at which the beam centre propagates, relative to the z-axis, and θ 0 is just an arbitrary fixed phase that can be set to zero, or any other value, without any loss of generality. There are other options, and interpretations, depending on the application in mind and figure 33 contains an appropriate form for beams. For pulses, it is common to use the following notation and form (5. 5) in which , even though it is dimensionless, t is regarded as the local time, the ½ is in its ‘traditional position’. The general solution to equation (5.5) can be written down like the general solution to (5.4) except for a different interpretation. The first point to make is that equation (5.5) describes the motion of a pulse, relative to a frame of reference moving with a speed equal to the group velocity v g [see figure 31 for the relevant transformation]. The question of whether a laboratory frame, or a moving frame, is used is really relevant to the temporal (pulse) case. Figure 34 illustrates the frames of reference and emphasises that in the moving frame the measurement of time is local to the pulse i.e. it is not absolute, or global, time. Hence, we tend to put the pulse centre at

and let T range from

217

negative to positive values as the pulse is transversed in time. In the vast literature on this topic, the symbol t is usually used, as shown in equation (5.5), but it is really the local T, as emphasised in figure 34. The general solution is also shown in figure 34 and relates immediately to that shown in figure 33. If x0 = 0 in figure 33, then A Ω z of figure 34 is to be compared to ξ z in figure 33. For spatial solitons ξ is the angle the beam axis makes to the z-axis. For temporal solitons Ω is a frequency. A little consideration will show, noting the change in the position of the ½ and the 2 in equations (5.5) and (5.4), that the solutions given in figures 33 and 34 are identical.

Figure 34 Frames of reference.

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In dimensionless units [36], writing u = Asech

where q = AΩZ

,

where A is the amplitude and now Ω is a shift in reciprocal dimensionless velocity. The phase displacement is

is, in

fact, a frequency and the soliton velocity, relative to the already moving frame is So when Ω is zero, the soliton is at rest in the frame moving at the group velocity. This ½, versus 2, position accounts for the z being measured as z → 2 z in the spatial beam formula and leads to [note the plus sign etc] as opposed to [note the minus sign etc]. To recap, the interpretation is (1) ξ is the angle of beam propagation, where AΩ is the relative soliton pulse displacement (2) Ω is a frequency. In this formulation, for pulses, 2A is the mean soliton energy, Ω is a mean frequency and the mean time is q/A. Hence, [A, Ω , q, φ] characterise the temporal soliton at any point during its propagation. It cannot be overemphasised that, since we are dealing with the general formula it is important we are already in the moving frame of the group velocity, so quantities like Ω (a frequency displacement) are in addition to this and describe a soliton moving relative to this frame. In the spatial soliton case, ξ ≠ 0 means that the beam is pointing in a certain direction and that is easy to understand. For pulses, however, usually something has to be happening to the pulse environment for pulses to drift relative to group velocity frame. Such events may be the presence of amplifiers, causing perturbations, or self-frequency shifts acquired when pulses get very short.

6. INVERSE SCATTERING TRANSFORM (IST) METHOD The generic equation for pulse and beam envelope soliton propagation is (6.1) Although Korteweg and de Vries managed to solve their particular nonlinear equation, attempts to formalise exact solutions immediately encountered the problem that the usual Fourier transform method does not work on such an equation. This is because of the presence of the nonlinear term 2|U|2 U. For most of this century this mathematical difficulty remained unresolved. Thirty years ago, however, Gardner, Greene, Kruskal and Muira [37] invented what is known as the inverse scattering method (IST) to extract soliton solutions from the Korteweg-de Vries shallow water problem. A beautiful extension by Lax to a wide class of nonlinear problems soon followed. The honours, however, go to Zakharov and Shabat for their discovery of how to deal with (6.1), so it is widely known as the Zakharov and Shabat problem [38]. At first sight, the IST method appears to be very abstract, especially to physicists. It even seems mysterious but, once mastered, the technique is very appealing and really does have a lot of practical uses. The idea at the heart of the method is to break up the nonlinear equation (6.1) into two ordinary linear differential equations that look like a scattering problem so dear to the hearts of physicists. This move takes us into familiar quantum mechanics territory, in so far as we have become accustomed to generating both eigenvalues and eigenfunctions from some kind of potential function that is acting as a scattering agent. Furthermore, if the number of eigenvalues reveals just how many solitons are present in an evolving beam, or pulse, this would be a very strong bonus, indeed. If 219

these same eigenvalues told us the amplitude and velocity of the solitons present, then this would be even better. Finally, if the scattering potential function is actually the initial value, U(0,S) then this would be an excellent outcome! It turns out that this is precisely the case. To achieve such a remarkable mathematical coup was always going to be a difficult task and, indeed, when the first case applied to the Korteweg-de Vries (KdV) equation, was presented, it was regarded as possibly just luck! Zakharov and Shabat changed that view and now the IST is applicable to quite a broad class of evolution equations. Possibly more than 100 equations. The main problem [2,9,39] is to find linear operators that permit the creation of an auxiliary eigenvalue problem to replace the nonlinear partial differential equation. There is still a strong measure of intuition needed here, even today. The auxiliary problem uses the pulse or beam shape U(Z = 0,S) as a potential. For this choice of linear operator the eigenvalue must remain fixed as U(Z = 0,S) evolves to U(Z,S), provided that U(Z,S) satisfies (6.1). In fact, given U(Z = 0,S), the eigenvalue problem becomes an ordinary (direct) scattering problem so the evolution of the eigenfunctions, with Z, becomes trivial. Given the eigenfunctions at Z ≠ 0 the function U(Z,S) can be found by inverting the problem; just like taking an inverse Fourier transform, only more complicated. It is interesting that Gelfand, Levitan and Marchenko [2,9,39] had provided a method for doing such as inverse, quite a few years before it was needed for soliton problems. As will be seen below, we will not need to go all the way to the inverse because even at the eigenvalue equation stage, very powerful quantitative statements about the nonlinear system can be extracted. Having stated in words what the idea is, let us now be more specific. Let us be clear, first of all, that Fourier analysis does not apply to a nonlinear problem but it is important to appreciate how beautifully simple Fourier analysis is in the linear case. Suppose that the nonlinear term 2|U| 2 U is absent, then (6.1) becomes (6.2) For a plane wave where k is wavenumber and ω is angular frequency, the dispersion relation is k = ω 2 and, from the Fourier transform, (6.3) (6.4) where (0,ω) is the Fourier transform of the INITIAL (input) function at Z = 0. Here then is a major clue to generalising the well-known Fourier method to nonlinear problems, because the evolution of U(0,S) to U(Z,S) is trivially asserted by equation (6.3). In other words, given U(0,S) (0, ω) can easily be found and once (0, ω ) is well-known (6.3) evolves the information to yield U(Z,S). In practice, (6.3) is, of course, the familiar INVERSE TRANSFORM i.e. in mathematical language, the inverse mapping of (0, ω) onto U(Z,S). This little example is very helpful, however, in enabling the task to be seen. If only we could use just U(0,S), coupled to a simple evolution like (6.3) then even if the inverse is more complicated than (6.4) the nonlinear task would be complete. How is it possible then to proceed? The first point is that using a traditional Fourier decomposition is out, but if the nonlinear differential equation could be reduced to linear calculations, using only

220

U(Z = 0,S) then the calculation could proceed. The result of such n o n l i n e a r decomposition will be, as stated earlier, to make the problem like a traditional, linear, scattering problem of the type that appears in undergraduate quantum mechanics texts. Indeed, it is well known that if we could collect enough scattering data, such as reflection and transmission coefficients for any wave scattering off a potential well or bump then a reconstruction of what is actually doing the scattering is possible, in principle. This is working backwards [INVERSELY] from the scattered field to a knowledge of the scattering object. It is like hearing the sounds from a distant drum and using these sounds to reconstruct the shape of the drum. In our case the “drum”, i.e. the potential, is the initial value U(0,S). From U(0,S) we ought to be able to construct U(Z,S) by some form of inversion. We will not perform the inversion, however, because it turns out that the initial break down into linear equations yields a pair of equations that permits a very powerful and straightforward assessment of soliton presence. The problem, then, is even easier, in principle, than it looked at first sight. Unfortunately, breaking down equations like (6.1), in the first place, is not easy and required a burst of inspiration from Zakharov and Shabat, through a consideration of the following. Zakharov and Shabat recast equation (6.1) by introducing operators and , with the properties (6.5) where

and

are linear operators. Zakharov and Shabat showed that

(6. 6)

where 0 < p < 1. At this stage, things do not look too bad! Furthermore (6.7) where ψ 1, ψ2 are eigenfunctions and ξ is an eigenvalue. In other words (6.7) is solved just like a scattering problem in quantum mechanics but note that ξ remains constant, with respect to Z, i.e. if the eigenvalue can be found for U(Z = 0,S) it remains the same for all U(Z,S). Equations (6.7) yield the linear coupled equations [37] (6.8a) (6.8b) where Z i is the initial value Z = Zi , and the eigenvalue is, in general, complex. The beautiful thing about ξ, however, is that (for pulses) its real part ξ r gives the velocity of the soliton, relative to the frame of reference moving at the group velocity and ξ i gives the amplitude i.e. [40]

221

(6.9)

The interpretation of (6.8a, 6.8b) is as follows. U(Z,S) is the starting (initial ) value of the nonlinear solution of (6.1) before any evolution to further points along Z is achieved. We usually set out at Z = 0, so that (6.10a) (6.10b) An example, which will serve to illustrate the power of equations (6.10) will now be given. It shows how to determine the ‘soliton content’ of an input pulse or beam that has a rectangular shape i.e.

(6.11)

According to equations (6.10), and writing Λ = -iξ , the solutions ψ 1, ψ 2 are (6.12)

(6.13)

(6.14) where k2 = N2 - Λ 2 and g is some constant. Note that U(0,S) = 0 for S < - ½ and S > ½ and that equations (6.10) reveal that either ψ 1 or ψ 2 is zero in these regions. In other words, to get functions that do not diverge [‘blow up’], ψ1 ≠ 0, ψ 2 = 0 in the region S < - ½ and ψ 1 = 0, ψ 2 ≠ 0 in the region S > ½. Because the coefficient of ψ 1 in the region S < - ½ is normalised to unity, the coefficient of ψ 2 is equal to some constant value, which we call g. In the region - ½ < S < ½, ψ 1 ≠ 0, ψ 2 ≠ 0 but ψ 1 = 0, precisely, at S = ½ and ψ 2 = 0, precisely, at S = ½, to avoid divergences.

222

Figure 35 Dispersion relation for scattering off the rectangular distribution (6.11), using the coupled linear equations (6.8).

The boundary conditions at S ± ½ give

(6.15a) (6.15b) (6.15c) From equations (6.15a) and (6.15c)

(6.16)

223

and substitution of equation (6.16) into (6.15b) gives the dispersion equation (6.17)

and Equation (6.17) can be solved graphically as shown in figure 35. The ψ 1 , ψ 2 solutions for the three eigenstates, predicted by assuming that figure 36.

Figure 36

224

Eigenstates for N = 2π (1.7)

are shown in

Figure 37 Deducing the soliton content of Nrect(S)

The crossing points on the Λ = 0 axis of ( Λ ,k) plots are found from (6.18) and these occur at (6.19) On the Λ = 0 axis, however, k = N so equation (6.18) tells us the critical values of N needed to get n = 1, n = 2, n = 3,.… crossing points of the curves i.e. it reveals the values of N needed to generate 1,2,3,… solitons from a given Nrect(S) input condition. Figure 37 illustrates this answer for two crossing points and sets out the soliton existence conditions. The procedure described above permits the determination of whether a given input will lead to solitons, as the input evolves, during propagation. It also determines how many solitons lie “buried” in the input condition. The eigenvalue equations (6.10), on working backwards yield only the unmodified nonlinear Schrödinger equation yet this can still be used to determine soliton content [40,41] when extra terms, needed to account for 225

damping for instance need to be added to modify the original nonlinear Schrödinger equation. All we need to do is to use the exact numerically determined solution of the modified NLS at any point Z = Z 1 and use that in place of the function U(0,S) in equations (6.10). The value of the eigenvalues ξ will then reveal, given the starting function U(Z 1,S), a certain soliton content at that point of the evolution. What this means is that if any point during the evolution is used as an input to an unmodified nonlinear Schrödinger equation then the ξ = ξ r + ξ i values show how many solitons could emerge and what they will be like in terms of amplitude and velocity. In other words, at every point along the propagation direction the soliton content of a pulse, or beam, can easily be found. Indeed, it is rather easy to program (6.10) with modern mathcad software. Plots of ξ, as a function of Z, show just when the system is capable of supporting solitons and a precise way of finding out when the pulse or beam finally becomes devoid of soliton content. The power to determine this comes from the direct scattering equations (6.10). There is no need to worry about taking the inverse. A great deal can be learned about the system at this crucial, eigenvalue, stage. The algorithm is illustrated in figure 38.

Figure 38 How to deduce if an evolving pulse has any soliton content.

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REFERENCES 1. 2. 3. 4. 5.

6. 7.

8. 9. 10. 11. 12.

13. 14. 15. 16.

17. 18.

19. 20.

21. 22. 23.

J.S. Russell, Report on waves, Proc. Roy. Soc. Edinburgh, 319-320, (1844). P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge University Press, Cambridge (1983). G.B. Airy, Tides and Waves, Encyc. Metrop., Fellowes, London (1845). G.G. Stokes, On the theory of oscillatory waves, Camb. Trans. 8, 441-473, (1847). J. Boussinesq, Théorie de l’intumescence liquid appelée onde solitaire ou de translation, ce propageant dans un canal rectangulaire, Comptes Rendus Acad Sci (Paris), 72, 755-778, (1871). Lord Rayleigh, On waves, Phil. Mag. 1, 257-279, (1876). D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag. 39, 422-443, (1895). N. Zubusky, and M. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys Rev Lett 15, 240-243, (1965). M. Remoissenet, Waves Called Solitons, Springer-Verlag, Berlin (1995). D.L. Lee, Electromagnetic Properties of Integrated Optics, John Wiley & Sons, New York (1986). A.C. Scott, F.Y.F. Chu and D.W. McLoughlin, The soliton: a new concept in applied science, Proc. IEEE 61, 1443-1483, (1973). (a) B.M. Oliver, Bell Telephone Laboratories Technical Memorandum MM-51-15010, Case 33089, March 8, (1951). (b) S.C. Bloch, Introduction to chirp concepts with a cheap chirp radar, Am. J. Phys. 41, 857-864, (1973). R. Dawkins, The Blind Watchmaker, Penguin, London (1986). G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego (1995). A.K. Zvezdin and A.F. Popkov, Contribution to the nonlinear theory of magnetostatic spin waves, Sov. Phys. JETP 57, 350-355, (1983). J. S. Aitchison, Y. Silberberg,, A.M. Weiner, D.E. Leaird, M.K. Oliver, J.L. Jackel, E.M. Vogel and P.W.E. Smith, Spatial optical solitons in planar glass waveguides, J. Opt. Soc. Am. B. Opt. Phys., 8(6), 1290-1297 (1991). J.S. Aitchison, K. Al-Hemyari, C.N. Ironside, R.S. Grant and W. Sibbett, Observation of spatial solitons in AlGaAs waveguides, Electron Lett., 28, 1879-1880, (1992). (a) Y. Silberberg, Spatial optical solitons in Optical Solitons, Ed. J Satsuma, SpringerVerlag, Berlin (1992). (b) P.V. Mamyshev, A. Villeneuve, G.I. Stegeman and J.S. Aitchison, Steerable optical waveguides formed by bright spatial solitons, Electronics Letters 30, 726-727, (1994). A. Villeneuve, J.S. Aitchison, J.U. Kang, P.G. Wigley and G.I. Stegeman, Optics Letters 19, 761-763, (1994). G.I. Stegeman, A. Villeneuve, J.S. Aitchison and C.N. Ironside, Nonlinear integrated optics and all-optical waveguide switching in semiconductors, Fabrication, Properties and Applications of Low-Dimensional Semiconductors, Ed M Balkanski and I Yanchev, Kluwer Academic Publishers, Netherlands (1995). A.D. Boardman and K. Xie, Theory of spatial solitons, Radio Science, 28, 891-899, (1993). A.D. Boardman, K. Kie and A.A. Zharov, Polarisation interaction of spatial solitons in optical planar waveguides, Phys. Rev. A., 51, 692-705, (1995). A.D. Boardman and K. Xie, Dynamics of spatial soliton coupling. Studies in Classical and Quantum Nonlinear Optics. Ed Ole Keller, 2-30, Nova Press, New York (1995).

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24. A.D. Boardman, K. Xie and A. Sangarpaul, Stability of scalar spatial solitons in cascadable nonlinear media, Phys. Rev. A 52, 4099-4106, (1995). 25. A.D. Boardman and K. Xie, Magnetic control of optical spatial solitons, Phys. Rev. Letters, 75, 4591-4594, (1995). 26. A.D. Boardman and K. Xie, Waveguide-based devices: linear and nonlinear coupling, Low-Dimensional Semiconductor Devices, Ed. M. Balkanski, Kluwer Publishers, Amsterdam, (1996). 27. J. Boyle, S.A. Nikitov, A.D. Boardman, J.G. Booth and K.M. Booth, Nonlinear selfchannelling and beam shaping of magnetostatic waves in ferromagnetic films, Phys. Rev. B, 53, 1-9, (1996). 28. A.D. Boardman and K. Xie, Spatial solitons in χ (2) and χ(3) dielectrics and control by magneto-optic materials. Proceedings of Minnesota International Mathematics Workshop, Springer-Verlag (1997). 29. Boardman, A.D. and Xie, K. Soliton-based switches, logic gates and transmission systems. Ed. M. Balkanski, Kluwer Publishers, Amsterdam (1997). 30. S.A. Akhmanov, A.P. Sukhorukov and R.V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium, Sov. Phys. Usp., Engl. Transl., 93, 609-636, (1968). 31. M. S. Sodha, A.K. Ghatak and V.K. Tripath, Self-focusing of laser beams, Tata McGraw-Hill, New Delhi (1974). 32. O. Svelto, Self-focusing, self-trapping and self-phase modulation of laser beams, Progress in Optics, 11, 1-51, (1974). 33. F.A. Jenkins and H.E. White, Fundamentals of Optics, McGraw-Hill, New York (1950). 34. A. Barthelemy, S. Maneuf and F. Froehly, Propagation et autoconfinement de faisceaux laser par non-linearite de Kerr, Opt. Comm. 55, 201-206, (1985). 35. A.D. Boardman, S.A. Nikitov, K. Xie and H.M. Mehta, Bright magnetostatic spinwave envelope solitons in ferromagnetic films, JMMM, 145, 357-378, (1995). 36. J.P. Gordon and H.A. Haus, Random walk of coherently amplified solitons in optical fiber transmission, Optics Letters 11, 665-667, (1986). 37. C.S. Gardner, J.M. Green, M.D. Kruskal and R.M. Miura, Method for solving the Korteweg de Vries equation, Phys. Rev. Lett. 19, 1095-1097, (1967). 38. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34, 62-69,(1972). 39. G.L. Lamb, Elements of Soliton Theory, John Wiley & Sons, New York (1980). 40. A.N. Satsuma and G.M. Dudko, Initial value problems of one-dimensional selfmodulation of nonlinear waves in dispersive media, Prog. Theor. Phys. Suppl. 55, 284306, (1974). 41. V.V. Afanasjev, J.S. Aitchison and Y.S. Kivshar, Splitting of high-order spatial solitons under the action of two-photon absorption, Optics Comm, 116, 331-338, (1995).

228

SPATIAL SOLITONS IN QUADRATIC NONLINEAR MEDIA

Lluís Torner Laboratory of Photonics Department of Signal Theory and Communications Universitat Politècnica de Catalunya Gran Capità UPC-D3, Barcelona, ES 08034, Spain

1

Introduction

Self-focusing and self-trapping of light have been investigated since the early days of nonlinear optics. 1,2 Interest in this field has been maintained by the fascinating range of new phenomena encountered and their potential applications, such as soliton propagation, all-optical switching, and logic for ultrafast signal processing devices. For many years such effects have been pursued using the optical Kerr effect in cubic nonlinear media, 3 – 1 0 and since more recently using the photorefractive effect. 11 However, selfinduced trapping of light also occurs in quadratic nonlinear media (hereafter referred to as χ ( 2 ) media). 12 In this case, both spatial and temporal solitons form through the mutual focusing and trapping of the waves parametrically interacting in the nonlinear medium. In these lectures we focus on spatial solitons that form with cw light signals in planar waveguides and in bulk crystals, but most of the mathematical results hold also in the case of temporal solitons. Also, following the usual convention, throughout these lectures we make no distinction between the solitary waves that we study and mathematical solitons, referring to both of them as solitons. Self-focusing effects in quadratic nonlinear processes were long known to be possible under specific conditions, namely when the parametric interaction is very weak and yields an effective third-order effect for the pump wave. 1 3 , 1 4 By and large, however, the full extent of the self-focusing and its implications were not fully appreciated until recently. 1 5 The remarkable exception is the theoretical work of Karamzin and Sukhorukov more than twenty years ago, 12 who investigated the mutual focusing of beams in parametric processes and identified its implications for the formation of solitons. In the last few years, spatial soliton propagation has been observed experimentally in secondharmonic generation settings by Torruellas and co-workers in bulk potassium titanyl

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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phosphate (KTP), 16 and by Schiek et. al., in planar waveguides made of lithium niobate (LiNbO 3 ). 17 Generation of strings of solitons through the modulational instability of wide intense beams has been observed experimentally by Fuerst and co-workers.18 Solitons in quadratic nonlinear media form in planar waveguides and in bulk media, and the lowest-order bright solitons are stable on propagation in both cases. Stable light bullets are also theoretically possible. Those solitons might have important applications to different optical devices, including switching and routing devices and optical cavities containing χ ( 2 ) crystals. The aim of these lectures is to present the background of the topic in an unified point of view. We shall only discuss the basic properties of the solitons in the simplest configuration, namely second-harmonic generation, and we shall concentrate in Type I phase-matching. Naturally, the families of solitons are richer in the case of Type II phase-matching geometries because of the additional degree of freedom that this phase-matching offers, and that is important regarding potential applications of the solitons, but Type I geometries are simpler and capture the main features of the solitons. Hence, except for the last part of the lectures where we discuss a few applications of the solitons, we shall focus on Type I phase-matching. Also, we shall only study bright solitons. The lectures are organized as follows. In Section 2 we shall describe the physical setting considered and the evolution equations used to model the light propagation in quadratic nonlinear crystals. Section 3 is devoted to the basic properties, including stability, robustness and excitation, of the simplest families of bright solitons that exist in the absence of Poynting vector walk-off. In Section 4 we shall meet the solitons that exist in the presence of Poynting vector walk-off and discuss briefly their salient properties. Section 5 is devoted to the dynamics on evolution of beams with topological phase dislocations, or optical vortices. In this part we shall meet the modulational instabilities existing in quadratic nonlinear media and we will discuss how different combinations of topological charges of the input light signals produce certain patterns of bright solitons. To end the lectures, in Section 6 we shall discuss briefly discrete solitons and in Section 7 we shall summarize our main conclusions.

2 2.1

Physical Setting Evolution Equations

We consider cw light beams travelling in a medium with a large, non-resonant χ (2) nonlinearity under conditions of second-harmonic generation. We focus on spatial solitons in Type I phase-matching settings. The electric field of each of the waves is written in the form E (r, t) = A ( r) exp[ikz – iω t], and we consider experimental conditions for which both the scalar and the slowly varying envelope approximations for the fields hold. These conditions are expected to be fulfilled under most relevant experimental situations, but they might fail for narrow beams, very high input powers, and in highly anisotropic crystals.

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We shall study the formation of solitons in planar waveguides and in bulk media. Most of the generic expressions shall be written for the bulk case, but it is assumed that they hold in both cases. Throughout the lectures we use the subscripts 1 and 2 for quantities associated with the fundamental and second-harmonic waves, respectively. The appropriate χ ( 2 ) nonlinear coefficients involved in the two-wave mixing process are included in the normalization of the fields. Under such conditions, the beam evolution can be described by the reduced equations 19

(1) where a1 and a2 are the normalized envelopes of the fundamental and second-harmonic waves, ξ is the normalized propagation coordinate, r = –1, and α = – k 1 / k 2 . Here k 1,2 are the linear wavenumbers at both frequencies. In all realistic situations α – 0.5. The parameter β is a measure of the linear phase mismatch between the fundamental and second-harmonic waves and is given by β = k 1 η2 ∆ k, where ∆ k = 2 k 1 – k 2 , is the wave vector mismatch and η is a characteristic beam width. The parameter δ and the unit vector characterize the magnitude and direction of the Poynting vector walk-off due to the fact that the energy and wave fronts generally propagate in different directions in a birefringent medium. One has, δ = k 1 ηρ w , with ρw being the walk-off angle, and we note that walk-off is absent for propagation along the principal optical axes of the crystal. Finally, the transverse coordinate is given in units of η , and the propagation coordinate is normalized in such a way that z/l d = 2 ξ, with l d = k 1 η2 /2, being the diffraction length of the fundamental beam. In the one-dimensional case, equations (1) also hold for pulsed light. Then, diffraction is replaced by dispersion, Poynting vector walk-off is replaced by group velocity mismatch, and r and α are given by the group velocity dispersions. 2.2

Limit of Large Wave vector Mismatch

The self-focusing nature of the wave-mixing process at the regime of large phase-mismatch between the waves (β >> 1) and small conversion to the second-harmonic, is simply exposed by noticing that in such conditions the governing equations approximately reduce to the nonlinear Schrödinger equation (NLSE)2 0 – 2 2 (2) which in one-dimensional geometries is known to have stable soliton solutions. The properties of the χ (2) solitons are impacted by this fact. 19 However, the NLSE does not allow stable solitons for bulk geometries and a question might arise about the implications of this fact to χ (2) trapping . Actually, the beam evolution in the χ ( 2 ) medium quickly violates the approximations required to derive the NLSE, and stable solutions do exist in bulk quadratic nonlinear media.

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However, an important point must be emphasized. Most experimentally relevant solitons in quadratic media form for small and negative wave vector mismatches, at exact phase-matching, and in general under other conditions where (2) does not hold. Therefore, such solitons exhibit new properties, dynamics, and features. Thus, they ought to be treated accordingly. 2.3

Experimental

Values

We shall shortly be giving some values of β and δ that are relevant to experimental conditions, but the impact of these parameters on the formation of solitons is best elucidated by recalling their relation to the linear lengths that characterize the lowpower beam evolution in the χ (2) medium, namely the diffraction length l d , the coherence length lc = π / |∆ k|, and the walk-off length l w = η /ρ . One has (3) Non-critical or temperature-tuned phase-matching configurations that yield a negligible 0; by contrast, δ ≠ 0 occurs when angle-tuning phasewalk-off correspond to δ matching is used and walk-off is present. When the diffraction and walk-off lengths are comparable one has δ ~ 1. Most crystals with large χ(2) nonlinear coefficients are also highly birefringent, as is the case for most organic materials, and for that case walk-off would dominate diffraction and large values of δ would result for angle-tuned configurations. Under such conditions, equations (1) may not be valid. The role of the parameter β is exposed by recalling that for focused beams the value of ∆ k gives only partial information about whether or not it corresponds to near phase-matching. This is because focused beams contain a broad spectrum of transverse wave vectors, and each spectral component of the beam experiences a different wave vector mismatch. The effective phase mismatch, as measured by the second-harmonic generation efficiency, depends on the diffraction properties of the beam, and for the case we are studying here it is given by the value β . Outside exact phase-matching, which 2 –1/2 corresponds to β = 0, it is useful to write β = ( η/η0 ) , with η 0 = (k 1  ∆k ) being a characteristic width. Thus, for a given input beam width η , a wave vector mismatch such that η ≈ η0 , effectively corresponds to near phase-matching, whereas a larger ∆ k , with |β| >> 1, corresponds to a large phase mismatch. The mutual trapping of beams in the χ (2) crystal is governed by the interplay between the linear lengths contained in β and δ, in addition to the nonlinear length determined by the light intensity and by the input conditions. Regarding the linear lengths, favourable conditions for self-trapping occur when all lengths are comparable, so that β ~ ±3, and δ ~ ±1. Typical experimental conditions that yield these values, and which are representative of the actual parameters involved in the experimental observation of solitons in KTP cut for Type II phase-matching, 16 are η ~ 15 µm, and ρ w ~ 0.1° – 0.5°. For such parameters, the value of δ falls in the range 0.3 – 1.5, and one needs a coherence length of some l c ~ 2.5 cm, to have β ~ ±3. For typical materials and wavelengths, say

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–4

λ ~ 1 µm, this coherence length corresponds to a refractive index difference ∆ n ~ 10 between the fundamental and second-harmonic waves. The above parameters yield a diffraction length of l d ~ 1 mm, so that ξ in the range 0 – 20, corresponds to a few cm. Large effective wave vector mismatches, in the sense mentioned above, correspond to β ~ 30, whereas exact phase-matching corresponds to β = 0. Large values of δ, say δ ~ 5, which for the above parameters is obtained in configurations with a walk-off angle of some ρ w ~ 1.5°, correspond to a moderate Poynting walk-off. They are representative of the experimental conditions encountered, e.g., in organic materials with very large quadratic nonlinearities, at appropriate wavelengths close to those where non-critical 23,24 phase-matching occurs. To grasp the order of magnitude of the optical intensities associated to the normalized quantities used along the lectures and that are required to form solitons in quadratic nonlinear media, notice that for a beam width η ~ 20 µ m and the typical nonlinear coefficients of KTP, a normalized power (defined below) of some I ~ 50, leads to an actual peak intensity of the order of ~ 10 GW/cm2 . Existing inorganic materials (e.g., potassium niobate), semiconductors (e.g., gallium arsenide) and organic materials (e.g., N-4-nitrophenil-(L)-prolinol, NPP, or dimethyl amino stilbazolium tosylate, DAST), with larger quadratic nonlinear coefficients would require lower power requirements. 2 3 , 2 4 No need to say that, this is so provided that the appropriate nonlinear coefficients are accessible at suitable wavelengths and phase-matching geometries with a reasonable walk-off and low absorption, and provided that long enough, mechanically stable, high quality samples can be made. Quasi-phase-matching of the largest nonlinear coefficients of suitable materials, e.g., LiNbO 3 , also leads to reduced power requirements and to suitable operating conditions, so that it holds promise for future use. 2 5 – 2 7

2.4

Conserved Quantities

Central to the beam evolution described by equations (1) is the fact that they constitute an infinite-dimensional, Hamiltonian dynamical system.1 0 , 28 To expose this fact, it is convenient to rewrite the equations in terms of the new fields A 1 = a1 , and A2 = a2 exp(–i βξ ) . T h e n , the conserved Hamiltonian writes

(4) The governing equations can now be written in the canonical form (5) where the symbol δ F indicates a Fréchet or variational derivative and Ã2 = We shall also make use of two additional conserved quantities: the total power or energy flow given by the Manley-Rowe relation (6)

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and the total transverse beam momentum (7) In Section 5 we shall consider light beams without azimuthal symmetry, and in such cases it is useful to monitor the evolution of the longitudinal component of the total angular momentum of the light beams given by (8) with being the transverse beam momentum density in expression (7). In the absence of Poynting vector walk-off, L is also a conserved quantity of the beam evolution. When walk-off is present, one readily finds that (9) For the right-hand-side of this equation to vanish in the presence of walk-off, the transverse momentum of the second-harmonic beam has to be parallel to Otherwise, the total angular momentum is not conserved in the presence of Poynting vector walk-off. It is also important to examine the rate of energy exchange between the fundamental and second-harmonic waves. Writing the fields in the form a 1,2 = R 1,2 exp( i φ 1,2 ), where R 1,2 and φ 1 , 2 are real quantities, one arrives at (10) Therefore, to cancel the energy exchange between the fundamental and second-harmonic beams, their wave fronts, including the nonlinear wave vector shifts induced by the wave interaction, ought to verify φ 2 ( ξ , r ⊥ ) = 2φ1 ( ξ, r ⊥ ) + β ξ. This is what happens when a soliton is formed: the transverse complex shapes of the two interacting beams induce the appropriate wave front distortions to cancel diffraction and Poynting vector walk-off, while the energy exchange between the waves is also cancelled. It is worth emphasizing an obvious but important fact: such process occurs dynamically, and it takes an infinite distance to form a true stationary soliton out of input conditions that do not coincide with such a soliton, as it is always the case in practice. 3 3.1

The Simplest Bright Solitons Families of Stationary Solitons

We first consider stationary soliton solutions in the absence of Poynting vector walk-off (i.e., δ = 0). The simplest solutions have the form (11) with κ v being the wave number shifts induced by the nonlinear wave interact ion. For the solutions to be stationary one needs κ 2 = 2 κ1 + β. According to (10) this relation

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Figure 1: Typical shape of (1+1) solitons for different values of the wave vector mis32,33 match. The solitons correspond to I = 30. ensures that there is no power exchange between the waves. The coupled equations obeyed by U1 and U2 are given by

(12) By and large, these equations have a rich variety of solutions, including solutions with different dark shapes, solutions with multiple peaks, and solutions with exotic shapes. In these lectures we are only interested in the lowest-order solutions. In the case of (2+1) solitons in bulk media (i.e., solitons with two transverse coordinates plus one longitudinal coordinate), those correspond to nodeless, radially symmetric solutions with no azimuthal angular dependence. In equations (12), r, α and β contain material and linear wave parameters, while the nonlinear wave number shift κ1 parametrizes the families of solutions. In all plots presented in these lectures r = –1, and α = –0.5. In the case of (1+1) solitons, one analytical solution with a bright shape is known. Namely, 12,19,29–31 (13) where s is the transverse coordinate. This solution occurs at β = –2 ( α – 2r), with the special value κ1 = –2 r. The whole family of solutions for different values of κ 1 and at other values of β can be found by solving equations (12) numerically using a shooting or a relaxation algorithm. Figure 1 shows the typical shape of a few representative solitons. In the case of (2+1) solitons no analytical solutions are known, but whole families of solitons are readily obtained by solving the equations numerically. 32–35 3.2 Similarity Rules Even though the families of stationary soliton solutions are only known numerically,

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important information about their properties can be obtained analytically by examining the scaling properties of eqs. (12). One readily finds that such equations are invariant under the similarity transformations (14) with µ ≠ 0 being an arbitrary parameter. Thus, the soliton solutions can be transformed into each other following such rules. This self-similarity has important consequences. For example, when doing the numerical calculations it can be used to solve equations (12) in an efficient way. More important, it has direct physical consequences. In particular, it shows that the general features of the solutions at either side of phase-matching are similar. Similarly, it implies that at phase-matching ( β = 0), all the solitons in the absence of Poynting vector walk-off are self-similar. Therefore, at phase-matching the relation between the amplitude and the width of the family of solitons is amplitude × (width)² = constant.

(15)

Outside phase-matching, the stationary solutions exhibit different amplitude-width relations. We shall examine them below. 3.3

Stability

One crucial property of the families of solitons is their stability under propagation. Here we refer to the stability of the solitons when propagated in the system with the same number of “transverse dimensions” than the system where they have been found. In other words, modulational instabilities against higher-dimensional perturbations are not considered. Otherwise, solitons in quadratic nonlinear media are known to be modulationally unstable in both (1+1) and (2+1) geometries if the corresponding perturbations are allowed to grow in the experimental setting considered.36,37 We shall discuss this issue in Section 5. The stability of the families of stationary solutions can be elucidated by using different methods.38—42 In these lectures we shall only discuss a geometrical approach. 42 Such geometrical derivation of the stability criterion is useful by itself, because in many cases it gives direct insight into the stability of the solitons, and also because it shows the universality of the stability criterion for similar systems. One first finds that the stationary solutions with the form (11) occur at the extrema of the Hamiltonian for a given energy flow, i.e., they correspond to (16) Now, the stability of the stationary solutions can be elucidated by noticing that the global minimum of H gives stable solutions, whereas local maxima yield unstable solutions. This is a consequence of Lyapunov theorem about dynamical systems applied to this case.10,28 Therefore, to elucidate the stability of the families of solitons one has to plot the curve H = H (I) and identify its lower and upper branches. The condition of

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Figure 2: Hamiltonian and wave number shift versus energy for the families of (1+1) solitons. (a) and (c): phase-matching and positive β; (b) and (d): negative β. Solid lines: stable solitons; dashed lines: unstable solutions. 42 marginal stability, that separates stable from unstable solutions, is given by the point that separates the lower from the upper branches of the curve H = H (I ). There are different ways to determine that point. For example, examination of the corresponding curves, shown in Figure 2 for the case of (1+1) solitons, leads immediately to the so-called Vakhitov-Kolokolov criterion, 43 given by (17) The mathematical proof is given by Whitney’s theorem about two-dimensional maps applied to this case. 44 In the case of spatial solitons in quadratic nonlinear media, the lower branches of the curves in Figure 2 are found to correspond to the global minimum of H , and the upper branches to local extrema. Therefore, criterion (17) holds. Physically, the main conclusion to be raised from Figure 2 in the case of (1+1) solitons, and from the similar plot but for (2+1) solitons,34,35 is that at positive phasemismatch and at phase-matching, all the lowest-order stationary solutions in the absence of walk-off are stable. Very narrow regions of solutions that would be unstable exist at negative phase-mismatch near the cut-off condition for the soliton existence, but those have a very limited physical relevance to the experimental formation of solitons. This is so because of several reasons. First, because solutions near cut-off correspond to increasingly broader beams; second, because in reality the input beams never match exactly the shape of the unstable stationary solutions; and third, because above the threshold light intensity for the existence of solitons there is always a stable soliton. Therefore, the excitation of the stable solitons that exist reasonably far from cut-off is what dominates the dynamics of the beam evolution.

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3.4

Two Useful Properties

The families of stationary soliton solutions contain important information. In these lectures, we shall only discuss two of them, that have direct experimental implications. One important property of the families of solitons is their amplitude-width relations. At phase-matching the amplitude-width relation is given by (15), and otherwise it has to be calculated numerically. Figures 3(a)-(b) show the outcome in the case of (2+1) solitons at β = ±3. 35 One important consequence of the existence of soliton families with a given amplitude-width relation is that under the influence of any small perturbations which lead to adiabatic changes of the energy flow along the propagation direction, like small material or radiative losses or gain, the shape of the solitons tends to adiabatically evolve following the corresponding relation. Figures 3(a)-(b) show that in the case of solitons in quadratic nonlinear media, at reasonably large amplitudes the width increases rather slowly as the amplitude of the solitons decreases. Therefore, such solitons retain their shape a great deal while they propagate under the influence of the non-conservative perturbations, provided that they are small. It is worth noticing that the behavior shown in Figs. 3(a)-(b) is consistent with the behavior of the soliton width for different soliton energies that were observed experimentally.16 Another important feature of the family of solitons is the fraction of the total energy that is carried by each of the waves, fundamental and second-harmonic, that form the soliton. Figure 3(c) shows how this fraction depends on the wave vector mismatch for two representative values of the total energy flow of the soliton. 33 This particular plot corresponds to (1+1) solitons, and it shows that at positive phase-mismatch most of the energy is carried by the fundamental wave, whereas at phase-matching the energies are comparable and at negative β the largest amount of energy is carried by the secondharmonic. Naturally, all this has important implications when it comes to the excitation of solitons with different input light conditions.

Figure 3: (a) and (b): width of the (2+1) solitons as a function of their amplitude.35 (c): fraction of energy carried by each wave forming the (1+1) solitons as a function of the wave vector mismatch. Solid line: I = 18; dashed line: I = 30. 33 238

Finally, notice that expression (16) provides the starting point of a variational approach to obtain approximate analytical expressions of the families of solitons. One needs to calculate I and H for given beam shapes and then optimize their parameters to minimize H. See ref. 45. 3.5

Excitation, or “Oscillating Solitons”

In practice, the input light conditions do not coincide with the shape of the stationary solitons. Therefore, the excitation of solitons with arbitrary input light beams is a central issue, that has to be investigated in detail. This is particularly true in the case of solitons that form in the presence of walk-off, because walk-off poses severe restrictions to the formation of solitons in realistic experimental conditions. Also, notice that at present, the longest χ (2) crystals available are a few centimeters long, which corresponds to a few tens of linear diffraction lengths, so that in single pass experiments only the soliton evolution during a relatively short distance is relevant. In general, one has to elucidate both the soliton content of the input conditions considered, and the dynamics of the evolution of such input conditions. In the case of mathematical solitons, namely soliton solutions of so-called completely integrable evolution equations, a great deal of this information can be obtained using the tools provided by the Inverse Scattering Transform; in particular, the soliton content of the input conditions can be determined a priori. However, this is not so for solitons of nonintegrable evolution equations, as it is the case of the system (1). Therefore, the study of the dynamics of the soliton excitation relies heavily on numerical experiments. Those are performed by solving the evolution equations (1), typically with a split-step Fourier or a finite-difference standard scheme, for given arbitrary input conditions. Motivated by its experimental relevance, one might consider sech-like, or Gaussian input beams with the form (18) with A and B being the amplitudes of the fundamental and second-harmonic beams, respectively. The excitation of bright solitons has been examined for a wide variety of input conditions, both in the case of (1+1) and (2+1) geometries. The numerical experiments show that solitons emerge from the input beams in a wide variety of input conditions, in terms of wave vector mismatches and input light shapes and intensities, not necessarily close to those given by the stationary soliton solutions. Solitons also emerge with inputs which fall very far from those solutions indeed, and in particular when only the fundamental beam, or only the second-harmonic plus a small fundamental seed, is supplied at the input face of the χ (2) crystal. In such a case, the input beams reshape, exchange power and adjust themselves through radiation of dispersive waves, and then they mutually trap. See refs. 46 and 47 for details. Here we shall only recall one point that is found: the excitation of solitons with arbitrary inputs, that hence contain one stationary soliton plus some amount of radiation,

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Figure 4: Excitation of solitons with the input conditions (18). Left: Peak amplitude of the fundamental beam, scaled to the input value, as a function of propagation distance. The plot is for (2+1) propagation. Right: Detail of the beam evolution in the case A = 20, B = 0.47 produces “oscillating states”. The amplitude of the oscillations decreases as the beams shed dispersive waves away, but after the initial reshaping process the leak is extremelly small. Figure 4 shows the typical beam evolution during the first tens of diffraction lengths. The plot corresponds to propagation in bulk media. The oscillations might have several experimental consequences. For example, when only fundamental beams are initially launched with energy well above the threshold for the existence of stationary solitons, the numerical simulations predict fast oscillations that produce sharp intermediate stages. 47 Under those conditions the peak power at the centre of the beams reaches very large values. Such values might be high enough to damage the crystal. Also, at such values equations (1) might not be valid, and the actual beam evolution inside the crystal might uncover new important features that ought to be investigated. 3.6

Solitons

in

Quasi-Phase-Matched

Samples

By and large, quasi-phase-matching (QPM) offers an attractive approach to produce highly-efficient parametric wave-mixing interactions in quadratic nonlinear optical media (for a comprehensive review, see refs. 25-27). Conventional phase-matching techniques used to compensate for the wave vector mismatch between the waves of difference frequencies that interact in the nonlinear medium are based on the birefringence and thermal properties of the materials involved. As a consequence, the overall efficiency of the interaction can be limited by a variety of effects, mainly Poynting vector spatial beam walk-off due to the different propagation directions of energy and phase fronts in anisotropic media, non-convenient operation temperatures or crystalline optical axes orientations, and the value of the quadratic nonlinear coefficients accessible through the polarization of the waves involved in the interaction.

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With QPM, the coupling between the fundamental and second-harmonic waves can be chosen so that non-critical phase-matching can be achieved at a suitable temperature, without Poynting vector spatial walk-off between the interacting beams, by using the largest second-order coefficients of the material employed, and with an optimized overlap between the guided modes involved in waveguide devices. Such advantages hold very promise for many applications, in both waveguide and bulk settings, and in particular to the formation of solitons. QPM can be potentially used for a variety of materials and operating frequencies. Typical experimental conditions give coherence lengths l c = π /|∆ k| in the range 2 – 20 µ m, with the actual value depending on the material involved and the wavelength used. In the normalized units of equations (1), one has β ~ 500 – 1000. For the typical values of the other various involved parameters, given in Section 2, β ~ 500 corresponds to a coherence length of some 7 µm. These are the conditions encountered, e.g., in the QPM of the diagonal d 33 nonlinear coefficient of LiNbO3 at λ ~ 1.5 µ m. QPM relies on the periodic inversion of sign of the nonlinear χ( 2 ) coefficient at given multiples of the coherence length l c , and the so-called m -th order QPM corresponds to a periodic domain inversion with period m = 2ml c (see refs. 25-27 and references therein, and Marty Fejer’s lectures). In general, the biggest nonlinearity is obtained for 1-st order QPM. In such a case, the corresponding evolution equations at leading-order, after averaging over the periodicity of the QPM structure, are analogous to equations (1), but with effective nonlinear coefficients 2/π smaller than the actual material coefficients and a global phase shift between the fundamental and second-harmonic waves. Hence, 48 so are the solitons. Higher-order corrections have been considered in ref. 49. Here we only wish to discuss the robustness of the soliton formation and evolution against random deviations of the domain length. Such random deviations occur as a consequence of the fabrication tolerances of the QPM domains. The physical nature and statistical properties of the random deviations from the nominal lengths depend a great deal on the specific fabrication technique used to implement the QPM sample. Here we only discuss the so-called duty-cycle errors, 25 that occur when the periodicity of the domains is very well controlled, but the positions where the domain walls actually form differ from the nominal ones. In the resulting structure, the ending wall of the n -th where domain is located at the position is the nominal length of each domain and Rn is the random shift. Such errors are short-range correlated along the longitudinal coordinate. Therefore, they have a small, adiabatic impact on the solitons, because when the correlation length is much shorter than the characteristic soliton length the stochastic effects can be averaged out to a large extent over every characteristic soliton length. On the contrary, long-range correlations might be far from being averaged out over a soliton period, hence they impact more strongly the soliton evolution. This is the case of the so-called random-walk errors that occur when the ending wall of the n -th domain is located at Figure 5 shows the typical outcome of the excitation of a soliton in a QPM sample with duty-cycle domain length errors. The plots show the fundamental beams propa-

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Figure 5: Excitation of a soliton in QPM samples with duty-cycle domain-length errors. Dashed lines: case of an ideal structure; solid lines: case of structures with different random domain errors, with typical deviation: in (a) 10 % ; in (b) 20 %. 48 gated some 20 diffraction lengths. The dashed lines show the soliton in the case of an ideal structure and the solid lines correspond to structures with random errors with a Gaussian distribution around the nominal domain length. On average the duty-cycle random errors induce small radiative losses in the solitons. Because losses are small, once the soliton is formed after the input beams reshape slightly, they evolve adiabatically following the amplitude-width relation of the family of solitons. 48,50

4 4.1

Walking Solitons Motivation

By their very nature, solitons in quadratic nonlinear media are made out of the mutual trapping of several waves. Here we consider the formation of spatial solitons under conditions for second-harmonic generation, therefore the solitons exist due to the mutual trapping of the fundamental and second-harmonic beams. In general, except under suitable conditions, in the low-power regime the beams propagate along different directions due to the Poynting vector walk-off present in anisotropic media and this fact has important experimental implications when it comes to the choice of suitable materials, input light wavelength and general conditions suitable to the formation of solitons. However, when a soliton is formed the interacting waves mutually trap and in the presence of Poynting vector walk-off the beams drag each other and propagate stuck, or locked together. Such a beam locking opens the possibility to specific applications of the solitons, 51–56 and it also poses new challenges to the understanding of the soliton formation. This is so because the “walking” solitons existing in the presence of walkoff exhibit new features in comparison to the non-walking solitons. Investigation of these new features is important regarding their potential applications, but also from a fundamental viewpoint because the approach and outcome have implications to walking solitons existing in other analogous but different physical settings.

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4.2 Families of Solitons Next, we examine stationary solutions of equations (1), with δ ≠ 0, describing mutually trapped beams walking off the ξ axis. Those have the form (19) where U and φ are real functions, η = s – v ξ is the transverse coordinate moving with the soliton, and φ v (ξ,s) = κ v ξ + ƒv (η). Here v is the soliton velocity, and ƒv ( η) are the wave fronts of the solitons. According to (10), to avoid energy exchange between the waves one needs κ 2 = 2 κ1 + β , as for non-walking solitons. However, one also needs that the wave fronts verify ƒ 2(η) = 2ƒ 1 ( η ) everywhere, or alternatively U v ( η ) and ƒ v ( η) ought to be symmetric and anti-symmetric functions of η, respectively. The solitons that exist in the absence of walk-off fulfil the former condition, whereas with this exception all the walking solitons fulfil the latter. Substitution of (19) into (1) yields the system of coupled nonlinear ordinary differential equations fulfilled by the functions U v ( η ) and ƒ v ( η). They can be solved numerically to obtain the families of walking solitons.57–60 Recall that κ 1 and v parametrize the families of walking solitons. Experimentally, such parameters correspond to the light intensity and to the angular deviation of the solitons from the longitudinal propagation axis. Walking soliton solutions exist for values of κ 1 and v such that solitons are not in resonance with linear dispersive waves. The resonance condition can be readily 57 calculated to obtain (20) Figure 6 shows the typical amplitude and wave front shape of a walking soliton. As shown in the plot, the walking solitons have curved wave fronts. Such curvature depends on all the parameters involved, including the wave vector mismatch, the walk-off measure, the soliton energy and the soliton velocity. 57–60 As it is the case of the non-walking solitons studied previously, important information about the families of walking solitons can be obtained from the conserved quantities of the wave evolution, as follows. One first finds that the stationary walking solitons with the form given by (19) occur at the extrema of the Hamiltonian for a given energy flow and a given transverse momentum, i.e., they occur at (21) This is an important result that has important implications to the soliton stability. One can also use it to find approximate analytical expressions of the shape of the walking solitons using a variational approach. The walking solitons have curved wave fronts and therefore their transverse momentum is not simply proportional to their velocity. The actual relation between the velocity and the momentum for the walking solitons can be elucidated by examining the evolution of the energy centroid of the bound state constituted by the fundamental and second-harmonic beams propagating stuck together. One finds 51

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Figure 6: Typical shape of a (1+1) walking soliton. The plots is for β = –3, δ = 1, and I = 30. In (a)-(b): v = –0.5; in (c)-(d): v = 0.5. 57

(22) In contrast to this result, the travelling-wave solutions that occur in the absence of Poynting vector walk-off have a flat wave front and a tilt given by the soliton velocity. In such conditions, one finds the particle-like results (23) 2

where H v = 0 is the Hamiltonian of the zero-velocity solitons and the term Iv /2 represents the kinetic part of the Hamiltonian of the solitons walking with velocity v. However, such is not the case of the solitons that exist in the presence of Poynting vector walk-off. 4.3 Stability The stability of the families of walking solitons can be elucidated by different ways. Here we shall use the geometrical approach discussed previously for the case of non-walking solitons. Once again, our starting point is the variational expression (21). Because the families of stationary walking solitons realize the extrema of H for given I and J, one concludes that solutions that realize the global minimum of H are stable, whereas those that realize a local maxima are unstable on propagation. Therefore, we ought to examine the surface H = H(I, J) and identify its lower and upper sheets (Fig. 7). In the case of “smooth” H = H (I, J) surfaces, the curve that separates the lower and upper sheets of the surface can be determined by noticing that over the curve the vector normal to the surface is contained on the horizontal plane. A similar procedure holds in the case of sharp surface foldings. Sheet-crossings ought to be treated separately. The vector normal to the two-parametric surface is given by the expression (24)

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Figure 7: Sketch of the procedure to determine geometrically the condition of marginal stability of the walking solitons. is contained where the symbol ∂ ( F, G ) / ∂(a,b) stands for a Jacobian matrix. The vector on the horizontal plane when the last term in the RHS of (24) vanishes, so that (25) This is the condition of marginal stability for the families of walking solitons that is derived using a linear stability approach.59 To identify the stable and the linearly unstable walking solitons one has to evaluate the condition (25) for the families of walking solitons. Actually, only the families that exhibit multi-valued surfaces H = H (I, J) need to be studied in detail. See ref. 60 for the details. The main conclusion is that, similarly to the families of solitons that exist in the absence of walk-off, narrow regions of unstable solutions exist near cut-off, but with such exceptions all the walking solitons are stable on propagation. 4.4 Excitation The excitation of walking solitons with different input beams is governed to a large extent by Eq. (22). For our present purposes it is better to write it as (26) This expression has to be used with caution because it holds for the families of stationary walking solitons, but not for the input light conditions. The difference is that unless the input conditions exactly match the shape of the walking soliton solutions, the beam dynamics and reshaping towards the formation of a walking soliton always produces some radiation that takes energy and momentum away. However, when the radiation produced is small Eq. (26) provides a direct estimation of the velocity of the walking soliton that eventually gets excited. The ratio I 2 /I depends strongly on the linear wave vector mismatch between the waves and also on the total energy flow. In particular, at positive β the solitons have small I 2 /I, and they walk slowly. At phase-matching and at negative β the solitons have larger I 2 /I, thus they walk faster.

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5 5.1

Vortices Motivation

Vortices, or topological wave front dislocations, are ubiquitous entities that appear in many branches of physics. 61 Regardless the physical setting considered, the vortices display fascinating properties and a strikingly rich dynamical behavior. Optical vortices are not an exception. They are spiral, or screw dislocations of the wave front that has a helical phase-ramp around a phase singularity. They appear spontaneously in speckle-fields, in the doughnut laser modes, and in optical cavities, and otherwise they can be generated with appropriate phase masks or by the transformation of laser modes with optical components. Optical vortices have been investigated in linear media, cubic nonlinear media, and photorefractive crystals. 62–70 In this section we shall examine the phenomena generated by intense beams containing optical vortices in bulk quadratic nonlinear media under conditions for secondharmonic generation. With moderate input powers and wide beams, when only the fundamental pump beam is initially launched light undergoes frequency doubling together with the generation of a phase dislocation nested in the second-harmonic beam. However, with intense, tightly focused beams soliton effects become crucial and a whole new range of phenomena appears. In particular, the composite states of mutually trapped beams containing the phase dislocations self-break inside the quadratic crystal into separate beams, that then form sets of spatial solitons. 71–73 Such a behavior defines the principle of operation of a new class of devices that can process information by mixing topological wave front charges and producing certain patterns of spatial solitons. The number of output solitons can be controlled by the value of the “array” of topological charges of the input light signals. Next, we shall present the basic ingredients needed to understand the device behavior. Namely, the existence of solitary-wave vortices and their instability, and the dynamics generated by the mixing of beams with different topological charges. 5.2 Bright Vortex Solitary Waves We examine families of stationary solutions of equations (1), with δ = 0, that correspond to solitary-wave vortices. Specifically, we consider topological phase dislocations nested in the centre of beams with a bright, Gaussian-like shape. Such solitary waves appear as higher-order solutions of the governing equations. They have the generic form (27) where ρ is the radial cylindrical coordinate, and ϕ is the azimuthal angle. In the case of solutions with a phase dislocation nested in the centre of the beams, m v are the topological charges of the dislocations and sgn(mv ) their chirality. The transverse profiles U v are assumed to be real, radially symmetric functions. To avoid power exchange between the fundamental and second-harmonic waves one needs κ 2 = 2κ1 + β, and m 2 = 2m1.

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Figure 8: Typical shape of bright vortex solitary waves. Solid lines: fundamental beam; dashed lines: second-harmonic. Conditions: β = 3, κ1 = 3. 72 The coupled equations obeyed by U1 and U 2 are

(28) These equations have a rich variety of solutions. In particular, there are families of bright, higher-order solutions of two types. Namely, higher-order solutions without vorticity, that are characterized by the number of nodes, or field zeroes of the beams, and solutions without nodes, but with phase dislocations of increasingly higher charge. Here we only consider the latter. Figure 8 shows the typical shape of a solitary-wave vortex with charge m1 = 1 and with charge m1 = 2. The whole families of solutions in the case of Type I geometries can be found in ref. 72. Type II geometries produce analogous results. 73 However, in the case of Type II the problem is richer, because it involves three waves (see appendix A). In particular, the families of solitary-wave vortices occur for the combinations of topological charges that verify m 2ω = m o ω + m e ω ,

(29)

where moω and me ω correspond to the ordinary-polarized and the extraordinary-polarized fundamental beams. The important point for our present purposes is that the bright vortex solitary waves are unstable. In general, such instability produces the self-breaking of the beams along the azymuthal direction. Such self-breaking is due to the modulational instability of the top of the ring-shaped beams, somehow analogous to other spatial and temporal modulational instabilities arising in χ (2) media,36,37,109–111 and similar to azimuthal modulational instabilities that occur in χ ( 3 ) media.6 3 , 6 5 , 6 7 , 7 4

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Figure 9: (a) and (b): Growth rate of perturbations with different azimuthal indices for vortices with m1 = 1. (c): Typical decay of the solitary-wave vortex in Fig. 8(a). 72 5.3 Azimuthal Modulational Instability (AMI) To examine the stability of the solitary-wave vortices against azimuthal perturbations one can seek solutions of the form

(30) Inserting (30) into (1), and linearizing around the perturbations, one obtains the set of four coupled linear partial differential equations obeyed by ƒv ( ρ,ξ) and gv ( ρ,ξ). Such equations have many possible types of solutions, but now we are only interested in those that display exponential growth along ξ. To obtain them one can use the method described in refs. 74 and 75. Figure 9 shows the typical outcome of such an stability analysis. See refs. 72 and 73 for the details. The main result predicted by the plots is that the larger the parameter κ1 , hence the light intensity, the stronger the instability. For vortices with charge m1 = 1 , t h e perturbations with n = 3, together with and n = 2 exhibit the largest growth rate. For m1 = 2, the perturbations that tend to dominate are n = 5, together with n = 4. Therefore, under ideal conditions when all the perturbations are excited with equal strength, the vortex solitary waves tend to split into the corresponding number of beams. This is so at the initial states of the evolution, where (30) is justified. By and large, numerical simulations confirm such predictions. Figure 9 (c) shows the result of a representative simulation.71 5.4

From Topological Charge Information to Sets of Solitons

By now we have introduced the ingredients needed to describe the principle of operation of a class of devices that mix wave front topological charge dislocations nested in focused light beams and produce certain patterns of bright spatial solitons. The devices can operate in different regimes, as follows. Let us consider Type II geometries.

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Figure 10: Sets of solitons obtained with different combinations of the topological charges of the input light. In a first regime only the fundamental beams are input in the crystal, with a high enough intensity to form spatial solitons. Then, a second-harmonic is generated with the topological charge m 2 ω = m oω + me ω , and the beams self-split into a certain number of solitons because of the AMI of the solitary-wave vortices. Ideally, the number of output solitons is given by the index of the azimuthal perturbations with the highest growth rate, or by the interplay between different perturbations when there are some with similar growth rates. In practice, the several existing asymmetries in the experimental set-up, including Poynting vector walk-off, can seed a dominant perturbation. In a second regime, a coherent second-harmonic seed is present at the input together with the two fundamental beams. When m 2 ω = mo ω + me ω the beam evolution is similar to that of the regime mentioned above. A totally new situation is encountered when the topological charges of the input light are chosen to verify m 2 ω ≠ m o ω + me ω . Then, the number of output solitons is dictated by the different dynamics experienced by the different azimuthal portions of the beams.76 Down-conversion schemes, where an intense second-harmonic light beam is input in the crystal together with noise at the fundamental frequency, are another possibility. All such processes are driven by the azimuthal dependence of (10), that in the case of Type II phase-matching writes (31) The central result is that the “information” coded in the value of the input array [moω , me ω , m2 ω ] is transformed into a certain number of output soliton spots. Figure 10 shows the outcome of typical numerical experiments with different combinations of topological charges and energies of the input light beams. 249

6

Discrete Solitons

6.1 Motivation Discrete solitons on nonlinear lattices are a subject of intense research and continuously renewed interest, as they appear in many models of energy transport in a variety of physical, chemical and biological scenarios (see, e.g., refs. 77-82, and references therein). Discrete solitons form as a consequence of the balance between linear coupling that tends to spread the excitation across the lattice, and nonlinearity. Different types of nonlinearities, and among them cubic nonlinearities that yield evolution equations belonging to the family of the NLSE, have been investigated for many years. However, discrete solitons also exist in quadratic lattices. Besides its potential application to arrays of optical waveguides, discrete solitons on quadratic lattices might have applications to mathematically analogous, but physically different systems. As a matter of fact, parametric interaction of modes in general is a universal phenomenon, hence similar discrete solitons might also exist in other branches of nonlinear science. Naturally, the same is true for the continuous solitons. 6.2 A Quadratic Lattice Consider the differential-difference normalized evolution equations

(32) which come from the standard discretization of equations (1). In general, these equations might hold as a model for the parametric interaction of two generic modes in different scenarios, not only in Optics. In any case, β is the phase-mismatch between the modes, and n the position on the lattice. The parameters α 1 and α 2 give the strength of the linear spreading effects. In the numerics we set α 1 = 0.5 and α 2 = 0.25, to allow comparison with the spatial solitons discussed previously in the continuous case. For later use it is convenient to introduce the quantities Q n = An , Pn = B n exp(–iβ ξ) . We shall make use of two conserved quantities of the corresponding evolution equations, namely the norm (33) and the Hamiltonian

(34) Those are the discrete versions of the continuous quantities (4)-(6). However, notice that one crucial difference between the continuous and the discrete systems is that no analog to the transverse momentum (7) is known in the discrete case.

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Figure 11: Typical shape of two discrete solitons ( β = 0). (a): κ 1 = 1; (b): κ 1 = 5. 6.3

The Simplest Discrete Solitons

Let us consider solutions of (32) with the form A n = q n exp( iκ 1 ξ), B n = p n exp( i κ2 ξ) , where for simplicity here the numbers q n , pn are assumed to be real. As always, A n and B n ought to be phase-locked, so that κ 2 = 2 κ 1 + β. One readily finds that such stationary solutions realize the extrema of the Hamiltonian for a given norm, with κ 1 being the corresponding Lagrange multiplier. The difference equations obeyed by the series of amplitudes {qn} and {p n } are

(35) Similarly to other discrete systems, eqs. (35) have several different types of localized solutions. Here we only consider the simplest ones, namely those that have a bright shape and a constant phase across the lattice (i.e., “unstaggered”), and that have a single maximum that is located in one lattice site (i.e., “on-site” solitons). To elucidate the existence of such solutions one might solve numerically (35) searching for converging series {q n }, { pn }, with the symmetry conditions q o > q n for n > 0, and q n = q – n , and similarly for {pn }. Other families of solutions, including “inter-site” (i.e., qn = q– n + 1 ) and “staggered” solitons, also exist. Figure 11 shows the typical shapes of two discrete solitons. The most interesting is the strongly localized soliton in Fig. 11(b). By and large, the properties and features of discrete solitons can be drastically different from those of their continuous counterparts. Solitons of the continuous and the discrete NLSE provide a paradigmatic example. Discreteness modifies the shape and confinement features of the solitons, and modifies the existence conditions and properties of walking solitons that move across the lattice. In particular, this includes the amplitude-width relations of the families of solitons, and in the case of solitons supported by quadratic nonlinearities, the fraction of energy (or norm) in each mode forming the soliton, that we discussed in 3.4 for the continuous solitons. As an illustrative example, let us examine the ratio of the peak amplitudes of the two modes forming the discrete soliton, i.e. q 0 /p 0 . For the continuous families of solitons

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such ratio is given by 32,33 (36) According to the similarity rules (14), at phase-matching ( β = 0) all the continuous solitons are self-similar. Thus, the ratio q(0)/p (0) is identical for all members of the 1.2069. The discreteness of (35) breaks such soliton family. Numerically, q (0)/ p(0) similariry rules, so that in particular q0 / p0 is no longer constant at β = 0. For example, in the case of the weakly-localized soliton of Fig. 11(a) one has q0 / p 0 1.20, whereas for the strongly localized soliton of Fig. 11(b) one finds q 0 / p0 1.33. Discreteness introduces a variety of other differences. Other discretizations of (1) different from (32), and the corresponding discrete solitons, might be also of interest.

7 Concluding Remarks In these lectures we have only examined the basics of solitons in quadratic nonlinear media. Specifically, we only considered the simplest bright solitons, mostly in Type I phase-matching geometries. However, much more is known.8 3 Important investigations have been reported by many authors about Type II geometries, 8 4 – 8 6 dark-like and symbiotic solitons, different types of temporal solitons,1 9 – 2 1 , 2 9 – 3 2 , 8 7 – 9 1 including Thirring and gap solitons, 9 2 – 9 5 higher-order soliton solutions, 9 6 – 9 9 soliton interactions and collisions, 1 0 0 – 1 0 3 the effects of higher-order nonlinearities that may be present in addition to the quadratic nonlinearity,1 0 4 – 1 0 9 magneto-optic effects, 1 1 0 modulational instabilities in various geometries and dimensionalities,3 6 , 3 7 , 7 1 – 7 3 , 1 1 1 – 1 1 3 some of which have been observed experimentally,1 8 and so forth.1 1 4 – 1 1 6 A few applications of the solitons, mainly those potentially relevant to all-optical switching schemes but also other applications, have been examined and some experimentally observed. 52,53,117 The field is growing vigorously, theoretically and experimentally, and new areas where spatial, temporal and spatio-temporal solitons in quadratic nonlinear media might have important applications are emerging already. Optical cavities containing quadratic nonlinear crystals and quantum optical devices are fascinating and promising examples. In principle, new and existing materials with very high quadratic nonlinearities hold very promise for the future to reduce the power requirements to form solitons and hence render them closer to practical applications. From a broader viewpoint, it is worth stressing that solitons in quadratic nonlinear media can potentially have important implications not only to various parts of nonlinear optics, but also to other branches of nonlinear science. That is so because parametric wave mixing is a universal phenomenon. 118 Therefore, the formation of stable multidimensional soliton entities in mathematically similar, but physically different settings is a potentially important possibility that has to be explored.

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Acknowledgements This work has been supported by the Spanish Government under grant PB95-0768. My own work in the topic addressed in these lectures has been done in close collaboration with many colleagues, including George I. Stegeman, Curtis R. Menyuk, Dumitru Mihalache, Dumitru Mazilu, Juan P. Torres, Nail N. Akhmediev, William E. Torruellas, Ewan M. Wright, Dmitrii V. Petrov, José-Maria Soto-Crespo and Maria C. Santos. I am most grateful to all of them. The numerical work has been carried out at the C 4Centre de Computació i Comunicacions de Catalunya. Support by the European Union through the Human Capital and Mobility Programme is gratefully acknowledged.

A

Type II

The normalized evolution equations for the slowly-varying field envelopes in Type II phase-matching geometries for second-harmonic generation can be written as

(37)

where a 1, a 2 and a3 are the normalized envelopes of the ordinary polarized fundamental beam, the extraordinary polarized fundamental beam and the second-harmonic beam, respectively. In the case of spatial solitons α 1 = –1, α 2 –1, and α 3 –0.5. The equations for Type I are obtained from these equations by setting a 2 = a 1 = a ω , α1 = α 2 = r, α 3 = α, δ 2 = 0, δ 3 = δ, and a 3 = a 2 ω . Equations (37) have several conserved quantities, including the corresponding Hamiltonian. For our present purposes, we only need the energy flow given by the Manley-Rowe relation (38) the unbalancing between the energies of the two fundamental waves Iu = I 1 – I 2 , and the transverse beam momentum

(39) Non-walking solitons of (37) are a two-parameter family, whereas walking solitons constitute a three-parameter family. Physically, such parameters are related to the total energy flow, to the unbalancing energy and to the soliton velocity. The extra degree of freedom relative to Type I geometries, namely the unbalancing energy, has important implications. 84 In particular, it can be used to control the velocity of the walking

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solitons. The analogous of Eq. (26) but for Type II geometries writes (40) Changing the polarization of the input light at the fundamental frequency modifies the ratios I 2 /I and I 3 /I, hence it changes the velocity of the excited walking soliton.5 3 , 5 5 The properties of the families of solitons in Type II geometries can be found in refs. 84 and 85. Walking solitons have been also studied. 86

References [1] R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964). [2] P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1964). [3] V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972). [4] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973). [5] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). [6] A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985). [7] J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990). [8] Y. Silberberg, in Anisotropic and Nonlinear Optical Waveguides, C. G. Someda and G. Stegeman eds., chapter 5, pp. 143-157 (Elsevier, Amsterdam, 1992). [9] A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, 1995). [10] N. N. Akhmediev and A. Ankiewicz, Solitons (Chapman and Hall, London, 1997). [11] M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992). See Moti Segev’s lectures for a comprehensive review. [12] Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976). [13] L. A. Ostrovskii, JETP Lett. 5, 272 (1967). [14] C. Flytzanis and N. Bloembergen, Prog. Quantum Electron. 4, 271 (1976). [15] G. I. Stegeman, M. Sheik-Bahae, E. VanStryland, and G. Assanto, Opt. Lett. 18, 13 (1993). [16] W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995). [17] R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996). [18] R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78 , 2756 (1997).

254

[19] C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994). [20] Q. Guo, Quantum Opt. 5, 133 (1993). [21] R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993). [22] A. G. Kalocsai and J. W. Haus, Phys. Rev. A 49, 574 (1994). [23] I. Ledoux, C. Lepers, A. Perigaud, J. Badan, and J. Zyss, Opt. Commun. 80, 149 (1990). [24] C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995). [25] M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. QE-28, 2631 (1992). [26] M. Houe and P. D. Townsend, J. Phys. D: Appl. Phys. 28, 1747 (1995). [27 ] L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995). [28] E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986). [29] K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993). [30] M. J. Werner and P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993); Opt. Lett. 19, 613 (1994). [31] M. A. Karpierz and M. Sypek, Opt. Commun. 110, 75 (1994). [32] A. V. Buryak and Y. S. Kivshar, Opt. Lett. 19, 1612 (1994); Phys. Lett. A 197, 407 (1995). [33] L. Torner, Opt. Commun. 114, 136 (1995). [34] A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995). [35] L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, Opt. Commun. 121, 149 (1995). [36] A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981). [37] S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995); Phys. Rev. E 51, 4994 (1995). [38] D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995). [39] S. K. Turitsyn, JETP Lett. 61, 469 (1995). [40] A. D. Boardman, K. Xie, and A. Sangarpaul, Phys. Rev. A 52, 4099 (1995). [41] L. Berge, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, Phys. Rev. A 52, 28 (1995). [42] L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995). [43] M. G. Vakhitov and A. A. Kolokolov, Sov. Radiophys. Quantum Electron. 16, 783 (1973).

255

[44] H. Whitney, Ann. Math. 62, 374 (1954). Other applications can be found in the books about Catastrophe Theory. See also F. Kusmartsev, Phys. Rep. 183, 1 (1989). [45] V. V. Steblina, Y. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995). [46] L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994); J. Opt. Soc. Am. B 12, 889 (1995). [47] L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 13 (1995); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996). [48] L. Torner and G. I. Stegeman, “Soliton evolution in quasi-phase-matched second-harmonic generation,” J. Opt. Soc. Am. B. [49] C. B. Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997). [50] C. B. Clausen, O. Bang, Y. S. Kivshar, and P. L. Christiansen, Opt. Lett. 22, 271 (1997). [51] L. Torner, W. E. Torruellas, G. I. Stegeman, and C. R. Menyuk, Opt. Lett. 20, 1952 (1995). [52] W. E. Torruellas, Z. Wang, L. Torner, and G. I. Stegeman, Opt. Lett. 20, 1949 (1995). [53] W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996). [54] L. Torner, J. P. Torres, and C. R. Menyuk, Opt. Lett. 21, 462 (1996). [55] G. Leo, G. Assanto, and W. E. Torruellas, Opt. Commun. 134, 223 (1997). [56] A. D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, and G. F. Nalesso, IEEE Photonics Technol. Lett. 9, 602 (1997). [57] L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996). [58] D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997). [59] C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 6155 (1997). [60] L. Torner, D. Mihalache, D. Mazilu, M. C. Santos, and N. N. Akhmediev, “Spatial walking solitons in quadratic nonlinear crystals,” J. Opt. Soc. Am. B. [61] J. F. Nye and M. V. Berry, Proc. R. Soc. London A 336, 165 (1974). [62] See, e.g., G. Indebetouw, J. Mod. Opt. 40, 73 (1993). [63] V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985). [64] I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993). [65] N. N. Rosanov, in Progress in Optics XXXV, E. Wolf ed. (Elsevier, Amsterdam, 1996).

256

[66] G. A. Swartzlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992). [67] V. Tikhonenko, J. Christou, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995). [68] Y. S. Kivshar and B. Luther-Davies, Phys. Rep. (1997). [69] A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997). [70] Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, Phys. Rev. Lett. 78, 2948 (1997). [71] L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997). [72] J. P. Torres, J.-M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary wave vortices in quadratic nonlinear media”, J. Opt. Soc. Am. B. [73] J. P. Torres, J.-M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary wave vortices in Type II second-harmonic generation”, Opt. Commun. [74] J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, Phys. Rev. A 44, 636 (1991). [75] N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, Sov. Phys. JETP 61, 62 (1985). [76] L. Torner and D. V. Petrov, J. Opt. Soc. Am. B 14, 2017 (1997). [77] J. C. Eilbeck, P. S. Lomdahl, and A. C. Scott, Physica D 16, 318 (1985). [78] M. Peyrard and M. D. Kruskal, Physica D 14, 88 (1984). [79] A. S. Davydov, Solitons in Molecular Systems (Reidel, Dordrecht, 1985). [80] A. C. Scott, Phys. Rep. 217, 1 (1992). [81] D. K. Campbell and M. Peyrard, in Chaos, D. K. Campbell ed., (AIP, New York, 1990). [82] A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 55, 1172 (1996). [83] G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996). [84] A. V. Buryak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Y. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997). [85] U. Peschel, C. Etrich, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 7704 (1997). [86] D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, “Walking solitons in Type II second-harmonic generation,” Phys. Rev. E (1997). [87] K. Hayata and M. Koshiba, Phys. Rev. A 50, 675 (1994). [88] A. V. Buryak and Y. S. Kivshar, Opt. Lett. 20, 1080 (1995). [89] A. V. Buryak and Y. S. Kivshar, Phys. Rev. A 51, 41 (1995).

257

[90] R. Schiek, Int. J. Electron. Commun. 51, 77 (1997). [91] T. Peschel, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 4730 (1997). [92] C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 78, 2341 (1997). [93] H. He and P. Drummond, Phy. Rev. Lett. 78, 4311 (1997). [94] S. Trillo, Opt. Lett. 21, 1111 (1996). [95] B. A. Malomed, P. Drummond, H. He, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E. [96] D. Mihalache, F. Lederer, D. Mazilu, and L.-C. Crasovan, Opt. Eng. 35, 1616 (1996). [97] H. He, M. J. Werner, and P. D. Drummond, Phys. Rev. E 54, 896 (1996). [98] A. V. Buryak, Phys. Rev. E 52, 1156 (1995). [99] M. Haelterman, S. Trillo, and P. Ferro, Opt. Lett. 22, 84 (1997). [100] D.-M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995). [101] C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, 3444 (1995). [102] C. B. Clausen, P. L. Christiansen, and L. Torner, Opt. Commun. 136, 185 (1997). [103] G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7 (1997). [104] S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Commun. 122, 200 (1996). [105] A. V. Buryak, Y. S. Kivshar, and S. Trillo, Opt. Lett. 20, 1961 (1995). [106] M. A. Karpierz, Opt. Lett. 20, 1677 (1995). [107] K. Hayata and M. Koshiba, J. Opt. Soc. Am. B 12, 2288 (1995). [108] O. Bang, J. Opt. Soc. Am. B 14, 51 (1997). [109] L. Berge, O. Bang, J. J. Rasmussen, and V. K. Mezsentsev, Phys. Rev. 55, 3555 (1997). [110] A. D. Boardman and K. Xie, Phys. Rev. E 55, 1899 (1997). [111] S. Trillo and M. Haelterman, Opt. Lett. 21, 1114 (1996). [112] H. He, P. D. Drummond, and B. A. Malomed, Opt. Commun. 123, 394 (1996). [113] A. De Rossi, S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Lett. 22, 868 (1997). [114] R. Mollame, S. Trillo, and G. Assanto, Opt. Lett. 21, 1969 (1996). [115] W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 55, 6134 (1997). [116] P. Algin and G. I. Stegeman, Appl. Phys. Lett. 69, 3996 (1996). [117] R. A. Fuerst, B. L. Lawrence, W. E. Torruellas, and G. I Stegeman, Opt. Lett. 22, 19 (1997). [118] D. J. Kaup, A. Reiman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).

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PHOTOTOREFRACTIVE SPATIAL SOLITONS

Mordechai Segev¹ , Bruno Crosignani² , Paolo. Di Porto², Ming-feng Shih¹ , Zhigang Chen¹ , Matthew Mitchell¹ and Greg Salamo¹ ¹Electrical Engineering Department, Princeton University, Princeton, New Jersey 08544 ²Dipartimento di Fisica, Universita' dell'Aquila, 67010 L'Aquila, Italy and Fondazione Ugo Bordoni, Roma, Italy ³Physics Department, University of Arkansas, Fayetteville, Arkansas 72701

INTRODUCTION The advent of Nonlinear Optics, mainly due to the work of Bloembergen and coworkers at the beginning of the '60, has opened the way to a number of fundamental discoveries and applications [1]. No matter how sophisticated the microscopic or phenomenological approach adopted to deal with this topic, one of the central problems to the complete understanding of nonlinear processes is always the solution of the associated wave propagation equation, which is, of course, intrinsically nonlinear. This circumstance can be considered in many cases as an obstacle to a full comprehension of the problem, due to the well-known mathematical difficulties usually encountered when trying to solve this type of equations. However, it is precisely the nonlinear nature of the wave equation which gives rise to a wealth of solutions, and thus of possible behaviors of the propagating field, and makes nonlinear optics much more interesting that its linear counterpart. A typical situation is the one associated with the so-called optical Kerr effect. In this case, the third-order polarizability takes a particularly simple form and the nonlinearity is characterized by a contribution to the refractive index n proportional to the intensity I of the propagating field, that is n=n1 +n2 I. Inserting this expression into the wave propagation equation allows one to deduce the equation of evolution of the field amplitude in the form of a spatio-temporal partial differential equation, usually referred to as nonlinear Schrödinger equation (NLSE), whose solutions have been investigated in great detail starting with the pioneering work of Chiao et al. [2] and of Zakharov and Shabat [3]. In

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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particular, this analysis has shown the existence of a peculiar kind of solutions (solitons), able to propagate without distortion (or periodically recovering their initial shape), and has given rise to an entirely new field of applied research. [4] [5]. It is now well established that solitons are a general phenomenon in nature: they appear in many fields in which waves propagate and can exhibit both dispersion and nonlinearities, including surface waves in fluids (where they were initially discovered), volume fluid waves (deep sea waves), charge density waves in plasma and in solid state, etc. [6]. In Optics, two distinct types of solitons are known: temporal solitons [4] and spatial solitons [5]. Temporal solitons are pulses of very short duration that maintain their temporal shape while propagating over long distances. They are now routinely generated [7] and are the backbone of future highspeed telecommunication links. Conversely, it is much more difficult to generate and observe spatial solitons, in particular those that resulted from the Kerr effect (so-called Kerr-type solitons), which are optical beams that propagate without diffraction (stationary solutions of the NLSE), for several reasons. First, because the nonlinear change in the refractive index scales with the optical intensity and since the values assumed by n 2 a r e -16 characteristically very small (e.g., in silica glasses n2 is of the order of 10 cm² /Watt), very large optical power densities are required [8]. Second, Kerr-type spatial solitons are stable only in two-dimensional systems, i.e., one longitudinal dimension along which the beam propagates and one transverse dimension in which the beam diffracts or self-traps (this configuration is often referred to as a (1+1) D system). Full 3D optical beams undergo catastrophic self-focusing [9], and (1+1) D beams become transversely unstable [10] when they propagate in self-focusing Kerr-like nonlinear media. This means that Kerr-type spatial optical solitons can be observed in single-mode waveguides, as demonstrated in [8]. In this respect, there is an apparent asymmetry between temporal and spatial solitons, notwithstanding the intrinsic spatio-temporal symmetry of the wave equation, due to the additional dimensionality in the latter case. Optical spatial solitons have undergone a sharp and dramatic conceptual change during the last few years. It has been driven by the discovery of three distinct nonlinear mechanisms that have been shown to support three dimensional [i.e., (2+1) D] solitons [11], that is, beams that are self-trapped in both transverse dimensions: the photorefractive nonlinearity [12,13], the quadratic nonlinearity [14], and the vicinity of an electronic resonance in atoms (or molecules) [15]. This article is dedicated to solitons in photorefractive media. The photorefractive nonlinearity occurs in high-quality lightly-doped electro-optic crystals, such as BaTiO 3 (barium titanate), LiNbO3 (lithium niobate), SBN (strontium barium niobate). The photorefractive effect [16] was originally interpreted as an "optical damage" of the crystal provoked by the optical beam. It exhibits a reversible variation of the refractive index induced by the spatial variation of the optical intensity. This mechanism, which is, by comparison to standard nonlinear optical effects, rather slow but is effective at extremely small optical powers, possesses the capability of recording in real time the information encoded in the spatial modulation of the beam and, because of this capability, has been widely used in the frame of real-time holography and optical phaseconjugation. [17], [18],[19]. These properties suggested to look for propagation of spatial solitons in photorefractive materials, hoping that, in the nonlinear regime where the induced refractive index variation affects the very beam which has produced it, the beam diffraction could be compensated by its self-focusing and self-trapped (non-diffracting) propagation would result [12]. In fact, this hope has been more than justified by the great deal of theoretical and experimental results which have been obtained in the last five years and which have shown the existence of one and two-dimensional photorefractive spatial solitons, bright and dark, of vortex solitons, of photovoltaic solitons, together with a

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number of interesting applications (like, e.g., soliton-induced waveguides in bulk photorefractive media) and of self-trapping of incoherent light beams. In this review, our scope is to introduce the general topic of nonlinear propagation in photorefractive media and focus more specifically on photorefractive solitons. We will start with the basic band transport model of the photorefractive effect [20,21], which suffices for treating most of the situations we will be dealing with. As a first application, this will allow us to consider the concept of photorefractive gratings and to introduce the formalism of two-wave mixing as the simplest example of nonlinear propagation. We will then consider the propagation of a generic beam in a photorefractive crystal and introduce the formalism for describing this kind of propagation. Following that we will examine the conditions under which it should be possible to find self-trapped solutions, that is, particular transverse beam profiles which propagate without distortion (photorefractive solitons). After reviewing the main theoretical results which have been established up to now, we will present some of the impressive experimental evidence which confirms the great possibilities of application of the PR effect and discuss interactions between (or among) solitons. THE PHOTOREFRACTIVE EFFECT The photorefractive nonlinearity gives rise to light-induced changes in the refractive index of certain types of (typically non-centrosymmetric) crystals. More specifically, the spatial ditribution of the intensity of the optical beam (or beams) gives rise to an inhomogeneous excitation of charge carriers, which in turn produces, by redistributing themselves through diffusion and drift, a space charge separation whose associated electric field modifies the refractive index of the crystal via the Pockels effect. This modification, the photorefractive effect, can be qualitatively described by means of the simple band transport model introduced in [21]. More precisely, let us refer to Fig. 3.1 of Ref. [18], in which the energy diagram of a typical photorefractive material is shown. A large concentration (1018 -10 19 cm - 3 ) of identical, uniformly distributed, donor impurities, whose energy state lies somewhere in the middle of the bandgap, can be ionized by absorbing photons. Correspondingly, the generated electrons are excited in the conduction band where they are free to diffuse or to drift under the combined influence of self-generated and external field (if any). During this process, some of the electrons are captured by ionized donors neutralizing them, the successive ionization rate being proportional to the local illumination. In this way, the ensemble of the non-mobile donors acquires an inhomogeneous charge distribution which tends to be positive in the illuminated regions and to vanish in the dark regions. The combined presence of this charge distribution, of the electrons in the conduction band and of a number of ionized acceptors present in the crystal [22], gives rise to a low-frequency electric field (space-charge field, E). It is this field that, through the standard mechanism of Pockels' effect (δn α r E, where r is some electro-optic coefficient of the crystal), produces the refractive index variation responsible for the photorefractive (PR) effect. In order to translate in quantitative terms the above considerations, we need to write down the equations necessary to determine the space charge field E. The first one is the rate equation describing the donor ionization rate as a result of the competition between thermal and light induced ionization and recombination with free electron charges, that is

(1)

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where I is the optical intensity, ND the donor density , the ionized donor density, Ne the electron density, β the thermal generation rate, s the photoionization cross-section and γ the recombination rate coefficient. The second equation is the Gauss Law , where

(2)

is the low-frequency dielectric constant and the charge density ρ is given by ,

(3)

-q and N A representing the electron charge and the acceptor density, respectively. The last relevant equation is the continuity equation (4) where the current density (5) is the sum of the drift and diffusion contributions, µ representing the electron mobility. The set of Eqs. (1)-(5) has to be supplemented by the appropriate boundary condition: ,

(6)

where is the distance between the two crystal faces to which the external bias static potential V is applied. We will first consider the set of Eqs.(1)-(6) as if the optical light intensity, associated with the field Eop propagating at optical frequencies were a prescribed function. The main task of determining Eop will then be faced by solving in a self-consistent way the pertinent wave equation, written in the presence of the induced tensorial refractive index contribution associated with the linear electrooptic (Pockels) effect . More precisely, by choosing x,y,z along the principal dielectric axes of the crystal, for the typical values assumed by the space-charge field, it is possible to write [17] (7) where rijk is the linear electrooptic tensor (of rank three). THE SPACE-CHARGE FIELD We start dealing with the steady-state situation in which the time derivatives in Eqs.(1) and (4) can be assumed to vanish (actually, as we shall see in Sect.4, this corresponds to consider time larger than the so-called dielectric relaxation time). In this asymptotic

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regime, it is possible to show the possibility of achieving self-trapped beam propagation (screening solitons). The derivation presented in the next Sections closely follows that worked out in [23]. Let us first eliminate between Eqs.(1) and (2). With the help of Eq.(3), we obtain, in terms of suitable normalized quantities and assuming Ne << N A ,

(8) Eq.(8) expresses the relation between the normalized electron density and the normalized space-charge field, respectively given by

(9) and Q= (I+Id )/I d , that is

(10)

where (11) having introduced α=(N D - NA )/N A (typically a number much larger than one), β 1 = β /γ , s 1 = s/ γ , the dark intensity I d =β 1 /s1 , the normalized intensity |u|²=I/I d , the D ebye length

and the diffusion field E D = (K B T/q)k D .

Equation (8) can now be inserted into Eq.(4), which yields, after taking advantage of Eq. (5), a single equation connecting Y to Q, that is (12) Equation (12) can be cast in a simplified if approximate if the quantity Y is neglected with respect to one (and thus to α which is typically much larger than one), an hypothesis which needs to be verified a posteriori after Y is explicitly found. With this assumption, Eq. (12) becomes (13) In the following, we will consider situations in which the light intensity I is independent from z (or, at least, its scale of variation over z is much larger than those over x and y), so that Y and Q can both depend on the transverse coordinates x and y (2-D case) or on x alone (1-D case).

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In the 1-D case, Eq.(13), together with the proper boundary conditions Ey =E z =0 for x= ± /2 and E (x= /2)=E (x=- /2)=E DY0 , admits of the simple solution x x (14) where In the 2-D case, the most general solution of Eq.( 13) can be written, with the same boundary conditions as in the 1-D case, in the form

(15) where are unit vectors in the x and y direction and f(ξ,η ) is an arbitrary function which can be determined by imposing the condition (the electric field has to be conservative) (16) By doing that, we obtain an equation for f( ξ,η ) which reads (17) which has to be solved with the appropriate boundary conditions at zero and infinity. NONLINEAR OPTICAL BEAM PROPAGATION IN PHOTOREFRACTIVE MEDIA In the previous derivations, |u|2 has been assumed to be a prescribed function. In general, once in possession of the expression of Y, i.e., of E, the next step is to solve in a self-consistent way Helmholtz’s equation for the propagating optical (high-frequency) field in the presence of the refractive index determined by Eq.(7). Actually, in most cases, Eq.(7) can be reduced to a scalar form by noting that very often one of the diagonal components r ≡ r

= r

3 3 3 ) is much larger than all other components, xx x of the electro-optic tensor ( say so that if both the applied bias field E and the optical field E op are directed along the x-axis (parallel to the crystalline c-axis), the problem becomes a scalar one. This, in turn, allows to describe optical propagation by using the scalar Helmholtz equation for the x-component of E op . Furthermore, if we write

(18)

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where r ⊥ =(x,y) and k=( ω /c)n 1 , and assume the validity of the slowly-varying approximation for the field amplitude A, Helmholtz’s equation can be approximated by the parabolic (Fock-Leontovich) wave equation (19) where It is convenient at this point to introduce suitable adimensional variables defined through the relations (20) In these new variables, and recalling Eq.(15), Helmholtz’s equation takes the form (21)

f and we have neglected where the diffusive term which is responsible for self-bending effects [24]. Equation (21) has to be solved together with Eq.(17), which can be rewritten in the form (22) The set of coupled equations (21) and (22) provides the analytical description of nonlinear propagation in the general 2-D case.[23] SELF-TRAPPED BEAM PROPAGATION In the present Section, we examine the possibility of propagating self-trapped beams inside the PR crystal by exploiting an exact compensation between diffraction and nonlinear self-focusing induced by the PR effect. The results of the previous Section, showing that the diffusion term in the space-charge field is responsible for an energy transfer between two different plane waves while the nondiffusive one induces a phase change but no energy transfer, indicates that the presence of the bias field E0 is fundamental for achieving selftrapped propagation. In fact, diffraction involves accumulation of phases that are linear with the propagation distance to each individual plane-wave component of the beam which cannot be compensated by power exchange but, eventually, by a strong phase coupling. It is therefore desirable to eliminate the power-exchange term in the space charge field altogether. Although this cannot be accomplished in full (diffusion always exists), in all practical situations of self-trapped propagation of a single beam, the diffusion field (the first term on the RHS of Eq. (14)) is very small as compared to the "screening field" (the second term on the RHS in Eq. (14)), which is proportional to E 0 and gives rise to solitons. This approximation is valid for all types of PR solitons found thus far and occurs whenever the beam diameter is larger than ~5 µm [the diffusion field is ~ 50 V/cm whereas the trapping field E 0 is typically larger than 1 kV/cm]. Accordingly, we shall consider in the following

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situations where the first term on the RHS of Eq.(14), which is independent from E0, can be neglected. [24]

Figure 1. Formation of a bright photorefractive screening soliton.

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Spatial screening solitons in one transverse dimension Screening solitons were first predicted in [25],[26] and independently in [27]. At present, most of the experimental studies with PR solitons employ the (1+1) D screening solitons and they are considered best understood among all other types of PR solitons. An extensive review of the theoretical derivation of screening solitons (including those at very high optical intensities) can be found in Ref. [26], along with all a discussion on the physical materials parameters and proposed applications. Here we summarize the main steps in the theoretical formulation of screening solitons. Intuitively, one may view the formation of bright (low intensity) screening solitons in the following manner. A narrow light beam propagates in the center of a biased dielectric medium. As a result of the illumination, in the illuminated region the conductivity increases and the resistivity decreases. Therefore, the voltage drops primarily in the dark regions (a simple voltage divider) and this leads to a large space charge field there. The refractive index changes in proportion with the space charge field. If δn is positive, this process results in an antiguide (a large positive index change in the dark regions) which would strongly defocus the beam. However, if δn is negative, the large negative index change in the dark regions creates a "graded index waveguide" that guides the beam that generated it, in a self-consistent manner as illustrated in Fig. 1. For a rigorous treatment of the underlying theory, we recall the results of Sections 3 and 4. The wave equation for the field amplitude u reads (23) We look for self-trapped (stationary) solutions, which correspond to propagation without change in the beam transverse profile, of the form (24) where β is real, and we now defined having introduced the background illumination Ib which represents the intensity of a background optical beam used to illluminate the crystal uniformly (thus artificially increasing the dark current Id ). By assuming to be a real function (which means that we do not analyze here grey solitons, which were discussed in Ref. [27], or higher-order solitons), and indicating with a prime the derivative with respect to the argument , Eq.(23) becomes (25) Equation (25) respectively describes the case of a 1D beam which goes asymptotically (actually, on a transverse spatial scale much shorter than the distance between the electrodes) to zero (u0 =0, bright soliton) or to a constant value u0 different from zero (dark solitons). Note that in both cases it reduces, for u<<1, to the well-known nonlinear

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Schrödinger equation describing propagation in the presence of the optical Kerr effect [26]. It can be integrated by quadrature, thus transforming it in a first-order equation, by means of the change of variable (26) which implies (27) so that it becomes (28) For bright solitons one requires the boundary condition u(±∞ )= 0, while for dark solitons

Let us first consider dark solitons. Since the structure of Eq.(25) implies an asymptotic behavior of the kind

Figure 2. Width (FWHM of the intensity, u2, profile) of a bright low-intensity soliton (upper curve) as a function of u(0) and of a dark low-intensity soliton (lower curve) as a function of u(∞).

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(29) the only possibility consistent with the boundary condition is β+s=0. Furthermore, since the dark soliton condition requires u(0)=0, Eq.(25) yields u"(0)=0, which is consistent with an odd solution , corresponding to in the neighborhood of = 0. On the other hand, in the same zone the second term of Eq.(25) is negligible with respect to the third one (in fact, ), so that s has to be negative, i.e., s=-1, and, consequently, β =1. Finally, by imposing in Eq.(28) the condition p(u=±u 0 )=0, it is possible to find p(u=0) so that Eq.(28) can be rewritten as (30) Let us now consider bright solitons. In this case, the asymptotic behavior furnished by Eq.(29) implies β +s>0 . Besides, if we put p(0)=0 in Eq.(28), its asymptotic behavior for yields β = –s{ln[1+ u 2 (0)]}/ u 2 (0). Thus, the requirement β + s > 0 is possible = 0 , we obtain only if s=1. If we now specializes Eq.(25) around which is consistent with an even solution. By imposing in Eq.(28) the condition p[u=u(0)]=0, it is possible to find p(u=0) and rewrite it as (31)

We are now in the position to check the validity of the approximation underlying the derivation of Eq.(25). To this end, we use Eq. (14) (neglecting the diffusion field) in order to write ∇ . Y in the form (32) If we now introduce Eqs.(30) or (31) into Eq.(32), it is possible to check in a selfconsistent way the validity of the approximation leading to Eq.(25), that consisted in neglecting dYX/dξ with respect to unity. The integration of Eq. (31) can be accomplished only numerically. The problem has been extensively treated in [25], [26] and [27] and the soliton waveforms are shown there. Here, we show only the corresponding soliton existence curves, that is the relation which has to be satisfied between the full width at halfmaximum (FWHM) and u(0) (bright) or u0 (dark) in order to assure the existence of the soliton. The existence curves of both bright and dark solitons are shown in Fig. 2. Spatial screening solitons in two transverse dimensions We look for self-trapped solutions of Eq. (21), that is, for solutions whose transverse profile remains the same during propagation (spatial solitons), of the kind (33)

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With an argument similar to that adopted for the 1-D solitons, it is possible to show that s=1 , β = p 2 – 1 and s=-1 , β =1 for bright and dark solitons, respectively. [23] If we now introduce polar coordinates ρ and θ thorough the relations

and

write, respectively for

bright and dark vortex solitons, or = w(ρ ) exp (im θ ), where w is a real function and m is the so called topological charge (which can assume only integer values, [12]), Eq.(21) takes in the two cases the form (34)

and

(35)

where W 0 is the boundary value of w at large distance. They have to be solved together with Eq.(22), that is (36) where Q = 1 + w 2 and g has to go to zero for large ρ. The problem of finding cylindrically symmetric solutions of this set of equations has been extensively treated in [23] and the interested reader is referred to it..

DIFFERENT KINDS OF SPATIAL SOLITONS : QUASI-STEADY-STATE AND PHOTOVOLTAIC SOLITONS In the previous sections, we have investigated the possibility of generating spatial solitons in the asymptotic temporal regime in which the space-charge field has reached steady-state (see Sect. 4). On its way toward equilibrium, the space-charge field can assume , in a quasi-steady-state time-window, values and spatial distributions such that it is possible to propagate spatial solitons. These possess properties different from those of screening solitons, like, e.g., their independence on the absolute light intensity and its ratio to dark irradiance. While a large amount of experimental evidence shows the existence and allows to determine the properties of these quasi-steady-state spatial solitons [13],[28][34], their analytical description is still based on a phenomenological approach [12],[35],[36] and a recent study of a numerical nature [37] in the frame of (1+1) D theory. We refer the interested reader to a recent review [38] in which quasi-steady-state photorefractive solitons have been described in detail. In some photorefractive crystals, such as LiNbO3 , BaTiO3 and LiTaO3 , uniform illumination can generate (without any external field) a bias field driven by the bulk photovoltaic (or photogalvanic ) effect [39],[40]. The bulk photovoltaic effect arises from

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free charge carriers that are optically-excited from deep donor levels that reside in noncentrosymmetric potential (i.e., dopants that substitute a particular atom in the crystalline matrix of a non-centrosymmetric crystal). The photo-excited carriers possess excess kinetic energy in a preferential crystalline direction (the direction of the lower potential) and, before they "thermalize" (scatter from the lattice and/or from other electrons, impurities, etc.), give rise to a polar current. The photovoltaic current J p v depends on the optical polarization and intensity I through a third rank tensor κ and gives rise to an additional term in the current density of Eq.(5) of the form (37) In ferroelectric photorefractive crystals, the photovoltaic current is predominantly oriented (note that along the ferroelectric axis (c) and simplifies to photovoltaic currents driven by circular polarization also exist, as described in Ref. [40]). The bulk photovoltaic effect gives rise to a new kind of soliton, namely the photovoltaic soliton [41], both in the open and closed circuit configurations [42],[43]. Photovoltaic dark solitons and vortex solitons have been recently observed experimentally [44],[45] and a recent report [46] has presented a z-scan study of the photovoltaic self-defocusing nonlinearity. EXPERIMENTS WITH SCREENING SOLITONS Photorefractive screening solitons have been extensively investigated, both theoretically and experimentally, during the last three years. At present, the existence of both (1+1) D and of (2+1) D is well established, in both bright and dark realizations. In the last Sections, we provide an overview of the most significant experimental work on screening solitons, along with more advanced topics such as soliton interactions ("collisions"), and selftrapping of incoherent light beams. Observations of bright screening solitons In Sects. 5.1 and 5.2 we have analytically investigated the possibility of propagating selftrapped beams (both in 1 and 2 dimensions), in the asymptotic time regime where the space-charge field has reached its asymptotic value. We recall that the main feature of these PR screening solitons is the existence of a unique relation among the soliton width, the trapping voltage and the intensity ratio (the ratio of the soliton peak irradiance to the sum of the equivalent dark irradiance and uniform background irradiance), a large deviation from this relation (which we call existence curve , see Fig. 2) proving unsuitable for soliton propagation. In particular, the most favorable operation point (i.e., the point on the existence curve at which the narrowest soliton can be trapped with a given nonlinearity ) is at the minimum of the existence curve, that is, it is obtained at intensity ratio roughly equal 2.4 [26], which means that the solitons peak intensity should be roughly 2.4 times the sum of the dark plus background intensities. The dark irradiance is very small in all dielectric photorefractive media (on the order of mWatt/cm² µWatts/cm² ), so at the absence of background illumination solitons can, in principle, be observed with nWatt (or less) optical powers. However, operation at ultra-low optical powers poses a real experimental challenge, because the soliton formation time is of the order of the dielectric relaxation time, which in turn is inversely proportional to the soliton intensity. For very long (hours) formation times, small vibrations can eliminate any

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observation by distributing the space charge field over a region (in space) much broader than the optical beam. A second alternative for generating bright screening solitons corresponds to operating at very large intensity ratios while keeping the soliton intensity within a reasonable range. But, from the existence curve it is apparent that this option requires very large applied fields. For example, a ~ 1 µWatt soliton is more intense than the dark irradiance by at least 10 6 times, which implies a required trapping voltage of more than 10 6 Volt, to be applied across 5-mm electrodes - also unphysical (leads to surface arcing and at higher field to dielectric breakdown). The third choice is to generate "artificial" dark irradiance, using uniform background illumination, and controlling its magnitude relative to the soliton peak intensity. The background beam is typically a laser beam polarized orthogonal to the soliton beam and co-propagates with it. In this manner, both beams experience almost the same absorption (thus the soliton peak amplitude, u 0 , remains unchanged throughout propagation despite the absorption), nonetheless, the beams can be easily separated from each other after the crystal by use of a simple polarizer. This configuration also compensates for absorption, that must be fundamentally present in all PR media, because both the soliton and background beams experience roughly the same absorption so that the ratio between them is maintained, enabling soliton propagation over large distances (many absorption lengths are possible; in fact, the limiting factor is possible dichroism, that gives rise to a small difference between the absorption constants of both polarizations). This is the commonly used method for generating bright screening solitons. It was first used by Iturbe-Castillo et al. [47] for making the first observation of steadystate self-focusing effects in biased photorefractive crystals. An identical configuration was used by Shih et al. to observe the first screening solitons [48], and has been used since then for all experiments with bright screening solitons. At present, both (2+1) D [48],[49] and (1+1) D [50] bright screening solitons have been observed. A typical top view photograph on a (2+1) D screening soliton in SBN is shown in Fig. 3 (top), where, for comparison, we also show the normally-diffracting beam when the applied voltage is set to zero (bottom). It is well established now that both (2+1) D and (1+1) D cases exhibit stable self-trapped propagation over many (>10) diffraction lengths and robustness against localized perturbations (of a transverse scale much smaller than the soliton width) that could have fairy large amplitudes [49],[50]. The solitons are also stable against considerable (~10%) deviations in their initial conditions, i.e., the input beam profile, width, and phase profile [50]. In characterizing the soliton stability properties, it turns out that the existence curve plays a far more important role in soliton formation than initially thought. As evident from a series of papers on (1+1) D screening solitons (the first one being Ref. [50]), all these observations were made in 3D bulk crystals. This is despite the fact that any (1+1) D beams propagating in a threedimensional (bulk) Kerr-like nonlinear medium should be transversely unstable [10]. Obviously, (1+1) D photorefractive solitons behave in a manner different than their Kerr counterparts. From many experiments with photorefractive screening solitons (in fact, with any photorefractive solitons), it is apparent that, as long as the parameters of the beam and nonlinearity are "on", or close to, the existence curve, and as long as the intensity ratio is greater than ~ 0.1 (i.e., the screening solitons are not in the Kerr regime but rather the nonlinearity is saturated, at least in the center of the beam [26]), the transverse modulation instability is "arrested" to a degree that it is not observed even within a propagation length of > 15 diffraction lengths. (Many examples for this "instability arrest" can be found in the references in the next sections on soliton interactions and collisions). However, if the parameters deviate considerably from the existence curve, transverse instabilities become dominant and the one-dimensional beam becomes distorted and tends to break up into

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multiple filaments (see experiments in Ref. [50]). In this respect, it is instructive to mention experiments and numerical work that has reported just the opposite and has claimed that the (1+1) D screening solitons (bright and dark) always suffer from transverse instabilities [51]. It appears that the results presented in that paper were in error, as it contradicts a large amount of experimental data obtained by several groups. The mistake in [51] has been recently found by Infeld et al. [52], who have shown that bright (1+1) D screening solitons propagating in a 3D medium are transversely stable, in the conditions discussed above are indeed satisfied. At present (2+1) D screening solitons have been observed experimentally by several groups and their existence is well established (see, e.g., Refs. [48],[49] and many other references in the next sections). Experimental work has shown that they are robust and stable against small perturbations (such as material inhomogeneities) as long their intensity ratio is roughly larger than unity. Furthermore, as shown in [49], a Gaussian beam evolves into a soliton even if its initial parameters deviate from those that give rise to a soliton by ~10%. In a manner similar to the (1+1) D screening solitons, it is evident that the parameters (2+1) D solitons must obey an existence curve. As shown in [49] and [48], 1020% deviations from this curve (typically in the

Figure 3. A top view photograph of a 10 µm wide spatial soliton in a 5 mm long photorefractive crystal (top)), and, for comparison, the same beam diffracting naturally when the nonlinearity is « turned off » bottom.

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applied in the voltage) lead to elliptical self-trapped beams, whereas large enough deviations lead to instability and beam breakup [49]. However, at present the full theory for such solitons is not available yet, so a direct comparison (via the existence curve) between theory and experiments has not been demonstrated. In fact, early numerical work on (2+1) D beam propagation employing the photorefractive screening nonlinearity has steered much controversy, as it failed to find (2+1) D solitons [53]. That paper states explicitly that the evolution of a Gaussian-type beam in a biased photorefractive crystal is characterized by oscillations of its diameters in both transverse axes and by spreading, uncharacteristic of solitons. The simulations in [53] have predicted oscillations in the beam diameter from 35 µm to 17.5 µm every 2 mm and that these oscillating-beams cannot be axially symmetric. The experimental results(e.g., [49]), on the other hand, prove that the soliton beam maintains an axially-symmetric constant diameter and no oscillations in its diameter were observed. It is obvious that the experiments contradict those numerical simulations. The reason for this contradiction is that solitons exist for a specific set of parameters: those that are "on" the soliton existence curve. This means that, similar to the (1+1) D case, in a given crystal, a soliton of a specific diameter (at a given wavelength) and at a given value of intensity ratio, exists at a single value of the external field. For example, in the 1-D case "optical beams that significantly differ from soliton solutions tend to experience cycles of compression and expansion" (see [24]). The parameters used in the numerics of [53] were simply too far from those that can support a soliton. Furthermore, similar numerical calculation with (1+1) D screening solitons (for which the theory is well established) have failed to find solitons [54], for the very same reason: the parameters did not correspond top a point "on" the existence curve. Altogether, this is not surprising given the extensive literature on solitons in Kerr media: Burak and Nasaski [55] have shown that a Gaussian beam propagating in Kerr media is also characterized by oscillations in its diameter and that the beam converges a soliton only if the parameters are close enough to the soliton parameters. This means that finding the set of parameters that supports a soliton is therefore the critical issue: knowing them, one can simulate numerically the evolution of a Gaussian beam into a soliton, as done for the (1+1) D case. The soliton parameters, however, require an analytic treatment (it is an eigen-value problem) and are very hard to find using multiple numerical simulations. A first analytic treatment was recently given in Ref. [23], but, as explained there, the existence curve has been thus far found explicitly only for (2+1) D dark (vortex) solitons for intensity ratios < 2. Finally, it is worth noting that both (1+1) D and (2+1) D screening solitons self-bend towards a preferential direction (typically the c-axis in uniaxial PR crystals). This effect is driven by the first correction to the space charge field, which is the first term on the RHS of Eq.(14). Self-bending was first predicted in Ref. [24] and first observed in [49]. Observations of dark screening solitons Experiments with dark screening solitons fundamentally differ from those with bright solitons in several important items (apart from the sign of the applied field, which controls the nature of the nonlinearity: self-focusing or self-defocusing). First, the existence curve of (1+1) D dark screening solitons is a monotonically decreasing function of intensity ratio and converges asymptotically to a constant value. This implies that the "most favorable observation point" (the point on the existence curve at which the narrowest soliton can be trapped with a given nonlinearity, is obtained at the largest intensity ratio. The theoretical

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upper limit on this ratio is set by the validity of the approximation for the field at the electrodes (see Appendix of [26]), whereas the experimental limit is obtained in the absence of background illumination. Of course, this implies that, unlike their bright counterparts, dark screening solitons can be observed without additional background illumination. This is true only if the dark soliton is borne on an infinite beam, i.e., the notch-bearing solitonforming beam covers the entire face of the crystal . This means that to observe dark PR screening solitons one needs to launch a "true" dark soliton [56], rather than launching a dark notch borne on a finite beam (which will form an "approximate" dark soliton whenever the beam diameter is much larger than the width of the notch), as commonly done with Kerr-like dark solitons [57],[58]. The second item is that fundamental dark solitons require a π phase shift in their center. Combing these two poses an experimental challenge: to obtain a narrow dark notch of a few µ m width, in which the transverse phase jumps by π at its center, on the background of a uniform beam that covers the entire input face of the photorefractive crystal. The first attempt to generate dark screening solitons [59] employed the "conventional" method of using a tilted glass plate [57],[58], in which the "aspect ratio" between the width of the notch and the diameter of the beam is restricted by the thickness of the glass. Accordingly, this attempt has indeed demonstrated selfdefocusing but self-trapping of the notch was not conclusive [59]. Later on, Chen et al. have used a different technique to obtained the necessary waveform [56]. They have used a λ /4 step mirror made of an InP wafer, of which one half is etched to a λ /4 depth. This stepmirror was illuminated by a collimated beam and its reflection provided a dark notch (due to the π phase jump) in a broad beam of 1 cm in diameter. When this notch-bearing beam passed through a properly biased crystal, the formation of a fundamental dark soliton in

Figure 4. Typical experimental results of photographs and intensity profiles taken at the exit of a SBN crystal The figure shows an-odd -number sequence of dark (1+1)D soilitons that have all evolved from a single notch borne on an otherwise uniform beam. The applied field (and thus the nonlinearity) increase with increasing number of dark solitons. This figure is taken from Ref. 61.

steady-state was observed. The shape-preserving behavior of the dark soliton was confirmed in our earlier experiments by measuring the beam profile as it propagates

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throughout a specially-cut (wedged) crystal and by guiding a beam of a different wavelength [56]. Later experiments have demonstrated “higher-order” or “multiple” dark solitons, all evolving from a single input notch. In general, dark spatial solitons can be generated by launching an optical beam with two different initial conditions. One is the “odd” initial condition, which provides a π phase shift into one half of the beam (known as phase discontinuity or phase jump). The other is the “even” initial condition, which provides an amplitude depression at the center and an even-symmetry in the phase of the beam (known as amplitude discontinuity or amplitude jump). For PR screening solitons, if the initial width of the dark stripe is small, only a fundamental soliton [56] or a Y-junction soliton [60] is generated, corresponding to the lowest order in the odd- or even-number soliton sequence. As the initial width and the bias field are increased, a progressive transition from a lower-order soliton to a sequence of higher-order multiple solitons is observed. Odd initial conditions always generates an odd number of dark solitons, whereas an even initial condition gives rise a an even number of dark solitons (in this respect, the PR screening solitons are similar to Kerr-like dark solitons). A detailed review of multiple dark screening solitons can be found in Ref. [61]. Typical experimental results of photographs and intensity profiles taken at the exit face of a SBN crystal are shown in Fig. 4. The figure shows an odd-number sequence of dark (1+1) D solitons that have all evolved from a single notch borne on an otherwise uniform beam. The applied field (and thus the nonlinearity) increase with increasing number of dark solitons. This figure is taken from Ref. [61]. For fundamental dark screening solitons, the analytic theory (presented in the previous sections and in [25],[26],[27]) has predicted the existence curve given in Fig. 2. A detailed experimental study of this curve [56] has shown good agreement with the theory. For higher-order dark solitons, the comparison between theory and experiments was based more on numerical methods, as an analytic theory for higher order dark screening solitons is not available yet (in fact, such theory will be, most probably very challenging as this saturable nonlinearity is not integrable). Nevertheless, the comparisons between experiments and numerical simulations on high-order dark screening solitons were shown to be in good agreement [61]. As for observation of (2+1) D dark screening solitons (namely, vortex solitons), the same group that has tried to find bright screening solitons using numerical methods only (and did not succeed, see Ref. [53]), has attempted to find numerically vortex solitons, with no success either. This has lead them to believe that singly-charged vortex screening solitons cannot exist [62]. However, a recent paper by Chen et al. [63] has demonstrated just that: a self-trapped singly charged vortex screening soliton. They key idea that has enabled this observation was, once again, to launch a vortex nested on a broad beam that covers the entire PR crystal. In fact, the vortex soliton can be generated only when it is borne on an “infinite” beam, and it breaks up when it is nested on a finite donut-shaped beam. When the vortex beam is a donut-shaped narrow beam, it breaks up into two elongated “slices” (with a self-defocusing nonlinearity) or into two focused “filaments” (with a self-focusing nonlinearity) [63]. Typical experimental results showing the input (left), diffraction output (middle) and vortex screening soliton output (right), intensity distributions in a SBN crystal are shown in Fig. 5. WAVEGUIDES INDUCED BY PHOTOREFRACTIVE SCREENING SOLITONS

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Optical solitons are created when diffraction is balanced by self-focusing, that is, the optical beam modifies the refractive index of the medium in such a way that diffraction is eliminated. In essence, a soliton forms when the beam induces a waveguide (via the nonlinearity) and, at the same time, is guided in the waveguide it has induced. This means that the soliton is (must be) a guided mode of the waveguide it induces. Although this “selfconsistency” idea is fairly old [64], it has gained momentum in the last few years as Snyder and co-workers were able to use it for making predictions [65],[66],[67] on soliton interactions, collisions, stability and other properties in a general nonlinear medium (apart from Kerr medium, for which analytic results are available [3]). Experimentally, waveguides induced by (1+1) D Kerr solitons [68],[69],[70] and by (2+1) D vortex (dark) Kerr solitons [71] have been demonstrated. Following the first observation of photorefractive solitons [13] it was clear that such solitons should also exhibit waveguiding properties. Indeed, soon thereafter waveguides induced by quasi-steady-state photorefractive dark solitons have been demonstrated [33].

Figure 5.Typical experimental results showing the input (left), diffraction output (middle) and vortex screening soliton output (right).

Since the properties of all photorefractive spatial solitons greatly differ from those of Kerr-type solitons, one expects that the waveguides they induce will exhibit new features as well. First, photorefractive spatial solitons are stable when trapped in either one or two transverse dimensions. This implies that they can induce two-dimensional waveguides,

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which indeed has been demonstrated by Shih et al. [72] with (2+1) D photorefractive screening solitons. Second, the required optical power for formation of photorefractive solitons is very low (as low as 1 micro Watt, compared to 100 kWatt for optical Kerr solitons). This means that the waveguides can be induced by ~1 µWatt solitons [72]. Finally, since the response of photorefractive media is wavelength-dependent, one can generate a soliton with a very weak beam and guide in it a much more intense beam of a wavelength at which the material is less photosensitive [33]. This enables steering and controlling intense beams by weak (soliton) beams. Therefore, photorefractive spatial solitons seem very promising for various applications, such as light controlling light, optical wiring, near-field multi-channel to multi-channel interconnects, and frequency conversion in the soliton-induced waveguides (see discussion on potential applications in Ref. [26]). A recent paper [73] has presented a detailed experimental and theoretical study of the waveguiding properties of (1+1) D photorefractive screening solitons. It has shown that the number of possible guided modes in a waveguide induced by a bright screening soliton depends on the intensity ratio of the soliton. In particular, the number of guided modes increases monotonically with increasing intensity ratio from a single mode. When intensity ratio is much smaller than unity (the Kerr-limit [26]), the waveguide can support only one mode for the guided beams of the wavelengths equal to, or longer than, that of the soliton. At large intensity ratios the number of possible guided modes can be large. One can adjust the numerical aperture of the soliton-induced waveguide while keeping the soliton beam at a fixed shape by simply “walking along the existence curve”, that is, varying the intensity ratio and maintaining a soliton by adjusting the applied voltage according to that curve. On the other hand, waveguides induced by dark screening solitons can support only a single guided mode for all intensity ratios. For two-dimensional waveguides induced by (2+1) D screening solitons, it has been found [72] that the same principle holds, although no accurate comparison could be made between theory and experiments since the existence curve is available only for a limited regime. It was also shown [72] that the 2D induced waveguide is roughly isotropic as long as the (2+1) D soliton is circular. On the other hand, when the self-trapped beam has an elliptical profile, the induced waveguide is anisotropic in its modal properties [74]. Finally, these studies of the properties of the soliton-induced waveguide have proven to be useful for understanding and predicting interactions between solitons (collisions),. as explained in the next section. SOLITON COLLISIONS Amongst all soliton properties, interactions (commonly referred to as “collisions”) between solitons are perhaps their most fascinating feature, since, in many aspects, solitons interact like particles. As early as 1965, Zabusky and Kruskal have shown [75] that two interacting (colliding) Kerr solitons always conserve their individual energies and linear momentum, just like particles do. For this reason, Kruskal has named these nonlinear creature “solitons” to emphasize the similarity between these localized wave-packets and particles. Although it may seem rather logical that solitons possess this kind of behavior, it is by no means trivial. Keeping in mind that solitons result from nonlinearities, there is no seemingly straightforward reason as to why two such colliding solitons would not simply take out of balance the entire nonlinear system that supports them and disintegrate into small fragments that have no particular order. For many years soliton interactions were

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understood only in the framework of sophisticated numerical or analytical methods (e.g., inverse-scattering transform [3]). The more intuitive insight suggested recently by Snyder [66], which employs the idea that solitons must be guided modes of the waveguides they themselves induce, simplifies the understanding of soliton interactions by considering light propagation in waveguides at close proximity. Using this method, Snyder’s group was capable of predicting new phenomena, such as fusion and fission of solitons upon collision [76], and soliton spiraling [77]. In any case, before 1990, all soliton interaction studies were limited to collisions in a single plane, since practically all of the observed stable bright solitons before that time were one dimensional [(1+1) D]. The discovery of stable (2+1) D bright solitons (including photorefractive solitons which are described in detail in section 7.1, quadratic solitons [14], and solitons in resonant atomic vapor [15]), has initiated the exploration of full 3D interactions between bright solitons. During the last 2-3 years the ideas of soliton fusion, fission and spiraling upon collision were all demonstrated with photorefractive screening solitons (see Refs. [78], [79] and [80], respectively). Similar observations in other nonlinear media (e.g., [15]) confirm that these intriguing phenomena are not restricted to photorefractive solitons but are rather general features of interacting solitons.

Figure 6. Illustration of coherent and incoherent interactions between solitons.

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To intuitively understand soliton collisions, one needs to consider two optical waveguides that are brought to close proximity. The optical beams guided in these waveguides (in the form of guided modes) overlap, primarily in the "center" region between the waveguide where the evanescent "tails" of the modes coexist. Now, there are two possible scenarios for these self-trapped beams to interact: coherent versus incoherent interactions. Both types of interactions are illustrated in Fig. 6. Coherent interactions occur when the nonlinear medium can respond to interference effects between the overlapping beams. They occur in all nonlinearities with an instantaneous (or extremely fast) time response (such as the optical Kerr effect and the quadratic nonlinearity). For all other nonlinearities that have a fairly long response time (e.g., photorefractive and thermal), the relative phase between the interacting beams must be kept stationary on a time scale much longer than the response time of the medium. When this occurs, the material responds to interference between the overlapping beams. When the beams have a zero relative phase ("in-phase"), they interfere coherently and the intensity in the center region between the induced waveguides is increased. In a selffocusing medium, this leads to an increase in the refractive index in that region, which in turn, attracts more light to the center, moving the centroid of the solitons towards it and hence the solitons appear to attract each other. When the interacting beams are π out of phase from each other, they interfere destructively and the index in the center region is lower that it would have been if the beams were far away from each other, As a result, the solitons appear to repel each other. In fact, both "attraction" and "repulsion" between solitons are actually due to asymmetries in their induced waveguides that is caused (via the nonlinearity) by the close proximity of the beams. Incoherent interactions occur when the relative phase between the (soliton) beams varies much faster than the response time of the medium. In this case, the medium cannot respond to inteference effects but responds only to the time-averaged intensity (average taken over a time longer than material response time), which is identical to a simple superposition of the intensities. Therefore, irrespective of their relative phase, the intensities of the beams add up and the intensity in the "center" region between the solitons is increased (as compared to a single isolated beam). Since these solitons propagate in a self-focusing medium, this leads to an increase in the refractive index in that region. As a result, more light is "attracted" towards the center region and the solitons appear to attract each other. Such an incoherent "interaction force" is always attractive (for bright solitons), since the intensity in the center region cannot decrease by merely the coexistence of two soliton beams at close proximity. Collisions in Kerr media. The soliton interaction forces are, in principle, the same for all nonlinear media that can support solitons. There are, however, several very important differences in the outcome of collision processes between Kerr-type and saturable nonlinear media. First, in Kerr media all solitons are (1+1) D and the collisions are bound to occur in one single plane. In addition, in Kerr media, all collisions are fully elastic, which implies that the number of solitons is always conserved. Furthermore, the system is integrable, and therefore no energy is lost (to radiation waves) but rather conserved in each soliton. In addition, the trajectories and "propagation velocities" of the solitons recover to their initial values after each collision (whether attractive or repulsive). This equivalence between solitons and particles is the reason for the term "soliton". In an attractive coherent collision in Kerr media in which the solitons' trajectories are separated by a large angle, the solitons simply go through each other and remain virtually unaffected by the collision process (apart from a tiny displacement and a small change in absolute phase). When the

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attractive collision occurs at small angular separation between the solitons, they move towards each other, combine and separate periodically. On the other hand, in a repulsive Kerr collision the solitons simply move away from each other. Coherent collisions in Kerr media were demonstrated in Refs. [82] and [83]. Collisions in photorefractive and in other saturable nonlinear media are, in many aspects, much richer than those in Kerr media and therefore more interesting. First, saturable nonlinear media can support (2+1) D solitons and therefore collisions can occur in full 3D, giving rise to new effects that simply cannot exist in Kerr media. Second, as explained above, the self-induced waveguides in saturable nonlinear media can guide more than one mode. This gives rise to new phenomena, including soliton fusion, fission, and annihilation. In 1992 Gatz and Herrmann have found [84] numerically that solitons in saturable nonlinear media which undergo a coherent collision at shallow relative angles can fuse to each other. One year later,. Snyder and Sheppard have shown theoretically that colliding solitons can undergo "fission", that is, generate additional soliton states upon collision, or, in other cases, annihilate each other.[85] Their explanation was simple and elegant: since both solitons induce waveguides, one needs to compare the collision angle to the critical angle for guidance in these waveguide (that is, to the angle above which total internal reflection occurs and a beam is guided in the waveguide). If the collision occurs at an angle larger than the critical angle, the solitons simply go through each other unaffected (the beams refract twice while going through each other's induced waveguide but cannot couple light into it). If the collision occurs at "shallow" angles, the beams can couple light into each other's induced waveguide. Now if the waveguide can guide only a single-mode (a single bound state), the collision outcome will be identical to that of a similar collision in Kerr media. However, if the waveguide can guide more than one mode, and if the collision is attractive, higher modes are excited in each waveguide and, in some cases, the waveguides merge and the solitons fuse to form one soliton beam. Such a fusion process is always followed by some small energy loss to radiation waves, much like inelastic collisions between real particles. Experimentally, fusion of solitons was observed in all kinds of saturable nonlinear media: atomic vapor, photorefractives, and quadratic. Specifically with photorefractive screening solitons, fusion of two colliding soliutons was observed in incohertent [78],[81] as well as in coherent [79],[86],[87] interactions. Fission of photorefractive screening solitons was observed by Krolikowski and Holmstrom.[79] To illustrate these processes, we show the experimental results of collisions between photorefractive solitons in Figs. 7 and 8. Figure 7 shows a top-view photograph of an attractive incoherent collision in which the solitons pass through each at a large angle. An example of fusion in Fig. 8 shows the intensity distribution a long distance after a collision in which the same solitons collide at shallow angles and fuse to form a single beam. Soliton “collisions” for phase differences intermediate between 0 and π lead to energy exchange between the solitons. The interference pattern formed by the overlap of the “tails” of the solitons is intermediate in phase to that of both of the solitons so that power is scattered from one soliton into the other. The higher intensity soliton narrows in space, and the weaker one broadens. These effects have been observed with screening solitons, as descibed in Ref. [86]. Since saturable nonlinear media can support (2+1) D solitons, one can also look at collisions of solitons with trajectories that do not form a single plane. When the solitons are individually launched, they move in their initial trajectories. When they are launched simultaneously, they interact (attract or repel each other) via the nonlinearity and their trajectories bend. If the soliton attraction exactly balances the “centrifugal force” due to rotation, the solitons can "capture" each other into orbit and spiral about each other, much

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like two celestial objects or two moving charged particles do. This idea was suggested by Mitchell, Snyder and Polodian [77] in the context of coherent collision. Recently, Shih, Segev and Salamo have demonstrated such spiraling-orbiting interaction employing an incoherent collision between photorefractive screening solitons. [80] Under the proper initial conditions of separation and trajectories, the solitons capture each other into an elliptic orbit. This is shown in Fig. 9. If the initial distance between the solitons is increased, the solitons' trajectories slightly bend toward each other but their "velocity" is larger than the escape velocity and they do not form a "bound pair". On the other hand, if their separation is too small, they spiral on a "converging orbit" and eventually fuse. An effect similar to the spiraling-fusion part of the experiment was also observed in Ref. [88] with atomic vapor.

Figure 7. Top - view photograph of an (attractive) incoherent collision between two photorefractive screening solitons in which the solitons pass through each other at a large angle.

Figure 8. Fusion between the same solitons when the collision occurs at a shallow angle. Shown are the intensity profiles and photographs of beams A and B at (A) the entrance plane, (B) each individual soliton at the exit plane when the other is absent, and © the fused beam at the exit plane.

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The observation of spiraling brings about an interesting question: Do interacting spatial solitons also conserve angular momentum? Indeed, recent calculations by Y. Kivshar's group in Australia in collaboration with Segev's group confirm [89]that spiraling solitons conserve angular momentum whenever there is no power exchange between the interacting (coherently or incoherently) beams, i.e., the beams do not fuse and do not undergo fission [82]. The reason for these exceptions is intuitive: fusion and fission are always followed by "loss" of energy which is radiated away from the solitons (via radiation modes of the respective induced waveguides). Therefore, if energy (or power) is not conserved, angular momentum does not have to be conserved either. In the true spiraling process however, (i.e., excluding the fusion part of Ref. [80]), preliminary results indicate that all the power is conserved and therefore so is the angular momentum. It seems that this property is general, and should exist in any 3D nonlinear system of self-trapped wavepackets in nature. SELF-TRAPPING OF INCOHERENT LIGHT BEAMS As the final section of our review on photorefractive solitons, we wish to introduce a new, perhaps somewhat futuristic, direction into which solitons seem to be evolving: incoherent solitons. Until 1995, all soliton experiments and theories in nature employed a coherent "pulse", either in space, time, or both. In other words, given the phase at a given location on the pulse (space or time) one can predict the phase anywhere on that selftrapped pulse. However, pulses or wave-packets do not necessarily need be coherent. For example, one can focus into a narrow spot a light beam from a natural source such as the sun or an incandescent light bulb. Can such beam self-trap in a nonlinear medium?

Figure 9. Spiraling of two colliding photorefractive screening solitons with initial trajectories that do not lie in the same plane. Shown are the photographs of the optical beams. (a) Beams A and B at the input plane, (b) the spiraling soliton pair after 6.5 mm of propagation and (c) the spiraling pair after 13 mm of propagation. Note that occurs in elliptical beams. The triangles indicate the conters of the corresponding diffracting beams. After 6.5 mm the solitons have spiraled about each other by 270° and after 13 mm the spiraling angle doubles to 540°.

Last year, Segev's group at Princeton has demonstrated self-trapping of beams (spatial "pulses") upon which the phase varied randomly in time/space across any plane.[90] In the

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first experiment, a quasi-monochromatic partially spatially-incoherent light beam was employed. The beam originated from a laser and passed through a rotating diffuser that introduced a new (random) phase pattern every 1 microsecond. The beam was launched into a slowly-responding photorefractive crystal and, under appropriate conditions, the envelope of this beam self-trapped into a single non-diffracting narrow filament. Earlier this year, in a subsequent experiment Mitchell and Segev have demonstrated that an incoherent white light beam, i.e., a “pulse” that is both temporally and spatially incoherent, selftraps. [91] In this experiment the self-trapped beam originated from a simple incandescent light bulb which emitted light between 380-780 nm wavelength [see Fig. 10]. To understand the new ideas involved, we need first to explain some aspects of incoherent light. A spatially-incoherent beam is nothing but a multi-mode (so-called "speckled" beam) whose structure varies randomly with time. The beam consists of bright and dark "patches" (thus the notion "multi-mode") that are caused by a random phase distribution, which varies randomly with time. The envelope of this beam is defined by the time-averaged intensity. To illustrate this, consider a detector array (e.g., human eye) which monitors the beam. When this detector responds much slower than the characteristic

Figure 10. Self-trapping of an incoherent white light beam.

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phase fluctuation time, all it will "see" is the time-averaged envelope. Typically,such an incoherent beam diffracts much more than a coherent beam, since every small bright "patch" (speckle) contributes to the diffraction of the envelope. In the limiting case of the speckles being much smaller than the beam size, diffraction is dominated by the degree of coherence, i.e., the size of the speckle, rather than the diameter of the beam's envelope. Instantaneous nonlinearities cannot self-trap such a beam. If an incoherent beam is launched into a self-focusing nonlinear medium that responds instantaneously (e.g., the Optical Kerr effect), each small speckle forms a small "positive lens" and captures a small fraction of the beam. These bright-dark features on the beam change very fast throughout propagation and these tiny induced-waveguides intersect and cross each other in a random manner. The net effect is beam breakup into small fragments and self-trapping of the beam’s envelope will not occur. For self-trapping of an incoherent beam (an incoherent soliton) to exist, several conditions must be satisfied. First, the nonlinearity must be non-instantaneous with a response time that is much longer than the phase fluctuation time across the incoherent beam. Such a nonlinearity responds to the time-averaged envelope and not to the instantaneous "speckles" that constitute the incoherent beam. Second, the multi-mode (speckled) beam should be able to induce a multi-mode waveguide via the nonlinearity. This is achievable in any saturable nonlinear media. Third, as with all solitons, self-trapping requires self-consistency: the multi-mode beam must be able to guide itself in its own induced waveguide. Theory of incoherent solitons was presented in two recent papers by Christodoulides and Segev groups.[92],[93] From the theory it is apparent that the selftrapping process re-shapes the statistics of the incoherent beam. For example, incoherent sources (e.g., the sun) have non-localized statistics: the correlation length (loosely defined as the distance beyond which two points are no longer phase-correlated) does not depend on the absolute location. In the incoherent soliton, however, the correlation length has a different value at the center of the beam and at its margins. Furthermore, it is possible to "engineer" (at least to some extent) the coherence properties of an incoherent beam by the self-trapping process. The rapid progress in this direction brings about many interesting fundamental ideas (such as coherence control) and possible applications (e.g., using selftrapped beams from incoherent sources, such as, Light Emitting Diodes) for reconfigurable optical interconnects and beam steering. CONCLUSIONS The last few years have witnessed a great renewal of interest in the photorefractive effect thanks to the realization of the possibility of propagating stable 2D spatial solitons at very low optical powers in PR crystals. This result, besides possessing a great scientific relevance in itself, associated with the first experimental demonstration of a non-diffracting two-dimensional pencil of light, has also an interesting applicative potential thanks to the guidance properties of the waveguide associated with the soliton, which survives in the dark also after the soliton has been turned off. In this paper, we have tried to present a selfcontained approach to the theory of self-trapped nonlinear propagation, which is far from being definitive, together with the most important experimental demonstrations obtained till now. The novelty of the field, and the fact that it is still undergoing a rapid growth, has made our task not an easy one and we apologize to the readers for the many inevitable omissions both in the subjects we have chosen to emphasize and in the references.

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We cannot conclude this review without mentioning several important developments which started to be investigated during the completion of this paper. The first concerns self-trapping of optical beams in biased photorefractive semiconductors, such as InP:Fe, in which both electrons and holes participate in the formation of space charge field [94],[95]. Interestingly enough, the self-focusing effects undergo a large enhancement when the rate of optical excitation of holes is close to (but smaller than) the thermal excitation rate of electrons. When the optical excitation of holes exceeds the thermal excitation rate of electrons, self-focusing turns into self-defocusing, i.e., the sign of the optical nonlinearity can be reversed by all optical means. These type of solitons seem important for applications because they are sensitive in the near-infrared wavelengths range and because they respond much faster than all other photorefractive materials (with the same intensities). The second progress has to do with the existence of solitons in centro-symmetric photorefractive media [96], which fundamentally do not possess quadratic nonlinearities. The change in the refractive index that gives rise to these solitons is driven by the dc Kerr effect, which is similar to Pockels' effect but δn is now proportional to (E s c) ² and thus to 1/(I+Id ) ² . Recent experiments performed by Del Re, Crosignani and collaborators at the Burdoni Institute and University of L'Aquila (Italy) have demonstrated these solitons. Finally, in the last two years several groups have predicted the existence of photorefractive vector solitons [97] and two-component solitons [98],[99],[100]. Of particular interest are the incoherent soliton pair (which form a system that resembles Manakov's solitons. These were predicted by Christodoulides et al. [101] and subsequently observed by Chen et al. at Princeton [102],[103],[104]. Acknowledgments We would like to dedicate this review to our friends and collaborators who have gone with us a long way over the last few years, exploring these new and exciting field of physics: Demetri Christodoulides, George Valley, Yuri Kivshar, Marty Fejer, Matt Bashaw, Minoru Taya, Galen Duree, Mark Garrett, Toni Degasperis, Stefano Trillo, Matthew Chauvet, Tamer Coskun, and Amnon Yariv. REFERENCES 1.Bloembergen N., Nonlinear Optics, Addison-Wesley, Reading, MA (1991); Shen Y.R., The Principles of Nonlinear Optics Wiley, New York, (1984). 2. R.Y. Chiao, E. Garmire and C. H. Townes, Phys. Rev. Lett. 13:479 (1964), . 3. Zakharov V.E. and Shabat A.B., Sov.Phys. JETP, 34:62 (1972) . 4. see, e.g., Agrawal G.P., Nonlinear Fiber Optics, Academic Press, New York (1989). 5. see, e.g., Hasegawa A. and Kodama Y., Solitons in Optical Communications, Clarendon Press, Oxford (1995). 6. see, e.g., Infeld E. and Rowlands G., Nonlinear Waves, Solitons and Chaos, Cambridge University Press, Cambridge (1990). 7. Temporal solitons were first demonstrated by Mollenhauer L. F., Stolen R. H., and Gordon J. P., Phys. Rev. Lett. 45:1095 (1980). 8. see., e,g., the first observation of spatial solitons in solids, Aitchison J. S., Weiner A. M., Silberberg Y., Oliver M. K., Jackel J. L., Leaird D. E., Vogel E. M. and Smith P. W., Opt. Lett. 15:471(1990). 9. Kelley P. L., Phys. Rev. Lett. 15:1005 (1965). -

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10. Zakharov V.E., and Rubenchik A. M., Sov. Phys. JETP, 38:494 (1974). 11. We use the term "soliton" in conjunction with non-diffracting self-trapped optical beams, i.e., we use the broader definition of solitons that includes these in nonintegrable systems, as defined by Zakharov V. E., and Malomed B. A., in Physical Encyclopedia, Prokhorov A. M., Ed., Great Russian Encyclopedia, Moscow (1994). 12. Segev M., Crosignani B., Yariv A., and Fischer B., Phys.Rev.Lett. 68:923(1992). 13. Duree G.C., Shultz J.L., Salamo G. J., Segev M., Yariv A., Crosignani B., Di Porto P., Sharp E.J., Neurgaonkar R.R., Phys.Rev.Lett. 71:533 (1993) . 14. Torruellas W. E., Wang Z., Hagan D. J., Van Stryland E. W., Stegeman G. I., TornerL., and Menyuk C. R., Phys. Rev. Lett., 74:5036 (1995). 15. Tikhonenko V., Christou J., and Luther-Davies B., Phys. Rev. Lett. 76:2698 (1996) ; one related obserevation was reported much earlier: Bjorkholm J. E., and Ashkin A., Phys. Rev. Lett. 32: 129( 1974) . 16. Ashkin A., Boyd G.D., Dziedzic J.M., Smith R.G., Ballman A. A., Levinstein J. J. , and Nassau K. , Appl. Phys. Lett. 9:72 (1966). 17, Yariv A, Optical Electronics, 5th ed. Wiley, New York (1995). 18. Yeh P., Introduction to Photorefractive Nonlinear Optics Wiley, New York (1993). 19. Solymar L., Webb D.J., and Grunnet-Jepsen A. The Physics and Applications of Photorefractive Materials, Clarendon Press, Oxford (1996). (1975) . 20. Vinetski V.L., and Kukhtarev N.V., Sov. Phys. Solid State, 16:2414 21. Kukhtarev N.V. , Markov V.B., Odulov S.G., Soskin M. S., and Vinetski V.L., Ferroelectrics 22:961 (1979). 22. The necessity of having a number NA of acceptors impurities, which permanently trap a corresponding number of electrons, is due to the role they play in determining the Debye length of the crystal. In fact, if not for the presence of acceptors, the Debye length would be too large for the purpose of recording optical information. 23. Crosignani B., Di Porto P., DeGasperis A., Segev M., and Trillo S., "Threedimensional Optical Beam Propagation and Solitons in Photorefractive Crystals", to be published (accepted) in J. Opt. Soc. Am. B, November, (1997). 24. This term can be shown to be responsible for a self-bending of the self-trapped beam which does not alter its shape; see, e.g., Singh S.R. and Christodoulides D.N. , Opt.Comm., 118:569 (1995). 25. Segev M., Valley G., Crosignani B., Di Porto P., and Yariv A., Phys.Rev.Lett.,73:3211 (1994). 26. Segev M., Shih M. and Valley G., J. Opt. Soc. Am. B 13:706 (1996) . 27. Christodoulides D.N., and Carvalho M. I., J. Opt. Soc. Am. B 12: 1628 (1995) . 28. Segev M.,Yariv A., Salamo, G., Shultz J., Crosignani B., and Di Porto P., Opt. Photonics News 4(12): 8 (1993) . 29. Duree G.C., Morin M., Salamo G.J., Segev M., Crosignani B., Di Porto P., Sharp E., and Yariv A., Phys.Rev.Lett. 74: 1978 (1995) . 30. Segev M., Salamo G.J., Duree G.C.,Morin M., Crosignani B., Di Porto P., and Yariv A., Opt.Photonics News 5(12):9 (1994) . 31. Duree G.C., Salamo G. J. , Segev M.,Yariv A., Crosignani B., Di Porto P., and Sharp E., Opt.Lett. 16:1195 (1994) . 32. Segev M., Crosignani B., Di Porto P., Yariv A., Duree G.C., Salamo G.J., and Sharp E., Opt.Lett. 17:1296 (1994) . 33. Morin M., Duree G., Salamo G., and Segev M., Opt.Lett. 20:2066 (1995). 34. Stepanov's JOSA B,(Dec. 1996). 35. Crosignani B., Segev M., Engin D., Di Porto P., Yariv A., and Salamo G., J.Opt.Soc.Am. B 10:446 (1993).

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36. Christodoulides D. N., and Carvalho M. I., Opt. Lett. 19:1714 (1994). 37. Fressengeas N., Maufoy J., and Kugel G., Phys.Rev. E 54: 6866 (1996). 38. Segev M., Crosignani B., Salamo G., Duree G., Di Porto P., and Yariv A., Photorefractive Spatial Solitons, Chapter 4 in Photorefractive Effects and Materials Edited by Nolte D., Kluwer, Boston (1995). 39. Glass A. M., von der Linde D., and Negran T. J., Appl. Phys. Lett. 25:233 (1974). 40. Sturman B. I., and Fridkin V. M., The Photovoltaic and Photorefractive Effect in NonCentrosymmetric Materials, Gordon and Breach, Philadelphia (1992). 41. Valley G. C., Segev M., Crosignani B., Yariv A., Fejer M.M., and Bashaw M., Phys.Rev. A 50:R4457 (1994). 42. Segev M., Valley G.C., Bashaw M.C. Taya M, and Fejer M.M., J. Opt. Soc. Am. B 14:1772 (1997). 43. Taya M., Bashaw M.C., Fejer M.M., Segev M., and Valley G.C., Phys.Rev.A 52:3095 (1995). 44. Taya M., Bashaw M.C., Fejer M.M., Segev M., and Valley G.C., Opt.Lett. 21:943 (1996) 45. Chen Z., Segev M., Wilson D. W., Muller R. E., and Maker P. D., Phys. Rev. Lett. 78:2948 (1997). 46. Bian S., Frejlich J., and Ringhofer K. H., Phys. Rev. Lett. 78:4035 (1997). 47. Iturbe-Castillo M. D., Marquez-Aguilar P. A., Sanchez-Mondragon J. J., Stepanov S., and Vysloukh V., Appl. Phys. Lett. 64:408 (1994) . 48. Shih M., Segev M.,Valley G.C., Salamo G., Crosignani B., and Di Porto P., Electron.Lett. 31:826 (1995). 49. Shih M., Leach P., Segev M., Garret M.H., Salamo G., and Valley G.C., Opt.Lett. 21:324 (1996). 50. Kos K., Meng H., Salamo G., Shih M., Segev M., and Valley G.C., Phys.Rev. E 53:R4330 (1996). 51. Mamaev A.V., Saffman M., Anderson D.Z. and Zozulya A.A., Phys.Rev. A 54:870 (1996). In this paper, it is claimed that both bright and dark photorefractive solitons suffer from transverse instabilities and are unstable. But the experimental results presented in the same paper clearly show “self-trapped channel of light” that do NOT exhibit transverse instabilities (Figs. 5d and 14b). Instability was observed only when further “increasing the nonlinearity” (see discussion on page 874 there). The similarity between the experimental and the numerical results showing the development of the instability in that paper is misleading: the experimental results (Figs. 5 and 14) show sequences of output intensity distributions (at a fixed propagation distance) for different levels of applied voltages (i.e., different levels of nonlinearity), ONLY ONE of which corresponds to that of a soliton (is "on" the existence curve), whereas the simulations (Figs. 1 and 9) show the near field intensity evolution for different propagation distances for a fixed set of parameters (i.e., at a fixed voltage) that greatly DIFFER from that of a soliton. 52. Infeld E., and Lenkowska-Czerwinska T., Phys. Rev. E 55:6101 (1997). 53. Zozulya A. A, and Anderson D. Z., Phys. Rev. A 51:1520 (1995). 54. Zozulya A. A, and Anderson D. Z., Opt. Lett. 20837 (1995). 55. Burak D., and Nasalski W., Appl. Opt. 33:6393 (1994). 56. Chen Z., Mitchell M., Shih M., Segev M., Garret M.H., and Valley G.C., Opt.Lett. 21:629(1996). 57. Swartzlander G.A., Andersen D.R., Regan J. J., Yin H. and Kaplan A.E., Phys. Rev. Lett. 66:1583 (1991) .

288

58. Allan G.R., Skinner S.R., Andersen D.R. and Smirl A.L., Opt. Lett. 16:156 (1991) . 59. Iturbe-Castillo M. D., Sanchez-Mondragon J. J,, Stepanov S. I., Klein M. B., and Wechsler B. A., Opt. Comm. 118:515 (1995). 60. Chen Z., Mitchell M., and Segev M., Opt.Lett. 21:716 (1996). 61. Chen Z., Segev M., Singh S. R., Coskun T., and Christodoulides D. N., J. of Opt. Soc. Am. B 14, 1407 (1997). 62. Mamaev A.V., Saffman M., and Zozulya A.A., Phys. Rev. Lett., 77:4544 (1996). 63. Chen Z., Shih M., Segev, Wilson D. W., Muller R. E., and Maker P. D., Steady-state vortex screening solitons formed in biased photorefractive media, submitted to Opt. Lett., August (1997). 64. Askar'yan G. A., Sov. Phys. JET, 15:1088 (1962). 65. Snyder A. W., Mitchell D. J., Poladian L., and Ladouceur F., Opt. Lett. 16:21 (1991). 66. Snyder A. W., Mitchell D. J., and Kivshar Y. S., Mod. Phys. Lett. B, 9: 1479 (1995). 67. Mitchell D. J., Snyder A. W., and Poladian L., Opt. Comm. 85:59 (1991). 68. De La Fuente R., Barthelemy A., and Froehly C., Opt. Lett. 16:793 (1991). 69. Luther-Davies B., Yang X., Opt. Lett. 17:496 (1992); Opt. Lett. 17: 1755 (1992). 70. Bosshard C., Mamyshev P. V., and Stegeman G., Opt. Lett. 19:90 (1994). 71. Swatrzlander G. A., and Law C. T., Phys. Rev. Lett. 69:2503 (1992). 72. Shih M., Segev M., and Salamo G., Opt.Lett. 21:931 (1996). 73. Shih M., Chen Z., Mitchell M. and Segev M., Waveguides induced by photorefractive screening solitons, to be published (accepted) in J. Opt.Soc. Am. B, October, 1997. 74. Lan S., Shih M., and Segev M., Self-trapping of 1D and 2D Optical Beams and Induced Waveguides in Photorefractive KNbO3, to be published (accepted) in Opt. Lett., 1997. 75. Zabusky N. J., and Kruskal M. D., Phys. Rev. Lett. 15:240 (1965) . 76. Snyder A. W., and Sheppard A. P., Opt. Lett. 18:482 (1993). 77. Mitchell D. J., Snyder A. W., and Polodian L.,, Opt. Comm. 85:59 (1991). 78. Shih M., and Segev M., Opt. Lett. 21:5318 (1996). 79. Krolikowski W., and Holmstrom S. A., Opt. Lett. 22:369 (1997). 80. Shih M., Segev M., and Salamo G., Phys. Rev. Lett. 78:2551 (1997). 81. Shih M., Chen Z., Segev M., Coskun T., and Christodoulides D. N., Appl. Phys. Lett. 69:4151 (1996). 82. Barthelemy A, Maneuf S., and Froehly C., Opt. Comm. 55:201 (1985); F. Reynaud and A. Barthelemy, Europhys. Lett. 12: 401 (1990). 83. Aitchison J. S., Weiner A. M., Silberberg Y., Oliver M. K., Jackel J. L., Leaird D. E., Vogel E. M., and Smith P. W., Opt, Lett. 15:471 (1990); ibid 16:15 (191). 84. Gatz S., and Herrmann J., IEEE J. Quant. Elect. 28:1732 (1992). 85. Snyder A. W., and Sheppard A. P., Opt. Lett. 18:482 (1993). 86. Meng H., Salamo G., Shih M., and Segev M., Opt. Lett. 22:448 (1997). 87. Garcia-Quirino G. S., Iturbe-Castillo M. D., Vysloukh V., Sanchez-Mondragon J. J., Stepanov S. I., Lugo-Martinez G., and Torres-Cisneros G. E., Opt. Lett. 22:154 (1997) 88. Tikhonenko V., Christou J., and Luther-Davies B., Phys. Rev. Lett. 76:2698 (1996). 89. Shih M., Segev M., Salamo G., and Kivshar Y., Do interacting spatial solitons conserve angular momentum?, to be published, Opt. and Phot. News, December, 1997. 90. Mitchell M., Chen Z., Shih M., and Segev M., Phys.Rev.Lett. 77:490 (1996).

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91. Mitchell M., and Segev M., Nature, 387:880 (1997); See also commentary on this article in the News and Views section of the same issue: Boardman A., Nature, 387:854 (1997). 92. Christodoulides D.N, Coskun T.H., Mitchell M., and Segev M., Phys.Rev.Lett. 77:490 (1996). 93. M. Mitchell, M. Segev, T. Coskun and D. N. Christodoulides, Theory of self-trapped spatially-incoherent light beams, submitted to Physical Review Letters, June, 1997. 94. Chauvet M., Hawkins S.A., Salamo G. J., Segev M., Bliss D.F., and Bryant G., Opt.Lett. 21:1333 (1996). 95. Chauvet M., Hawkins S.A., Salamo G. J., Segev M., Bliss D.F., and Bryant G., Appl. Phys. Lett. 70:2499 (1996). 96. Segev M., and Agranat A., Opt. Lett. 22:1299 (1997). 97. Segev M., Valley G. C., Singh S. R., Carvalho M. I., and Christodoulides D. N., Opt. Lett. 20:1764 (1995). 98. Singh S. R., Carvalho M. I., and Christodoulides D. N., Opt. Lett. 20:2177 (1995). 99. Carvalho M. I., Singh S. R., and Christodoulides D. N., Phys. Rev E, 53:R53 (1996). 100. Krolikowski W., Akhmediev N., and Luther-Davies B., Opt. Lett. 21:782 (1996). 101. Christodoulides D. N., Singh S. R., Carvalho M. I., and Segev M., Appl. Phys. Lett. 68:1763 (1996). 102. Chen Z., Segev M., Coskun T., and Christodoulides D. N., Opt. Lett. 21:1436(1996). 103. Chen Z., Segev M., Coskun T., Christodoulides D. N., Kivshar Y., and Afanasjev V. Opt. Lett. 21:1821 (1996). V., 104. Chen Z., Segev M., Coskun T., Christodoulides D. N., and Kivshar Y., Coupled photorefractive spatial soliton pairs, to appear in J. Opt. Soc. Am. B, Nov. 1997.

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SUB-CYCLE PULSES AND FIELD SOLITONS: NEAR- AND SUB-FEMTOSECOND EM-BUBBLES

A. E. Kaplan, S. F. Straub * and P. L. Shkolnikov Electrical and Computer Engineering Department The Johns Hopkins University Baltimore, MD 21218

ABSTRACT We demonstrate the feasibility of strong (up to atomic fields) and super-short (few- or even sub-femtosecond) sub-cycle (non-oscillating) electromagnetic solitons -- EM bubbles (EMBs) in a gas of two-level atoms, as well as EMBs and pre-ionization shock waves in classically nonlinear atoms. We show that EMBs can be generated by existing sources of radiation, including sub-picosecond half-cycle pulses and very short laser pulses. We investigate how EMB characteristics are controlled by those of originating pulses. Our most recent results are focused on the related transient phenomena, including EMB formation length, multi-bubble generation and shock-like waves. We also develop the theory of the diffraction-induced transformation of sub-cycle pulses in linear media. * Also with Abteilung für Quantenphysik, Ulm University, Ulm, Germany

1 . INTRODUCTION Contemporary optics usually operates with almost-harmonic, multi-cycle oscillations modulated by an envelope much longer than a single cycle of the oscillations. In fact, any narrow-line radiation is an envelope signal, be it a coherent radiation of a laser, or an incoherent light filtered through a spectroanalyzer. This is also true for any optical pulse, including self-induced transparency (SIT) solitons in two-level systems (TLS) [1], described by Maxwell-Bloch or sine-Gordon equations; mode-locked laser pulses [2] due to multi-mode cavity interaction with laser medium; and optical-fiber solitons [3] due to Kerr-nonlinearity, described by a nonlinear Schrödinger equation [4],etc. To describe any of those pulses, slow-varying envelope approximations are used in both the propagation (by reducing Maxwell equations to a parabolic partial differential equation) and the material response (rotating-wave approximation in constitutive equations). Due to the

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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availability of very short laser pulses (down to ~ 6ƒs length [5a] and even below 5ƒs [5b]), with just a few laser cycles, the efforts are made to improve the envelope approximation at least for linear propagation (see e. g. [6]). A lot of new experimental techniques and applications, however, such as time-domain spectroscopy [7] of dielectrics, semiconductors and flames [8], and of transient chemical processes, e. g. dissociation and autoionization [9], new principles of imaging [10], and atomic physics by means of photoionization [11], would greatly benefit from the availability of short and intense electromagnetic pulses of non–oscillating nature, i. e. sub-cycle (almost unipolar) "half-cycle" pulses (HCPs). The spectra of currently available HCPs generated in semiconductors via optical rectification, reach into terahertz domain; these HCPs are ~400 – 500 ƒs long, with the peak field of 150 – 200 KV/cm [9]. In our recent work [12-14], we have proposed two new different principles of generating much shorter (down to 0.1 ƒs = 10 –16 s) and stronger (up to ~ 1016 W/cm 2 ) pulses. One of these principles is based on stimulated cascade Raman scattering and would result in the generation of an almost periodic train of powerful sub-femtosecond pulses [12], while the other relies on the generation of powerful "EM-bubbles" (EMBs) [13,14], sub-cycle solitary pulses of EM radiation propagating in a gas of two-level or classically nonlinear atoms. The latter effect would allow one to generate a single EMB, or a few EMBs with controllable parameters, each of EMBs propagating with different velocity such that one can easily separate them into individual pulses. In this paper we review our recent research on EM-bubbles and present new related results on transient processes, in particular, on the formation length of EMBs, the generation of multi-EMBs, and the formation of shock and shock-like waves. Such super-short and intense sub-cycle pulses might be of great interest for the host of applications (see below). Especially significant are non-oscillating solitary waves that are able to propagate over substantial distances with unchanged shape and length. The exact soliton-like solutions for the nonlinear propagation of unipolar pulses in the stronglydriven two-level system (TLS), described by full Maxwell + full Bloch equations, were found quite a while ago [15]. The solutions have a familiar, 1/cosh, profile, with its duration and velocity related to its amplitude. At that time the authors of Refs. [15] did not believe that these nonlinear pulses were feasible; the main stumbling point they saw was 2 that the pulse intensities would exceed ~ 1014 W/cm , the level unaccessible then. Now optical fields a few orders of magnitude larger are available; however, one of the major problems in the generation of such short (and intense) pulses lies in that the TLS model used in the theory [15] (and in some more recent research [16,17]) will be stretched far 14 2 beyond its limitations, since intensities above ~ 10 W/cm cause very fast over-thebarrier ionization. What are the largest intensities (and thus the shortest lengths) of these pulses that can still be supported by atomic gasses? What are new properties of these pulses beyond the TLS approximation? Fortunately enough, these and other questions about high-intensity super-short pulses, can be addressed using the very fact that the atom is so strongly excited that one can use again its classical (as opposed to quantum) description [13]. In the intermediate domain, a multi-level quantum approach has to be used. We show here that EMBs are not only feasible but natural for many nonlinear system, both quantum and classical. Their length may range from picoseconds to subfemtoseconds, depending on their intensity. We call them EM-bubbles to stress their nonenvelope nature. We demonstrate that field ionization, a fundamental factor not considered previously, imposes an upper limit on the EMB amplitude and a lower limit on its length; after an EMB reaches its shortest length at some peak amplitude, further increase of the amplitude results in EMB broadening. At some threshold amplitude, the EMB degenerates into a shock wave that is a precursor of a dc ionizing field -- a new feature which is not present in TLS model. Furthermore, we show that even at much lower peak intensities,

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when TLS model may still be valid, the initially smooth HCP may drastically steepen to form a shock-like wave which then breaks in a multi-EMB solution [14]. Unlike a dcionization precursor shock wave, this shock-like wave can appear far below the ionization. EMBs can potentially be as short as 10 – 0.1ƒs, with the amplitudes approaching the atomic field. These super-short and intense sub-cycle pulses might be of great interest for the host of applications. They can be used for a "global" spectroscopic technique based on a shock-like excitation across the entire atomic spectrum (to the extent similar to passing atoms through a foil), including normally prohibited transitions. The ionization by a pulse shorter than the orbital period may bridge a gap between conventional photoionization and collisional ionization by a particle [11], with the important difference being that EM pulses offer a control of the quantum state of the atom during the entire process, and hence a control of its final state. This, in turn, has far-reaching implications for applications in timeresolved spectroscopy of transient chemical processes occurring on a femtosecond time scale, e. g. dissociation and autoionization (see e. g. [9]), especially for quantum control of chemical transformations (see e. g. [18]). These new pulses may expand time-domain spectroscopy of dielectrics, semiconductors, and flames [8] from presently available THz domain [7-11] to optical frequencies. One can also envision their applications to probing high-density plasmas, testing the speed of light, imaging molecules and atoms at surfaces, and for an order of magnitude frequency up-conversion due to the large Doppler shift of a counter-propagating coherent light backscattered by EMB, etc. A train of sub-femtosecond pulses with very high repetition rate (~ 125 THz, or with the spacing ~ 8 ƒs), feasible in cascade stimulated Raman scattering [12], can be used for the stroboscopy of atomic motion in a molecule (e. g. during its dissociation). Another property of EMBs, which may be greatly instrumental in their applications, is their extremely broad spectrum, which ranges ideally from radio-frequencies to visible or even ultra-violet domains. A single pulse of such nature would have a continuous power spectrum from zero frequency to the highest (cutoff) frequency of the pulse, (1.1) where t p is the pulse duration (evaluated at half-intensity). For example, with τ =0.2 ƒs, the cutoff wavelength, λ cut = 2 πc/ωcut ~ 2.4 c tp , is ~1440°A , i. e. in the far UV. It would be seen by a human eye as an extremely short and powerful burst of white light. Even the spectrum of a much longer, 1ƒs pulse, with λ cut ~7200°A, would still cover the infrared, millimeter, microwave, and rƒ domains. Thus the propagation of EMB would be greatly sensitive to a material in which they propagate. The EMB spectrum will be affected strongest by metallic particles or any other good conductors (the part of the spectrum below the respective plasma frequency will be absorbed), or by the presence of water or other substance having strong absorption bands, especially in infrared. Designating the EMB radiation here "S-rays" (where "S" stands for "sub-cycle" or "sub-femtosecond") in analogy to recently demonstrated "T-rays" [10] (THz pulses, see below), we note that the fact that different materials have different transparency for S-rays, suggests a great number of possible applications utilizing EMBs to emulate X-rays without X-ray-induced ionization damage. These S-rays can be used e. g. to monitor processing of high-density computer chips, screening food products at the food-processing facilities, luggage and concealed weapons in the airports, etc. One can also envisage applications of S-rays, similar to T-rays, but in the new frequency domain and with orders of magnitude broader spectra, for medical imaging, in particular, for a new kind of tomography, "S-tomography", with an additional possible advantage of positioning an S-ray source inside a human body. S-rays can also be a useful tool for the diagnostic of high-density fusion plasmas. This paper is structured as follows. Section 2 discusses a general relationship between the field and polarization, which results in a solitary wave as a solution of full

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Maxwell equation + arbitrary constitutive equations. Section 3 addresses an exact EMbubble solution of full Maxwell+Bloch equation for a two-level system. Section 4 is on EM-bubbles and shock waves which are due to a classical anharmonic potential with ionization. Section 5 discusses various approximate approaches to the transition processes in nonlinear EMB propagation. Section 6 concentrates on EMB generation by half-cycle pulses. Section 7 elaborates on multi-EMB solution; Section 8 discusses shock-like wave fronts due to multi-EMB formation, and Section 9 gives an example of EMB formation by short laser pulse. In Section 10, we develop the theory of diffraction-induced transformation of sub-cycle pulses. In conclusion, we briefly discuss future research on and physical ramifications of EMBs.

2. MAXWELL EQUATIONS AND GENERAL SOLITARY WAVE CONDITION Maxwell equation for the electric field is:

of a plane EM wave propagating along the z-axis, (2.1)

where is polarization density. Considering a pulse that propagates with a constant veloand imposing a steadycity, c β EMB , introducing retarded variables, state condition, (2.2) we reduce Eq. (2.1) to the "solitary wave (EMB) Maxwell equation": (2.3) where (2.4) is an EMB’s normalized relativistic "momentum". Stipulating now that an EMB car(a so called ries finite energy per unity area of cross-section, i. e. that bright soliton condition), and integrating Eq. (2.3) twice, we obtain a universal "EMBreplication" relationship between and (2.5) For Note that Eq. (2.5) is valid regardless of constitutive relationship between and our further calculations, we assume the field is linearly polarized, so that the wave equation can be reduced to scalar equation, and introduce dimensionless variables: field,ƒ, polarization, p, time, where ω0 is a characteristic frequency of the system, and distance, as well as dimensionless particle density, Q. All these variables and parameters are defined below for quantum and classical models separately; using them, we write Maxwell equation as (2.6) and EMB-replication relationship (2.5) as (2.7)

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3. EM-BUBBLES IN TWO-LEVEL SYSTEM Consider first the pulse propagation in a medium of quantum TLS characterized by the dipole moment, and resonant frequency, ω0 . We introduce normalized variables: the field (3.1) where Ω R is Rabi frequency; the polarization per atom, p = ρ12 + ρ 21 ; the population difference per atom, η = ρ 11 – ρ 22 , where ρ jk (j, k =1, 2) are density matrix elements of a TLS and time τ = (t – z/βEMB c) ω0 , to write full Bloch equa(with ρ 11 +ρ 22 =1 and tions as: (3.2) where the overdot designates ∂/ ∂τ ; we use the notation of [19], which addressed high harmonics generation in a super-dressed TLS. Note that (3.2) is not based on rotating wave approximation. Relaxation is not included in (3.2) since we consider pulses much shorter than TLS relaxation times. The first integral of Eq. (3.2) is square of the Rabi sphere radius, (3.3) The polarization density here is parameter Q in (2.6) is as:

where N is the density of particles; therefore, the (3.4)

where e is the electron charge, λ 0 = 2 πc/ω0 and is the fine structure constant. To find an EMB solution for TLS, we substitute the condition (2.7) (with unknown at this point M or β EMB ) into (3.2). Having in mind the invariant (3.3) for atoms being initially at equilibrium, η → 1 at | τ| → ∞, such that the first of Eqs. (3.2) gives us η ( τ ) = 1 –ƒ2/(2QM), we obtain from the second of Eqs. (3.2) a nonlinear equation for the EMB field, ƒ( τ ), as: (3.5) which is a so called Duffing equations. Its first integral is (3.6) (the integration constant C = 0 under the bright soliton condition), which determines a separatrix in the phase plane, and ƒ, starting and ending at the point = ƒ = 0. The next integration gives us finally an EM bubble, a solitary, non-oscillating wave: (3.7) the polarization and population are then: (3.8) In Eq. (3.7), the amplitude of EMB and its length are respectively: (3.9) Dimensional EMB amplitude, EEMB , by the definition of ƒ, Eq. (3.1), is (3.10) (For EMB length, T, defined at a half–peak field, i. e. T ≈ tEMB /1.32 = τ EMB /1.32ω0 , we have E EMB

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Instead of having ƒ EMB as function of M or β E M B , we can express β EMB in terms of ƒEMB : (3.11) or, if Q << 1 (see below, Section 5), (3.12) Shorter EMBs have higher amplitudes and move faster, approaching the vacuum speed of light. The lowest allowed speed of a bubble is (3.13) which corresponds to a linear propagation of an adiabatically slow pulse. The Fourier spectrum of EMB, (3.14) spreads from zero to the cutoff frequency, (3.15) Phase-portrait considerations show that with ƒ =p=1– η =0 a t | τ | → ∞, t h e non–oscillating EMB (3.7) is the only soliton supported by the system. Therefore, surprisingly, regular SIT envelope solitons [1], which have been obtained in the rotating-wave approximation, are inconsistent with the exact solution (3.7) based on full Bloch equations (3.2). This indicates that higher-order approximations may render SIT solitons unstable at long enough distances. EMB (3.7) may be regarded as a "full-Bloch" 2π -soliton; by introducing phase (or area) (3.16) we get φ R(∞) =2 π, which points to a "full-Bloch" area theorem. A similar EMB solution, (3,7), holds also for amplifying TLS media with the inversed population, η (| τ | → ∞ ) = – 1 . In this case, however [13], (3.17) Since a TLS with η ∞ = –1 is a non-equilibrium system storing pumping energy, β E M B here is not the speed of energy propagation, so that the fact that (i. e. the EMB moves faster than light) is not incompatible with special relativity. More intense EMBs here move slower, approaching the speed of light from above as their amplitude increases.

4. EM-BUBBLES AND SHOCK WAVES IN A CLASSICAL POTENTIAL The solution (3.7) is valid within the limitations of our TLS model. In particular, the EMB duration, must be shorter that all the atomic relaxation times, which still allows for EMBs as long as ~ 10 – 9 s, with longer EMBs having lower peak amplitude, Eq. (3.8), and moving slower, Eq. (3.9). It is instructive to consider an example of Xe, with ω0 – 8.44eV, effective dipole size, d/e ~ 7°A (based on the "super-dressed TLS" data –2 for high-harmonic generation in Xe [19]), and N ~ 10 19 cm –3 (Q ~10 ). For a 10ps long EMB, we have E p k ~10 3 V/cm. Longer pulses can be considered within the TLS model with relaxation. Of a particular interest, however, are the shortest and most intense pulses. When the EMB field approaches the atomic field (~108 –10 9 V/cm), the EMB formation is affected mainly by the atomic ionization potential, which limits EMB length and

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on EMB within a classical 1-D model of an atom, with a strongly nonlinear potential, U(x) , limited at |x| → ∞, to allow for ionization; here x is the electron displacement. Then Bloch equations (3.2) are replaced by a classical normalized equation for the electron motion: (4.1) with the dimensionless variables and parameters of the system defined as (4.2) where x 0 is an atomic characteristic size, and U0 is a characteristic energy (e. g., the ionization potential) [20]; m e is the mass of electron. The polarization density here is P=Nxe=Nex 0 p. Note that for EMB, TLS Bloch equations (3.2) reduce to a simple Duffing equation for, e. g. p, (4.3) with A = QM – 1 and B = (QM) 2 / 2, which is equivalent to Eq. (4.1) (with ƒ =pMQ ) for the simplest classical anharmonic potential, with

a = const > 0.

(4.4)

Hence, the potential (4.4) can give rise to the same EMB, Eq. (3.7). For an arbitrary potential u(p), the family of EMB solutions, p(τ), is found from Eq. (4.1) through the quadrature [13]: (4.5) A "bright" solitary solution to Eq. (4.5) exists, however, only for particular nonlinearities. For example, for (4.4) the nonlinearity must be "positive", a > 0 [21]. In general, if u is a smooth, monotonically increasing function of p 2 , the "bright" solitary solution exists only if near p = 0 (4.6) This requires the atomic potential to have sufficiently "hard walls", which holds for some model potentials [22] (but not for a "soft" potential as e. g. u = – (1 +p 2 ) –1/2 [22]). An example of a model potential that allows for an explicit analytic solution of Eq. (4.5) is: with

b = const >0 .

(4.7)

To illustrate the limitations imposed by over-the-barrier ionization, consider first a classical "box" potential, u(p) = 0 for |p| < 1, and u(p) = 1 otherwise, in which case the EMB field is (4.8) with ƒEMB ≤ 2, and M =ƒ EMB /Q. (We presume here that an electron always starts its motion at p = 0.) Thus, the maximal field strength, E max , and shortest EMB length, t min , are: (4.9) where U 0 is an ionization limit, and 2x 0 is the total box width. Emax is of the same nature as an atomic field, E a t =E max / 2, i. e. the atom is ionized (in classical terms) by a pulse of a certain shape [here, Eq. (4.8)], if its peak amplitude exceeds E max ; t min is the time required for such a field to pull an electron out of the potential well. (With U 0 =20eV and x 0 =1°A, this results in Em a x ≈ 2 10 9 V/cm, and t m i n ~ 0.4 10 – 1 6 s.) To make a connection to atoms with Coulomb long-range attraction, consider now a potential (4.10) with u – 1 ≈ p when

–1

, at |p | → ∞ It has a single well and satisfies hard-wall condition only For a given U0 and atomic number, Z, we have 297

(4.12) here r e = e 2 /m e c 2 is the classic electron radius. As an illustration, consider a limiting case with b = 0. Small-amplitude EMBs are governed again by a Duffing equation, its solitary solution being (Fig. 1, curve 1).

Fig. 1. Normalized field amplitude, ƒ, vs time, τ, for steady-state EMB (curves 1-3) and a shock wave (curve 4) due to ionization potential. Curves: 1 -- MQ = 0.12, 2 -- MQ = 0.187, 3 -- MQ = (MQ) ion – 1 0 –5 ; 4 -- MQ = (MQ) ion ≈ 0.3403.

Here and therefore, β cr = 0, i. e. small-amplitude EMBs here can move very slowly, a typical feature of any potential with du(0) /d(p 2 ) = 0. The EMB peak amplitude is Hence, as its amplitude increases, an EMB moves faster, and shortens. However, at pp k ≈ (8/45) 1/4 ≈ 0.65, ƒ pk ≈ 0.122, EMB length (at the half-peak amplitude) reaches its minimum, τA min ≈ 5.3 (at the half-peak amplitude, Fig. 1, curve 2) or τI min ≈ 2 (at the half-peak intensity). Assuming U 0 ≈ 2 4eV and Z = 2, as in He, one obtains the shortest EMB length: (4.13) (Significantly shorter EMBs can be attained with ionized atoms, e. g. ion beams, which may have ionization potential, U0 , orders of magnitude larger.) As the field amplitude continues to rise, EMB begins to broaden, becoming a flat-top pulse (Fig. 1, curve 3). Finally, at a threshold amplitude, p p k ≈ 1.245, ƒ p k ≈ 0.42, it becomes a shock (anti-shock) wave whose single leading (trailing) edge is a front of an ionizing (de-ionizing) cw field (Fig. 1, curve 4) [23]. The amplitude front rises (falls) as e x p ( τ / τ ion ), w i t h τ ion ≈ (MQ) –1/2 ≈ 1.7. This shock wave is typical to any hard-wall potential with ionization. Our preliminary results indicate, though, that a single-front shock wave becomes unstable, producing a short precursor that travels as a pilot EMB at a faster speed ahead of the group of other, longer and closely spaced EMBs, which merge into a cd field far behind the precursor. This pattern persists if one accounts for the plasma due to ionization behind the pilot group of EMBs. In a more detailed picture of a shock wave, the classical over-thebarrier ionization near the threshold must be modified by quantum tunneling.

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5. VARIOUS APPROXIMATE APPROACHES TO EMB PROPAGATION To demonstrate the existence of EMBs (in both quantum and classical eases) most rigorously, we have used so far a "double-full" approach: full Maxwell equation (2.1) + full constitutive equations (3.2) or (4.1) (i. e. without rotating wave approximation). The problem with this "double-full" approach is that at this point we do not have a mathematical theory which would allow us to handle a general solution of the problem (including the case of an arbitrary initial/boundary conditions) with the same degree of confidence and insightfulness as the inverse scattering theory provides for the so called fully integrable partial differential equations, like Kortdeweg-de-Vries, nonlinear Schroedinger, and some other equations. There are no results regarding the full integrability of "double-full" equations, nor even about them being of the same class of equations that can lend themselves to the inverse scattering theory. In physical terms, the very fact that the full nonlinear Maxwell equation allows for the coupling between forward and backward propagating waves, creates a significant complicating factor. Hence, in theoretical consideration of the propagation, as well as in numerical simulations, in particular for all kind of transient problems, we need to look for connections to some better understood equations, at least in certain meaningful limits. Closer consideration shows, fortunately, that "double-full" equations can often be reduced to much simpler equations (with some of them being fully integrable), while keeping them free from a rotating wave approximation and hence open to broad-spectrum solutions. Our computer simulations have shown that at low density, Q << 1 (e. g. in gasses, –1 –4 where typically, Q ~ 10 -10 ), Maxwell equation can be reduced to approximate firstorder equation without losing any significant feature of nonlinear propagation. In particular, the EMB solution have the same form, as for full Maxwell equation. This is explained by the fact that when Q << 1, the propagation velocity approaches the speed of light, 1 – β = O(Q) Q << 1. and any retroreflection can be neglected. Assuming now that the wave propagates only in one direction (e. g. positive ), using retarded variable τ = τ – / βLN , and keeping in mind definition (3.13), we transform Maxwell equation (2.6) to the equation: (5.1) Neglecting in it the term (which is small since the pulse changes relatively slow as it propagates along the axis), and eliminating one derivative, ∂ / ∂τ, by integrating the resulting equation over ∂τ, we can write now: (5.2) By rescaling the propagation coordinate,

we finally obtain: (5.3)

The physical implication here is that nonlinear retroreflection is neglected; the counterpropagating waves are decoupled. The validity of the reduced Maxwell equation can be verified by e. g. using it instead of Eq. (2.6) to obtain EMBs in either quantum and classical limits, as well as by numerical simulations of the transient propagation [14,13]. We have found also that Eq. (5.3) can still be used even if Q is not small, if the field spectrum does not stretch beyond ω0 . In order to describe the studied process by even simpler equations, and especially by fully integrable ones, one can work now on the simplification of constitutive equations. A major step in this direction is based on the observation that for the most of nonlinear gasses of interest, in particular, for noble gasses, the TLS frequency of the first transition, ω0 , is

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extremely high, so that even near-femtosecond pulses and laser oscillations are relatively slow compared with a cycle of that frequency. How slow is "slow" in this case? Xe has the lowest energy of the first excited level among the noble atoms; with ~ 8.5 ev, one cycle of ω 0 is 2 π /ω 0 ~ 0.5fs. For He, with ~ 20 ev, one cycle is ~ 0.2 fs. For any available half-cycle pulses (HCP), the HCP length, t 0 , is is three orders of magnitude longer; even the full cycle of e. g. Ti:Spph laser oscillation is ~ 2.7 fs -- still much longer than 2 π /ω 0 in these gasses. Another important parameter is the dimensionless amplitude of the incident field ƒ 0 , (3.1), which is also related to the time scale of the nonlinear motion in TLS, ƒ 0 = O( τ EMB ). Thus, introducing a parameter, we can see that it is very small for HCPs available now or in the foreseeable future. For noble gasses, for example, and with E 0 ~ 2 MV/cm, which is an order of magnitude higher than presently available HCP, we have ƒ0 ~ 10–2 ; with t 0 ~ 400 ƒs, we also have Hence, if ε << 1, we can use a "slow motion" (but no envelopes!) approximation, whereby ƒ(τ), p(τ) and η ( τ ) have their Fourier frequencies much smaller than ω0 . As a first step, "instantaneous weak response", we neglect in the second of equations (3.2), so that p 1 =ƒ η, and substitute it in the first of equations (3.2). By integrating it and having in mind the invariant (3.3) (i. e. η = 1 at we obtain (5.4) , and neglecting again in (3.2), by assuming now that we obtain in the next approximation: This, after evaluating from the latter equation and substituting it into the former one, results in integration of which yields ∆η ≈ For ƒ – p we have now: Writing

(5.5) Two last terms in the rhs of (5.5) reflect the Rabi dynamics, without which EM bubbles would not exist. Rhs of Eq. (5.5) is O( ε3 ); the next approximation correction is O( ε 5 ). In the limit ε → 0, Eq. (5.5) can be further simplified by noticing that since as well as and we can write, still with the precision O( ε5 ): (5.6) Eqs. (5.3) and (5.6) yield a single self-contained abridged Maxwell-Bloch equation: (5.7) It can be readily shown that solution (3.7) of the full Maxwell-Bloch equations are also solutions of Eq. (5.7). (5.7) is one of the so called Modified Kortdeweg-de Vries (MKdV) equations. The MKdV solutions could be associated [24] with a regular KdV equation, where in the second term, instead of ƒ 2 , one has ƒ. MKdV is fully integrable by using inverse scattering method and has an infinite number of invariants [24]. Similar equation can be obtained for a classical anharmonic oscillator (4.1) if the amplitude is not large, i. e. when approximation (4.4) with the coefficient of first-order nonlinearity, a, can be used. In this case, similarly to (5.6), we can write (5.8) and the self-contained wave equation, similar to (5.7), will be again MKdV: (5.9) If the incident field is due to a laser, and is, therefore, oscillating and strong, we may have ε >> 1. Presuming that TLS model and Bloch equations (3.2) are still valid, the Maxwell-Bloch equations can be reduced to another well-explored equation. Since we

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expect the driven polarization, p, to vary rapidly, we can drop p from the second of equations (3.2), assuming that Solving (3.2) in this approximation, we readily obtain: (5.10) where φR is Rabi phase (3.16); in this approximation, equation (5.3) can then be written as

The reduced Maxwell or (5.11)

With a proper choice of retarded coordinate, τ1, it can be reduced to even simpler equation, (5.12) These are different forms of so called sine–Gordon equation, which is fully integrable. It has again the same soliton solution, Eq. (3.7), as general Maxwell+Bloch equations do. While using Eqs. (5.10)-(5.12), one has to be cautious about choosing boundary conditions; only those functions ƒ(τ ) at ζ = 0 that satisfy a condition on the area of the pulse (see below), S ≡ φ R (∞ ) = 2 π n, where n is integer, are applicable for simulations. (The best choice would be S = 0, since it would allow one to vary the amplitude of the incident field once the shape of the field is chosen). If S is not integer of 2 π, this approximation will be inconsistent with the physics in the sense that it may happen that neither area S( ζ ) of the propagating pulse nor its energy W are invariants. Fortunately, the condition S = 0 can easily be satisfied for an oscillating field, which is exactly the area of the intended applicability of (5.10)-(5.12). However, even more stringent conditions may be imposed by the fact that TLS model is invalid when the Rabi frequency, Ω R, exceeds the TLS frequency, ω 0, i. e. when ƒ >> 1. Some other approximations that result in fully integrable equations can be found in [15-17]. We have to note that at this point no mathematical proof exists that in the general, "double-full" formulation, EMBs are real solitons in the sense of full integrability of the full Maxwell + full constitutive equations, and that, therefore, EMBs are absolutely stable. Our numerical simulations for both TLS and nonlinear classical potentials show that small EMB due to reduced Maxwell equation (5.3) are stable against both small and large (e. g. collision with another EMB) perturbations, which is consistent with the results of Ref. [15] for TLS. Large EMBs (approaching the ionization threshold) may become unstable and break down into smaller EMBs. In a related simulation, we have discovered that significantly below the ionization threshold the EMBs are remarkably stable upon temporal or spatial changes of medium parameters. In particular, when the gas density, N, was changed by two orders of magnitude along the path of propagation, the EMB profile and its length remained stable; only its velocity, βEMB , was adjusting to a varying density, such that N(z ) M( β EMB ) = inv .

(5.13)

An EMB generated e. g. in a gas jet can therefore "slide" into vacuum without distortion. Finally, it is worth noting that a very interesting recent work [25] suggested generation of non-oscillating or unipolar EM solitons and shock-like waves in nonlinear dielectrics due to collective effect (phonons) in a crystal lattice. The time scale of these solutions is much larger than those discussed here, with the soliton length tp being much longer than a cycle of the transverse optic lattice resonant frequency, ωOT, i. e. would be no shorter than a few picoseconds for the best of materials. The nonlinearity in [25] scales as E 2 , with the single soliton having the profile of cosh –2 ; the important fact is that, as shown in [25], the full nonlinear Maxwell equation for the case in consideration (based on simplified constitutive equation that uses the assumption of low frequencies and relatively weak field) can be reduced to a fully integrable Boussinesq-like equation.

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6. EM-BUBBLE GENERATION BY HALF-CYCLE PULSES One of the major avenues of EMB-generation [13] is to use existing half-cycle pulses (HCPs) [7-11] to launch much shorter EMB pulses in a nonlinear medium via a transient propagation process. At first, it is important to obtain 50 ƒs to ~ 5 ƒs long EMBs, thus attaining one to two orders of magnitude enhancement over available HCPs. In our computer simulations [13,14], we found that distinct individual EMBs could be obtained with a HCPs. In these simulations, we used incident HCPs (i. e. the solution ƒ( ζ , τ ) at ζ = 0 ) o f various profiles resembling a typical experimental profile, in particular, Gaussian, (6.1) and 1/cosh profile, i. e. the same as EMB, (3.7), but with its amplitude, ƒ0, unrelated to its length, τ0 : (6.2) All of them show very similar behavior in the formation of EMBs; in this paper we will address 1/cosh profile (6.2) as the only one that can bring up exact analytical results related to the formation of multiple EMBs; however, most of our results here on the parameters of the leading (i. e. largest and fastest) EMB, "EMB-precursor", in particular its amplitude, length, and formation distance, will be valid for any incident HCP. The TLS approximation is valid with great margin, if the instantaneous Rabi frequency is relatively small, ƒ << 1, which, as was shown in the previous section, is the case for the available HCPs; for instance, in noble gasses, even with still unavailable E = 2 MV/cm, ƒ ~ 10 –2. For a given length of the incident HCP, τ 0 , HCP’s threshold (minimal) amplitude required to attain a single EMB, provided the HCP has a profile (6.2), according to (3.9) and (3.10), is: (6.3) In most of our runs, we used τ0 = 4000, which corresponds to t 0 ~ 313 ƒs (or 413 ƒs at pulse’s half-amplitude), for Xe in this case, ƒthr = 1 0 –3 and E thr ≈ 60 KV/cm. Typical patterns of EMB formation are shown in Figs. 2 and 3.

Fig. 2. Double-EMB formation by HCP with ƒ0 = 2ƒthr . 302

Fig. 2 depicts a double-EMB formation for ƒ0 = 2ƒthr ; the larger EMB here is 3ƒthr = (3/2)ƒ0 → 180 KV/cm, and of the weaker one, ƒthr ; they are respectively 104 ƒs and 313 ƒs long. Fig. 2 can also be seen as a collision of two EMBs, with each of them coming out unaffected by the collision (aside from slight shift of their center lines of propagation); this can be shown by retracting the plot back in ζ < 0. For larger ƒ0 , more EMBs are formed and the strongest EMB moves faster than the rest of the pack, leading the train as a precursor. The "density plot" showing the linear trails of individual EMBs moving with different velocities, with the front trail being due to EMB-precursor, is depicted in Fig. 4. As ƒ0 increases, the precursor is growing stronger and shorter, and the distance, ζEMB , for it to break away from the mother-HCP, is decreasing. Fig. 3 depicts multi-EMB formation for E 0 =2 MV/cm (ƒ0 ≈ 3.3 × 10 –2 = 33ƒ thr ), expected to be available in the near future. In this case, z EMB is estimated (see end of Section 8) from ζ EMB ~ 1.23 × 10 5 ; for Xe at 10 atm (Q ~ 0.57 ), it translates into z EMB ~ 12.5 cm. The precursor here is 4.8 ƒs long, two orders of magnitude shorter than available HCPs.

Fig. 3. The formation of multi-EMBs and a shock-like wave front for as the wave propagates in ζ; inset: superimposition of the field profiles at different ζ illustrating the front formation.

Experiment- and application-wise, it is important to know how the properties of EMBs are controlled by the incident HCP. In this respect, one has to answer a few important questions: given the amplitude, E0 , and length, t 0 , of the incident HCP, (i) what are the leading EMB’s amplitude and (ii) its length? (iii) how many EMBs (per one HCP) can be

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generated? (iv) what are amplitudes and lengths of these EMBs? and (v) what is the formation distance for the first EMB? Although the equations governing the wave propagation are fully integrable in certain approximations (see the preceding section), the above questions cannot in general be answered analytically. Our combined numerical and analytical efforts [14] allowed us, however, to obtain remarkably simple results, which could be summarized as follows: the amplitude of EMB, EEMB , is proportional to (and larger than) the amplitude E 0 of an incident HCP; the EMB’s length is inversely proportional to E EMB ; the number of the EMBs is proportional to the area of the incident HCP. We have also discovered that when multiple EMBs are generated, they form at some point a shock-like wave front, its its formation being is proportional to We have shown that the distance of the first EMB formation is proportional to if E 0 is relatively small; for sufficiently large E0 , this process coincides with the shock-like wave formation. The main good news is that very short EMBs can be generated by a long HCP with sufficiently large amplitude.

Fig. 4. Density plot for the propagation shown in Fig. 3; note that the "trails" of individual EMBs make straight lines, i. e. each one of them propagate with its individual constant velocity, βEMB , (3.11).

In our computer simulations [14], using HCPs of various profiles [in particular, Gaussian (6.1), and cosh –1 (6.2)], we found that an EMB-precursor shows a linear dependence of its amplitude, ƒEMB , on the incident amplitude, ƒ0 , regardless of the profile: with a = const ~ 2.

(6.4)

For the profile (6.2), Eq. (6.4) becomes exact with a = 2, so that ƒEMB =2ƒ0 – ƒthr ,

(6.5)

see Fig. (5). This also gives the precursor’s length: (6.6) Due to (6.5) and (6.6), the amplitude and length of the largest EMB tend to constants as the HCP’s length τ0 increases (Fig. 6). Hence, to attain large and short EMB, there is no need to use a short incident pulse; the only prerequisite is a sufficiently high amplitude.

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To explain these results and to find other characteristics of the EMBs, in particular their formation distance, we use here the approach reminiscent of that developed in the theory of modulation instability in self-focusing and in propagation of pulses in nonlinear optical fibers. Approximating an initially long and smooth HCP by an almost dc wave with the amplitude of the incident slow pulse, and evaluating the propagation characteristics of this wave, we analyze the behavior of small perturbations of this wave.

Fig. 5. The EMB’s amplitude, ƒEMB , vs the amplitude, ƒ0 , of the incident HCP (both normalized to ƒthr ) . Curves: solid -- EMB-precursor; broken -- higher order EMBs.

Fig. 6. The EMB’s amplitude, ƒEMB , vs the length, τ 0 , of the incident HCP (normalized to ƒthr and τ thr , respectively). Curves: solid -- EMB-precursor; broken -- higher order EMBs.

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Linearizing the original equations with respect to these perturbations, and deriving a dispersion equation for their spectral components, we find then the spectral component with the fastest phase change. The speed of propagation of this unique component, its frequency, as well as a spatial scale (the shortest of all the components) at which a sufficient phase accumulation occurs, -- all this point to an EMB-precursor that will develop from this component. Note that this approach allows us to still work with full Maxwell+constitutive equations; to demonstrate it, we show here how the meaningful results can be obtained for full Maxwell+Bloch equations (2.6) and (3.2); we will also keep track of simplifications stemming from reduced equations (5.3) and (5.6), (5.7). Assuming a field with the amplitude ƒ0 = const in Eqs. (2.6) and (3.2), one obtains that the population difference, polarization per atom, "momentum" parameter, and the speed of this "dc" field are respectively: (6.7) where (6.8) is the Stark-shifted frequency of TLS due to the field effect. Solving now (2.6) and (3.2) for small perturbations of this solution, and representing these perturbations in terms of spectral components, we obtain the dispersion relationship between the wave number of the perturbation component q and its frequency Ω:: (6.9) In the linear (ƒ0 → 0), low frequency (Ω → 0) limit we have: q LN = Ω ( 1 + Q) 1/2 .

(6.10)

The part of q which is due to both the nonlinearity and dispersion if Q << 1 is thus: (6.11) The lowest ∆q( Ω) < 0 corresponds to the fastest perturbation. Looking for the minimum of ∆ q, we obtain the frequency, Ω = Ω fast , of this component as: (6.12) hence if

we have (6.13)

Substituting Ω = Ω fast into (6.9), we evaluate qfast and the phase velocity β fast ≡ Ω fast /q fast , of this component in the case Q << 1 as: (6.14) Comparison with (3.11) shows that a matching EMB, βEMB = β fast , has an amplitude: (6.15) or, for ƒfast ≈ 2ƒ0

(6.16)

Eq. (6.16) confirms the linear dependence between ƒ0 and ƒfast in (6.4) and fits perfectly the coefficient a = 2 in (6.5). Note that in an ideal dc field, τ0 → ∞ and thus ƒthr → 0, which explains the difference between (6.5) and (6.16). The same approach can be used to estimate the precursor formation distance, but only in the limited range of the parameters, since in general depends on total area of HCP (see below). To still use perturbation approach, we substitute again Ω = Ω fast i n t o

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(6.9) and estimate ζ EMB as a distance at which a certain change of phase, φ = 0(2 π ), is accumulated (the best fit is provided by In particular, if ƒ 0 << 1, ∆ qfast ≈ and [14] (6.17) (curve 1 in Fig. 7). Thus, ζ EMB , estimated by a phase change, scales as pares well with the distance of first appearance of a saddle point, in Fig. 7), in the numerically obtained field profile up to (ƒ0 /ƒthr )cr = Ncr ~ 4.

Fig. 7. The normalized formation distance, Curves: 1 -(6.17), field profile.

This com(dots

of EMB-precursor vs normalized incident amplitude, (6.18); dots -- first saddle point appearance in a

For larger ƒ0 , when multiple EMBs are generated (see below), right before the EMBprecursor breaks away, the initially smooth HCP drastically steepens to form a shock-like wave (Fig. 3), which, unlike a dc-ionization shock wave (section 4), can appear now far below ionization. Its formation distance, ζ sh (curve 2 in Fig. 7), that can be analytically calculated based on the theory of shock-like wave, section 8 below, is as [14]: (6.18) which scales as

now.

7. MULTI-BUBBLE SOLUTION When the incident amplitude of HCP, ƒ0 , sufficiently exceeds the threshold of EMB formation, ƒthr , more than one EMB will be generated, as one can see from Figs. 2 and 3. In the limit ƒ0 << 1, when the propagation is described by modified KdV (5.7), one can develop the analytical theory on N-bubble solutions for the profile (6.2). The results of our theory, to be published elsewhere, are based on invariants of MKdV [24], and are briefly summarized here. Total number of EMBs, NEMB , is:

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(7.1) where L(x) the largest integer not greater than x. For N0 >> 1, NEMB is proportional to the incident HCP area, ƒ0 τ0 , With the EMB-precursor designated by number 1, the amplitude, ƒn of the n -th EMB is given by an amazingly simple formula: (7.2) such that the decrement, (7.3) is independent of n. Since each bubble with the amplitude ƒEMB carries an energy: (7.4) which is proportional to its amplitude, ƒEMB , a unique quality of the function (6.2) is that the energies of the bubbles generated by it, are equidistant, or quantized, in the way reminiscent of the energy spectrum of a linear oscillator, with the "quantum" (or "quton") of their energy spectrum being ∆ Wq = 2 Wthr . It is worth noticing that when N 0 is an integer, the lowest energy of a bubble is exactly Wthr = ∆ Wq /2, which again is reminiscent of the energy of the ground level of a linear oscillator, being equal to half the quantum of excitation. For the EMB-precursor, n = 1, Eq. (7.2) coincides with Eq. (6.5), as expected. If ƒ0 i s an integer of ƒthr (6.3), the incident HCP gives rise to an exact N-bubble solution. Otherwise, a part, ∆ Wrad , of its incident energy, W 0 , is radiated away into non-trapped modes; their relative impact decreases rapidly as the total number of EMBs increases: (7.5) Thus ∆Wrad is always smaller than the critical (smallest) energy, Wthr of a bubble (for the fixed τ0). Furthermore, as the incident amplitude increases, the relative maximum energy of un-trapped radiation greatly decreases: (7.6)

8. SHOCK-LIKE WAVE FRONTS When the incident HC-pulse is sufficiently strong to generate many EMBs (N0 >> 1) , right before the EMB-precursor is formed, the initially smooth HCP drastically steepens and forms a shock wave at the front of the pulse. The formation of the EMB-precursor coincides with the point in space at which the shock wave is steepest; this front is about τEMB long. After this point, the shock wave breaks into the train of EMBs. To investigate this shock wave formation and estimate the location of the breaking point, we make further approximation, which may be called "instantaneous reaction", by dropping the higher derivative terms in the constitutive equations of the system. In the limit Q << 1, Eq. (5.3) is replaced in the case of TLS by (8.1) and in the case of the anharmonic classical oscillator (5.3) -- by (8.2) If the HCP amplitude is small, ƒ << 1, Eqs. (8.1) and (8.2) are further simplified to

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(8.3) with a being the same as in Eq. (4.4) for classical anharmonic potential; a = 1/8 for TLS. Any equation in the form (8.4) where F(ƒ) is some smooth function, has a general solution whereby each point of the solution, ƒ = ƒ1 , moves with the fixed velocity determined by ƒ1 : (8.5) 2 –3/2

where in the case of Eq. (8.1), F(ƒ) = 1 – (1 +f ) , and of Eq. (8.3), -- F(ƒ) = 12aƒ2 . Evaluating now the derivative ∂f/∂τ at a point ζ, we find (8.6) such that for any nonzero there will be a point ζ, at which ∂ ƒ/∂τ → ∞, which signifies the formation of shock wave. Since the formation of the shock wave will is maximal, we find that for the profile be arrested at the amplitude ƒ1 , at which (6.2), such a point is at c osh or at with Hence the distance of formation of the shock wave is (8.7) which in the case of TLS (i. e. when a =1/8 and τ0 = 4/ƒthr ) gives Eq. (6.18). At the point of shock wave formation, the full constitutive equation will prevent the discontinuity of the exact solution, and break the shock wave into the train of solitons; the length of the steepest rise of the shock wave is thus determined by the EMB-precursor time, (3.9). Consider an example, E 0 = 2 MV/cm with t 0 ~ 313 ƒs (or 413 ƒs at pulse’s halfamplitude), in Xe. In this case ƒthr = 10 –3 and Ethr ≈ 60 KV/cm (ƒ0 ≈ 3.3 × 10 –2 = 33ƒthr ) , and the formation distance is estimated, Eq. (6.18), as ζ EMB ~ 1.54 × 10 6 , which under 10 atm pressure (Q ~ 0.57 ) translates into z sh ~ 12.5 cm. The EMB-precursor here is 4.8 ƒs long, two orders of magnitude shorter than available HCPs. Note that in all these examples with HCPs, the field ƒ << 1 (Ω R << ω 0 ) is much below the super-dressed regime of TLS, and therefore far from the ionization. The distance of the shock wave (and first EMB) formation can be shortened, if its leading front is sharpened (e. g. by a shatter), such that τlead < τ0. The distance ζ sh can then be evaluated by multiplying (8.7) by a factor τlead / τ0 ; in the above example, if the HCP leading front is shortened down to ~ 40 ƒs, the shock formation distance reduces to ~ 1.25 cm.

9. EM-bubbles generation by a short laser pulse Even the highest realistically expected fields of HCPs are still much lower that the amplitudes readily attainable in lasers. The possibility of the EMB formation in each laser cycle, therefore, increases tremendously, although the ensuing picture becomes more complicated due to the multiple EMB interactions, when the regular laser radiation with many oscillations in the envelope is used instead of HCPs. Indeed, since the laser cycle is much shorter (e. g., the cycle duration for the radiation with λ=0.9 µ m is ~ 3 ƒs ), with the laser intensities of ~ 10 14 W/cm 2 (which corresponds to the field ~ 2.7 × 10 8 V/cm), the EMB formation distance reduces to less than 1 mm, and the EMB becomes an order of magnitude shorter than the optical cycle. Fig. 8 shows a group of EMBs developing from a very short (6 ƒs) laser pulse with the relatively low peak intensity 6.8 × 1012 W/cm 2 . One can see that

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the EMB formation length is about 0.1 – 0.2 mm, and the length of the EMB-precursor is about 0.5 ƒs. There is a distinct possibility that the very high-order harmonics generation [26] in noble gasses might be to a substantial degree attributed to the multiple EMB formation, which would explain many major features of the HHG phenomenon, such as its puzzling insensitivity to the phase mismatch at different high harmonics, broadening and shift of harmonic spectra, etc.

Fig. 8. EMB formation from an oscillating laser pulse (tp ≈ 6 ƒs). Inset -- a final cross-section magnified to show EMBs.

10. DIFFRACTION-INDUCED TRANSFORMATION OF SUB-CYCLE PULSES [27] So far we were focusing on nonlinear propagation of sub-cycle pulses, being mostly interested in formation of solitons and EM-bubbles in nonlinear media. >From the application point of view it is important also to know what would happen with a sub-cycle pulse when it is emitted into a linear medium (e. g. air or vacuum). It is clear that the diffraction will immediately affect not only spatial profile of the pulse, but also its temporary profile, since different Fourier components of such a broad-band pulse diffract differently. Lowfrequency components diffract most drastically, almost as the radiation of a point source, while very high-frequency components may propagate almost without diffraction like geometro-optical rays. Thus, on-axis radiation will be loosing the low-frequency part of its

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spectrum, while the off-axis radiation will be loosing its high-frequency part, which will result in a peculiar transformation of both of them. In particular, we show below that the on-axis pulse in the far field area will be mimicking the time-derivative of the original pulse (thus formating "full-cycle" pulse out of half-cycle pulse), while in addition to that, the off-axis propagation also results in the lengthening of this pulse. In this Section, following our recent work [27], we develop an analytic approach to the theory of linear diffraction transformation of pulses with super-broad spectra and arbitrary time dependence, in particular half-cycle (unipolar) pulses. Since this theory has much broader applications than nonlinear processes, we will develop it for spatially 3D case (or precisely speaking, 2+1+1D case, i. e. with the spatial cross-section of the beam having two dimensions, and other two dimensions formed by the axis of propagation and time). We found close-form solutions for pulses with initially Gaussian spatial profiles having either cosh– 1 -like or Gaussian time dependence. The far-field propagation demonstrates time-derivative behavior regardless of initial spatio-temporal profile. Diffraction is one of the fundamental manifestations of the wave nature of light. The diffraction theory of monochromatic light has been developed in great details (see e. g. [28]). In general this theory is heavily loaded with various special functions, but the development of lasers advanced the use of Gaussian beams, which are auto-model solutions of a so called paraxial approximation (PA) and allow one to handle the diffraction of spatially-smooth optical beams in a very simple way (see e. g. [29]). Recent developments [7-11,18] in optics resulted in the generation of short and intense EM pulses of non–oscillatory nature, or almost unipolar "half-cycle" pulses (HCPs), with extremely broad spectra that start at zero frequency; even existing pulses have many exciting applications (see introductory section and Refs. [7-11,18]). The spectra of currently available HCPs generated in semiconductors via optical rectification, reach into terahertz domain; they are ~400–500 ƒs wide, with the peak field up to 150– 200 KV/cm. In our recent work part of which is described in this paper (see also Refs. [12-14]), we proposed new different principles of generating much shorter (down to 0.1 ƒs = 10 –16 s) and stronger (up to ~ 10 16 W/cm 2 ) HCPs. As it was pointed out above, different frequency components in HCPs diffract differently, far away from the source HCPs propagate with significant dispersion and distortion even in free space [711,18,30,31]. This phenomenon calls for the diffraction/transformation theory of pulses with super-broad spectra, preferably comparable in its simplicity and insights with that of Gaussian beam diffraction of monochromatic light. Such a theory could also apply to other fields of wave physics: acoustics, solid-state physics and quantum mechanics. The work [31] analyzed asymptotic field behavior in far-field area of a beam with an (initially) Gaussian spatial profile and found time-derivative behavior in that area. Ref. [31] considered only the fields with an also (initially) Gaussian temporal profile. However, even for that profile, no global analytic solution (even for on-axis field) for the field behavior along the entire propagation path was found, leaving the theory without the major advantage of a standard theory of monochromatic Gaussian beams. The ability of theory to globally address the propagation, in particular, in between near- and far-field areas, is essential, since in practice, that intermediate area could be of most significance, with the diffraction distance for the highest spectral frequency, (where 2t 0 is a pulse time-width, and 2r0 a pulse transverse size), being considerably large. For 2t0 ~ 400 ƒs and 2r 0 ~ 1 cm [3], one has z d ~40cm, the same as e. g. for 2t 0 ~ 4 ƒs and 2r0 ~ 1 mm. We derive here a simple equation for on-axis field (with a Gaussian initial spatial profile) valid for an arbitrary temporary profile and for any distance from the source. Using that equation, we obtain close-form solutions for the field transformation due to dif–1 fraction for some temporal profiles, in particular cosh -like and Gaussian profiles, and show that in far-field area pulses demonstrate time-derivative behavior regardless of their 311

frequency limit, we find a general solution valid for any spatia1 and temporal profiles of the field, which also explains in simple antenna terms the nature of time-derivative behavior; this solution is also valid in far-field area for any frequency. Consider now a pulse propagating along the z axis and having an arbitrary time dependence and a known transverse profile, at the point z =0. We assume at this point a high-frequency limit, meaning that the shortest temporal scale of the pulse, t0 (in the extreme case of non-oscillating, half-cycle pulse, it is its initial half-timewidth, see below), and its respective longitudinal scale, ct0 , are much shorter than its transverse radius, r0 , (10.1) The frequency components of the largest part of its spectrum, will propagate with relatively small diffraction, so that one can apply a standard paraxial approximation (PA) to each one of them. Within PA, the diffraction of a monochromatic field, E ω exp[– i ω (t – z/c )], in a free space, is described using a PA wave equation similar to a Schrödinger equation for a free electron: (10.2) where is a transverse Laplacian; note that PA allows one to neglect polarization of the field and reduce the problem to a scalar one. We will also assume the field cylindrically symmetric in its cross-section, so that where r is the radial distance from the axis z in the cross-section. With the most of the available or to be available sources of HCPs, one can assume that the field at the source has a plane phase front for all the spectral components, so that their waists are located at the source. The spatiallyGaussian field at the source, z=0, can then be written as where r0 is the Writing the radius of the spatial field profile at the level exp(– 1/2) of peak amplitude, solution of PA equation (10.2) as a Fourier transform: (10.3) where as:

is a retarded time, we have the field spectrum S(ω,r,z ) for a Gaussian mode (10.4)

Here

is the spectrum of original pulse, and (10.5)

is a diffraction factor due to PA. For the on-axis field, r=0, we have By substituting this into (10.3), and introducing a dimensionless retardation time, and propagation distance, we derive a simple equation for the temporal dynamics of an on-axis field at any point ζ: (10.6) If the full energy of the field at source is finite, the solution for the on-axis field is: (10.7) where s(x ) = –1 if x < 0, and s(x) = 1 otherwise. One can see that, as expected, in a nearfield area, ζ << 1, the original temporal pulse profile is almost conserved, Eon ≈ E 0 ( τ). T h e most interesting and universal (see below) pulse transformation occurs in far-field area, ζ >> 1. In this case, E on can be expanded as (10.8) 312

so that as ζ → ∞, the on-axis far-field replicates time derivative of the original pulse: (10.9) All of the results (10.6)-(10.9) are true for an arbitrary initial temporal profile, E0 ( τ). I n particular, any HCP is transformed in the far-field area into a single-cycle pulse. Writing Eq. (10.6) in real time, where we notice that it coincides with an equation for the voltage UR ∝ E on at a resistor R in a series RC circuit (high-pass filter) driven by a source US ∝ E0 so that the circuit relaxation time is T=RC. Since the only parameter with the dimensionality of resistance in a free-space propagation is the wave impedance of vacuum (R = 120 π ohm ), the circuit capacitance is then which is consistent with a capacitor formed by electrodes having the pulse waist area ∝ and spaced by z and thus provides an interesting and simple interpretation of the nature of pulse transformation in free space. for

A simple example of the field evolution along the the entire path of propagation (i. e. an arbitrary ζ ), is given by a smooth bell-shaped initial profile with exponential tails at | τ | → ∞. Eq. (10.7) yields then: (10.10)

at ζ = 1 and τ > 0, the rhs of (10.10) is e x p ( – τ )( 1 – 2 τ 2 )/4. A familiar profile E 0 = ξ 0 /cosh( τ) does not behave as nicely; in this case, a solution (10.7) in elementary functions exists if ζ is any rational number, but its form is different for different ζ’s. At ζ =1, one has The solution (10.7) for an Gaussian initial temporal profile (here t 0 is the pulse half-width at exp(–1/2) peak amplitude), having the spectrum is handled analytically for any ζ: (10.11) where per unity area,

As the pulse propagates, its total on-axis energy

(10.12) which in the limit ζ → ∞ yields w( ζ ) → (2 ζ2 )–1 , as expected. The evolution of the profile and spectrum of the on-axis Gaussian HCP as it propagates away from the source, is shown in Fig. 9; one can clearly see that the pulse sheds off lower frequencies to finally form almost exact mimic of the time-derivative of the original Gaussian pulse. The zero point (i. e. the moment τ z where E = 0), is moving closer to τ = 0 as the distance ζ >> 1 increases. Using first two terms in the expansion (10.8), with the zero point found from , we have τz ≈ ζ –1 . The lower-frequency radiation diffracts stronger and hence is found mostly off-axis, where, by the same token, the higher frequencies are weaker. all of which results in the lengthening of the pulse. The smaller the diffraction angle at the cut-off frequency, θ d = ct0 /r 0 (<<1 due to (10. 1)), the stronger this effect is pronounced. In far-field area, ζ >> 1, we introduce the angle of observation, θ = r/z, and an angular factor due to diffraction, and approximate the spectrum (10.4) of the pulse as: Soƒƒ ( ω, θ, ζ) ≈ which in the case of Gaussian initial temporal profile, yields an off-axis pulse in far-filed area: (10.13)

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(10.13) resembles (10.9) with the main difference being that off-axis pulse stretches in time by the factor Θ and its spectrum is respectively "squeezed" by the same factor.

Fig. 9. The evolution of the on-axis temporal profile (normalized field, Eon 0 vs normalized time, τ, and the normalized amplitude spectrum, | S | 0 t 0 , vs normalized frequency, v ≡ ωt 0 (inset), of the initially Gaussian half-cycle pulse, as it propagates along the axis Curves: 1, ζ = 0; 2, ζ = 0.25; 3, ζ = 0.5; 4, ζ = 1; 5, ζ = 2 ; 6, ζ = 4. For the sake of comparison, each curve in the main Fig. is scaled up by –1/2 (ζ). the factor w

In the so called low-frequency limit r0 << ct 0 , opposite to (10.1), with the source size being much smaller that the wavelength λ = 2πc/ ω of any frequency component, all the components have the same dependence on the angle of propagation; also, the initial spatial profile of the field becomes unimportant. The radiation pattern at each frequency is then determined by an elementary (i. e. point-like) dipole formed by the field distribution, (t,x,y). At the distance from this point-like source (i. e. away from the very small near-field area ρnear << λ), and assuming that the field is linearly polarized, the spectrum of radiative waves is: (10.14) where θ is now the angle between the axis z and the direction of propagation, in the plane of the vector of polarization, and the observation point.

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The Fourier transform of (10.14) produces the same time-derivative profile everywhere, (10.15) where is the polarization vector of radiative field, = cos θ. E q . (10.15) explains pulse transformation in simple terms of elementary dipole antennae driven by a current which is induced by the dynamics of one of the dipole electrical "charges" q0 originated by the source field, E 0 ; hence time-derivative temporal profile. Bearing in mind that for a Gaussian beam, Eq. (10.15) at θ =0 is consistent with the Gaussian on-axis far-field (10.9), indicating that the results (10.6-10.12) for the on–axis field are valid regardless of the condition (10.1). Furthermore, in far-field area, Eq. (10.15) describes an on-axis field for any distribution, regardless of whether it is Gaussian or not. The dispersion and transformation of the pulse due to the propagation and diffraction can to great degree be reversed. The feasibility of that is related to the time/space reciprocity manifested here by Eq. (10.6) being invariant to the simultaneous sign reversal of time τ and distance ζ . (The same is true for the solution (10.7), if E0 ( τ ) is a symmetric function, see Eqs. (10.10)-(10.12).) In practice, the diffracted HCP can be transformed back almost into its original temporal profile (except for its cw component) by reflecting its diffracted wave front e. g. from a spherical concave mirror, if the angular aperture of the mirror is significantly larger than the diffraction angle, θd . If such a mirror has the radius of curvature R m and is situated sufficiently far from the source, with the distance between them being ƒ 1 >> z d , the pulse is focused again into a tight spot at the distance ƒ2 , determined by a If ƒ2 <ƒ 1 , the area of this spot is smaller standard optical mirror formula, that that of the original spot, and the amplitude of the focused pulse is larger by the factor ƒ1 /ƒ 2 . The residual distortion of the pulse (in particular, slight bipolarity of initially unipolar HCP) will be due to lower-frequency diffraction losses at the mirror; the larger the mirror size, the smaller this effect. In the case of point-like source, pulse restoration can be achieved by using a full ellipsoid of revolution, with the source and observation points situated at the foci of the ellipsoid.

11. CONCLUSION In conclusion, we have theoretically demonstrated feasibility of powerful, near- and sub-femtosecond sub-cycle EM pulses and solitary waves, EM bubbles, supported by both quantum and classical nonlinear media. We have shown how their maximum amplitude and minimum length are limited by the atomic ionization. It follows from our theory that 10 – 0.1 ƒs EMBs can be generated by the available half-cycle pulses and short laser pulses; 14 the peak EMB intensity can reach ~ 10 – 10 16 W/cm 2. Those results represent only the very first steps in the exploration of the new time domain. Our hope is that EMBs will be experimentally observed in the near future. This will pose new set of problems, such as EMB detection and characterization, separation, gating, control, focusing and guiding, and exploring various EMB applications. In a transverse-limited EM field, a zero-frequency spectral component of the incident HCP will not propagate beyond the near-field area, and in the far-field area, EMB will assume a modified profile. Using analytic approach to the diffraction-induced transformation of pulses with arbitrary temporal profiles, including half-cycle pulses, we found close-form solutions for the propagation of most commonly used initial spatio-temporal profiles, and explained the nature of time-derivative transformation in far-field area for arbitrary pulses.

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The new time domain, being largely an uncharted territory, holds a lot of promises for the physics of field-matter interaction. The most familiar nonlinear effects and parameters associated with coherent light-matter interactions (harmonic generation, self-induced transparency, photon echoes, soliton generation and propagation, saturation of all kinds, n2 , χ (3) , etc.) are likely to take on entirely different forms, or may even cease to exist. One of the most fundamental and intriguing phenomenon is the field ionization of atoms, molecules, and semiconductor quantum wells by a super-short pulse with the amplitude comparable to or larger than the ionization threshold. Such pulses could cause a substantial "shake-up" excitation or ionization of an atomic system within the time much shorter than any characteristic time of the system. In our most recent research [33] we showed that a few ƒs long and unipolar EMB acting upon a semiconductor quantum well, can cause both forward and backward field ionization, with the photoelectrons emitted in both directions (i. e. not only in the direction of the ionizing unipolar field) with comparable intensities. Even more fundamental and exciting results are obtained for the hydrogen atom hit by a sub-cycle pulse with an sub-atomic unit amplitude. We also observed that the ionization response of the atom consists of a sequence of well-separated peaks resulting in strong spatio-temporal inhomogeneity of the photoelectron cloud, and found an explanation of such a behavior. This work is supported by AFOSR. The work by SFS is in part supported by the Deutsche Forschungsgemeinschaft. AEK is a recipient of the Alexander von Humboldt Award for Senior US Scientists of AvH Foundation of Germany.

REFERENCES 1. 2. 3. 4. 5.

6.

7. 8. 9. 10. 11. 12. 13.

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S. L. McCall and E. L. Hahn, Phys. Rev. Lett. 18, 908 (1967). P. W. Smith, Proc. IEEE, 1342 (1970) and references therein. A. Hasegava and F. D. Tappert, Appl. Phys. Lett. 23, 142 (1971). V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972). (a) R. L. Fork, C. H. Brito Cruz, P. C. Becker, and C. V. Shank, Opt. Lett., 12, 483 : (1987); (b) M. Nisoli, S. De Silvestri, O. Svelto, R. Szipocs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, Opt. Lett. 22, 522 (1997). K. E. Oughstun and H. Xiao, Phys. Rev. Lett. 78, 642 (1997); K. E. Oughstun and G. C. Sherman, Electromagnetic pulse propagation in casual dielectrics (Springer, Berlin, 1994). P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE JQE, 24, 255 (1988). D. Grischkowsky, S. Keidin, M. van Exter, and Ch. Fattinger, JOSA B, 7, 2006 (1990); R. A. Cheville and D. Grischkowsky, Opt. Lett. 20, 1646 (1995). J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990); B. B. Hu and M. S. Nuss, Opt. Lett. 20, 1716 (1995); R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993). A. E. Kaplan, Phys. Rev. Lett. 73, 1243 (1994); A. E. Kaplanand P. L. Shkolnikov, JOSA B 13, 412 (1996). A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995); also in Int. J. of Nonl. Opt. Phys. & Materials, 4, 831 (1995).

14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24.

25. 26.

27. 28. 29. 30.

31. 32. 33.

A. E. Kaplan, S. F. Straub and P. L. Shkolnikov, Opt. Lett., 22, 405 (1997); also to appear in JOSA B 14 (1997). R. K. Bullough and F. Ahmad, Phys. Rev. Lett. 27, 330 (1971); J. C. Eilbeck, J. D. Gibbon, P. J. Caudrey, and R. K. Bullough, J. Phys. A 6, 1337 (1973). E. M. Belenov, A. V. Nazarkin, and V. A. Ushchapovskii, Sov. Phys. JETP 73, 423 (1991). A. I. Maimistov, Opt. & Spectroscopy, 76, 569 (1994) and 78, 435 (1995). B. Kohler, V. Yakovlev, J. Ghe, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. Yan, Phys. Rev. Lett. 74, 3360 (1995) A. E. Kaplan and P. L. Shkolnikov, Phys.Rev. A. 49, 1275 (1994). with For a harmonic oscillator with a frequency ω0 , it is natural to choose , where is the Compton wavelength. If nonlinearity is negative, a < 0, one can expect formation of "dark" EMB (a solitary "hole" propagating on a cw field background): ƒ( τ) ∝ƒ0 tanh( τ ƒ 0 ), ƒ 0 =const. D. G. Lappas, M. V. Fedorov, and J. H. Eberly, Phys. Rev. A 47, 1327 ( 1993); J. H. Eberly, Q. Su, and J. Javanainen, JOSA B 6, 1289 (1989). Recent studies of shock-like envelope fronts can be found, e. g. in S. R. Hartmann and J. T. Massanah, Opt. Lett. 16, 1349 (1991); E. Hudis and A. E. Kaplan, Opt. Lett. 19, 616 (1994); W. Forysiak, R. G. Flesh, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 76, 3695 (1996). R. M. Miura, J. of Math. Physics, 9, 1202 (1968); R. M. Miura , C. S. Gardner and M. D. Kruskal, ibid, 1204 (1968); M. Wadati, J. Phys. Soc. Japan, 32, 1681 (1972); ibid, 34, 1289 (1973) L. Xu, D. H. Auston, and A. Hasegava, Phys. Rev. A45, 3184 (1992). A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIntyre, K. Boyer, and C. K. Rhodes, JOSA B 4, 595 (1987); M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompre, G. Mainfray, and C. Manus, J. Phys. B 21, L31 (1988); A. L’Huillier and Ph. Balcou, Phys. Rev. Lett. 70, 774 (1993); L’Huillier, A, Lompre, L. A., Mainfray, G., and Manus, C., in Atoms Intense Laser Field , Ed. M. Gavrila (Acad. Press, Inc., Boston, 1992), p. 139-206. A. E. Kaplan, to appear in JOSA B. Max Born and Emil Wolf, Principles of Optics, 6-th edition (Pergamon Press, NY, 1980). A. Siegman, Lasers (Univ. Science, Mill Valley, CA, 1986); A. Yariv, Quantum Electronics (Wiley, NY, 1989). M. van Exeter and D. R. Grischkowsky, IEEE Trans. Microwave Theory Techn., 38, 1684 (1990); J. Bromage, S. Radic, G. P. Agrawal, C. R. Stroud, Jr., P. M. Fauchet, and R. Sobolevski, Opt. Lett. 22, 627 (1997). R. W. Ziolkowski and J. B. Judkins, JOSA B 9, 2021 (1992). I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, NY, 1980). A. E. Kaplan, S. F. Straub and P. L. Shkolnikov, to be published; first reported in Quant. Electr. & Laser Science Conf., v. 12, 1997 OSA Techn. Digest Series (OSA, Washington, DC, 1997), p. 31.

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NONLINEAR WAVEGUIDING OPTICS

R. REINISCH Laboratoire d'Electromagnétisme, Microondes et Optoélectronique, Unité Mixte de Recherche 5530 du Centre National de la Recherche Scientifique, Institut National Polytechnique de Grenoble - Université Jopseph Fourier, ENSERG, 23 ave des Martyrs, BP 257, 38016 Grenoble Cedex, France. email : [email protected]

I. INTRODUCTION Nonlinear waveguiding optics is concerned with nonlinear optics in devices which support guided waves or surface plasmons (or any combinations of them)[1]. These waves may be resonantly excited using prism or grating couplers. The theories which allow the study of such nonlinear optical interactions are the total field analysis[2] and the coupledmode approach[3,4]. In the total field analysis no hypothesis is made regarding the transverse field map (i. e. perpendicular to the direction of propagation of the guided waves). Thus this is a general method which also applies to non-guiding structures. But its generality does not allow an easy insight in the underlying physics. In the coupled-mode approach the electromagnetic (EM) field is expanded on the guided modes and on the radiation fields of the guiding device. This method is convenient when guided modes are excited. However this is not always the case : when considering prism or grating couplers or Fabry-Perot's no guided modes are present because these devices behave as "open" resonators. Then one is left with the radiation fields that is to say with an integral representation of the EM field which does not lead to an easy analysis of "open" resonators. But it is known that there exists a close link between EM resonances and poles[5,6] arising from the solution of what is called the homogeneous problem. Once the importance of poles in some suitable complex planes is recognized, it is tempting to solve the homogeneous problem in order to use these poles for the study of nonlinear optical interactions in "open" resonators. It is at this level that the difficulty appears : due to the "open" feature, the EM resonances involves leaky modes[5,7]. These modes, when derived from the solution of the homogeneous problem, are found to exist by themselves and consequently, due to their leaky feature, to diverge at infinity. This difficulty comes from the fact that the solution of the homogeneous problem yields the poles but tells nothing about the way they contribute to the EM field. In other words the solution of the homogeneous problem, although useful, is not sufficient to get a complete solution of this kind of problems. When looking for the expression of the EM field due to a source (incident beam(s) and/or nonlinear polarization), it is found that leaky modes have no individual existence : they always constitute a portion of the EM field i. e. they only exist in a finite region of space preventing in this way any divergence at infinity[8]. As already mentioned, the coupled-mode formalism[3,4] only involves the radiation fields but does not exhibit explicitely the leaky modes although these modes constitute a powerful tool for the study of "open" resonators in nonlinear optics. However no coupled-mode formalism has been developed bringing into play leaky modes. In this chapter we show how leaky modes can be simply used for the study of nonlinear optical interactions in "open" resonators. This requires to go somewhat in the detail of the Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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theory of leaky modes, to show that they arise from poles which are called "improper" (guided modes are associated to "proper" poles), to explain that several expansions of the EM field are possible. Passing from one of these expansions (the transverse one) to another one (the leaky mode one) leads to a relation between leaky and radiation modes. This relation serves as a basis to derive the equation of evolution of a leaky mode amplitude. Cerenkov second harmonic generation (SHG) is chosen as an example to illustrate the interest of leaky modes in guided wave nonlinear optics. II. LEAKY MODES Various representations of the EM field will be considered : the first one is the longitudinal, or Fourier, representation from which two other expansions are derived : the transverse representation which displays the guided modes and the radiation field(s) supported by the structure and the leaky mode representation which is appropriate when considering "open" or leaky resonators. Section 2 is devoted to the derivation of the equation of evolution of a leaky mode amplitude. 1. Different representations of the EM field in linear planar structures The structure of interest is a linear planar multilayer configuration illuminated by a light beam (time dependence e –iω t ) under incidence θi . Figure 1 shows an example corresponding to a system involving three media 1,2,3 with relative permittivity ε1 , ε 2 , and ε 3 with ε2 >( ε 1 , ε3 ) in order to allow for guided modes.

Figure 1. The structure of interest.

For simplicity, we assume : i) lossless media, Thus the EM field

ii) a two-dimensional situation where depends on x and y (i. e. of the electric field

is either TE polarized (specified by the z-component or TM polarized (specified by the z-component of the

magnetic field The analysis closely follows refs. [3,8-10].

320

which

1.1 The longitudinal representation. The solution Fourier integral :

(x, y) is expressed as a

(1) Let α q denote the y-component of the wavevector in the outside media q (q=1,3). The quantities αq and γ are related by : (2a) or (2b) with: (c : speed of light in vacuum)

(3)

In eq. 1 the integration is performed along the path P corresponding to the real γ -axis. Hence the name of longitudinal representation given to this expansion which includes imaginary values of α q since γ varies from −∞ to +∞. Equation 1 represents Φ z (x,y) as a continuous infinite spectrum of waves : each elementary spectral component is described by ϕ z ( γ ,y)e i γ x and corresponds to a plane wave solution with longitudinal wavevector component γ and transverse field map ϕ z ( γ ,y). It is the quantity ϕ z ( γ ,y) which depends on the particular structure of interest : multilayer configuration, gradient index device… The advantage of eq. 1 is its relative simplicity. The drawback comes from the fact that this expression does not exhibit explicitely possible guided modes which, as is known, may be present in the structure fig. 1. 1.2 The transverse representation : guided modes and radiation fields. This representation is obtained from eq. 1 by performing a contour deformation where the path P of integration along the real γ -axis is transformed into a path P γ in the complex γ -plane. Such a procedure is valid provided the function ϕ z ( γ ,y)eiγ x is analytical[11] in region C bounded by P and P γ (Fig.2). This requirement implies a search of the singularities of ϕ z (γ ,y)e i γ x . These singularities are of two types : * poles which contribute to the integral in the complex γ -plane through the residue theorem, * branch points, due to the square root in eq. 2b, which require cuts in the complex γ -plane in order to avoid multivalued functions. Concerning the branch points, some care must be exerted. From equation 2b, it is seen that there are branch points at ± k q (q=1,3). In medium 2 passing from one determination of the square root of the transverse components of the wavevectors eq. 2b to the other one does not give a different EM field[9]. Thus the determination chosen for the square roots in this medium is not of importance. For q=l,3 the choice of the branch cuts is rather arbitrary. However convenient cuts are those for which the radiation condition at infinity is fulfilled on the entire top sheet of the four-sheeted Riemann γ -plane. The radiation condition writes : Im( αq ) > 0

(q= 1,3)

(4a)

321

and the cuts correspond to : Im( α q ) = 0

(q=1,3)

(4b)

With these cuts the entire top sheet of the four-sheeted γ -plane is mapped on the upper half of the complex αq -planes (q=1,3). The image of the cuts eq. 4b in the complex γ -plane corresponds to the hyperbolas : γ ' γ " = k' q k" q

(q= 1,3)

(4c)

with :

γ = γ' + iγ "

(4d)

k q =k'q +ik"q

(4e)

In eqs. 4c-e and in fig. 2, the outside media exhibit infinitesimal losses in order to avoid cuts located on the γ " and γ ' axis. The resulting contour of integration Pγ is illustrated in fig. 2.

Figure 2. Top sheet of the complex γ-plane. The bold lines represent the branch cuts.

When γ is swept around the cut, αq remains real and varies from +∞ to - ∞ . In the region x>0, the semi-circle at infinity does not contribute to the integral. For x<0, this semicircle is in the lower half of the top sheet of the γ -plane. All these mathematical considerations lead to the following transverse representation of Φ z (x,y) which results from a deformation of the contour from P to Pγ and from a change of variable of integration:

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(5) Since the entire integration is carried out in the top sheet of the complex γ -plane, all the poles which are captured when deforming P into Pγ are located only in the upper half of the αqplanes (q=1,3) and consequently comply with the radiation condition eq. 4a. These poles, which are called "proper", are associated to non-diverging EM fields obtained solving the homogeneous problem, i. e. without excitation. Thus the corresponding residues account for guided modes described by the summation over m in eq. 5. The two continuous spectra (Q=I,II) come from the paths around the cuts and describe two types of radiation fields which include imaginary values of γ : the class I radiation field arises from the branch point kq=1 whereas class Q=II is due to the branch point kq=3 . The term α Q denotes α1 for class Q=I and α3 for class Q=II. These so-called radiation modes correspond to the solution obtained when illuminating the system by a plane wave of unit amplitude incident on the interfaces y=-t and y=0 for Q=I and Q=II respectively. Figure 3 is a possible representation of the radiation modes of class Q=I,II[3].

Figure 3. Class I and class II radiation fields : ρI,II and τ I,II denote respectively the amplitude of the reflected and transmitted fields for the two classes of radiation fields.

As compared to the longitudinal representation eq. 1, expression 5 of Φ z(x,y) exhibits the guided modes and the radiation fields. In expansion 5, a coefficient cm corresponds to the amplitude of the mt h guided mode and cQ ( αQ) to the amplitude of a spectral component of the radiation field of class Q; ϕ m,z(y) and ϕ Q,z( αQ ,y) represent the associated transverse field map which depends on the guiding structure. It is seen that the transverse representation corresponds to the expansion of the EM field used in the coupled-mode formalism[4]. 1.3 The transverse representation : orthogonality relation[3]. The modes of the transverse representation (i. e. guided or radiated) have orthogonality properties which are a consequence of the Lorentz reciprocity theorem[12]. It is assumed that the permittivity tensor [ε] and the permeability tensor [µ] are symmetrical tensors of rank 2. Let and be the EM fields for two different modes m and n (guided or radiated). Use of the Lorentz reciprocity principle leads to the following orthogonality relation when i) at least one of the modes is a guided one :

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(6a) In eq. 6a δm,n denotes the Kronecker symbol with the following meaning : (6b) Therefore notice that δm,n = 0 when γ n = γ m . ii) two radiation modes α and β are involved, the orthogonality relation writes : (7a) In eq. 7a δ[γ ( α) + γ (β)] is the Dirac-δ function : two radiation modes α and β for which γ(α) + γ(β) ≠ 0

(7b)

are orthogonal. In eq. 6a and 7a, 〈....〉 means an integration in the cross-section plane :

and = (1,0,0). Equations 6a and 7a show that only the transverse components of the EM field enter the orthogonality relations. The interest of the orthogonality relations eqs. 6a and 7a is that they allow to determine the unknown coefficients occuring in the transverse representation of the EM field :

(8)

According to eqs. 6 and 7, it is worth noting that in order to derive the amplitude of a given mode one has to use the orthogonality relations with a mode having the same absolute value of the longitudinal wavevector component but propagating in the opposite direction. Stated differently, one has to associate two counterpropagating modes. Proceeding along these lines, the following equations yield the amplitude (the superscript ± stands for forward (+) and backward (-) modes) * of a guided mode : (9a) with (9b)

32 4

* of a radiation mode : (9c) with (9d)

where

(10a)

(10b)

1.4 The transverse representation : equation of evolution of a mode amplitude[3]. Let us consider now the situation where a nonlinear polarization is present. This leads to x-dependent modes amplitudes in eq. 8. The aim is to derive the equation obeyed by these coefficients. The starting point is the Lorentz reciprocity relation used under the following conditions NL : one of the EM solution is generated by a nonlinear polarization (x, y) whereas the other one corresponds to an eigenmode n (guided or radiated). Thus : (11) In eq. 11

is the EM fields arising from the nonlinear polarization

NL

(x,y).

Since the

field tends to zero at infinity, eq. 11 leads to :

(12) Equations 9a and 9b show that the following equations are obeyed by * the amplitude of a guided mode : (13a) or (13b)

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* the amplitude of a radiation mode : (14a) or (14b) It is worth noting that eqs. 13, 14 are first order differential equations although no slowly varying envelope approximation has been made. The important consequence of the orthogonality relations eqs. 6,7 is that guided modes cannot be excited by a beam incident on the device from the substrate or the superstrate. The excitation of guided modes is possible either using the but-coupling technique or "opening" the structure. This leads us to the third representation. 1.5 The leaky mode representation. The transverse expansion eq. 5 is of great interest when considering situations where guided modes exist. But "open" resonators do not exhibit proper poles. Thus eq. 5 only includes the radiation fields which often describe an EM field traveling along an oblique direction with respect to the x-direction. For these structures eq. 5 is not more helpful than eq. 1 is. Such situations are conveniently handled introducing another complex plane which we call the complex w-plane. Let us consider the EM field Φ z(x,y) in medium q=3. The integration in eq. 1 is carried out in the complex wplane : w = u + iv,

(15a)

through the following change of variables : γ = k 3 sin w

(15b)

α3 = k3 cos w

(15c)

together with the use of polar coordinates : x = r sin θ

(15d)

y = r cos θ

(15e)

where r and θ are shown in fig. 4a.

Figure 4a. The change of variables eqs. 15d,e. A leaky mode only contributes to the raditation field within the shaded region.

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It should be noted that the transformation eqs. 15 b,c maps the sheets of the γ-plane into a strip of width 2π in the complex w-plane avoiding, in this way, the branch cut associated to the determination of α3. This mapping plots the four quadrants of the top and bottom sheets of the γ-plane into the regions denoted Ti and Bi (i=1,2,3,4) respectively (fig. 4b). The path Pw in fig. 4b is the image of the original contour of integration P, used in the integration of eq. 1, through the transformation eqs. 15b, c.

Figure 4b. The complex w-plane. Solid curve : SDP (θ=0), dashed curve : SDP (θ= π /4), dotted-dashed curve : SDP (θ= π/2).

In view of an asymptotic evaluation of Φ z(x,y) in medium 3, Φ z(x,y) is written (according to eqs. 15) : (16a) The leaky mode representation involves a contour deformation from path Pw to the steepest descent path (SDP). The calculation shows that there is a saddle point at w=θ and that the SDP obeys the equation : cos(u - θ )cosh v = 1

(16b)

Before going further on some comments are in order. i) According to eq. 16b, the position of SDP in the complex w-plane depends on the direction of observation θ i. e. SDP=SDP(θ ). The SDP curve crosses the v=0 axis at u= θ. ii) Only those poles located between paths Pw and SDP contribute to the integral eq. 1. Since SDP generally passes through strips B1 or B3, poles located on the bottom sheet of the γ plane may be captured. These poles, which do not fulfill the radiation condition eq. 4a, are called "improper", and correspond to leaky modes whose amplitude diverges at infinity. But fig. 4b shows that an improper pole contributes provided the angle of observation θ is greater than a critical angle, θ c,l (the index l labels a pole), the value of which depends on the

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location of the pole in the w-plane (in the example fig. 4b θ c,l = π /4). Stated differently, the improper poles contribution only occurs within a finite angular region corresponding to θ > θc,l (shaded region of fig. 4a) preventing, as y → + ∞, any divergence in the EM field representation. iii) These leaky modes are characteristic of open resonators : the improper feature of pole γ l comes from the EM coupling with the outside. These modes are obtained solving the homogeneous problem. iv) It was pointed out in section 1.2 that improper poles cannot be captured within the transverse expansion. Thus leaky modes are part of the radiation field. Finally it is seen that the EM field eq. 1 is the sum of the space wave Φsw,z,3 resulting from the integration along SDP and of the residue contributions coming from the poles located between Pw and SDP : (17a)

The discrete summation in eq. 17a is carried out considering all the poles located between P w and SDP : the leaky ones and the "proper" ones within the angular regions where they contribute. The asymptotic integration of eq. 16a along SDP leads to : (17b) In eq. 17b asymptotic means k3 r >>1. The far-field corresponds to a cylindrical wave which decreases as r -1/2 together with an angular dependence ϕz( θ ) cos θ independent of r. This is a classical result of far-field radiated by antennaes[13]. To summarize for large y (x constant), leaky modes do not contribute and the space wave eq. 17b constitutes a good approximation of the radiation field. In, or very close to, the guiding layer (medium 2), it can be shown[3,9] that the leaky modes provide a good description of the radiation field. Besides it is known[5] that improper poles are intimately linked to EM resonances. Therefore in the resonance domain it is possible to replace the radiation fields involved in the EM resonance process by the associated leaky mode. This important result greatly simplifies the study of optical resonators in nonlinear optics as we show in the next section. 2. Equation of evolution of a leaky mode amplitude The starting point is constituted by the transverse and leaky mode expansions (eqs. 5 and 17a respectively). In this section the following hypothesis are assumed which correspond to usual situations when interested in "open" resonators : a) we look for the EM field in, or in the vicinity, of the guiding layer (medium 2), b) their exists isolated improper poles c) the working point remains in the vicinity of such a pole i. e. of the order of the distance from the pole to the real γ -axis. When points a-c apply, eqs. 5 and 17a yield : (18) According to equation 18, a leaky mode constitutes a good approximation of the radiation field of class Q involved in the resonant process. Use of eqs. 14b and 18 leads to :

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(19) NL In eq. 19 the first term arises from the existence of the nonlinear polarisation whereas the second one describes the in-coupling of possible incident beam(s) on the system. We now show that, thanks to hypothesis a-c, eq. 19 can be greatly simplified.

2.1 Simplification of equation 19. It is worth noting that

and

have different expressions. Indeed is the solution of the homogeneous problem whereas is the solution corresponding to an incident plane wave. This means that includes a resonant denominator contrary to which does not. Besides the comparison of and is possible only in the vicinity of an isolated pole i. e. close to resonance. Let be the transverse field map at resonance defined as follows : (20a) In order to exhibit the resonant term in we notice that it is possible to express and of the transmission with in terms of and (fig. 3) : (20b) In the following A i =1, hence : (20c) Since the functions and are defined within a multiplicative constant, the simplest choice, which will be done throughout this chapter, is : (20d) and (20e) Besides : (20f) Provided hypothesis a-c apply, the constant K is close to 1 : K≈1

(20g)

Let us consider now successively the two terms of eq. 19.

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2.2 Simplification of eq. 19 : the influence of the nonlinear polarization. Equation 19 becomes :

(21a) Hence from eq. 20f : (21b) To avoid unecessary complicated notations, we pursue considering forward propagating modes and we drop the superscript +. It is convenient to rewrite eq. 21b under the form :

(22a) with (22b)

The knowledge of NQ,rad requires the calculation of : (23) The term is calculated in ref. [3] : When dealing with nondegenerated radiation modes : (24a) where : denotes the reflectivity for a radiation mode of class Q=I,II, (24b) ηQ is the constant value of η (y) in the outside medium 1 for class I and medium 3 for class II. For symetrical structures :

(25a) where + and - stand for even and odd modes respectively. If in addition the system is lossless then :

330

(25b) Moreover close to an improper pole γl [5] : (26a)

(26b) In eq. 26b γ z,Q denotes the zero[5] of the reflectivity for class Q radiation modes. According to eqs. 26 and 25b, eq. 25a can be rewritten as : (27a) with (27b) In eqs. 26 and 27 t Q, r Q et n Q do not depend on αQ. Equation 26a shows that

is a

lorentzian centered at γ 'l , with a width 2 γ " l and of peak value | tQ | ² . This lorentzian is associated to the EM resonance giving rise to the improper pole γl . Thus the link between improper poles and EM resonance appears markedly. The integral eq. 23 only depends on α Q and can always be calculated numerically. In fact, according to the expressions of τQ ( α Q ) and N Q( α Q) on the one hand, assuming that points a-c apply on the other hand, IQ can be easily calculated using the theorem of the residues[11]. Keeping in mind points b) and c), a tedious but straightforward calculation yields * for lossless symetrical structures : (28a) * in the general case : (28b) with (28c) (28d) In eqs. 28 α Q ,l denotes α 1, l for Q=I and α 3, l for Q=II.

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2.3 Simplification of eq. 19 : the in-coupling process. One has to consider the second term of eq. 19. Because no source term is assumed in this section, a coefficient c Q does not depend on x : (29) Equations 20b and 20f show that : (30) where A i ( αQ ) is the spectral component of the incident beam in the transverse representation. Use of eq. 30 shows that eq. 29 writes : (31) Let a i(x) be the expression of the incident field at y=0 : (32a) and (32b) equations 26a and 32 lead to : (33) 2.4 General equation of evolution of a leaky mode amplitude. When a nonlinear polarization is present together with incident beams, the amplitude of a leaky mode obeys the following equation derived from eqs. 22a and 33 :

(34a) According to eq. 20g, eq. 34a writes :

(34b) In eqs. 34 γ i is the longitudinal wavevector component of the excitation (nonlinear polarization and/or incident field(s)). 2.5 Alternative derivations of N Q,rad : total field method, "closing" procedure. Other methods allow to derive N Q,rad : these are the total field method and the “closing” procedure. In both cases the validity of eqs. 34 is assumed. The problem which remains to be solved is the determination of γl , t Qand N Q,rad : the solution of the homogeneous

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problem yields γl , t Q is easily obtained from the knowledge of the peak value of the resonance curve eq. 26a, concerning NQ,rad two methods have been employed. In the first one it is noticed that NQ,rad only involves the transverse (i. e. along y) field maps. This suggests to derive the expression of NQ,rad through a comparison with the result obtained from the total field method[2] using the undepleted pump approximation (UPA) and a simple x-dependence of the nonlinear polarisation of the form eiγ x . This was done in refs. [14,15]. The drawback is the necessity of solving a linearized nonlinear problem. Another method, which is an approximate one, has also been used. It relies on the fact that high finesse resonators are considered for which the EM leakage is small. Thus NQ,rad i s derived by “closing” the leaky resonator. This means that for a Fabry-Pérot the reflectivity of the mirrors is 100%, for a prism coupler the prism is removed, for a grating coupler the leakage due to the radiated diffracted orders is neglected. Figures 3 and 4 of ref. [16] and fig. 2 of ref. [17] show that the “closing” method yields results in good agreement with those derived from the total field analysis. This “closing” procedure has also been used in ref. [18]. The main advantage of such a method, as compared to the total field calculation, is its simplicity since it only involves quantities which can be obtained through a linear calculation. When comparing NQ,rad derived from the theory developed in section 2.2 with its corresponding value obtained with the closing method the agreement is * excellent : of the order of 5 10-2 for prism couplers (guided waves or surface plasmons), * good : of the order of 10-1 for dielectric Fabry-Perot’s. The fact that the agreement is less good for dielectric Fabry-Perot’s than for prism couplers is presumably due to the low finesse of such Fabry-Perot’s. In that case the transmission can no longer be approximated by a lorentzian, eq. 20g does not hold and one has to calculate K in eq. 20f. 2.6 Brief discussion. The comparison of eqs. 19 and 34 clearly shows the interest of leaky modes : a leaky mode replaces the “packet” of radiation modes involved in the resonant process i. e. for which : (35) 2 γ "l "measures" the aperture of the nonlinear optical resonator. Thus leaky modes appear as well suited candidates for the study of nonlinear interactions in nonlinear optical resonators. It is seen that the difficulty of the divergence of the leaky modes is an apparent one : in fact, leaky modes have no individual existence, they contribute to the EM field only in a finite region of space. Use of eqs. 34 requires the determination of NQ,rad . There are three possibilities : eqs. 28a,b, the total field method or the “closing” procedure. The latter yields accurate results. The theory developed in section 2, which allows the demonstration of eq. 34, can be considered as a justification of the “closing” procedure used in previous studies with no demonstration. The main interest of the “closing” procedure and of eqs. 28 yielding NQ,rad is their simplicity since they rely on a calculation of linear optics. For NQ,rad it suffices to know the improper pole αl , the zero γz,Q and tQ . Finally the method of ref. [19] allows to relate the expression of the EM field outside the guiding layer to the known one inside the guiding layer. Having demonstrated the equation of evolution of a leaky mode amplitude and explained how NQ,rad can be calculated, let us choose, as an example, Cerenkov SHG. This type of SHG has been recently reconsidered [20,21].

III. AN EXAMPLE : CERENKOV SHG The Cerenkov regime is interesting because it gives the opportunity to use different methods to derive the EM field : the total fied analysis, the transverse representation (or coupled-mode formalism), the leaky mode representation. Figure 5 represents a typical

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Cerenkov configuration : the fundamental frequency ω field is guided and the SH field is radiated.

Figure 5. A typical Cerenkov configuration

In this section we only consider the total fied method and the leaky mode representation. The transverse representation involves the radiation field and therefore leads to tedious calculations. 1. The total field method This method has been used in refs. [22-24]. The interest is in the EM field at the SH frequency 2 ω. The starting point is Maxwell equations at 2ω : (36a) (36b) Solving Maxwell equations in media 1,2 and 3 yields : (37a) (37b)

(37c) In eq. 37b the terms with amplitudes A2 and B 2 correspond to the free solution with +i2 γ ω x longitudinal wavevector component γ 2ω and ℑ NL (y)e describes the driven EM field with longitudinal wavevector component 2 γ ω . The expression of ℑ NL (y) is not of importance here. The boundary conditions at y=0,e i) require that the longitudinal wavevector components of the free and driven waves at 2 ω b e equal yielding :

334

γ ω = 2γ ω 2

(38a)

ii) lead to : (38b) Equation 38a is nothing but than the nonlinear Snell’s law[25]. In eq. 38b, is a column matrix whose elements are the unknown amplitudes A1, A2 , B 2 , A 3 of the free SH fields in media 1, 2, 3 and

is the known column matrix associated to the driven

solution. The solution of eq. 38b shows that the amplitudes of the free waves are inversely proportional to a quantity which is the determinant ∆ of matrix

. The important point

is that ∆ has zeros in the complex γ2ω -plane. These zeros of ∆ are poles of A 1, A2, B 2 and A3 . In the vicinity of such a pole, assumed to be isolated, a very good approximation of 1/ ∆ is provided by the first term of the Laurent expansion[11]: (39a) Since in Cerenkov SHG the SH field is radiated, the poles of 1/∆ are improper poles (see section 1.5). Equation 39a shows that the peak value of |1/∆|2 is reached when : (39b) Phase matching is achieved when γω fulfills eq. 39b. It is worth noting that eq. 38a does not imply eq. 39b. In other words eq. 38a does not correspond to phase matching with an eigenmode at 2ω. The Cerenkov phase-matching condition is intimately linked with a of ∆ : resonance occurs when eq. resonance effect associated to the complex zeros γ l, 2ω 39b holds. The curve |1/∆ |2 is a Lorentzian provided the improper poles i) are spread out in the complex γ2ω -plane without clustering and ii) are not too close from cut-off (not too close means that the distance to cut-off is large as compared to the width at half-maximum of |1/∆|2). In order to vizualise the Cerenkov phase matching process, it is convenient to plot |1/ ∆ | 2 in the complex neff2 -plane

This is done in figs. 6 with

the numerical values of ref. [20] : e=4.90784 µm n 2 ( ω)= 1.6992 n 1 ( ω)=n 3 ( ω)=1.6875 n 2 (2 ω) = 1.69772 n 1 (2 ω )=n 3 (2 ω)= 1.71.

335

Figure 6. The Cerenkov phase-matching surface.

The peaks correspond to the improper poles of 1/ ∆ . In an SH experiment the longitudinal wavevector of the fundamental frequency guided mode is real. Thus only the real axis (n"eff2 = 0) is accessible : when sweeping neff1 , phase matching takes place for the values n eff1,res of n e f f 1 fulfilling eq. 39b. Clearly the smaller n" e f f 2 at the peak position, the more efficient the associated EM resonance. Figures 6, which display the complex poles of 1 /∆ , represents the Cerenkov phase-matching surface : this surface clearly demonstrates the existence of phase-matching in Cerenkov SHG. 2. Leaky mode analysis Since the Cerenkov regime involves radiation fields, we know from section 1. 5 that the associated poles at the SH frequency are improper poles describing leaky modes at 2 ω. Thus Cerenkov SHG can be considered i) either as a guided mode at ω - radiation fields at 2ω interaction or ii) as a guided mode at ω - leaky mode at 2 ω interaction. In this section scheme ii) is considered[21].

336

The starting point is eq. 34b at frequencies ω and 2ω. The guided mode amplitude c1 (x) at ω and the leaky mode amplitude c2 (x) at 2ω obey the following set of equations (only forward modes are considered) : (40a) (40b) with : (40c)

(40d) The " - " denotes the complex conjugate. Let us pursue in the framework of the UPA. Equation 40b yields : (41a) Hence from eq. 41a, the EM field Φ z (2 ω, x, y) in the waveguide is given by (see eq. 17a) : (41b) Equation 41b shows that Cerenkov SHG involves two types of waves : a leaky wave, with complex longitudinal wavevector , which decays on a length characterized by the out-coupling length

c

: (41c)

and a wave with real longitudinal wavevector component 2γω. For x>> c, the SH intensity is constant and its value at y=y 0 is given by : (42a)

Equation 42a describes the same resonant process than that corresponding to eq. 39a. The SH intensity is a Lorentzian centered at with width and peak value :

(42b)

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The width of the Lorentzian may be interpreted as a measure of the amount of radiation modes involved in the resonant excitation of the leaky mode , . The smaller the EM leakage, the longer c and the larger the enhancement of the SH efficiency resulting from this resonance process. According to eq. 42a the peak value of I2is obtained when : (43) Equation 43 is identical to eq. 39b and expresses the resonant excitation of a leaky mode at 2ω. Consequently this equation constitutes the phase matching relation for Cerenkov SHG. Clearly eq. 43 is not automatically satisfied: a phase matching condition exists with one of the leaky modes of the structure. Figure 7 is a plot, with the numerical values of ref. [20], of I2 as a function of the effective index neff1 at ω ( n eff1

The intensity at 2ω has been derived in two ways:

from eq. 42a with ξ 2 calculated using the "closing" method (dashed line) and from the total field analysis (solid curve).

Figure 7. Plot of I 2 as a function of the effective index n e f f 1 at ω.

The resonance effect, and as a result the existence of a phase-matching condition, clearly appears. The agreement with the total field calculation can be considered as excellent. This shows the interest of the "closing" procedure: the knowledge of the SH intensity requires the determination of

and ξ 2 . The values of these

parameters are obtained solving the homogeneous problem. With the numerical values used in fig. 7 : n eff1 = 1.69666, n eff2, = 1.69654+ i5.071610 –4, ξ 2 = 7 .4610–21 + i8.3810 –20 . Leaky modes play an important role in Cerenkov SHG, It is seen that the situation is very similar to SHG at prism or grating couplers which, as is known, behave as leaky resonators : in Cerenkov SHG the leaky waveguide is equivalent to a resonator with a high finesse because the SH field is at grazing incidence. The description of guided wave SHG

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corresponding to the guided mode at ω - guided mode at 2 ω interaction is obtained from = 0. Thus guided wave SHG and Cerenkov SHG are similar. Cerenkov SHG letting The only difference is that for the former the waveguide is equivalent to a closed optical resonator whereas for the latter the SH interaction takes place in an open, i. e. leaky, resonator. From this point of view there is no difference between Cerenkov SHG and SHG at distributed couplers and the phase-matching condition which exists for the latter is also present for the former. IV. CONCLUSION The equations of evolution of a leaky mode amplitude (eqs. 34) apply to a wide class of nonlinear optical effects at leaky resonators : χ2 , χ 3 . . at prism or grating couplers, FabryPerot's, multilayer devices…. It is not the geometry and the specificities of the resonator which are important but the set of parameters characterizing the EM resonance brought into play: γ l or α l , γ z , Q , tQ . In this sense the theory presented here constitutes a unified approach of leaky resonators in nonlinear optics. The development of the leaky mode formalism is an opportunity to present and discuss several expansions of the EM field together with the link between them. The longitudinal representation constitutes the starting point of two types of coupled-mode analysis : the transverse expansion which corresponds to the "usual" coupled-mode theory[3,4] and the leaky mode representation whose main interest comes from the fact that a leaky mode replaces the "packet" of resonantly excited radiation modes. The resulting simplification allows an easy insight in the physics of nonlinear optics in "open" resonators. Thus leaky modes provide a convenient framework and a powerful tool for the study of nonlinear optical interactions in such devices. Acknowledgment I wish to thank Dr. G. Vitrant, Prof. M. Nevière and Dr. E. Popov for helpful discussions. REFERENCES 1. See, for instance, G. I. Stegeman, Introduction to nonlinear guided wave optics in : Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, Eds., NATO ASI Series, Kluwer Publ., Dordrecht (1992). 2. V. C. Y. So, R. Normandin, and G. I. Stegeman, Field analysis of harmonic generation in thin film integrated optics, J. Opt. Soc. Am. 69: 1166 1979). 3. C. Vassalo, Théorie des Guides d'Ondes Electromagnétiques, Tomes 1 and 2, CNETENST, Eyrolles, Paris (1985). 4. H. Kogelnik, Theory of dielectric waveguides in : Integrated Optics, ed. T. Tamir, Springer-Verlag, New-York (1975). 5. M. Neviere, The homogeneous problem in : Electromagnetic Theory of Gratings, ed. R. Petit, Springer-Verlag, New-York (1980). 6. M. Nevière, E. Popov, R. Reinisch, Electromagnetic resonances in linear and nonlinear optics : phenomenological study of grating behavior through the poles and zeros of the scattering operator, J. Opt. Soc. Am. A12:513 (1995). 7. T. Tamir, Leaky waves in planar optical waveguides, Nouv. Rev. d'Optique 6:273 (1975). 8. T. Tamir, A. A. Oliner, Guided complex waves, fields at an interface, Proc. IEE 110:310 (1963). 9. T. Tamir, L. B. Felsen, On lateral waves in slab configurations and their relation to other wave types, IEEE Trans. Antennas and Propag. 13:410 (1965). 10. V. V. Shevchenko, Continuous Transitions in Open Waveguides, The Golem Press, Boulder, Colorado (1971). 11. G. Arfken, Mathematical Methods for Physicists, Third Edition, Academic Press (1985).

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12. R. E. Collins, Field Theory of Guided Waves, IEEE Press (1991). 13. J. D. Kraus, Antennas, Mc Graw-Hill, New-York (1950). 14. G. Vitrant , M. Haelterman and R. Reinisch, Transverse effects in nonlinear planar resonators II. Modal analysis for normal and oblique incidence, J. Opt. Soc. Am B7: 1319 (1990). 15. R. Reinisch, G. Vitrant and M. Haelterman, Coupled-mode theory of diffractioninduced transverse effects in nonlinear optical resonators, Phys. Rev. B44:7870 (1991). 16. R. Reinisch, M. Neviere, E. Popov, and H. Akhouayri, Coupled-mode formalism and linear theory of diffraction for a simplified analysis of second harmonic generation at grating couplers, Opt. Comm. 112:339 (1994). 17. R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Neviere, and H. Akhouayri, Modal analysis of grating couplers for nonlinear waveguides, Nonlinear Optics 5: 111 (1993). 18. M. Haelterman, G. Vitrant and R. Reinisch, Transverse effects in nonlinear planar resonators I. Modal theory, J. Opt. Soc. Am B7: 1309 (1990). 19. R. Reinisch, M. Neviere, P. Vincent, G. Vitrant, Radiated diffracted orders in Kerr type grating couplers, Opt. Comm. 91:51 (1992). 20. G. J. M. Krijnen, W. Torruellas, G. I. Stegeman, H. J. W. Hoekstra, and P. V. Lambeck, Optimization of second harmonic generation and nonlinear phase-shifts in the Cerenkov regime, IEEE Journ. of Quant. Elect. 32:729 (1996). 21. R. Reinisch, G. Vitrant, Phase-matching in Cerenkov second harmonic generation : a leaky mode analysis, Optics Letters 22:760 (1997). 22. M. J. Li, M. de Micheli, Q. He, and D. B. Ostrowsky, Cerenkov configuration second harmonic generation in proton-exchanged lithium niobate guides, IEEE Journ. of Quant. Elect. 26: 1384 (1990). 23. N. Hashizume, T. Kondo, T. Onda, N. Ogasawara, S. Umegaki, and R. Ito, Theoretical analysis of Cerenkov-type optical second harmonic generation in slab waveguides, IEEE Journ. of Quant. Elect. 28:1798 (1992). 24. Y. Azumai, I. Seo, and H. Sato, Enhanced second harmonic generation with Cerenkov radiation scheme in organic film slab guide at IR lines, IEEE Journ. of Quant. Elect. 28:231 (1992). 25. N. Bloembergen, Nonlinear Optics, Benjamin Inc., New-York (1965), pp. 74-83.

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QUADRATIC CASCADING: EFFECTS AND APPLICATIONS Gaetano Assanto Department of Electronic Engineering Terza University of Rome Via della Vasca Navale, 84 Rome, ITALY 00146

INTRODUCTION In the search for a viable ultrafast, low power, lossless approach to all-optical processing of signals, quadratic cascading, i.e. the aimed sequence of two secondorder nonlinear processes, has emerged in the past few years as one of the solutions better suited for such a broadband task. Despite its foundation lies in the nonlinear optics (NLO) knowledge of the late 60's,1-6 quadratic cascading has stimulated a vast scientific effort in fundamental NLO and its applications only after extensive research in cubic materials with an intensity-dependent refractive index and the parallel development of material science in noncentrosymmetric materials for electrooptic and parametric effects. Following numerous theoretical 1,3,5,6,9,10,13,15-17 and experimental 4,7,8,11,12,14 reports on phase-effects through quadratic nonlinearities, the second-harmonic generation (SHG) experiment performed by DeSalvo et al. in a KTP crystal clearly demonstrated the implications of a cascading-induced nonlinear phase shift for all-optical processing. 18 Several steps forward have been accomplished since then, both theoretically and experimentally, exploring and/or discovering various areas in which cascading does or could play a leading role, from guided-wave all-optical switching devices to all-optical transistors or isolators, from self-guiding beams in space or time to gap solitons, from wavelength shifters to nonlinear competition. In this Chapter, for the sake of conciseness although at the expense of coverage, we will discuss illustrative features of cascading in bulk and in waveguide configurations, with specific reference to quasi-cw cases and excluding the important situations involving laser cavities and competing nonlinearities. The Chapter is organized in Sections dealing with plane waves, guided waves and self-confined beams, respectively: Section I introduces the model equations for cw propagation in both Type I and Type II SHG, discussing features of cascading phase-shift and amplitude modulation of a fundamental frequency wave. Section II illustrates various all-optical transistors/modulators, whereas

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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Section III examines several nonlinear integrated devices for all-optical switching. Finally, Section IV is devoted to some applicative features of spatial solitary waves through Type II SHG in bulk media.

I.

MODEL EQUATIONS

Quadratic cascading is based on the consecutive application of two nonlinear operators, namely second-order susceptibility tensors, to the electromagnetic field(s). Denoting by subscripts 1, 2 and 3 the three waves involved in the interaction, the simplest case of cascading is described by an upconversion process or a with ω 3= ω1 + ω2 followed by downconversion with process conveniently described by the product of susceptibilities . Its simplest implementation is in Type I SHG, with and the two fundamental frequency (FF) fields at ω oscillating with the same polarization. The other frequency degenerate case involves optical rectification through , i.e. the electrooptic effect. 3,22 The latter relies on the generation of a DC field and will not be discussed further. Using the standard slowly-varying-envelope approximation, in the presence of a quadratic nonlinearity and material dispersion such that a single three-wave interaction is nearly phase matched (i.e. the three waves have nearly the same phase velocities) and is therefore dominant, the evolution of the plane-wave amplitudes Ej (z) of the copropagating electric fields along z is described by:

(1)

with the wavevector or phase mismatch. In writing eqns. (1) the interaction has been reduced to a scalar one through the projections and similar ones, with e 1 , e 2 , e 3 the unit polarization vectors for the three fields. Taking a further step in order to simplify the notation, assuming all the wavelengths to be far from material resonances such that the frequency dependence in d (2) can be neglected (i.e. invoking Kleinman symmetry), and scaling the amplitudes according to such that (j=1,2,3) are the field intensities (W/m 2 ), eqns. (1) reduce to:

(2)

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with ζ =z/L and ∆ = ∆ kL the normalized propagation distance and mismatch, respectively, and the nonlinear coefficient. It is instructive to analyze separately the cases of Type I and Type II interactions, the first referring to complete degeneracy between the fields with subscripts 1 and 2, and the second to a "true" three-wave process. Type I Second Harmonic Generation When subscripts 1 and 2 effectively refer to identical parameters, i.e. the fields E 1 and E 2 are indistinguishable, taking into account the degeneracy factors eqns. (2) reduce to two coupled nonlinear ordinary differential equations describing the interaction between two waves of frequencies ω and 2ω:

(3)

with For SHG in the low depletion limit, i.e. assuming a negligible second and harmonic (SH) , the second of (3) can be integrated from 0 to ζ and substituted in the first of (3), to yield:

(4) with n2,eff ( ζ ) an equivalent z-dependent (focusing or defocusing) Kerr coefficient, and β eff ( ζ ) an equivalent two-photon absorption. The intensity dependence in the FF field evolution establishes a parallelism with the effects of a cubic nonlinearity, provided the z-variant feature is averaged out in the limit of large mismatches, i.e. when the low-depletion approximation holds. In this limit, the nonlinear absorption is zero and the Kerr coefficient takes the value: (5) This equivalent Kerr response, obtained through a quadratic (SHG) interaction in the limit of low conversion efficiency, encompasses a linear dependence on the usual figure of merit for quadratic materials, namely (d(2) )2 /n3, and an inverse dependence on the phase mismatch. In addition, the factor L/λ can be regarded as an enhancement term the role of which, in geometries involving a transverse resonance at the generated frequency, is played by the quality factor of the cavity. 23 Moreover, the effective change in index or, more appropriately, the nonlinear phase shift changes sign according to ∆ k. This implies that a focusing or defocusing behaviour depends on the material arrangement with respect to the incident field to be doubled, for instance orientation, temperature, periodic poling, 343

applied voltage, etc. Some of these important characteristics were verified in bulk by DeSalvo et al. in a 1mm KTP crystal at 1.064µm via Z-scan, 18 and later on by Danielius et al. in BBO via self-diffraction, 24 by Ou in potassium niobate inside a cavity, 25 by Fazio et al. in KDP, 26 by Wang et al. 2 7 in the organic NPP and by Vidakovic et al. in periodically poled lithium niobate (PPLN).28 The shown similarity between a phase mismatched SHG process and a cubic frequency degenerate interaction is, however, of limited use. Although it allows a rough estimate of the potential of a cascaded process in mimicking the Kerr effect in quadratic media, 29 it disregards some important features of cascading which, in a measurable sense, does not induce any actual change in refractive index. It is worthwhile, therefore, to examine the induced phase shift directly, dropping the low-depletion approximation and resorting to a simple integration of eqns. (3). Such integration can be performed exactly 30-31 or numerically 19,21 , with the aim of investigating the evolution of amplitude (or intensity) and phase shift of the FF field component. Figs.1a-b show the evolution of phase shift and normalized intensity versus propagation distance for various mismatches and a fixed excitation Γ | Φ 1 (0)|= 20. The FF intensity undergoes oscillations, recovering its initial amplitude with periodicity decreasing for increasing mismatch, as expected based on coherence length considerations. Similarly, the degree of conversion to second harmonic becomes higher when approaching the optimum condition, i.e. phase matching. It is rather more interesting the evolution of the FF phase, graphed in Fig.1a. The phase grows stepwise, with jumps corresponding to energy flowing back to the FF through downconversion, and plateaus separated by π /2 in the case of complete FF depletion. It is apparent that the linear

Fig. 1 a) FF nonlinear phase shift ϕNL in units of π and b) FF throughput T vs. propagation for various mismatches (solid lines). The dashed lines refer to an input SH seed of 0.001 in intensity and relative phase of π (discussed later in the text). Here

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behaviour discussed above in the framework of an undepleted FF is only adequate to describe the evolution of the phase in its initial stages or for high mismatches, when the corresponding FF amplitude remains nearly constant. In another limit, for intense excitations and/or large phase shifts, the dependence becomes linear in input amplitude, rather than in intensity. 19 An intuitive explanation of such a complex behaviour can be provided in terms of photons (or energy) which, partially converted from FF to SH, propagate at different velocities and acquire a phase difference before eventually recombining at the FF after a coherence length. 21,32 A phasorial representation of the field amplitudes in snapshots during propagation, as shown in Fig. 2, helps the intuitive understanding. The linear phase rotation caused by a nonzero mismatch on the SH phasor allows the downconverted photons to tilt the progressively more depleted FF phasor, until a ≈π/2 phase change is completed after one cycle.

Fig. 2 Phasorial representation of the cascaded FF phase rotation during the first cycle (see Fig. 1). At various propagation distances, the FF and SH components are graphed, the additional vectors being the previous- and present-step contributions to the FF field in each picture. Here ∆=π and

The evolution of the FF throughput versus input excitation is displayed (in phase and transmission) in Fig. 3, for a fixed mismatch. For increasing input, the phase plateaus become progressively extended, while the recursive FF depletion becomes more and more pronounced, according to a nonlinear improvement of wavevector matching. The latter effect can be better appreciated when plotting the FF throughput versus ∆ , as in Fig. 4. A stronger excitation tends to deplete more the FF, narrowing the center lobe and making the sidelobes grow deeper. The occurrence of flat phase-plateaus when the energy flows back into the FF allows the realization of phase and polarization interferometers, with reduced power requirements when a folded geometry or two crystals are employed. 33-34

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FF throughput (left axis) and phase (right axis) vs. excitation Fig. 3 for ∆ = π, and no SH seed (solid lines), SH seed 1% in intensity of the FF input and with relative phase of π (dashed lines).

FF throughput vs. mismatch for excitation for low excitation Fig. 4 Γ |Φ1 (0)|=1 and high excitation Γ |Φ1(0)|=4 without (solid line) and with (dashed line) an SH seed of intensity 1% than the FF and phase π.

The features illustrated so far refer to SHG, i.e. to a process with a single FF input. The interaction is, however, inherently coherent, because it does not depend exclusively on the field moduli. 21 This translates in the possibility of seeding the process with another input at the second harmonic, controlling the overall evolution by acting on its amplitude or on the relative phase between Φ 1 (0) and Φ 3(0). In order to illustrate this concept, various cases in the presence of a weak SH seed have been added to Figs. 1,3 and 4. Since the phase plays a major role even though the added input might be rather small, this opens up the way to various possibilities for phase-to-amplitude transducers or all-optical modulators/amplifiers, as discussed in Section II. Several schemes, based on reflecting configurations where the coherently generated second harmonic and the leftover fundamental propagate back into the crystal, can also be envisaged. 35

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Type II Second Harmonic Generation and Non-Degenerate Frequency Mixing When more (than one) inputs are required to generate the new frequency, extra degrees of freedom are available and can be exploited for cascading. This is the case of Type II phase-matching in SHG and, in general, of all three-wave mixing processes with distinguishable field components. 20 Eqns. (2) can be manipulated to give: (6) for Φ 1 , and similar for the others. Eqn. 6 shows that the evolution of each wave depends on the intensity of the others. Proceeding in analogy to the previous subsection (Type I SHG), let us assume that the third wave is entirely generated through the nonlinear interaction, i.e. Φ 3 (0)=0. If we let the two inputs to be in the low conversion approximation for Φ3 unbalanced, with eqn. (6) reduces to: (7)

which shows how the more intense wave induces a phase shift on the weaker component, even for a vanishing phase mismatch (i.e. |∆|→0). This observation, implicit in the non-degenerate analysis,20 was outlined in the context of SHG by Experimental verification of cascaded phase-shifts through Belostotsky et al.36 frequency non-degenerate interactions have been reported by Nitti et al. in the organic crystal MBA-NP 37 and by Asobe et al. in PPLN.38 Related effects have been discussed in [39]. For simplicity and without lack of generality, we will

FF phase shift vs. mismatch for Type I SHG with Γ 2 |Φ 1 (0)| 2 =150 Fig. 5 (solid line), and for Type II SHG (long and short dashes) with 2 2 2 2 2 Γ [|Φ1 (0)| + |Φ2 (0)| ]=150 and |Φ1 (0)| / |Φ 2 (0)| = 0.5. The short dashes represent the phase of Φ 2 .

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henceforth refer to SHG with orthogonal FF inputs, as this is the case of a few applications discussed later. Fig. 5 is a plot of nonlinear phase shift in the two unbalanced FF components versus mismatch, calculated from numerical integration of (2). A nonlinear phase-shift is induced on the weaker component even in perfect matching, whereas the phase of the stronger wave goes through zero in analogy to the Type I SHG case. As visible in Fig. 6, however, for a given intensity ratio ≠ 1, in contrast to Type I SHG the maximum phase jump is π, with a staircase evolution sharper for better wavevector matching and/or higher input imbalance (not shown) . 40 The corresponding FF throughput versus imbalance is graphed in Fig. 7, where a sharp transition can be observed as an input ratio ≠ 1 is introduced. This abrupt change in transmission, further discussed in the framework of all-optical transistors, can be related to a dramatic shortening of the characteristic length for energy exchange between the three waves near phase matching, 41-42 leading to FF modulation at the end of a finite length medium. For perfectly balanced inputs at FF, conversely, cascading via Type II and via Type I SHG present identical phenomenologies.

Fig. 6 Phase shift of Φ 1 v s . e x c i t a t i o n f o r T y p e I I S H G w i t h |Φ 1 (0)| 2 / | Φ2 (0)| 2 = 0.1 and various mismatches.

Fig. 7 Throughput T 2 of the strong polarization component Φ 2 v s . 2 ratio |Φ1 (0)| / |Φ2 (0)| 2 for various mismatches.

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In the next Sections, the phase shift and related amplitude/throughput modulation obtainable through cascading will be discussed in the framework of a few specific but paradigmatic applications. Phase conjugation, 43 four-wave m i x i n g , 44 laser mode-locking and pulse conditioning 29 v i a c a s c a d i n g a r e important effects and applications among those left out of this review. The following section is devoted to all-optical transistors, and the next deals with guided-wave geometries involving cascading.

II. ALL-OPTICAL MODULATORS VIA SHG CASCADING Fig. 8 shows a few basic schemes of all-optical transistors based on quadratic cascading via second-harmonic generation. Many of them have already been demonstrated experimentally in bulk crystals of KTP, and we will try to outline major advantages and drawbacks of each. Type I SHG transistor with SH input signal This implementation (Fig.8a) of an all-optical transistor is based on the intrinsic coherence of the quadratic SHG interaction, such that a weak input at the SH can alter the evolution of the two waves (FF and SH) along propagation, resulting in phase and/or amplitude modulation of the FF component. 21,45-46 For a fixed input FF intensity, a variation in the relative phase between the SH seed and the FF pump induces a shift in the overall phase matching response, as seen in Fig. 4. If the operating ∆ -point is chosen in order to maximize this phase-tothroughput modulation (i.e. along the slope of the main lobe), a large change in transmission can result from a small variation in input phase. Conversely, keeping the phase constant, the presence of an SH input determines the response of the device, i.e. results in an intensity controlled transistor.

Fig. 8 All-optical transistors based on SHG cascading: a) Type I SHG with SH signal; b) Type I SHG with SH pump; c) double Type I SHG with orthogonal FF inputs coupled to one SH wave; d) Type II SHG with orthogonal FF inputs; e) Type II SHG with SH pump; f) double Type II SHG with coupler and amplifier.

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Fig. 9 shows the results of the experiment performed by Hagan et al. in KTP, together with the numerical simulation. 46 The FF throughput, despite the use of temporal pulses with a Gaussian beam distribution, clearly exhibits a modulation versus input relative phase, with a contrast as high as 4.6:1 when employing SH pulses with energy 1.2% of the pump at 1.064µm. Notice that, although KTP is Type II phase-matched for SHG at this wavelength, the FF input was injected at 45° with respect to ordinary and extraordinary crystal directions, thereby resulting in standard Type I cascading. The major drawbacks of this scheme are the use of a signal of frequency twice the output, and the need for an interferometrically stable setup in order to properly control the phase. The first problem can be solved by working with an SH pump (Fig. 8b), i.e. in downconversion with input and output signals at the FF. The use of a short-wavelength pump, however, may be undesirable. The second problem can be solved by employing a Type II configuration, as discussed later.

Fig. 9 FF transmission vs. relative phase between SH seed and FF pump. The peak FF intensity is 20GW/cm 2 , and the seed energy fraction is 1.2%. (After Ref. 46)

Double Type I SHG transistor Crystal symmetries can, sometimes, allow more than one quadratic interaction to be nearly phase matched. Considering the additional possibilities offered by Quasi-Phase-Matching in ferroelectric crystals, this opportunity can be exploited in coupling two orthogonal FF waves (Φ1 and Φ2 ) through the nonlinear susceptibility by means of two distinct Type I SHG interactions (labelled "a" and "b"), i.e. via a single SH wave Φ 3 . 47 The resulting equations:

(8)

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encompass cross-phase modulation between Φ1 and Φ 2 , and the possibility of nonlinear polarization rotation, FF modulation and small-signal amplification when varying either FF input with respect to the other, either in phase or in amplitude. This scheme (Fig.8c), potentially feasible in a temperature tuned periodically poled lithium niobate crystal, 47 has not yet been demonstrated. Type II SHG transistor As outlined above, the use of a Type II phase-matching geometry allows to introduce more degrees of freedom in SHG cascading, with the extra advantage of phase insensitivity when inputting the FF only. This latter feature, on the basis of the input-output response shown in Fig. 7, stimulated the study of a frequency degenerate, phase insensitive, all-optical transistor (Fig.8d). 4 0 - 4 1 , 4 8 - 5 0 The transmission at FF at the end of the crystal, in either or both polarizations, will abruptly change when a small imbalance is introduced between the input FF components, provided the interaction is nearly phase-matched. This process does not depend on the relative phase between Φ 1 and Φ 2 , and can be exploited around an appropriate bias point in order to obtain small-signal amplification, as sketched in Fig. 10. In addition, through the phase shift experienced by the weakest input, a rotation of the polarization state by nearly π /2 at the output can be exploited for all-optical switching through an output analyzer. 50-51 Analytical expressions for the obtainable gain can be derived under plane wave and cw a p p r o x i m a t i o n s , 31,34,41-42 although the use of gaussian beams and pulses in e x p e r i m e n t s i s b o u n d t o d e g r a d e t h e p e r f o r m a n c e t o s o m e e x t e n t .4 8 - 5 0 Experiments performed at 1.064µm have demonstrated amplifications as high as 21.6dB in KTP crystals, 48 and a typical set of data together with a numerical simulation (from eqn. (2) including beam and pulse shapes) is shown in Fig. 11. 49 This transistor scheme, although phase insensitive and with inputs and output at the same frequency, relies on an input signal which, for best operation,

FF transmission of a Type II SHG transistor vs. input Fig. 10 imbalance. The two marked regions (dashes) identify the pre-amlifier or 2 2 2 2 c o u p l e r ( |Φ 1(0) | << |Φ2 (0)| ) and the amplifier (|Φ 1 (0)| ≈ |Φ2 (0)| ) , respectively. Notice that ∆Tc << ∆Ta for comparable changes in imbalance. I tot is the total FF input intensity.

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should have amplitude comparable with the other FF component.40-41 This limits its usefulness in any realistic application, because a weak signal to be amplified should be coherently superimposed to a large background in order to satisfy the small-imbalance requirement. A Type II downconversion (half-harmonic generation) process (Fig. 8e) is in principle able to solve this problem at the

Fig. 11 Experimental FF transmission for the extraordinary (squares) and ordinary (triangles) components and the total (filled circles) vs input imbalance through a 2mm-long KTP crystal. The solid line is calculated using eqns. (2) including beam and pulse profiles. Here Ito t =12GW/cm2 (peak value). (After Ref. 49)

expense of a strong pump at the second harmonic, because in this case a small input at FF in either polarization would trigger the parametric downconversion from SH and result in large gain at FF. An alternative way to employ cascading for the amplification of a small signal incoherent with the pump is described in the following paragraph. Finally, the injection of a weak coherent seed can, even for the Type II transistor, widen the range of possibilities available to control the interaction. 31 Double Type II SHG transistor This scheme (Figs. 8f and 12) employs two nonlinear stages pumped by the same strong FF wave: the first couples a weak FF signal to a large FF pump in the orthogonal polarization through Type II SHG, while the second operates with inputs of comparable intensities as described in the previous paragraph.52-53 Effectively, the coupler or preamplifier works in the initial (large imbalance) region of the characteristic of Fig. 10, in order to generate the coherent superposition required by the amplifier. The latter, in fact, is best suited to work in the central region (small imbalance) of Fig. 10. Although the experimental implementation is rather complicated due to the use of pulsed beams and the polarization constraints, preliminary results utilizing KTP crystals have demonstrated its feasibility with amplifications of the order of 7dB at 1.064µm.54

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F i g . 1 2 Schematic of a two-stage Type I I SHG transistor . PBS=polarizing beam-splitter, Coupler and Amplifier are Type II crystals for SHG. (After Ref. 52)

III. QUADRATIC CASCADING IN GUIDED-WAVE GEOMETRIES Dielectric waveguides for transverse confinement of the electromagnetic fields over propagation lengths not limited by diffraction are a convenient setting for the evaluation of nonlinear effects and the implementation of nonlinear integrated optics devices. 55 In the framework of SHG cascading, experimental measurements have been reported in KTP, 56-57 in lithium niobate 58-63 a n d i n DAN [4-(N,N-dimethylamino)-3-acetaminonitrobenzene], 6 4 and a number of guided-wave geometries and approximations have been investigated theoretically or numerically. Here, with specific reference to the simple case of cw excitation in lossless structures, we will briefly review the basic concepts involved in waveguide nonlinear optics through quadratic cascading and describe several alloptical devices demonstrated and/or proposed to date. The equations describing quadratic cascading in waveguides are substantially similar to those relative to plane waves, provided the overlap of the modal fields is properly taken into account and the mismatch calculated with the guided-wave wavevectors. Introducing pairs of (single) superscripts "l,k" identifying the channel (planar) waveguide eigenvalues and eigenmodes, normalizing the modal field transverse distributions (taken real) such that are guided powers (in units of W for channels, in W/m for planar guides), for Type I SHG eqns. (3) hold with

(9) (10) (p,q)

(m,n)

and β 1 β3 being the wavevectors of the eigenmodes at 2ω and ω , respectively. A straightforward extension is valid for Type II SHG. The overlap

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integral in (9) alters the bulk nonlinearity, and proper parity of the involved modes has to be selected in order to maximize the conversion efficiency. Single channels A small phase-mismatch is required for efficient cascading to take place, at least in type I SHG. In waveguides this requirement translates in proper orientation of the medium, polarization of the excited eigenmodes, temperature or electro-optical adjustements of the effective indices (i.e. wavevectors), variations of the nonlinearity along the propagation direction (QPM), etc.65-69 In many applications a single-mode waveguide at the FF is desirable, together with a limited number of modes at the higher frequency. When more modes are present at the SH, competition between them can take place provided they are nearly phase-matched to the same FF wave, as demonstrated by Treviño-Palacios et al. in a lithium niobate QPM waveguide. 6 2 More complicated geometries, with Cherenkov emission at the SH, have also been investigated both experimentally (in DAN) and theoretically, showing that large nonlinear phase shifts of the FF can be achieved in crystal cored fibers with the SH field trapped in the fiber cladding. 64,70-71 Lee et al. have investigated selfphase modulation in SH surface emitting geometries with counterpropagating FF w a v e s , 72 and Ueno et al. in semiconductor waveguides subject to large walk-off and losses. 73 Modulators. Single channels can be operated as amplitude or phase modulators controlled by the intensity of the incoming FF wave(s). This is indeed the signature of quadratic cascading, as it has been verified experimentally by Sundheimer et al. in KTP channels, 56-57 and by Schiek et al. in lithium niobate channels. 58-59 The experiments performed in KTP were carried out by measuring the spectral broadening of laser pulses through self-phase modulation due to cascading, whereas the lithium niobate waveguides were evaluated with an interferometric setup, and phase matched through temperature tuning with a nonuniform longitudinal profile (due to the oven geometry). The latter configuration, with lower temperature at the crystal ends, allowed efficient SHG conversion internally to the sample, with a resulting conspicuous phase shift (as high as 1.5π ) but low depletion of the FF wave at the output of the channels. This is a desirable situation for all-optical devices based on frequency conversion. Fig.13 shows the FF transmission and the corresponding nonlinear phase shift versus temperature. Notice that the phase matching temperature T P M for Type I SHG between the TM00 (ω ) and TE 00 (2 ω ) modes was 336.6°C. Temperatures higher (lower) than TPM induce a negative (positive) phase contribution, resulting in an overall large positive phase shift for T
(11)

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Fig. 13 Measured and calculated a) FF transmission and b) nonlinear phase shift vs. temperature in a 15µm-wide lithium niobate channel. Peak power is 60W and wavelength 1.319µm. (After Ref. 58)

with

Γ(ζ)

the

longitudinally varying nonlinearity, and the propagation-dependent mismatch. The ζ-variant linear contributions are linked to variations in the linear parameters, with (12) and similarly for δβ 3 ( ζ ). The ζ-variant nonlinearity includes quasi-phasematching, and can also be taylored to specific bandwidth-enhanced highand efficiency applications. 7 4 - 7 5 Defining the new amplitudes and the phase (13)

eqns. (11) can be recast in the form:

(14)

Solving the second equation in (14) in the undepleted FF approximation, the amplitude a 3 (1) results proportional to the Fourier transform of a function which incorporates the perturbation, i.e.: (15) with

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(16) This approach, although rigorously valid only for low conversions, allows to investigate and design tentative profiles for various operations. A linear variation was induced through the oven temperature profile in Ref. [58-59], and a nonlinear one through ion-exchange in Ref. [56-57]. Other interesting possibilities encompassed by (11) or (14) will be illustrated below. Finally, it is worth mentioning the possibilities offered by Type II and SH-seeded guided-wave interactions, in analogy to the plane-wave cases discussed above. 21,40-42,45-46 Wavelength Shifter. If one of the two processes involved in cascading is made non-degenerate in frequency, a small frequency shift can be obtained at the expense of a FF input pump through the cascaded interaction: (17) as demonstrated in bulk BBO by Tan et al ., 76 and proposed in a MQW waveguide with a vertical SH resonator by Gorbounova et al.77 and in a QPM lithium niobate channel by Gallo et al. 78 This configuration for wavelength shifting offers a large 3dB bandwith and amplification at the expense of the input pump, which can be within the useful spectral range of the Er-doped fiber amplifiers used in telecommunications. Fig. 14 shows the calculated response in the latter case for a QPM channel in lithium niobate, with an additional reflector at SH in order to maximize the useful interaction length. 78 Preliminary results in a 9.5mm-long waveguide (without reflector) are in qualitative agreement with the simulations.79

Fig. 14 a) Sketch and b) calculated conversion efficiency (ω –δ → ω+δ) vs. signal wavelength λ 2 f o r a λ -shifter in a QPM lithium niobate channel of length L and with a 2 ω-reflector to maximize the difference-frequency-generation.

If a ζ non-uniform perturbation, either linear or All-optical diode. nonlinear, is also non-symmetric with respect to the terminations of a channel, the quadratic interaction will induce a nonreciprocal response of the device upon excitation from different sides. This corresponds to a purely dielectric isolator or all-optical diode, which is - in principle - able to transmit the FF component only

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for increasing (or decreasing) ζ. A simple implementation of this concept has been reported by Treviño-Palacios et al. in a QPM channel partially overcoated with a thin photoresist film in order to introduce a change in wavevector mismatch. 6 3 The results - reproduced in Fig. 15 - demonstrate the expected nonreciprocity. For the case of nonlinear tayloring through the QPM periodicity, Fig. 16 shows the calculated response of a diode with a flat + linearly chirped mismatch profile (Fig.16a): the isolation effect is apparent at various input power levels, with a large extinction ratio when propagating in opposite directions (Fig.16b). .L

Fig. 15 SHG wavelength scan of a 7 µ m-wide 1cm-long QPM lithium niobate channel without (dashed line) and with (solid lines) a superimposed thin film to realize a non-uniform mismatch distribution. Open and filled circles refer to forward and backward propagation, respectively. (After Ref. 63)

Fig. 16 a) Chirped mismatch profile vs. length and b) FF transmittance vs. excitation for an alloptical diode with a flat+linear mismatch induced by varying the QPM periodicity. Forward (solid line) and backward (dashed line) FF transmissions show a good degree of isolation at given powers.

Integrated Mach-Zehnder Interferometer The integrated interferometer is, among various devices, the most obvious implementation of an all-optical switch based on a nonlinearly-induced phase357

shift.5 5 When employing cascading, a simple configuration with geometrically identical waveguides (Fig. 17a) characterized by opposite signs of phase-mismatch allows to minimize the power requirements, because the nonlinear phase shifts in the two arms will effectively combine to produce throughput switching.8 0 - 8 1 Moreover, the phase plateaus (versus excitation, see Fig. 3) allow this device to work with good switching contrast even in the presence of pulses with a continuous distribution of instantaneous power. An experimental demonstration has been reported by Baek et al. in a lithium niobate channel with a non-uniform temperature profile for phase matching, 60 and a typical set of experimental data with a numerical simulation is reproduced in Fig. 17b. Excellent performance has also been predicted for the Mach-Zehnder device with soliton-like pulses injected at FF.82

Fig. 17 Guided-wave Mach-Zehnder interferometer. a) sketch of the device; b) measured (solid line) and calculated FF transmittance vs. input power at 1.319µ m. (After Ref. 60)

Directional Coupler The directional coupler, widely studied in the context of Kerr nonlinearities, 55 has also been investigated in the case of cascading nonlinearities. 80,83-88 The pertinent equations describing a coupler with negligible overlap between the fields Φ 3 and Ψ 3 at the SH, and linear coupling strength κ between the fields Φ 1 and Ψ 1 at FF, are:

(18)

with

the linear detuning. Despite the fact that, due to the periodic character of the FF amplitude in propagation (Fig.1b), a "clean" switching in a uniform coupler with identical arms (Γ Φ = Γ Ψ , δ =0) is predicted only in the "cross" channel (Fig.18a), experimental

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results for a non-uniform phase-matching profile in lithium niobate channels have shown the feasibility of such a switch,55 as apparent in Fig. 18b. Conversely, a nonlinearly asymmetric but otherwise uniform coupler, with one arm substantially linear, is expected to exhibit good switching performances. 8 5 Directional couplers based on Type II SHG can be operated as logic AND/NAND gates or spatial demultiplexers. Notice that eqns. (18) can also describe the Mach Zehnder interferometer by setting κ to zero, or a counterpropagating geometry with distributed feedback coupling by changing the sign of ζ in the bottom equations. Lee et al. have investigated a geometry involving surface emitted secondharmonic, 8 7 whereas soliton-based directional couplers have been analyzed by 86 88 Schiek, 83 Karpierz, and De Angelis et al.

Fig. 18 Half-beat length symmetric directional coupler. a) Calculated FF transmission through bar ("=") and cross ("X") arms vs. excitation (∆ =8 π); b) measured and calculated FF transmission in a non-uniformly temperature-tuned (Τ=343.5° C) lithium niobate device operated at 1.319µ m. (Ref. 61)

Distributed Couplers Prisms or gratings for input coupling a radiation field into a planar waveguide need to be properly tuned to a geometric resonance in order to satisfy momentum conservation. In the presence of a quadratic process such as SHG in the waveguide, in-coupled power at FF can be converted to SH and cause a cascading phase shift, thereby altering this linear tuning and affecting the overall coupling efficiency through interference effects in addition to local energy conversion. This phenomenon, in the form of a travelling wave interaction, for a prism coupler (Fig. 19a) exciting the FF guided mode Φ 1 of eigenvalue β1( m )can be described by:

(19)

with t the coupling strength, δ = (ki n sinθ in − β1( m ) )L the linear detuning and 1ω , 1 2ω reirradiation lengths into the prism. Here Φ in f in ( ζ) represents the input field with its distribution, typically gaussian with size d z along z. A similar set of equations can be written for a grating coupler. Numerical solutions of (19), with optimized linear coupling conditions at FF (L = 0.87 d z , l ω = 0.74 d z ) have shown that a) a

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d e t u n i n g δ translates into a nonlinear phase shift of the FF mode through cascading, even under perfect SHG matching ( ∆ =0); b) an SHG mismatch can partially compensate the input detuning, maximizing the FF coupling efficiency at a given power level; c) a power/detuning scan of coupled FF power can provide information about nonlinearity and phase mismatch of the nonlinear structure. Fig. 19b shows the calculated response of a prism-waveguide system with a zero (dotted line, FF) or a small SHG phase mismatch (solid line FF, dashed line SH) and the other parameters optimized for best input coupling at low powers: compared to the perfectly matched case, cascading induces a secondary maximum in efficiency at high powers.8 9

Fig. 19 Distributed prism coupler: a) sketch of the geometry; b) calculated coupling efficiency at FF (solid and dotted lines) and transfer efficiency to SH (dashed line). The dotted line refers to a coupler in linear and SHG matching ( δ =∆ =0), whereas the other lines are calculated for δ=0, ∆ = 0 . 9 .

Based on a coupled-mode formalism and linear theory of diffraction, 90 cascading through second-harmonic generation in distributed couplers has also been shown to give rise to optical bistability.91 Mode Mixer Mode mixing devices allow the spatial routing of information in bimodal waveguides where interference between two FF modes ( Φ1 a n d Ψ1 ) takes place depending on input power or on the presence of a coherent seed (see sketch in Fig. 20a). The existence of two modes at FF implies, however, several modes at SH, with various Type I and Type II interactions potentially contributing to the cascading phase shift. To this extent, a channel waveguide in z-cut lithium niobate has been numerically investigated by De Rossi et al. for operation at λ =1.55 µ m, and the results show good switching contrast with low FF depletion.9 2 A typical set of calculated results is shown in Fig. 20b. Preliminary experimental results at 1.319 µ m in a temperature-tuned lithium niobate channel confirm the theoretical expectations.

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Fig. 20 Mode mixing device. a) Sketch of the waveguide with the two FF modes involved; b) calculated FF throughputs in the two channel halves (Right: dashed line, Left: solid line) vs. FF input excitation (equally distributed between the two modes). The dotted line is the total FF transmission.

Distributed Feedback Grating and Gap Solitons A distributed feedback grating (Fig. 21a) of period Λ is able to couple forward and backward waves at a given wavelength λ≅2 Λ in the medium, i.e. it opens a stop-gap in the dispersion diagram of the wave. When employed in a quadratically nonlinear structure for SHG, the grating can either induce a Bragg reflection at one or both frequencies, depending on dispersion, polarization, index/depth profile. Due to the coupling between counterpropagating components and the energy exchange between fields at different frequencies, a deformation of the stop-gap(s) is expected with input excitation, potentially leading to induced transparency and optical bistability. While in the Kerr limit ( | ∆ | → ∞) this is reminiscent of Kerr-induced energy localization in Bragg coupling structures,9 3 the situation becomes more complicated when FF depletion and stop-gaps at both frequencies are involved. In the limit of a single stop-gap at FF [see eqns. (18) with Ψ 1 a n d Ψ 3 backpropagating waves along – ζ ], Picciau et al. have identified a range of parameters for which localization actually takes place and is able to induce optical bistability with or without the injection of a weak coherent SH seed.9 4 - 9 5 Fig. 21b shows bistable loops for an unseeded case with a zero Bragg detuning, and various SHG mismatches, whereas Fig. 21c displays a loop obtained with an in-phase coherent seed input at SH with δ =∆ =0 (linear and SHG matching). When two stop-bands are considered (i.e. also Φ 3 a n d Ψ 3 are Braggcoupled), it is necessary to define the detunings δ 1 a n d δ 3 with respect to the bottom of the stop-bands at FF and SH, respectively, the mismatch and the curvatures ω"1 and ω"3 (in wavevector space) of the closest Bloch eigenvalues to FF and to SH. In this case the equations describing the FF (u 1 ) and SH (u 3 ) envelopes in the structure can be cast in the form:

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(20) with ρ 1 a n d ρ 3 overlap integrals between the pertinent Bloch eigenfunctions.96 Equations (20) are formally equivalent to those describing dispersive SHG after interchanging t and z and, therefore, can exhibit solitary-wave solutions, i.e. they lead to the existence of gap solitons in quadratic media.9 7 - 9 8 Such form of zlocalization, or mutual trapping of the FF and SH fields, is obtained through the counterbalance of grating dispersion and parametric mixing, provided the parity of the Bloch solutions is such that ρ 1 and ρ 3 do not vanish, i.e. the nonlinearity is effective. This condition corresponds to require that the SH is not close to the upper edge of its stop-band. Stable and stationary bright-bright and twin-hole dark solitons have been identified for the two envelopes, with several nonstationary and unstable solutions as well.97 The non-stationary solutions are able to travel at reduced speeds inside the structure, making ideal candidates for power-dependent delay lines or optical buffers. Since the dispersion normally available in uniform media is not large enough to permit the observability of temporal solitons of the quadratic nonlinearity (simultons), propagation in periodic structures, including the cases of out-gap frequencies,9 9 - 1 0 0 might actually open the way to their use in ultrafast all-optical devices for processing and switching. In this framework, for a singly FF resonant grating, Fig. 22 describes the excitation of a two-color solitary wave: an incident sech-shaped FF pulse reaches the

Fig. 21 Distributed feedback grating. a) Sketch of the doubly-resonant Bragg reflector operating at both FF and SH; b) calculated FF transmission vs. excitation for various SHG mismatches and exact Bragg resonance at FF only; c) FF transmission vs. in-phase SH seed in a coherently controlled device with no Bragg resonance at the second harmonic.

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nonlinear periodic medium (z>0), where the required SH is generated and locks with the FF to originate a slowly moving simulton (propagating at 30% of the natural FF group velocity). In Fig. 23 the FF reflectivity is calculated versus input peak intensity for two sets of detunings, only one of which allows the formation of a travelling simulton. In the latter case the reflectivity drops below 50% at high excitations. 1 0 0

Snapshot of the z-distributions of the reflected (z<0) and Fig. 22 transmitted (z>0) intensities at FF (thick solid lines) and SH (thin solid lines) after the nonlinear interaction has taken place. The dashed line represents the incident pulse, 100ps in duration. Intensities are normalized such that a peak value p=10 corresponds to about 500MW/cm² in KNbO 3 , whereas distances are expressed in grating coupling lengths.

Fig. 23 FF reflectivity vs. incident peak intensity in z=0, for a large positive mismatch (∆ =22 π) and for FF close to either the upper (dashed line) or the lower band edge (solid line). In the latter case, a simulton is generated at the interface and the reflectivity drops.

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IV. APPLICATIONS OF VECTORIAL SPATIAL SOLITARY WAVES VIA SHG Self-guided beams or spatial solitary waves (SSWs) in quadratic media are, in 29 For the framework of cascading nonlinearities, a major field of investigation. this reason, such mixed states of mutually trapped interacting fields are extensively discussed elsewhere in this Book. After their first obser vationin a crystal for Type II SHG ,101 however, they have gained considerable importance due to their potential applications. In this framework, and as a natural extension of some concepts introduced in Sections I and II, in this Section we will discuss a few applications of a specific class of SSWs emerging from Type II SHG. These SSWs contain an SH and two orthogonally-polarized FF components and, for the latter reason, are often referred to as vectorial SSWs. Self-guided beams in space originate in quadratic media through the balance of natural spreading and parametric gain in the presence of a nonlinear phasefront distortion due to cascading, provided the nonlinear length is shorter then the involved diffraction and walk-off lengths. 1 0 1 - 1 0 2 Their evolution, assuming standard slowly-varying envelope and paraxial approximations, is described by coupled equations which, for an "eoe" interaction and in the presence of birefringent walk-off, take the form:

(21)

and ρ ² ω the walk-off angles in the x-z plane and the wavevectors of Φ1 Φ 2 and Φ3 , respectively. Subscripts "e" and e 0 "o" refer to "extraordinary" and "ordinary", respectively. These equations, integrated numerically with a split-step propagator, can describe the formation of SSWs at large enough intensities and for a finite FF beam excitation. Due to the presence of walk-off and the extra degree of freedom introduced by unbalancing the FF inputs, however, they encompass a rich variety of phenomena ranging from angular steering to switching, from imbalance-controlled collisions to phaseand polarization-insensitive down-conversion SSWs, etc. We will briefly discuss some of these applications in the following. with

All-Optical Steering and Switching Fig. 24 shows the evolution in the x-z plane of an FF beam which, launched at z=0 without any SH and with , propagates in a KTP-like medium. The chosen intensities are large enough to induce self-guidance, and it is apparent that, once the injected FF waves have generated enough SH to sustain parametric gain and overcome linear diffraction and walk-off, the SSW propagates along a direction which depends on the prevailing input polarization component. When , the SSW travels along a direction comprised between the "e" axes

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Fig. 24 FF propagation maps of a spatial solitary wave in the x-z plane of a 1cm-long KTP-like crystal. Here ∆ =0.1 π and the input intensities in a 20µm-waist beam are (units are GW/cm²):

at FF and SH, whereas for the SSW evolves along the "o" axis. Finally,when if the power is sufficient to support two SSWs, the input beam feeds two diverging solitary waves (a slight asymmetry with respect to perferct balance is due to the presence of two "e" out of three interacting waves).1 0 3 This behaviour, inherently associated with a three-wave process and walk-off in Type II SHG, indicates the feasibility of all-optical angular steering, the latter controlled by the relative input intensities rather than the phase of a coherent SH seed. 104 Fig. 25 shows the results of experiments performed at 1.064µm in a 1cm-long KTP crystal: a single FF gaussian beam was launched at θ≈ 45° with respect to "e" and "o" axes, and a variation in θ allowed to vary the imbalance. 103 The demonstrated steering action can occur over a substantial range of imbalances spanning transverse distances much larger than the input waist, as shown in Fig. 26. Moreover, the transverse displacement can be enhanced/reduced by the introduction of an input tilt in the x-z plane, leading to amplified/compensated steering depending on the relative signs of tilt and birefringent walk-off. 102 Finally, an aperture (mirror) placed at the crystal output can convert the induced steering into switching (routing), making this configuration an ideal candidate for digital all-optical operations in bulk. In addition, in analogy to Type II SHG in the plane- or guided-wave approximations, the phase shift experienced by the weakest FF component over the strongest one leads to polarization rotation of the outcoming SSW beam, as investigated theoretically in [105-106] in the absence of walk-off.

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Measured x-profiles of the SSW emerging a 1cm-long KTP Fig. 25 crystal for various rotations of the λ /2 waveplate from the position. The input beam was delivered at 1.064µm in 35ps pulses, with a 12µm waist. (After Ref. 103)

Fig. 26 Sketch of an all-optical beam steering device based on vectorial solitary waves in KTP. The output position after 1cm propagation depends on the FF input imbalance. FF input waist is 20µm.

Collisional Interactions Parametric solitary waves tend to propagate without diffraction through gain and phasefront distortion. It is worth emphasizing that, unlike their Kerr counterparts, they are not associated to an actual change in refractive index. This characteristic leads to some peculiar effects when two (or more) SSWs collide, because the corresponding field distributions Φ1 , Φ 2 and Φ 3 , both in amplitude and in phase, can locally interact through interference depending on all parameters defining the SSW at any given point in space. l 0 7 - 1 1 4 Collisional

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interactions of vectorial quadratic solitary waves can, therefore, be controlled by their overall intensity/power, phase, polarization content, SH seed, etc. 113-114 After colliding, the quadratic SSWs can either cross, or coalesce, or repel, or form a bound state. The important feature of vectorial SSWs being the possibility offered by the FF input imbalance to control their evolution, it is rather appealing that the same parameter can also condition the outcome of their interaction, as shown in Fig. 27 for a few cases.113 Relying on relative intensity, rather than phase, in order

Fig. 27 Propagation of vectorial SSW pairs launched at an angle θ in various cases: a) | φ 1 (0)|² = 21.5GW/cm², |φ 2 (0)|² = 18.5GW/cm², θ=4ρω ; b) |φ 1 (0)|²=21GW/cm²,| φ 2 (0)| ² = 19 GWcm², θ =4ρ ω ; c) | φ 1 (0)|² = 20GW/cm², |φ 2 (0) |²=20G W/cm ² , θ = 4 ρ ω ; d) | φ 1 ( 0 ) | ² = 2 0 G W / c m ² = 2 0 G W / c m ² , θ =8ρ ω . x and z axes are in µm and in diffraction lengths, respectively, ∆=0.2 π and the input beams are identically gaussian with waist=20µm.

to regulate the outcome of SSWs’ collisions is definitely a step towards their applications to all-optical switching and processing. Finally, taking advantage of the asymmetry introduced by walk-off and imbalance, even pairs of SSWs launched parallel to each other can be made to collide or diverge depending on their input polarization state, as demonstrated in Fig. 28. Down-conversion switching A Type II phase-matched frequency-degenerate interaction can also be utilized in down-conversion, i.e. feeding energy through an SH input and converting it to the FF. Specifically, if the pump is associated to the Φ 3 beam, a small input in either Φ 1 or Φ 2 will promote the growth of the FF at the expense of the pump, allowing the formation of a self-guided beam. An example of such behaviour, calculated for a KTP-like medium, is shown in Fig. 29. While the pump normally diffracts in the absence of an FF seed (Fig. 29a), the latter will originate an SSW carrying a substantial portion of the input power (Fig. 29b-c). The solitary 367

Propagation of SSWs launched parallel to each other and 60µm Fig. 28 apart, but with different imbalances: a) Left and right beams: b) Left and right beam same as in a). Units are as in Fig. 27.

wave is, then, switched-on by a small FF input polarized along either the "o" or the "e" axes, irrespective of the input relative phase. In addition, for FF seeds substantially weaker than the pump, the undesired situation of both FF input polarizations present (which, based on a Type II three-wave interaction, would be expected to become phase-sensitive) does not alter the phenomenology except for the initial stages of the parametric interplay. 115 Finally, Fig. 30 shows the total FF power carried by the formed SSW versus a wide range of input FF signals and for various mismatches. Clearly the process appears rather "robust" with respect to variations in either parameters. Such results are in agreement with preliminary experimental demonstrations performed at 1.064µm in a 1cm-long KTP crystal. 116

and Evolution in the x-z plane of (d) for Fig. 29 down-conversion SSWs. In (a) and Φ 3 diffracts; in (b) thru (d) a weak FF seed has been introduced (either Φ 1 or Φ 2 ), with peak amplitude five orders of magnitude down with respect to the SH pump. Distances along x and z are in µm and in diffraction lengths, respectively.

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Fig. 30 Total FF power vs. input seed power normalized to the SH pump, after propagation in a 2cm KTP-like material (no walk-off), for various mismatches: ∆=0 (solid line), ∆=0.2 π (short dashes), ∆=-0.2 π (long dashes), ∆ = 2π (dash-dotted line), ∆ =-2π (dotted line).

CONCLUSIONS Today quadratic cascading is to be considered a well-established approach to all-optical processing based on nonlinear optics. Effects and applications of the phase shift intrinsic to sequential up- and down-conversion processes are numerous and diversified, ranging from bulk to guided-wave configurations for both analog and digital operations. The interplay between phase distortion and parametric gain can support the propagation of diffractionless beams, which can also be employed towards applications. The experimental results in known quadratic crystals have demonstrated the practical relevance of these classes of phenomena, paving the way to more efficient and low-power implementations in advanced materials with larger nonlinearities, low absorptions and limited walkoffs. The field, in its infancy from a historical perspective but rather mature in terms of accomplishments and understanding, is still rich of potentials in the framework of both fundamental science and optical engineering.

ACKNOWLEDGMENTS I am indebted to several colleagues and collaborators: G.I. Stegeman and E.W. Van Stryland (CREOL-UCF, Orlando), K. Gallo, G. Leo, and C. Conti (Terza Univ. Rome). Partial support was provided by the Italian Ministry for Research (MURST 40% "Photonic Technologies...") and the National Research Council (grants 96.01844.CT11 and 96.02238.CT07). REFERENCES 1. L.A. Ostrovskii, Self-action of light in crystals, JETP Lett. 5:272 (1967); ZhETF Pis 'ma 5:331 (1967)

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2. F. De Martini, Coherent Raman amplification, Il Nuovo Cimento 51 B:16 (1967); J.P. Coffinet, F. De Martini, Coherent excitation of polaritons in gallium phosphide, Phys. Rev. Lett. 22:60 (1969) 3. T.K. Gustafson, J.-P.E. Taran, P.L. Kelley, R.Y. Chiao, Self-modulation of picosecond pulses in electro-optic crystals, Opt. Commun. 2:17 (1970) 4. E. Yablonovitch, C. Flytzanis, N. Bloembergen, Anisotropic interference of three-wave and double two-wave frequency mixing in GaAs, Phys. Rev. Lett. 29:865 (1972) 5. J.-M.R. Thomas, J.-P.E. Taran, Pulse distortion in mismatched second harmonic generation, Opt. Commun. 4:329 (1972) 6. D.N. Klyshko, B.F. Polkovnikov, Phase modulation and self-modulation of light in three-photon processes, Sov. J. Quant. Electron. 3:324 (1974) 7. S.A. Akhmanov, A.N. Dubovik, S.M. Saltiel, I.V. Tomov, V.G. Tunkin, Nonlinear optical effects of fourth order in the field in a lithium formiate crystal, JETP Lett. 20:117 (1974). 8. G.R. Meredith, Cascading in optical third-harmonic generation by crystalline quartz, Phys. Rev. B 24:5522 (1981); Second-order cascading in third-order nonlinear optical processes, J. Chem. Phys. 77:5863 (1982) 9. R.C. Eckardt, J. Reintjes, Phase matching limitations of high efficiency second harmonic generation, IEEE J. Quant. Electron. 20:1178 (1984) 10. J.T. Manassah, Amplitude and phase of a pulsed second harmonic signal, J. Opt. Soc. Am.. B 4:1235 (1987) 11. P. Qiu, A. Penzkofer, Picosecond third-harmonic light generation in β-BaB 2 O 4 , Appl. Phys. B 45:225 (1988) 12. N.R. Belashenkov, S.V. Gagarskii, M.V. Inochkin, Nonlinear refraction of light on secondharmonic generation, Opt. Spectrosc. 66:806 (1989) 13. H.J. Bakker, P.C.M. Planken, L. Kuipers, A. Lagendijk, Phase modulation in second order nonlinear processes, Phys. Rev. A 42:4085 (1990) 14. M. Zgonik, P. Gunter, Second and third-order optical nonlinearities in ABO3 compounds measured in fast four-wave mixing experiments, Ferroelectrics 126:33 (1992) 15. J.H. Andrews, K.L. Kowalski, K.D. Singer, Pair correlations, cascading, and local-field effects in nonlinear optical susceptibilities, Phys. Rev. A 46:4172 (1992) 16. P. Pliszka, P.P. Banerjee, Self-phase modulation in quadratically nonlinear crystals, J. Mod. Opt, 40:1909 (1993) 17. A.G. Kalocsai, J.W. Haus, Self-modulation effects in quadratically nonlinear materials, Opt. Commun. 97:239 (1993) 18. R. DeSalvo, D.J. Hagan, M. Sheik-Bahae, G. Stegeman, E.W. Van Stryland, H. Vanherzeele, Selffocusing and self-defocusing by cascaded second-order effects in KTP, Opt. Lett. 17:28 (1992) 19. G.I. Stegeman, M. Sheik-Bahae, E. VanStryland, G. Assanto, Large nonlinear phase shifts in second-order nonlinear optical processes, Opt. Lett. 18:13 (1993). 20. D.C. Hutchings, J.S. Aitchison, C.N. Ironside, All-optical switching based on nondegenerate phase shifts from a cascaded second-order nonlinearity, Opt. Lett. 18:10 (1993) 21. G. Assanto, G.I, Stegeman, M. Sheik-Bahae, E. Van Stryland, Coherent interactions for alloptical signal processing via quadratic nonlinearities, IEEE J. Quantum Electron. 31: 673 (1995). 22. Ch. Bosshard, R. Spreiter, M. Zgonik, P. Günter, Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect, Phys. Rev. Lett. 74:2816 (1995) 23. J.B. Khurgin, Y.J. Ding, Resonant cascaded surface-emitting second-harmonic generation: a strong third-order nonlinear process, Opt. Lett. 19:1016 (1994) 24. R. Danielius, P. Di Trapani, A. Dubietis, A. Piskarskas, D. Podenas, G.P. Banfi, Self-diffraction through cascaded second-order frequency-mixing effects in β-barium borate, Opt. Lett. 18:574 (1993) 25. Z.Y. Ou, Observation of nonlinear phase shift in CW harmonic generation, Opt. Commun. 124:430 (1995) 26. E. Fazio, C. Sibilia, F. Senesi, M. Bertolotti, All-optical switching during quasi-collinear secondharmonic generation, Opt. Commun. 127:62 (1996) 27. Z. Wang, D.J. Hagan, E.W. Van Stryland, J. Zyss, P. Vidakovik, W.E. Torruellas, Cascaded second-order effects in N-(4-nitrophenyl)-L-prolinol, in a molecular single crystal, J. Opt. Soc. Am. B 14:76 (1997) 28. P. Vidakovic, D.J. Lovering, J.A. Levenson, J. Webjorn, P.St.J. Russell, Large nonlinear phase shift owing to cascaded χ ( 2 ) in quasi-phase-matched bulk LiNbO3 , Opt. Lett. 22:277 (1997)

370

29. G.I. Stegeman, D.J. Hagan and L. Torner, χ (2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons, J. Opt. Quantum Electron. 28:1691 (1996) 30. J.A. Armstrong, N. Bloembergen, J. Ducuing, P.S. Pershan, Interaction between light waves in a nonlinear dielectric, Phys. Rev. 127:1918 (1962) 31. A. Kobyakov, F. Lederer, Cascading of quadratic nonlinearities: an analytical study, Phys. Rev. A 54:3455 (1996) 32. G.I. Stegeman, R. Schiek, L. Torner, W.E. Torruellas, Y. Baek, D. Baboiu, Z. Wang, E.W. Van Stryland, D.J. Hagan, G. Assanto, Cascading: a promising approach to nonlinear optical phenomena, in Novel Optical Materials and Applications, I.C. Khoo, F. Simoni and C. Umeton eds., John Wiley & Sons, Interscience Div., New York, Ch.2 (1997). 33. S. Saltiel, K. Koynov, I. Buchvarov, Self-induced transparency and self-induced darkening with nonlinear frequency doubling polarization interferometer, Appl. Phys. B 63:371 (1996) 34. S. Saltiel, K. Koynov, I. Buchvarov, Analytical and numerical investigation of opto-optical phase modulation based on coupled second-order nonlinear processes, Bulg. J. Phys. 22:39 (1995); Analytical formulae for optimization of the process of low power phase modulation in a quadratic nonlinear medium, Appl. Phys. B 62:39 (1996) 35. K. Koynov, S. Saltiel, I. Buchvarov, All-optical switching by means of an interferometer with nonlinear frequency doubling mirrors, J. Opt. Soc. Am. B 14:834 (1997) 36. A.L. Belostotsky, A.S. Leonov, A.V. Meleshko, Nonlinear phase change in type II secondharmonic generation under exact phase-matched conditions, Opt. Lett. 19:856 (1994) 37. S. Nitti, H.M. Tan, G.P. Banfi, V. Degiorgio, Induced ‘third-order’ nonlinearity via cascaded second-order effects in organic crystals of MBA-NP, Opt. Commun. 106:263 (1994) 38. M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, All-optical switching by use of cascading of phase-matched sum-frequency-generation and difference-frequency-generation processes in periodically poled LiNbO 3 , Opt. Lett. 22:274 (1997) 39. A.V. Smith, M.S. Bowers, Phase distortions in sum- and difference-frequency mixing in crystals, J. Opt. Soc. Am. B 12:49 (1995) 40. G. Assanto, I. Torelli, Cascading effects in type II second-harmonic generation: applications to all-optical processing, Opt. Commun. 119: 143 (1995) 41. A. Kobyakov, U. Peschel, R. Muschall, G. Assanto, V.P. Torchigin, and F. Lederer, An analytical approach to all-optical modulation by cascading, Opt. Lett. 20: 1686 (1995) 42. A. Kobyakov, U. Peschel, F. Lederer, Vectorial interaction in cascaded quadratic nonlinearities an analytical approach, Opt. Commun. 124:184 (1996) 43. P. Unsbo, Phase conjugation by cascaded second-order nonlinear-optical processes, J. Opt. Soc. Am. B 12:43 (1995) 44. M. Zgonik, P. Günter, Cascading nonlinearities in optical four-wave mixing, J. Opt. Soc. Am. B 13:570 (1996) 45. P.St. J. Russell, All-optical high gain transistor action using second-order nonlinearities, Electron. Lett. 29:1228 (1993) 46. D.J. Hagan, M. Sheik-Bahae, Z. Wang, G. Stegeman, E.W. Van Stryland, G. Assanto, Phase controlled transistor action by cascading of second-order nonlinearities, Opt. Lett. 19: 1305 (1994) 47. G. Assanto, I. Torelli, S. Trillo, All-optical processing by means of vectorial interactions in second-order cascading: novel approaches, Opt. Lett. 19: 1720 (1994); All-optical phase controlled amplitude modulator, Electron. Lett. 30:733 (1994) 48. L. Lefort, A. Barthelemy, All-optical transistor action by polarisation rotation during type-II phase-matched second-harmonic generation, Electron. Lett. 31:910 (1995) 49. G. Assanto, Z. Wang, D.J. Hagan and E.W. VanStryland, All-optical modulation via nonlinear cascading in type II second harmonic generation, Appl. Phys. Lett. 67: 2120 (1995) 50. L. Lefort, A. Barthelemy, Intensity-dependent polarization rotation associated with type-II phase-matched second-harmonic generation: application to self-induced transparency, Opt. Lett. 20:1749 (1995) 51 I. Buchvarov, S. Saltiel, Ch. Iglev, K. Koynov, Intensity dependent change of the polarization state as a result of non-linear phase shift in type II frequency doubling crystals, Opt. Commun. 141:173 (1997) 52. Z. Wang, D.J. Hagan, E.W. VanStryland, G. Assanto, Phase-insensitive, single wavelength, alloptical transistor based on second-order nonlinearities, Electron. Lett. 32: 1135 (1996)

371

53. G. Assanto, D.J. Hagan, G.I. Stegeman, E.W. Van Stryland, W. E. Torruellas, Vectorial quadratic interactions for all-optical signal processing via second-harmonic generation, Optica Applicata V. XXVI: 285 (1996) 54. Z. Wang, D.J. Hagan, E.W. Van Stryland, G. Assanto, A phase-insensitive all-optical transistor using second-order nonlinear media, in Int. Quantum Electronics Conf., 1996 OSA 1996, Tech. Dig. Series, Washington D.C., 264 (1996) 55. G. Assanto, All-optical switching in integrated structures, in: Nonlinear Optical Materials: Principles and Applications, V. DeGiorgio and C. Flytzanis ed., IOS Press, Amsterdam (1995). 56. M.L. Sundheimer, Ch. Bosshard, E.W. Van Stryland, G.I. Stegeman, J.D. Bierlein, Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes, Opt. Lett. 18:1397 (1993) 57. M.L. Sundheimer, A. Villeneuve, G.I. Stegeman, J.D. Bierlein, Cascading nonlinearities in KTP waveguides at communications wavelengths, Electron. Lett. 30:1400 (1994) 58. R. Schiek, M.L. Sundheimer, D.Y. Kim, Y. Baek, G.I. Stegeman, H. Seibert, W. Sohler, Direct measurement of cascaded nonlinearity in lithium niobate channel waveguides, Opt. Lett. 19:1949 (1994) 59. Y. Baek, R. Schiek, and G.I. Stegeman, All-optical switching in a hybrid Mach-Zehnder interferometer as a result of cascaded second-order nonlinearity, Opt. Lett. 20:2168 (1995) 60. Y. Baek, R. Schiek, G. Krijnen, G.I. Stegeman, I. Baumann and W. Sohler, All-optical integrated Mach-Zehnder switching in lithium niobate waveguides due to cascaded nonlinearities, Appl. Phys. Lett. 68:2055 (1996) 61. R. Schiek, Y. Baek, G. Krijnen, G.I. Stegeman, I. Baumann and W. Sohler, All-optical switching in lithium niobate directional couplers with cascaded nonlinearity, Opt. Lett. 21:940 (1997) 62. C.G. Treviño-Palacios, G.I. Stegeman, M.P. De Micheli, P. Baldi, S. Nouh, D.B. Ostrowsky, D. Delacourt, M. Papuchon, Intensity dependent mode competition in second harmonic generation in multimode waveguides, Appl. Phys. Lett. 67:170 (1995) 63. C.G. Treviño-Palacios, G.I. Stegeman, P. Baldi, Spatial nonreciprocity in waveguide secondorder processes, Opt. Lett. 21:1442 (1996) 64. D.Y. Kim, W.E. Torruellas, J. Kang, C. Bosshard, G.I. Stegeman, P. Vidakovic, J. Zyss, W.E. Moerner, R. Twieg, G. Bjorklund, Second-order cascading as the origin of large third-order effects in organic single-crystal-core fibers, Opt. Lett. 19:868 (1994); W.E. Torruellas, G. Krijnen, D.Y. Kim, R. Schiek, G.I. Stegeman, P. Vidakovic, J. Zyss, Cascading nonlinearities in an organic single crystal core fiber: the Cerenkov regime, Opt. Commun. 112:122 (1994) 65. See, for example, Guided Wave Nonlinear Optics, D.B. Ostrowsky and R. Reinisch eds., Kluwer Academic, London (1992) 66. G.I. Stegeman, Overview of nonlinear integrated optics, in Nonlinear Waves in Solid State Physics, A.D. Boardman, M. Bertolotti, T. Twardowski eds., Plenum Press, New York (1990) 67. T. Suhara, H. Nishihara, Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings, IEEE J. Quantum Electron, 26:1265 (1990) 68. T. Gase, W. Karthe, Quasi-phase matched cascaded second order processes in poled organic polymer waveguides, Opt. Commun. 133, 549 (1997) 69. C. Kelaidis, D.C. Hutchings, J.M. Arnold, Asymmetric two-step GaA1As quantum well for cascaded second-order processes, IEEE Trans. on Quantum Electron. 30:2998 (1994) 70. G.J.M. Krijnen, W.E. Torruellas, G.I. Stegeman, P.V. Lambeck, H.J.W.M. Hoekstra, Optimisation of second harmonic generation and nonlinear phase shifts in the Cerenkov regime, IEEE J. Quantum Electron. 32:729 (1996); Cerenkov second-harmonic generation in the strong conversion limit: new effects, Opt. Lett. 15:851 (1996) 71. K. Thyagarajan, M. Vaya, A. Kumar, Coupled mode analysis to study cascading in the QPM Cerenkov regime in waveguides, Opt. Commun. 140:316 (1997) 72. S.J. Lee, J.B. Khurgin, Y.J. Ding, Self-phase modulation by means of resonant cascaded surfaceemitting second-harmonic generation, J. Opt. Soc. Am. B 12:275 (1995) 73. Y. Ueno, G.I. Stegeman, K. Tajima, Large phase shifts due to the χ ( 2 ) cascading nonlinearity in large walk-off and loss regimes in semiconductors and other dispersive materials, Jpn. J. Appl. Phys. 36:L613 (1997) 74. M.L. Bortz, M. Fujimura, M.M. Fejer, Increased acceptance bandwidth for quasi phase-matched second harmonic generation in LiNbO3 waveguides, Electron. Lett. 30:34 (1994) 75. K. Mizuuchi, K. Yamamoto, M. Kato, H. Sato, Broadening of the phase matching bandwith in quasi-phase matched second harmonic generation, IEEE J. Quantum Electron. 30:1596 (1994) 76. H. Tan, G.P. Banfi, A. Tomaselli, Optical frequency mixing through cascaded second-order processes in β-barium borate, Appl. Phys. Lett. 63:2472 (1993)

372

77. O. Gorbounova, Y.J. Ding, J.B. Khurgin, S.J. Lee, A.E. Craig, Optical frequency shifters based on cascaded second-order nonlinear processes, Opt. Lett. 21:558 (1996) 78. K. Gallo, G. Assanto, and G.I. Stegeman, Efficient wavelength shifting over the Erbium amplifier bandwidth via cascaded second order processes in Lithium Niobate waveguides, Appl. Phys. Lett. 71: 1020 (1997) 79, C.G. Treviño-Palacios, Private communication 80. G. Assanto, G.I. Stegeman, M. Sheik-Bahae, E. Van Stryland, All optical switching devices based on large nonlinear phase shifts from second harmonic generation, Appl. Phys. Lett. 62:1323 (1993) 81. C.N. Ironside, J.S. Aitchison, J.M. Arnold, An all optical switch employing the cascaded second-order nonlinear effect, IEEE J. Quantum Electron. 29:2650 (1993) 82. A. Laureti-Palma, S. Trillo, G. Assanto, A. D. Capobianco, C. De Angelis, All-optical switching via quadratic nonlinearities in a Mach-Zehnder device with soliton-like pulses, Nonlinear Optics 16: 303 (1996) 83. R. Schiek, Nonlinear refraction caused by cascaded second-order nonlinearity in optical waveguide structures, J. Opt. Soc. Am. B 10:1848 (1993) 84. R. Schiek, All-optical switching in the directional coupler caused by nonlinear refraction due to cascaded second-order nonlinearity, Opt. Quantum Electron. 26:415 (1994) 85. G. Assanto, A. Laureti-Palma, C. Sibilia, M. Bertolotti, All-Optical Switching via Second Harmonic Generation in a Nonlinearly Asymmetric Directional Coupler, Opt. Commun. 110:599 (1994) 86. M.A. Karpierz, Directional coupling for solitary waves in quadratic nonlinearity, Optica Applicata XXVI:391 (1996) 87. S.J. Lee, J.B. Khurgin, Y.J. Ding, Directional couplers based on cascaded second-order nonlinearities in surface-emitting geometry, Opt. Commun. 139:63 (1997) 88. C. De Angelis, A. Laureti-Palma, G.F. Nalesso, C.G. Someda, On the modelling of nonlinear Quantum Electron. 29:217 (1997) guided-wave optics for all-optical signal processing, Opt. 89. M.G. Masi and G. Assanto: “Prism coupling into second-order nonlinear waveguides”, Opt. Commun., submitted 90. R. Reinisch, M. Nevière, E. Popov, H. Akhouayri, Coupled-mode formalism and linear theory of diffraction for a simplified analysis of second harmonic generation at grating couplers, Opt. Commun. 112:339 (1994) 91. R. Reinisch, E. Popov, M. Nevière, Second-harmonic-generation-induced optical bistability in prism or grating couplers, Opt. Lett. 20:854 (1995) 92. A. De Rossi, C. Conti, G. Assanto, Mode interplay via quadratic cascading in a lithium niobate waveguide for all-optical processing, Opt. Quantum Electron. 29:53 (1997) 93. W. Chen, D.L. Mills, Gap solitons and the nonlinear optical response of superlattices, Phys. Rev. Lett. 58:160 (1987) 94. M. Picciau, G. Leo, and G. Assanto, A versatile bistable gate via quadratic cascading in a Bragg periodic structure, J. Opt. Soc. Am. B 13: 661 (1996) 95. G. Leo, M. Picciau, and G. Assanto, Guided-wave optical bistability through nonlinear cascading in a phase-matched distributed reflector, Electron. Lett. 31: 1661 (1995) 96. C. Conti, S. Trillo, G. Assanto, Bloch function approach for parametric gap solitons, Opt. Lett. 22: 445 (1997) 97. C. Conti, S. Trillo, G. Assanto, Doubly resonant Bragg simultons via second-harmonic generation, Phys. Rev. Lett. 78: 2341 (1997) 98. T. Peschel, U. Peschel, F. Lederer, B.A. Malomed, Solitary waves in Bragg gratings with a quadratic nonlinearity, Phys. Rev. E 55:4730 (1997) 99. R. Schiek, Soliton-like propagation and second harmonic generation in waveguides with second-order optical nonlinearities, AEÜ Int. J. Electron. Commun. 2:77 (1997) 100. C. Conti, G. Assanto, S. Trillo, Excitation of self-transparency Bragg solitons in quadratic media, Opt. Lett. 22:1350 (1997) 101. W.E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G.I. Stegeman, L. Torner, C. R. Menyuk, Observation of two-dimensional spatial solitary waves in a quadratic medium, Phys. Rev. Lett. 74:5036 (1995) 102. G. Leo, G. Assanto, and W. E. Torruellas, Beam pointing control with spatial solitary waves in quadratic nonlinear media, Opt. Commun. 134:223 (1997) 103. W.E. Torruellas, G. Assanto, B.L. Lawrence, R.A Fuerst, G.I. Stegeman, All-optical switching by spatial walk-off compensation and solitary-wave locking, Appl. Phys. Lett. 68:1449 (1996)

373

104. W.E. Torruellas, Z. Wang, L. Torner, G.I. Stegeman, Observation of mutual trapping and dragging of two-dimensional spatial solitary waves in a quadratic medium,Opt. Lett. 20:1950 (1995) 105. R. Mollame, S. Trillo, G. Assanto, Soliton polarization rotation and switching in type II secondharmonic generation, Opt. Lett. 21:1969 (1996) 106. A.D. Capobianco, B. Costantini, C. De Angelis, A. Laureti-Palma, G.F. Nalesso, Space and/or polarization diversity multiplexing with type II second harmonic generation, IEEE Photonics Tech. Lett. 9:602 (1997) 107. M.J. Werner, P.D. Drummond, Simulton solutions for the parametric amplifier, J. Opt. Soc. Am. B 10:2390 (1993) 108. D. Baboiu, G.I. Stegeman, L. Torner, Interaction of one-dimensional bright solitary waves in quadratic media, Opt. Lett. 20:2282 (1995) 109. A.V. Buryak, Y.S. Kivshar, V.V. Steblina, Self-trapping of light beams and parametric solitons in diffractive quadratic media, Phys. Rev. A 52:1670 (1995) 110. C. Etrich, U. Peschel, F. Lederer, B. Malomed, Collision of solitary waves in media with a second-order nonlinearity, Phys. Rev. A 52:3444 (1995) 111. M. Haelterman, S. Trillo, P. Ferro, Multiple soliton bound states and symmetry breaking in quadratic media, Opt. Lett. 22:84 (1997). 112. C. Balsev Clausen, P.L. Christiansen, L. Torner, Perturbative approach to the interaction of solitons in quadratic nonlinear media, Opt. Commun. 136:185 (1997) 113. G. Leo, G. Assanto, W.E. Torruellas, Intensity controlled interactions between vectorial spatial solitary waves in quadratic nonlinear media, Opt. Lett. 22:7 (1997); Collisional interactions of vectorial spatial solitary waves in type II frequency doubling media, J. Opt. Soc. Am. B, 14: in press (1997) 114. B. Costantini, C. De Angelis, A. Barthelemy, A. Laureti Palma, G. Assanto, Polarization multiplexed χ (2) solitary waves interactions, Opt. Lett. 22:1376 (1997) 115. G. Leo, G. Assanto, Phase- and polarization-insensitive all-optical switching by self-guiding in quadratic media, Opt. Lett., 22: 1391 (1997) 116. M.T.G. Canva, R.A. Fuerst, S. Baboiu, G.I. Stegeman, and G. Assanto, Quadratic spatial soliton generation by seeded down conversion of a strong pump beam, Opt. Lett., 22: (22) in press (1997)

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NONLINEAR OPTICAL FREQUENCY CONVERSION: MATERIAL REQUIREMENTS, ENGINEERED MATERIALS, AND QUASI-PHASEMATCHING

M. M. Fejer Edward L. Ginzton Laboratory Stanford University Stanford, CA 94305 [email protected]

1 . INTRODUCTION Since the first demonstration in 1961, quadratic nonlinear optical interactions have developed into widely used tools for the generation of coherent radiation. While more than three decades old, this field is still rapidly developing today. New generations of solid-state pump lasers and nonlinear optical materials have led to a renaissance in quadratic nonlinear optics over the past decade, leading both to practical sources of coherent radiation at wavelengths inaccessible to convenient laser transitions and to applications in areas such as signal processing, quantum optics, and cascade nonlinearities. The interactions considered here are those due to the lowest order nonlinear susceptibility, the quadratic polarization response to applied fields, given by P = ε0d : E 2. In the presence of applied fields at frequencies ω 1 and ω 2, the polarization response, and hence the generated field, contains components at the sum ( ω1 + ω 2 ), difference (ω 1 – ω2 ), and harmonic (2ω1 and 2 ω 2 ) frequencies, referred to as sum frequency generation (SFG), difference frequency generation (DFG), and second harmonic generation (SHG), respectively. Other important interactions include optical parametric amplification (OPA), in which a long wavelength signal field is amplified at the expense of a higher frequency pump

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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field, and optical parametric oscillation (OPO), where an OPA in a cavity with losses smaller the parametric gain generates outputs built up from the noise in the presence of only a high frequency pump wave. The first observation of a quadratic nonlinear interaction was by Franken in 1961 (Franken 1961), who generated the 347 nm second harmonic of a 694 nm ruby laser in a piece of crystal quartz. Despite a 3 J input pulse energy, the harmonic output was only about 10 nJ, a result of the small magnitude of the nonlinear susceptibility and the lack of phase velocity matching between the fundamental and harmonic waves, as was already understood in this first paper. Franken pointed out that the ratio of the nonlinear to the linear polarization should be on the order of the ratio of the applied optical electric field to the interatomic field in the solid, and hence be orders of magnitude smaller for reasonable applied fields. He further considered that the intensity of the generated field should be proportional to the square of the “volume of coherence”, and hence proportional to [λ /(n2 – n 1 )] 2 , where λ is the fundamental wavelength and n 2 and n 1 are the refractive indices at the second harmonic and fundamental wavelengths, respectively. He added that “the lateral extent of the coherence volume ... [is] determined by the coherence characteristics of the pump laser” and that “a maser of the gas discharge type is clearly more favorable in this respect than the ruby device.” While the evolution of laser technology has reversed the roles of solid state vs gas discharge lasers in this respect, it is remarkable that the major challenges of the following thirty years of research in nonlinear optical devices were already described in this two page letter: materials with larger nonlinear susceptibilities, materials allowing phase velocity matching, and pump lasers of high power and high spatial coherence. Subsequent progress was rapid. In 1962 the seminal paper of Bloembergen and coworkers presented a perturbation expansion for the nonlinear susceptibilities, quantitative analysis of the propagation effects associated with phase-matched and phase-mismatched interactions, and presented several schemes for accomplishing phase-velocity matching, including the quasi-phasematching scheme that will be the subject of the second section of this chaper. (Armstrong 1962) The most widely used phasematching scheme, based on orthogonally polarized waves in birefringent crystals, was not given in Bloembergen’s paper, but also appeared in 1962, independently in (Giordmaine 1962) and (Maker 1962). All the basic quadratic nonlinear phenomena, including SHG, SFG, DFG, OPA, and OPO were demonstrated by 1965. Since that time, progress in the field has been driven largely by improved nonlinear materials and pump lasers. Notable among current developments in pump lasers are high power diode lasers (1 W in a single transverse mode), diode-pumped solid-state lasers (1 - 10 W single-mode power available commercially at several near-IR wavelengths), rare-earth-doped fiber lasers (compact 30 W 1 µm lasers in Yb:glass, compact fs sources in Er:glass), and tunable/ultrafast lasers such as Ti:sapphire and Cr::LiSAF (fs pulses, TW peak powers, 400 nm tuning ranges). A good source of information on the current state of solid state laser technology is the series of Ref. (Pollock 1997).

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Developments in nonlinear optical materials and their applications are the subject of the remainder of this chapter, which is divided into two main components. The first deals with the basics of quadratic nonlinear optical interactions, with emphasis on issues relating to the necessary properties of nonlinear optical materials. The second addresses two closely related topics of growing practical importance, engineered nonlinear materials and quasiphasematching.

2. BASICS OF QUADRATIC NONLINEAR FREQUENCY CONVERSION

2.1 Plane Wave SHG In order to discuss the important properties for practical nonlinear materials, it is useful to first review the basics of nonlinear frequency conversion.(Shen 1984) Let us consider second harmonic generation as a prototypical interaction. Defining the envelope fields Ej (r) by

, where E j( r, t)is the total space and time dependent field at frequency ω j , and k j = ωj nj /c, the evolution of the second harmonic and fundamental fields is given by (1)

(2) where the k-vector mismatch is ∆k = k 2 – 2 k1 , and d eff is the effective nonlinear coefficient, a combination of components of the nonlinear susceptibility tensor discussed in section 2.4. In the low conversion limit, where depletion of the pump can be neglected, the solution for the second harmonic field in a crystal of length L is (3) Note that the integral takes the form of a Fourier transform of the nonlinear coefficient, a point that will be important in later discussions. If we further assume that the nonlinear coefficient is uniform throughout the crystal, we arrive at an expression for the efficiency η, (4) where I is the intensity, the nonlinear drive η0 = C 2L2 I1 is expresssed in terms of the material constant and the dephasing δ = ∆ kL/2. In the "phasematched" case, where δ = 0, the efficiency reduces to (5) so we see that the efficiency scales quadratically with the length of the crystal and the nonlinear susceptibility, and linearly with the input intensity. If we relax the assumption of 377

low conversion, but retain the assumption of phasematching, it can be shown that the efficiency obtained from the exact solution to Eqs. (1) and (2) is given by (6) In both cases, the efficiency depends only on the nonlinear drive; the properties of the material enter only through the parameter C2 . The simplest figure of merit for a nonlinear material is then d 2/n 3 . This is also one of the most widely used figures, but is not always the most relevant for practical applications. The tight grouping of materials used in commercial products in the low (< 1 pm/V), e.g. KDP, LBO, to moderate (< 5 pm/V), e.g. KTP, BBO, LiNbO 3, range of effective nonlinear susceptibilities, suggests that other considerations may dominate the actual utility of a nonlinear material. Elucidating these issues, which are dominated by the linear optical properties of the nonlinear materials, is the subject of section 3. For non-phasematched (∆ k ≠ 0) SHG in the undepleted pump limit, the factor sinc 2( δ) in Eq. (4) shows that the efficiency is sharply peaked around ∆ k = 0, falling to half of its peak value for δ = ±0.443 π or equivalently ∆ k = ±0.886 π / L. This behavior is a result of the periodic alternation of the direction of energy flow between the fundamental and second harmonic waves as their relative phase drifts due to the difference between their phase velocities. The distance for a half cycle of this oscillation is known as the coherence length, (7) For typical materials, the dispersion of the refractive index leads to coherence lengths in the range of several microns to several tens of microns for visble SHG. In order to obtain efficient SHG, the coherence length must exceed the length of the crystal. For 1 µm SHG in a 1 cm long crystal, this condition corresponds to matching the refractive indexes at the fundamental and the harmonic to better than 2x10-5 . Satisfying this strict criterion dominates much of the practical aspects of nonlinear frequency conversion. Before discussing the methods used to accomplish phasematching, it is worth discussing the solutions to Eqs. (1) and (2) in the presence of simultaneous high conversion and phasemismatch. The exact solutions can be stated in terms of Jacobian elliptic functions,(Armstrong 1962) but this level of detail is not necessary for the present discussion. One important point for subsequent analyses is that the central lobe of the phasematching curve narrows dramatically with increasing nonlinear drive. An approximate expression (valid for η 0 >> 1) for the first zero in the conversion efficiency (which occurs at δ = π in the low conversion limit) is (Eimerl 1987) (8) This narrowing of the phasematching peak, one manifestation of cascade nonlinearity, has a profound effect on the efficiency of high peak power SHG. Defining an effective an approximate expression (valid for η0 >> 1) phasemismatch for the efficiency within the first peak of the phasematching curve is (9) 378

Note that because of the increase in δeff with increasing η 0 at fixed δ, the efficiency is not a monotonically increasing increasing function of η0 unless δ = 0. For any finite value of δ, the efficiency increases with η0 (e.g. with increasing laser power) up to a maximum (always smaller than 1), and then decreases with further increase in the laser power. This point will be essential in the discussion of high peak power SHG in section 3.1. 2.2 Birefringent Media Phase velocity matching requires compensation for dispersion in the refractive index, which for typical materials leads to a difference of 0.01 – 0.1 between n2 and n 1 . The common method for meeting this requirement takes advantage of the difference in the refractive indexes for orthogonally polarized waves in birefringent crystals. As the effects of this birefringent phasematching are crucial to the understanding of mechanisms that practically limit the efficiency of nonlinear interactions, we briefly summarze the key points of propagation in birefringent media. In lossless nongyrotropic media, the dielectric tensor can be diagonalized with respect to an orthogonal coordinate system; in this principal axis representation, the three independent components of the dieletric tensor, εx x, ε y y , and ε zz , are real. In the simplest “uniaxial birefringence” case, i.e. for crystals of symmetry higher than orthorhombic, two of the principal values are degenerate (conventionally εxx = ε yy ) and are known as the ordinary component, ε o . The third component, εzz , is known as the extraordinary component, εe, and the z axis is known as the optical axis. For εe < εo , the crystal is called negatively birefringent, for εe > ε o positively birefringent. In a uniaxially birefringent crystal, light polarized perpendicular to optical axis propagates with a phase velocity c/n o , independent of propagation direction, where the ordinary refractive index no = εo1/2 . For light polarized in a plane containing the optical axis, the phase velocity is c/n e( θ ), and the extraordinary index n e (θ ) depends on the direction of propagation. For a given crystal, n e is a function only of the angle θ between the k-vector and the optical axis. The extraordinary refractive index varies continuously from no for θ = 0°, to n o = ε e1/2 for θ = 90°, (Yariv 1984) (10) where the mean index

= (n e + n o )/2, and δ n = ( no – ne )/2. The approximate form,

accurate to first order in the birefringence, is convenient for estimating bandwidths, but is not sufficiently accurate for the sensitive calculation of the phasematching angle itself. The propagation of an ordinary wave is essentially identical to that of a wave propagating in an isotropic medium, but the extraordinary wave differs in several ways. For our purposes, the most important of these is that the phase and group velocities are not parallel to each other. The “walkoff angle” ρ between the phase velocity, parallel to the wave vector k, and the group velocity, parallel to the Poynting vector S, also depends on θ . The exact dependence is straightforward (Yariv 1984); a useful approximate form is 379

ρ ≈ (2 δ n / ) sin(2 θ ). This expression, exact at θ = 0°, 45°, and 90°, and correct to second order in δ n/ elsewhere, shows that the walkoff angle vanishes for propagation along a principal axis, and takes its maximum value at θ = 45°. Note that this is also the direction for

which the extraordinary refractive index varies most rapidly with angle. For crystals of moderate birefringence δ n / ≈ 0.02, the maximum value of ρ is typically 4°, which will be seen to significantly affect the SHG efficiency. For crystals of orthorhombic and lower symmetry, all three principal values of the dielectric tensor are different, and the propagation effects for a general direction become substantially more complicated than in the uniaxial case. For propagation normal to one of the principal axes (in practice the situation in most cases of interest), the behavior is similar to the uniaxial case, with the wave polarized along the principal axis taking the role of the ordinary wave, and the wave polarized in the plane normal to the principal axis taking the role of the extraordinary wave. 2.3 Birefringent Phasematching In the simplest birefringent phasematching scheme, the pump is polarized as either an ordinary or an extraordinary wave, and the propagation direction is chosen to satisfy 2 k1 = k2 . For this "Type I" interaction, the phasematching condition can be restated as n 2 = n 1. For a negatively birefringent crystal, the second harmonic wave is polarized for the lower (extraordinary) index, and we have n e2 ( θ) = n o1. For a positvely birefringent crystal, the roles of the ordinary and extraordinary waves are of course reversed. In the slightly more complex “Type II” scheme, the pump is polarized to excite both an ordinary and an extraordinary wave. For example, in a neatively birefringent crystal,

or,

equivalently, It is clear that if Type II phasematching is possible in a given crystal, Type I is also. The reasons that Type II may be of interest are better use of the crystal propeties, e.g. a larger de f f or larger acceptance bandwidths, or technical advantages such as, for optical parametric oscillators, easier separation of the orthogonally polarized output beams, or better controlled tuning behavior near degeneracy. 2.4 Effective Nonlinear Coefficient The nonlinear susceptibility is a third rank tensor, whose form depends on the point group symmetry of the crystal.(Yariv 1984) Which combination of its components are pertinent to evaluating the effective nonlinear deff appearing in Eqs.(1) and (2) depends on the choice choice of polarization and propagation directions. Calculation of deff involves projecting the third rank tensor onto the unit vectors parallel to the three electric fields involved in the interaction. The details are straightforward, but tedious. Tabulations for the different possible phasematching schemes in each point group can be found, for example, in (Kurtz 1975). Note that since the k-vector mismatch depends on the even-rank dielectric tensor, it is even in θ ( ∆k(– θ) = ∆ k(θ )), while the effective nonlinear coefficient, which 380

depends on the odd-rank nonlinear susceptibility tensor, is not, (d e f f (– θ) ≠ de f f ( θ)). Thus, care must be exercised in choosing the quadrant of propagation to ensure not only phasematching, but also the largest possible de f f . 2.5 Phasematching Bandwidths The phasematching requirement, | ∆ k| ≤ 0.886 π / L sets several acceptance bandwidths for parameters such as propagation angle, temperature, and spectral bandwidth. If ∆k then the first term of a Taylor depends on a parameter ξ, and we define expansion around the phasematched point ξ = ξ 0 yields ∆ k(ξ) = β ξ δξ, where δξ ≡ ξ – ξ 0 . The FWHM acceptance bandwidth is then (11) Note that the bandwidth is inversely proportional to the length of the crystal, and depends on the material properties through β ξ . In the special case where the first derivative vanishes, the 2

2

2

second term in the Taylor series yields ∆ k ( ξ ) = (β ξ ξ /2) δξ , where β ξξ ≡ ∂ ∆ k/∂ ξ . The FWHM accepatance bandwidth is then (12) For such cases the bandwidth decreases only with the square root of the crystal length, substantially increasing the bandwidth compared to the critical case, a condition often termed “noncritical phasematching”. In the remainder of the discussion of bandwidths, we will round 0.886 ≈ 1, and 1.776 ≈ 2 to simplify the resulting formulae. The most important of the acceptance bandwidths is the angular acceptance. From the approximate form of Eq. (11), and Eq. (10) we can evaluate β θ . We find (13) Eq. (13) applies to Type I phasematching in a negative uniaxial crystal. Similar relations can be found for other phasematching cases. We see that the bandwidth is inversely proportional to crystal length, and is smallest for θ = 45 °. For typical cases, the acceptance bandwidths, normalized to the length of the crystal, are in the range 0.1 – 1 mrad-cm. The divergence in the bandwidth for θ → 0 reflects the approach to noncritical phasematching. With Eqs. (10) and (12), we find the bandwidth for θ = 0 (14) Typical bandwidths in this case, normalized to the square root of crystal length, are 10 – 30 mrad-cm 1/2, a significant practical advantage over the critically phasematched case. In order to take advantage of the attractive features of noncritical phasematching, a continuously tunable parameter other than angle is necessary to meet the phasematching condition. In general, this parameter is temperature. Following the same method as for angular acceptance, we find 381

(15) For typical thermooptic coefficients, the bandwidth is in the range of 0.1 – 10 K-cm. The temperature acceptance bandwidth can be especially important in high average power devices where absorbed optical power results in significant self-heating of the crystal. For tunable, pulsed, or spectrally noisy lasers, the wavelength acceptance bandwidth becomes an important parameter. Again following the same method as for the spectral acceptance, we find (16) where the second form uses the definition of the group index Typical wavelength bandwidths are 0.1 – 10 nm-cm. It is interesting to note that while ordinary phasematching requires only matching of the refractive indices, if, in addition, the group indices are matched at the fundamental and second harmonic frequencies, the denominator in Eq. (16) vanishes, and the interaction is wavelength noncritical. Many materials have at least one such wavelength pair. We will see the importance of this condition in section 3.2 on ultrafast nonlinear interactions. 2.6 Focused Interactions According to Eq. (4), the SHG efficiency is proportional to the pump intensity, suggesting that for a given available input power, focusing the input beam to a small spot will increase the efficiency. While this is true up to a point, the efficiency does not increase monotonically with reducing spot size; there exists an optimum spot size. This limit can be understood by again referring to Eq. (4). For a sufficiently tightly focused beam, the 2 diffraction length will be smaller than the length of the crystal, and the L scaling of the efficiency will no longer apply. The optimum spot size comes from this tradeoff between a tight focus for for high intensity and a loose focus for a long effective interaction length. For our discussion of focused SHG, it is necessary to first establish some basic properties of gaussian beam propagation. For a beam with a minimum waist w0 , the beam waist propagates according to

where the characteristic diffraction

length, known as the Rayleigh length is field,

the beam radius is nearly constant, while in the far field,

In the near the radius

grows essentially linearly with distance. The far field can thus be characterized by a diffraction angle, θ D = λ / πnw 0 . Note that the Rayleigh length decreases quadratically, and the diffraction angle increases linearly, with the spot size. For z R >> L, diffraction can be neglected, and the efficiency, defined here as the ratio of SH power out to fundamental power in, can be obtained by averaging Eq. (4) across the 2 profile of the gaussian beam. We find an effective area for the beam of πw 0 , and so for the near field efficiency, (17) 382

Detailed calculations for the efficiency of focused SHG (Boyd 1968) are beyond the scope of this review. Here we confine ourselves to motivating some basic results necessary for the remainder of our discussion. Consider first the case of noncritical phasematching. A reasonable guess for the optimal spot size is to choose it such that the entire crystal just fits in 2

the near field of the gaussian beam, i.e. 2z R = L, or equivalently, πw0 = L λ /2n, leading to an expression for the “confocal” efficiency (18) has units [%/W-cm]. Note that the pertinent

where the coefficient 2

material figure of merit is now d e2f f / n , and the efficiency scales with the inverse cube of the wavelength, rather than inverse square as in the plane wave case. In a material with deff = 5pm/V, and n=2, γ n c = 0.4 %/W - cm for 1 µ m SHG. For critical phasematching, an additional consideration comes into play, the Poynting vector walkoff that leads to non-parallel propagation of the fundamental and SH beams. A characterisitc distance for the beams to walk off each other is the aperture length, l a = w 0 / ρ . The optimum focusing remains close to confocal, but for L/ la > 1 the efficiency rapidly falls below the result in Eq. (17). The decrease in efficiency is conveniently described in terms of 1/2 a parameter B ≡ ρ (πnL/ 2 λ) , which is the ratio L/l a evaluated for a confocally focused beam. For B = 1, the efficiency is 0.58 η nc ; for B > 1.5 the efficiency for critical phasematching is accurately approximated by noncritically phasematched SHG is

In this latter limit, the efficiency for (19)

where the coefficient that the pertinent material figure of merit for this case is d

2 eff

has units [%/W-cm1/2 ]. Note 5/2 / n ρ, and that the efficiency

-5/2

scales with λ . For a material with the same parameters as the example for noncritical 1/2 phasematching, but with a walkoff angle ρ = 2 °, γ cr = 0.04 %/W-cm , an order of magnitude lower than in the noncritical case. Quantitative calculation of the efficiency for the general case of focused SHG can be accomplished by calculating the nonlinear polarization distribution for a given pump field, and treating it as a source term in an appropriate Green’s function for the generated field in an anisotropic medium. The results of this calculation are available in a readily used general form in Ref. (Boyd 1968). Their results for the general case are conveniently summarized as (20) where the dimensionless function h contains all the information on focusing and birefringence. Among the key results are that for noncritical phasematching and confocal focusing ( L/ zR = 2, B = 0) h = 0.8, and that the actual optimum efficiency occurs for tighter focusing ( L/ z R = 5.7, B = 0) h = 1.07, though this tighter focusing is often not used for practical reasons such as crystal damage effects.

383

3. MATERIALS ISSUES IN NONLINEAR FREQUENCY CONVERSION The basic results of the previous sections can be combined to elucidate the combinations of material parameters of particular importance in particular types of nonlinear interactions. The following sections several of these cases. It should be clear that these categories are not orthogonal, e.g. a device can operate simultaneously at high peak and high average power, so that more than one set of these criteria may apply, and in fact one may compound the difficulties presented by the other. 3.1 High Peak Power SHG Q-switched lasers are attractive pump sources for nonlinear frequency conversion, because, for a given average power, the increase in peak power leads to higher conversion efficiency. However, optimization of SHG for high peak power pulses involves subtleties, often overlooked, which lead to counterintuitive design rules. The origin of the complexity is in the narrowing of the phase matching acceptance bandwidths that accompanies high conversion efficiency SHG. The important quantity for pulsed SHG is the energy conversion efficiency, η E,defined as the ratio of the energy in the second harmonic output pulse to the energy input in the fundamental pulse. With nanosecond and longer pulses, for which dispersion is generally negligible, η E can be calculated by averaging the instantaneous conversion over the entire temporal pulse shape. For a pump that is Gaussian in space, one must also average over the radial intensity distribution. For a pulse that is gaussian in space and time, the ratio of ηE to the conversion efficiency at the peak of the pulse η p , is 1/2 in the undepleted pump limit. For example, for an energy conversion efficiency of 5%, the peak conversion must be 14%. This difference does not have serious implications in the low conversion limit, but as the conversion increases, the peak of the pulse is driven deeply into the saturated regime before the wings of the pulse reach significant conversion. Consider the following table showing for several values of the energy conversion efficiency, the corresponding peak conversion efficiency and the nonlinear drive (defined in Eq. (4)): ηE 25% 50% 85% 5% 75% ηP 14% 56% 87% 99.0% 99.9% η0 0.15 0.94 2.8 17.2 9.0 Note that for relatively modest energy conversion of 75%, the necessary peak conversion is already 99%, and the nonlinear drive is 9. With Eq. (8) we see that the phasematching peak has narrowed by approximately a factor of 3 in this case; for 85% energy efficiency, the peak is narrowed by a factor of 6. To understand the limits imposed by this intensity dependent narrowing of the angular acceptance, we must establish one further result. For a given input power, the nonlinear 2 2 drive, according to Eq. (4), is η 0 = C L P / πw2 , where w is the radius of the beam, while the dephasing due to the angular divergence of the beam is (Eq. (13)) δ = (L/2)β θ δ θ = (L/2) β θ λ / π nw, where the second form follows from the usual result for 384

the diffraction angle of a finite beam. Thus, the ratio η0 /δ2 is independent of both the length of the crystal and the focusing of the pump beam. This ratio is often termed the normalized 2 brightness, Φ = η 0 / δ . The importance of the normalized brightness is its connection to the maximum possible SHG conversion efficiency. From the solution for depleted pump SHG, (the exact Jacobian elliptic function solution or the approximate Eq. (9)) it can be shown that the maximum possible efficiency is a function only of Φ, given by (Eimerl 1987) (Beausoleil 1992) (21) and that this maximum efficiency occurs for an optimum value of the nonlinear drive (22) where K is the complete elliptic integral, and the approximate form holds for ηm < 0.95. It is important to note that for a given laser and a given nonlinear material, i.e. for a given Φ, η m from Eq. (22) is the highest efficiency possible. If this value is inadequate, then either a more powerful laser or a better nonlinear material is required. Given the optimum drive, we can calculate the optimum aspect ratio, for the beam, (23) Note that focusing more tightly than this condition for a given crystal length, or using a longer crystal than this for a given spot size will actually reduce the efficiency, even if the corresponding beam divergence does not exceed the conventional angular acceptance criterion. This is a nonlinear result, depending on the narrowing of the acceptance bandwidth at high conversion. It is conventional to present Φ in the form Φ = 4 πP/P t h , where the “threshold power” P t h is a material property defined by (24) The brightness must exceed 2, or equivalently the laser power must exceed Pt h / 2π in order to obtain 50% efficiency. Thus, the pertinent material figure of merit for high peak power SHG is d /n 2 β θ . This figure of merit diverges for a noncritically phasematched interaction; a similar analysis shows that in this case, any desired conversion efficiency can be obtained if 2

sufficiently long crystals are available.(Eimerl 1987) While this analysis was based on pulses uniform in space and time, it has been found that 2 they accurately predict the performance of real gaussian pulses if 1/e pulse lengths and beam radii are used to relate energy to power and power to intensity, respectively. 3.2 Ultrafast SHG For SHG with ultrafast pulses, the essential quantities are the energy conversion efficiency η E , defined as the ratio of the SH pulse energy out to the fundamental pulse energy in, and the distortion of the pulse shape/duration in the SHG process. It is convenient to

385

describe the efficiency (in the undepleted pump limit) by modifying Eq. (20) to (Arbore 1997) (25) where e 1 is the energy of the fundamental pulse, τ is the FWHM wifth of the pulse, g is a dimensionless function that accounts for the time dependence of the conversion efficiency and for spectral bandwidth effects. This decoupling of the spatial (h) and temporal (g) dependences is appropriate if the interaction is in the near-field limit or in the “quasi-static” (SHG acceptance bandwidth much greater than pulse bandwidth) limit. In the quasi-static regime, the required extension of the CW analysis is straightforward. At each instant in time the efficiency can be calculated according to the CW expression evaluated at the instantaneous intensity. In this case, (26)

where the normalized time is

= t/τ. For a gaussian pump pulse, g = 0.664; for a sech²

pulse, g = 0.589. If the pulse bandwidth approaches the acceptance bandwidth, the analysis becomes more complex. If we choose the crystal length so that the SHG acceptance bandwidth just equals the pulse bandwidth, we have with Eq. (16) (27) where ∆λ is the spectral FWHM of the pulse, and the group index mismatch is ∆ n g = n g2 – ng 2 . We can approximate the energy conversion efficiency in various cases by setting the crystal length L = l B W , and averaging the resulting instantaneous power conversion over the pulse shape. If the pulse is transform limited, and hence ∆λ is directly related to the pulse length , τ some general conclusions may be reached. We can write l BW in terms of τ (28) where q is a dimensionless factor dependent on the pulse shape, taking the value 1 for a gaussian and 1.4 for sech² . For typical materials, lτ is several mm for 100 fs near-IR pulses. It is worth noting that this length l τ can also be interpreted, in the time domain, as the length over which fundamental and SH pulses separate by a time equal to the pulse duration, due to their unequal group velocities. With L in Eq. (25) replaced by l τ from Eq.(28), the quasistatic approximation for the energy efficiency becomes (29) Note that the quantity in parentheses in Eq. (29) has dimensions of [%/nJ], i.e. the efficiency is a function only of the pulse energy, and not the pulse length. Typical values in the near-IR 386

are several %/nJ, unless a near degeneracy exists for the group velocities at the fundamental and SH. The material figure of merit for ultrafast SHG is seen to be . Evaluation of g without making the quasi-static assumption shows that the simple estimate of ηE for L = l τ from Eq. (29) overestimates the true value by a factor of 2 for sech² pulses, an indication that bandwidth limitations are coming into effect in this regime. These conclusions hold as long as the length is limited by bandwidth limitations, rather than by spatial walkoff effects, i.e. as long as l τ < l a . This condition always holds for noncritical phasematching, but may be violated for critical phasematching, especially for long pulses. In this case, the energy efficiency scales as (30) and is inversely proportional to the square root of the pulse length. Details are beyond the scope of this discussion.(Arbore 1997) It may be apparent from the previous discussion that a close analogy exists between time and space domain propagation effects in SHG, i.e. between group velocity mismatch and Poynting vector walkoff, and between diffraction and dispersion. This analogy can be made quantitative, so that analyses made in one domain are immediately transferrable to the other.(Akhmanov 1975)

3.3 SHG in Lossy Materials To this point the materials effects that we have been considering are connected to limitations imposed by phasematching. There are also adverse effects associated with optical losses. We consider here only linear loss mechanisms, which can be described by I (z) = I 0 exp(– κz), where the extinction coefficient κ is independent of the intensity I. The extinction generally is composed of two components, κ = α + σ, where α is the absorption, and σ is the scattering. The solution for plane-wave SHG in the presence of loss is straightforward,(Chemla 1987) and shows that the efficiency is significantly reduced compared to the lossless case when κ L > 1. For a crystal of the optimum length, , the efficiency is given, to a good approximation, by replacing L in Eq. (5) by When one of the extinction coefficients (κ > ) is much larger than the other, the optimum length approaches infinity, but the asymptotic limit of he efficiency is not significantly larger than that obtained for a crystal whose length is 1/κ > . The combination of material properties relevant for SHG in lossy media is then Similarly, for focused SHG, one can to a good approximation replace L in Eq. (18) by lk , in which case the pertinent combination of material properties is The importance of this ratio of nonlinear susceptibility to loss is the reason why the huge enhancements associated with resonant nonlinearities have not been used to produce efficient nonlinear devices. It is important to note that the analysis presented in this section applies only if it is possible to fabricate a crystal whose length is on the order of the extinction length. While this can be only several microns for resonant media, for low-loss dielectrics this length can easily 387

be several meters, impossible to obtain in any practical sense. However, it is possible by embedding the nonlinear crystal in a Fabry-Perot cavity to achieve efficiencies approaching those of a crystal of length lκ . 3.4 Resonator SHG Devices The single-pass SHG conversion efficiencies in the range of several tenths of one percent per watt available in typical nonlinear crystals are too low for use with most CW lasers. The high circulating fields available either inside a laser cavity or in a low-loss Fabry-Perot cavity can be used to significantly enhance the SH conversion efficiency. Because the field enhancement is inversely proportional to the losses in the crystal, these resonator applications depend on the availability of low-loss nonlinear materials. Consider externally-resonant SHG,(Kozlovsky 1988) (Schiller 1993) with a nonlinear crystal inside a ring Fabry-Perot cavity resonant at the fundamental frequency, with input mirror transmission T, and all other mirrors high reflectors for the fundamental and transmitting at the SH. We take the total cavity losses, exclusive of extinction in the nonlinear crystal, to be Aƒ . The matched resonator condition, choosing the input mirror transmission to be equal to the total cavity losses, i.e. T = κ 1 + Aƒ eliminates any reflection from the cavity so that all the input power is coupled into the cavity. For this condition, the ratio of the circulating power in the cavity to the input power is Pc / P1 = 1/( κ1 L + Aƒ ), and the efficiency is enhanced by the square of this factor. The optimum length for the crystal comes about from the tradeoff between increasing the single-pass efficiency with L at the expense of increased loss and decreased circulating power. Inserting Pc into Eq. (18) for noncritically phasematched SHG, and maximizing with respect to L, we find the optimum length is L opt = Aƒ / κ1 . For this choice, the efficiency is the same as a single pass device of length 1 / 4 κ1Aƒ , i.e. (31) and the material figure of merit is

For a crystal with κ1 = 0.3%/cm and a cavity

with Aƒ = 0.3%, the optimum length is 1 cm and the efficiency enhancement over a single pass device is 3 x 104. This analysis has neglected the loss that conversion to the SH represents to the circulating fundamental. When this conversion becomes comparable to the static losses in the cavity, the input coupling must be chosen equal to the total loss (linear and SH conversion) in order to achieve impedance matching. The analysis is more complicated, but the final result is simple.(Schiller 1993) To achieve 50% conversion to the SH, the necessary input power is (32) A similar analysis for a critically phasematched crystal (Bordui 1993) leads to (33) and hence a material figure of merit 388

For intracavity SHG, with the nonlinear crystal placed inside a laser cavity, the analysis is more complex, but similar ratios of nonliner susceptibility to loss emerge as the pertinent material parameters. 3.5 High Average Power SHG For SHG at high average power, the absorbed power deposited in the crystal as heat can significantly raise the temperature. Through the thermooptic effect, this temperature rise can spoil the phasematching as well as create a thermal lens, both of which effects can reduce the efficiency and distort the beam shape. For simplicity, consider a flat-top beam of radius w and average power Pa v passing through a crystal of radius R with absorption coefficient α and thermal conductivity k th , whose surface is held at a fixed temperature T0 . Solving the heat equation with a volume heat source αI(r), the temperature field in the crystal as a function of the radial coordinate r is found to be (34) The temperature drop from center-to-edge of the illuminated region is ∆T = αP av / 4πk th , independent of w and R, and is parabolic in form. The temperature drop from the center-toedge of the crystal is a factor 2 ln( R / w) + 1 larger, but is less problematic as it can usually be compensated by a change in the heat sink temperature T 0 . The parabolic temperature profile causes a radially varying phase mismatch across the beam, whose amplitude depends on the thermooptic coefficients of the crystal.(Okada 1971) (Gettemy 1988) While the average of this temperature rise can again be compenstaed by a shift in T0 , the radial variation cannot be corrected by simple means. This thermal phase mismatch is given by (35) The maximum allowable phase mismatch, ≈ π / 2, combined with Eq. (35), sets a limit on the allowable power in the crystal. Note that this limit scales inversely with the length of the crystal, and is proportional to Comparing Eq. (18) for the SHG efficiency in a noncritically phasematched interaction, and Eq. (35) for the thermal phase mismatch, we see that if absorption of the fundamental dominates the thermal loading, both are proportional to the product Pav L, and independent of spot size w, so that their ratio is a function only of material parameters (36)

The material figure of merit in this case is

If for a given material

this ratio is not adequate for a desired application, the only alternative, other than complex cooling geometries,(Eimerl 1987) is to use a different material. The commonly held notion that increasing the spot size can ameliorate the problem is counterproductive; the ratio η nc / δ t h 389

decreases monotonically with increasing spot size. A comparable expression can be developed for the case where SH absorption dominates, though the axial variation in the SH power, which leads to an axial variation in the temperature drop from center-to-edge of the crystal, must be taken into account. In addition to the thermal dephasing, absorbed power also leads to thermal lensing. The thermal phase shift across the beam takes a similar form to the thermal mismatch (37) This phase shift is distributed parabolically across the illuminated region (Eq. (34)), so to first order it simply produces pure lensing with focal length fth . The focal power 1/ f th is given by (38) which again scales with the product P av L. While the focal power is proportional to w ² , and hence can be reduced by increasing the spot size w, the thermal phase shift across the beam, often a better measure of the effect on beam propagation, cannot. It is worth noting that proper optical design can correct for the effect of a pure thermal lens, but for real (gaussian) beams the parabolic thermal field characteristic of idealized flat-top beams holds only near the axis. Higher terms in the power series representation of the exponential integral solution in the gaussian beam case lead to an aberrating lens that cannot be corrected without aspheric elements.(Stein 1974) 3.6 Optical Parametric Oscillators In the presence of a strong pump at frequency ωp , an input signal at frequency ωs both generates an “idler” wave at frequency ω i = ωp – ω s and experiences parametric gain. The small signal parametric gain g, defined as , near degneracy (ω s ≈ ωi ) is equal to the second harmonic conversion efficiency. For focused noncritically phasematched interactions, the parametric gain is again equal to SH conversion efficiency, so that with Eq. (20) we have

. For typical near-IR materials, the gain, like the SH

efficiency is in the range of 0.1%/W-cm. The Poynting vector walkoff associated with critically phasematched interactions affects parametric gain more than SH efficiency. In the limit of large walkoff angle ρ , the parametric gain reaches a limiting value given by independent of length for optimal focusing, where the effective length . For typical critically phasematched crystals, le f f can be less than 1 mm, imposing a severe limit on the gain available in such materials.(Byer 1975) If the parametric amplifier is placed in a cavity that resonates the signal wave, and the gain exceeds the loss, the output will build from the noise, creating an optical parametric oscillator. The oscillation condition is g = 2A s , where A s is the round trip power loss in the cavity at the signal frequency. For a gain of 0.2%/W and cavity losses of 1%, the threshold pump power is 10 W. If the cavity losses are dominated by mirror and interface losses, the 390

material figure of merit for OPO applications is d 2 / n2 for noncritical phasematching, and d 2 / n 3 ρ 2 for critical phasematching. At high intensities, the gain goes over to an exponential dependence, and I s ( L)/ Is (0) = For pulse lengths shorter than the build up time of the OPO cavity, the threshold intensity increases above the CW value, approaching a condition where the threshold intensity is inversely proportional to pulse length. In this limit, it is more useful to quote a threshold fluence (energy/area) than a threshold intensity. (Brosnan 1977) Since the allowable fluence is limited by surface damage effects, there exists a minimum crystal length for which threshold can be reached before damage occurs. In these cases, the critical parameter of the material is d 2 I dam / n 3 , where I dam is the surface damage threshold. Detailed discussion of optical parametric oscillators can be found in (Byer 1975) and (Byer 1977). A more recent review is in Ref. (Tang 1992) Two special issues on OPOs (Byer 1993) contain collections of current information, which can also be found in Ref. (Pollock 1997). 3 . 7 Waveguide

devices

Nonlinear frequency conversion in waveguides can have much higher normalized conversion efficiency than interactions in bulk media, because the diffractionless propagation available in channel waveguides eliminates the tradeoff between tight focusing for high intensity and loose focusing for long interaction lengths characteristic of confocally focused interactions. It can be shown that the efficiency of a waveguide SHG device takes the same form as for a plane wave interactions (39) where the normalized efficiency η nor , with dimensions [%/W-cm2 ], is given by η nor = C 2 / Aeff , where the effective area is, (Stegeman 1985) (Bortz 1994) (40) with propagation constants βj are normalized according to The modal fields where <ƒ> is defined as the integral over an infinite plane A ∞ normal to z, and we have allowed for the possibility that the nonlinear susceptibility is depth-dependent by defining a normalized (|g| ≤ 1) spatial dependence of deff by g ( x, y) = d eff ( x, y )/ dmax . For a weakly guiding waveguide, it can be shown that Aeff ∝ λ2 / n (nco – ncl ), where n co and n cl are the refractive indexes of the core and the cladding, respectively, so that overall the efficiency of 2 ( nco – ncl )λ4 / n For visible SHG in LiNbO3 waveguides, waveguide SHG scales as d eff typical values of ηnor are 100 – 1000 %/W-cm2.

391

3 . 8 Materials Requirement Summary As should be clear from the previous discussion, different applications put different demands on nonlinear materials; a number of material parameters other than the nonlinear susceptibility appear in the figures of merit applicable in these different cases. Note that these figures of merit address only the readily quantifiable properties of the nonlinear material. There are several other properties, less amenable to quantitative description, that are also important for determining the utility of a material. A number of these have to with fabrication issues. It is not uncommon for difficulties in the growth of crystals of adequate size and homogeneity to delay for many years, or in some cases prevent, otherwise attractive materials from moving from the laboratory into practical use. Similarly, problems with polishing (e.g. soft, easily cleaved, or hygroscopic materials) and coating (e.g. materials with strong anisotropy of thermal expansion) can complicate utilization of a material. Many nonlinear materials exhibit long-term aging phenomena, often due to photochromic or photorefractive responses, or sensitivity to environmental factors such as moisture, that degrade the properties of the material over time scales of tens to hundreds of hours. Such problems are often not apparent in laboratory device demonstrations, but can be prohibitive in practical applications. Aspects of these issues are addressed in Ref. (Bordui 1993). Given the variety and complexity of the requirements placed on nonlinear materials, it is not surprising that a large number of them have been explored over the years, and that no one material is suitable for all applications. The following section discusses some of the more commonly used nonlinear crystals in the context of these issues.

4.

NONLINEAR OPTICAL MATERIALS

The first NLO experiment used crystal quartz as the nonlinear material, a choice which severely limited the SHG efficiency due to the lack of adequate birefringence for phasematching. The first phasematched interactions used potasium dihydrogen phosphate (KDP) and ammonium dihydrogen phosphate (ADP), two materials produced in large quantities for piezoelectric applications, which fortuitously had adequate birefringence for phasematching SHG of 1 µ m lasers. Extensive materials research during the first decade of nonlinear optics research produced a number of new materials, including some still in use today. Two examples are the ferroelectric niobates, LiNbO3 and Ba2NaNb5 O15, which both have nonlinear susceptibilities an order of magnitude larger than those of KDP, and can noncritically phasematch SHG of 1 µ m radiation. Lithium iodate, whose large birefringence enabled phasematching to wavelengths as short as 380 nm, also emerged in this era. Ref. (Zernike 1973) contains an interesting discussion of the early nonlinear materials research. One of the factors stimulating the renewed activity in nonlinear frequency over the past decade has been the emergence of nonlinear materials with improved properties over those of the first generation materials. Here we briefly review the properties of some of these materials. More extensive discussion and references can be found in the review in Ref. 392

(Bordui 1993), and in Ref. (Dmitriev 1997). Ref. (Bordui 1993) also contains a tabular summary of the properties of several commonly used materials, and their figures of merit for various applications. 4 . 1 Currently Available Materials KDP remains the most widely used nonlinear material, though it is undistinguished except for the large size and high quality of crystals that can be grown from low temperature aqueous solution, and its high surface damage threshold. This observation reinforces the importance of crystal growth and control of extrinsic properties in a successful nonlinear material, and the inadequacy of simple figures of merit like d 2/n 3 for predicting a material’s ultimate practical utility. LiNbO3 is transparent from the mid-IR (5 µm) to the near UV (350 nm), has moderately large nonlinear susceptibility (5 pm/V), can be noncritically phasematched for SHG of 1 µm lasers, and can be grown in large, high quality boules by the Czochralski method. Its primary disadvantages are its relatively low surface damage threshold and its susceptibility to photinduced refractive index changes (photorefractive damage) for visible radiation. The 5%MgO:LiNbO 3 variant largely eliminates this latter limitation. K(TiO)PO4 (KTP) is transparent from the mid-IR (4.5 µ m) to the near UV (350 nm), has moderate nonlinear coefficients (3 pm/V), and birefringence adequate for nearly noncritical phasematching for SHG of 1 µ m lasers. In these properties, it is similar to LiNbO3, over which its primary advantages are resistance to photorefractive damage, high surface damage thresholds, and large temperature acceptance bandwidths. Its primary disadvantages are moderately difficult flux or hydrothermal crystal growth, high absorption in the mid-IR, and susceptibility to photo-induced absorption (grey-track damage). KNbO3 is transparent from the near-UV (400 nm) to the mid-IR (5.5 µm), has a large nonlinear coefficient (12 pm/V), can be noncritically phasematched into the blue (430 nm), and has good resistance to laser induced damage. Its primary disadvantages are difficult crystal growth and processing, and small temperature acceptance bandwidths. The borate family of materials, including BaB2 O 4 (BBO) and LiB3 O 5 (LBO) are characterized by small (0.8 pm/V, LBO) to moderate (2 pm/V, BBO) nonlinear susceptibilities, broad phasematching in the UV (fifth harmonic of 1 µm lasers in BBO, third harmonic generation of 1 µm lasers in LBO) and transparency to wavelengths below 200 nm, low optical loss and excellent resistance to laser damage (>l0 GW/cm2 for 10 ns pulses). Their major limitations are due to difficult crystal growth, large walkoff angles (BBO) and optical coating problems (LBO). The chalcopyrite materials, such as AgGaS2, AgGaSe2, and ZnGeP 2, are the primary materials for mid-IR nonlinear optics, with transparency extending to beyond 10 µm, and large nonlinear susceptibilities (18 pm/V AgGaS2, 33 pm/V AgGaSe2, and 70 pm/V ZnGeP2 ). They have various limitations, including low thermal conductivity (AgGaSe2) and moderately large IR losses (AgGaSe2 , ZnGeP2 ). In all three the growth and processing are difficult, and serious surface damage limitations exist, especially for AgGaS2 and AgGaSe 2. 393

4 . 2 Approaches to Finding Improved Nonlinear Materials There remain needs for better nonlinear materials in all spectral ranges. In the UV, materials combining non-critical phasematching, ease of crystal growth, and robustness with respect to aging effects are lacking. In the visible, materials combining adequately large nonlinear susceptibility and noncritical phasematching to enable efficient single-pass doubling of high-power CW lasers or CW optical parametric oscillators, with the thermophysical properties to allow support of multiwatt visible beams, are unavailable. In the mid-IR, current materials suffer from combinations of difficult growth, low surface damage theshold, limited non-critical phasematching, and high optical losses. For nontraditional applications like cascade nonlinearities or mixers and parametric amplifiers for communications and quantum optics, a generation of materials with an order of magnitude higher efficiency than those currently available are needed. There are several possible approaches to developing improved nonlinear materials. The most obvious, though often overlooked, of these is suggested by the observation that many of the problems of current materials, such as absorption, photorefractive and photochromic effects, limited homogeneity and size of available crystals are usually not intrinsic material properties. Thus, improving an old material is often the most direct approach to finding a “new” material. It is not uncommon for 10 – 20 years to elapse between the first laboratory demonstration of a new material, and its emergence in practical applications. Examples include the reduction of photochromic effects in KTP and photorefractive effects in LiNbO3 through the addition of suitable dopants, and the application of improved crystal growth methods to make available large crystals of the ZnGeP2 , a material first described more than 25 years ago. Finding wholly new materials can proceed by several paths. The first approach followed historically was to take advantage of materials that were already well-developed for other applications, primarily piezoelectricity, as it demands the same noncentrosymmetry requirement as nonlinear optics. This approach, which led to the early use of KDP and ADP, is attractive, as it takes advantage of years of preexisting materials development, but limited as only a small number of materials fall in this category. We will see the power of a variant of this approach in the discussion of engineered materials in the following section. The second approach followed historically was to search the mineralogical tables, guided by Miller’s rule, for crystals of high refractive index and large birefringence. This approach, followed in the 1960’s, led, for example, to the use of cinnabar (HgS) and proustite (Ag 3AsS 3 ), but appears to offer no current prospects. An important method is to search compounds isostructural to a known useful material. The KTP family is one such example, where K(TiO)AsO4 (KTA) was found to have similar properties to KTP, but with substantially lower mid-IR absorptivity. Similarly, a large number of crystals in the KDP family have been explored for advantageous phasematching properties. All these approaches have yielded useful results, but do not offer a systematic approach to designing a material suited to a particular interaction. Such design methods are the subject of the following section. 394

4 . 3 Engineered and Microstructured Materials In this context, the term engineered materials is applied to materials which can be tailored in a systematic fashion to suit specific applications, through, for example, control over their transparency range, nonlinear coefficient, or phasematching. The commonest approach to this goal has historically been through molecular engineering techniques. These have been widely applied to organic molecules, for which accurate techniques for design and synthesis of molecules with desired absorption edge and nonlinear susceptibility have been developed. Numerous examples of molecules with enormous nonlinear polarizabilities exist, some of which are discussed in other chapters of this volume. These techniques have been applied on a more limited scale to inorganic materials, especially the borate family, and again good results for design of the absorption edge and nonlinear susceptibility have emerged.(Chen, 1985) Far less successful have been efforts to design the birefringence for phasematching, which, as the small difference of large numbers is inherently more difficult, and the “growability” of these materials, which has precluded the use of otherwise attractive compounds. Techniques for achieving continuously tunable non-critical phasematching would be very powerful for optimizing materials for a variety of applications. One such approach is based on solid solutions such as AgGa1-xIn xSe 2 and K(TiO)P1-x As xO4 , which allow continuous interpolation of properties between those of one end compound and the other. If one of these compounds has too much birefringence for a given application and the other too little, in most cases a composition exists in between with the desired birefringence. While these techniques have been successfully demonstrated, they have not been widely exploited, because mixed crystals are almost always difficult to grow homogeneously, compounding an already challenging growth problem. An alternative approach, which doesn’t require solution of a new set of growth problems for each application, would be a very powerful simplification of the process of engineering nonlinear materials to suit particular interactions. An increasingly important family of these techniques are based on microstructured materials. The microelectronics industry provides a good model for the application of microstructured materials to nonlinear optics. In microelectronics, a broad range of devices, from high-current switches to microprocessors to RF amplifiers, are fabricated in one or two very well understood materials (Si and GaAs) by applying a small set of well-controlled processes steps, often lithography based. One such technique currently developing rapidly in the nonlinear optics field is quasi-phasematching, in which a periodic spatial variation imposed on the properties of a material allow a nonlinear interaction to proceed efficiently in the absence of phase-velocity matching. Other techniques are based on a variety of thin-film media, especially polymers and III-V semiconductors, in which, for example, waveguides with tailored dispersion and modal overlap for modal phasematching,(Wirges 1997) and multilayers with controllable form birefringence for birefringent phasematching (Fiore 1996) are being developed. The discussion in this chapter will focus on the quasi-phasematching technique. 395

5 . QUASI-PHASEMATCHING In a quasi-phasematched interaction, the phase error that accumulates between waves propagating with different phase velocities is periodically reset by a periodic variation in the properties of the nonlinear material. Taking SHG as an example, the relative phase of the the second harmonic field and the nonlinear polarization (proportional to deff E ² ) drifts by π 1

every coherence length l c (Eq. (7)). Because the direction of energy flow between the fundamental and the second harmonic depends on this relative phase, in a non-phasematched interaction the power that flowed into the SH in the first coherence length flows back to the fundamental over the next coherence length, so that there is no average growth of the second harmonic field. In a phasematched interaction, the relative phase is constant, so the second harmonic field grows linearly with distance. In the ideal quasi-phasematched interaction, the sign of the nonlinear polarization is changed every coherence length, so that the sign of the nonlinear polarization is reversed every coherence length, effectively adding the π shift necessary to properly rephase the nonlinear polarization and the SH field. The SH field then grows monotonically with distance, “quasi-phasematching” (QPM) the interaction. It is also useful to view this interaction in momentum space, rather than the real space picture presented above. In a phasematched interaction, 2k 1 = k 2 , i.e. the sum of the kvectors of the two fundamental photons destroyed equals the k-vector of the SH photon generated, conserving photon momentum. In a phase mismatched interaction, 2k 1 + ∆ k = k 2 , momentum is not conserved, and the efficiency is very low. If an additional structure with a spatial frequency K g = ∆ k exists in the medium, a sort of “quasi-momentum” conservation exists, and the interaction can proceed efficiently. This form of the analysis is clear from the integral form of the solution for the SH harmonic field, Eq. (3), which shows that a periodic variation in d(z) with spatial frequency equal to ∆ k will cancel the oscillatory term in the integrand, and lead to monotonic growth of the SH. 5.1 Basic Theory of QPM For quantitative analysis of QPM,(Fejer 1992) it is useful to write the spatially varying nonlinear susceptibility in the form (41) where d eff is the maximum value of the nonlinear coefficient, and g(z) is a variable normalized to  g(z) < 1 containing all the information about the spatial distribution of d(z). Substituting Eq. (41) for d(z) into Eq. (3), we can write the generated field as (42) where (43)

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Eq. (43) makes clear that the generated field depends on the Fourier amplitude of g( z) at the spatial frequency ∆ k. Note that if g = constant Eq. (43) simply reproduces the usual sinc² tuning for conventional SHG. If g(z) is periodic with period Λ, we can write g(z) as (44) where the mt h spatial harmonic of the grating is (45) Assuming that the m th spatial harmonic is much closer to ∆k than any of the others, that term will dominate in the integral in Eq. (43), and the generated SH field will be, neglecting an unimportant phase factor, (46) which is identical to that obtained for a conventionally phasematched interaction with a nonlinear coefficient d m = G m d eff and a phase mismatch shifted by an amount K m . If the grating is a rectangular wave of unit amplitude and period Λ , with positive sections of length l, the required Fourier coefficient G m can be expressed in terms of the duty cycle D = l/ Λ ; the resulting value of d m is (47) For an optimum choice of D, e.g. 50% for m odd, we find (48) For example, in periodically-poled lithium niobate (PPLN), with a nonlinear coefficient of d 33 = 27 pm/V,(Roberts 1992) the nonlinear coefficient for first order QPM is d 1 = 1 8 pm/V, while second and third order QPM have nonlinear coefficients of 9 pm/V and 6 pm/V, respectively, compared to the coefficient used for birefringent phasematching, d 31 = 4.3 pm/V. In the case of a quasi-phasematched waveguide interaction, the overlap integral includes the depth-dependent Fourier component of the nonlinear coefficient, i.e. the function g(x,y) in Eq. (40) is replaced in the overlap integral by Gm (x,y) of Eq. (44), which can have a major impact on the efficiency for common waveguide materials.(Bortz 1994). Summarizing, if we can construct a grating in the nonlinear susceptibility whose period is an integer multiple (m) of 2l c , a quasi-phasematched interaction proceeds essentially as a conventionally phasematched interaction with a nonlinear susceptibility reduced by a factor 2/m π. This discussion has focused on pure modulation of the nonlinear susceptibility, though QPM based on modulation of the linear properties is possible as well, generally at a significant reduction in efficiency.(Fejer 1992)

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5.2 Advantages of QPM By eliminating any dependence on birefringence for phasematching, QPM offers several major advantages compared to conventional phasematching techniques. QPM allows a material to be used for any interaction within its transparency range, and can even be applied in non-birefringent materials like the zincblende semiconductors. Noncritical phasematching is possible, eliminating the deleterious effects of Poynting vector walkoff, especially important for applications in optical parametric oscillators. Any combination of polarizations can be used, allowing, for example, parallel polarization of all the fields to take advantage of the often-large diagonal components of the nonlinear susceptibility tensor, inaccessible to birefringent phasematching. In LiNbO3 , the ratio d 33 / d 31 = 7, a factor that enters as the square in the calculation of the conversion efficiency. Perhaps most important in the long run is the ability QPM affords to systematically tailor one material for many applications. Despite these several advantages, QPM has not been widely used until the past several years, due to the difficulty in fabricating the necessary fine pitch (typically microns to tens of microns) gratings in the nonlinear susceptibility. 5.3

Early History of QPM

QPM was first discussed in the seminal 1962 paper by Bloembergen,(Armstrong 1992) and a patent issued to him in 1968. Despite its invention prior to that of birefringent phasematching, the latter has become the basis for almost all practical nonlinear frequency conversion devices. Early demonstrations of QPM were based on stacks of rotated plates of III-V and II-VI semiconductors. References to this early work can be found in (Fejer 1992). Development in this era was slow, due to the difficulty in fabricating the necessary stacks of thin plates, and losses associated with the many air-to-semiconductor interfaces. Practical QPM approaches awaited the advent of monolithically patternable media. A number of materials have been explored for monolithic patterning of the nonlinear susceptibility, including ferroelectrics, poled polymers, poled glass, patterned orientation growth of semiconductor films, and asymmetric quantum wells. By far the best developed of these are the periodically-poled semiconductors, which will be the subject of the remainder of this chapter. 5.4 Ferroelectrics for QPM Ferroelectrics are materials in which each unit cell develops a spontaneous electric dipole moment at temperatures below a critical “Curie” temperature.(Lines 1977) These dipoles form into macroscopic regions of aligned polarization, known as “domains”, separated by domain walls. The domains can be preferentially oriented by application of an electric field exceeding a value characteristic of the material, the coercive field. In many of these respects, ferroelectrics and electric fields are analogous to ferromagnets and magnetic fields. For a large class of ferroelectrics, there are two allowed orientations of the spontaneous polarization, 180° rotated with respect to each other. For the purposes of QPM, the important 398

point is that the sign of odd-rank material tensors, including the nonlinear susceptibility, changes from one domain orientation to the other. Thus, a periodic array of ferroelectric domains produces the periodic reversal in the sign of the nonlinear susceptibility necessary for QPM. The problem of creating a monolithic QPM structure is thus reduced to the problem of creating a micron-scale array of ferroelectric domains. This possibility was recognized early in the history of nonlinear materials development. As early as 1964, the effects of random arrays of domains in as-grown crystals was investigated.(Miller 1964) Subsequent to that, during the 1980’s, periodic arrays of domains were created by periodically perturbing the growth of bulk and fiber ferroelectric crystals. While interesting QPM SHG results were obtained, proving the potential of the method,(Xue 1984) (Jundt 1991) development was limited by the difficulty in creating adequately periodic structures over interaction lengths longer than about 1 mm. Note that the effective interaction length is limited to the distance over which accumulated drift in the domain location becomes on the order of one coherence length.(Fejer 1992) Widespread use of QPM techniques did not begin until lithographically controlled methods for patterning domains were devised. An important practical point is that at least two of the ferroelectrics widely used for optical applications, LiNbO3 and LiTaO3 , are also used for surface acoustic wave filters. This industry uses several tons (>106 wafers) per year, so the growth of these materials is very well developed, with high quality 3-inch diameter wafers available for prices in the range of $100. 5.5 Lithographic Patterning of Ferroelectric Domains The first lithographic domain patterning methods developed were based on patterned indiffusion of dopants into ferroelectric substrates.(Lim 1989) (Webjorn 1989) Initially, indiffusion of a Ti grating into LiNbO 3 was used, though techniques based on indiffusion of protons in LiTaO3 , (Mizuuchi 1991) and Ba and Rb into KTP (van der Poe1 1990) soon followed. The microscopic mechanisms causing the domain reversal in these methods are not well understood. These methods all produced shallow gratings, whose depth was on the order of the grating period (several microns), so that this material could be used only in waveguide interactions, in which the radiation is trapped in the waveguide region, of comparable depth to the domain pattern. Progress in these waveguide devices was rapid; SHG of visible light with efficiencies with normalized efficiencies in the range of 100%/Wcm² and several milliwatts of output power were demonstrated by 1992. Early work in patterned dopant QPM methods is reviewed in Ref. (Fejer 1992b). The next major step in the development of periodically-poled ferroelectrics was poling using electric fields applied through periodic electrodes patterned on the surface of the wafer.(Yamada 1993) This approach, surprisingly, created domains that propagated all the way through 0.5 mm thick substrates, allowing use as a bulk material as well as for waveguide applications. Initially developed in LiNbO3 , this technique was again rapidly applied to other ferroelectrics, including LiTaO3 , (Zhu 1995) and KTP (Chen 1994). Typical conditions for the poling of LiNbO3 are the application of voltages around 10.5 kV across a 399

0.5 mm thick wafer (fields of 21 kV/mm) for durations on the order of 100 ms, with an insulating film covering a periodic electrode on one face of the crystal, and a uniform ground electrode on the other face. The dynamics of the domain formation process are complex, but a model that explains important aspects of the poling behavior has been developed.(Miller 1996) A key point in explaining the high fidelity of the transfer of the pattern from the periodic electrode to the domain pattern is the strong dependence of the lateral velocity of the domain walls on the applied electric field. Measurements showed that the velocity changed by five orders of magnitude for a several percent change in the field in the vicinity of 21 kV/mm. When domains that nucleate under the electrodes begin to spread beyond the electrode and into the region covered by the insulator, the polarization charge deposited on the surface is uncompensated by free charge, reducing the average field across the wafer. Because of the steep dependence of the domain wall velocity on the electric field, these small changes are enough to stop the domains’ penetration under the insulating regions; The upshot is that it is possible to pole entire 3 inch (75 mm) diameter wafers with high quality volume domain gratings, with periods from the 30 µm range suitable for QPM of mid-IR generation, down to the 6 µm periods necessary for SHG of green light. Shorter period gratings are a topic of current research, with material suitable for operation into the blue spectral region fabricated,(Pruneri 1995) (Goldberg 1995) but over smaller areas than the longer pitch gratings. Shorter period gratings appear to be easier to pole in LiTaO3 and KTP than in LiNbO3 , with usable material fabricated with periods around 2 µm, and applied to the generation of UV radiation. (Mizuuchi 1997) (Meyn 1997)

6. CURRENT RESULTS FOR QPM DEVICES Fabrication and device application of quasi-phasematched materials is a very rapidly expanding field of research, with hundreds of papers published over the past several years. It is not possible to comprehensively review the work in this area in the space available. Here we focus on some key results in two areas, visible light generation and mid-IR OPOs. 6.1 Visible Light Generation Waveguide devices for visible SHG were the first to take advantage of electric-field-poled substrates. The major advantage over the chemically poled substrates was eliminating the difficult tolerances associated with overlapping the waveguide modes with the shallow, depth-dependent periodically-poled layer.(Bortz 1994) Efficiencies increased rapidly, again in all three common ferroelectrics, LiNbO3 , LiTaO 3 , and KTP. Normalized efficiencies for SHG of blue light of 1600%/W-cm² have been reported in LiNbO3 waveguides,(Kintaka 1996), 1500%/W-cm² in LiTaO3 ,(Yi 1996), both using annealed proton exchanged waveguides, and over >400%/W-cm² in KTP ion-exchanged waveguides.(Chen 1996). Relatively high powers have been generated in the waveguide devices, 25 mW of 490 nm output for 120 mW of 980 nm pump power in a LiNbO 3 waveguide,(Webjorn 1997), 23 400

mW output for 121 mW IR pump in an LiTaO3 waveguide,(Yamamoto 1992) and 12 mW output for 146 mW IR pump power in KTP.(Chen 1996) While these output levels appear to be stable, higher outputs (up to 100 mW) have been observed in LiNbO 3 waveguides,(Webjorn 1997b) though with evidence of photorefractive instabilities. An important step toward control of photorefractive damage has been the recent progress in poling MgO:LiNbO3 substrates, which are much more resistant to photorefractive damage than conventional LiNbO3 and LiTaO3 . An efficiency of 1200%/W-cm2 has been reported, with 5.5 mW of 434 nm light generated,(Mizuuchi 1997b). This device included a highindex top cladding layer of Nb 2O 5, which improved the efficiency by creating a more symmetrical waveguide, bringing the peaks of the fundamental and SH modes closer together, thereby increasing the overlap integral. Another device on MgO:LiNbO3 , generating 19 mW of 434 nm radiation at 600%/W efficiency using buried domains on an x-cut substrate, was reported.(Mizuuchi 1997c) Efforts to efficiently package QPM waveguide SHG devices have also been succesful, with cm3 devices including pump diode, coupling hardware, and the SHG chip, generating as much 15 mW blue output.(Webjorn 1997), (Kitaoka 1995) Scaling to higher powers than is practical in waveguides requires the use of bulk crystals. Ideally, these would have adequate conversion efficiency to operate in a single pass, thus avoiding the complexities associated with intracavity or externally resonant devices. In an ideal PPLN crystal, the normalized efficiency for SHG of green light, given the effective nonlinear coefficient of 18 pm/V, is γnc = 4%/W-cm. For a 5 cm long crystal, practical with current technology, a theoretical efficiency of 20%/W is obtained. For 1 µm pump laser with several watt output power, rather moderate by current standards, efficiencies in the saturated regime can be expected. The reduced susceptibility to photorefractive damage associated with the periodic variation in electrooptic and optogalvanic effects in periodically-poled crystals is an important practical advantage.(Taya 1996) (Sturman 1997) The highest CW output power reported to date was in a 5.3 cm long PPLN sample, whose 6.5 µm period domain grating was of sufficient quality to provide a nonlinear 2

susceptibility 78% of the ideal 18 pm/V. Temperature tuning showed a nearly ideal sinc response, indicating that the sample was uniform over its entire length, and a normalized efficiency of 10.5%/W. The maximum 532 nm output power, 2.7 W, was obtained at an internal 1064 nm pump power of 6.4 W, or 42% overall conversion efficiency. This is approaching the performance necessary for an ideal solid state replacement for the argon laser in many applications. At powers exceeding 2 W, significant thermal focusing effects were observed. Subsequent studies suggest that these are a result of a green-induced IR absorption phenomenon that must be reduced before higher single-pass powers are obtained in these long crystals.(Miller 1997) 330 mW of average green power was generated by SHG of a mode-locked 1 µm laser in PPLN, with an average conversion efficiency of 52%.(Pruneri 1996) Extension into the blue is challenging due to the difficulty in poling the necessary fine pitch (3 - 4 µm) period gratings, but recent results have been promising, with 49 mW of CW 401

473 nm radiation generated in a 6 mm long PPLN sample with near ideal 19 pm/V nonlinear susceptibility, and 3 K temperature tuning bandwidth.(Pruneri 1996b) Doubling of a 980 nm diode laser to produce 6.7 mW in bulk MgO:LiNbO 3 corona poled with a 5.3 µm period, for a normalized efficiency of 2%/W-cm has also been reported.(Harada 1996) 6.2 Optical Parametric Oscillators The noncritical phasematching available in PPLN, along with the large d 33 coefficient of LiNbO 3 , are particularly advantageous for use in OPOs, which have been perhaps the most impressive demonstrations of QPM materials to date. The first QPM OPO, based on an annealed proton exchanged waveguide in a dopant-diffusion-poled LiNbO 3 substrate, was demonstrated in 1995.(Bortz 1995) Shortly thereafter, the first QPM OPO in bulk material was demonstrated in electric-field poled LiNbO3 (Myers 1995) Progress since that time has been rapid. The gain of a bulk PPLN OPA at degeneracy, the same as the normalized SHG efficiency, is 4%/W-cm for 0.5 µm pumping, and, scaling as λ–3 , (Eq. (18)), is 0.5%/W-cm for 1 µm pumping. These high gains permit pumping with much lower power lasers than are possible with conventional materials. In the first bulk PPLN OPO, pumped with 7-ns pulses from a Q-switched diode-pumped Nd:YAG laser, the threshold for a 5 mm long PPLN crystal was 135 µJ, ten times lower than has been obtained with birefringently phasematched LiNbO3 , and more than 20 times below the surface damage fluence of 3 J/cm 2 , an important consideration for practical applications.(Myers 1995) In the same apparatus, a 15 mm long crystal and tighter pump focusing led to threshold of 12 µJ, well-matched to pumping with high-repetition-rate Q-switched diode-pumped lasers.(Myers 1995b) By use of multiple QPM gratings patterned on a single chip, it was possible to tune a 1-µm-pumped OPO from 1.36 to 4.8 µm, simply by translating the crystal to bring different gratings into the pump beam.(Myers 1996) (Myers 1995b) compares QPM OPOs vs birefringently phasematched OPOs, provides tabular data on the phasematching properties of PPLN for IR OPOs, and reviews pulsed singly-resonant and CW doubly-resonant OPOs in PPLN. An important application of PPLN, taking advantage of the high gain and low loss, has been in CW singly resonant OPOs. The first CW 1 µm pumped OPO, using a 5-cm-long PPLN crystal, demonstrated a threshold pump power of 3.5W, an output power at 3.25 µm of 3.5 W, and a pump depletion of 94% at the maximum pump power of 13.5 W, consistent with the high gain and low (<0.1 %/cm) mid-IR losses of PPLN.(Bosenberg 1996) Electricfield-poled PPLN has also been used in a waveguide OPO, with threshold of 1 W; design approaches to sub-100-mW thresholds seem realistic.(Arbore 1997b) Operation of PPLN OPOs has been extended to 0.5 µm pumps, including Q-switched (Pruneri 1995b) and CW (Batchko 1997) operation. The latter had a threshold of 0.9 W. Picosecond (Butterworth 1996) and femtosecond (Burr 1997) OPOs with IR pumps are another important current direction, taking advantage of the large ultrafast figure of merit (d 2/n 2∆ ng ).

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7. FUTURE DIRECTIONS The field of QPM nonlinear optics continues to grow rapidly. In addition to the wellestablished applications in mid-IR OPOs, applications are now growing in ultrafast nonlinear optics, especially for low pulse energy systems such as fiber lasers where the high figure of merit is of critical importance. Ultrafast SHG,(Arbore 1997) OPG,(Galvanauskas 1996) and OPOs (Butterworth 1996) (Burr 1997) have all been demonstrated. The use of Fourier synthesis techniques to design QPM gratings with desired spectral amplitude and phase response is just beginning to be investigated,(Arbore 1997c) and offers many exciting opportunities. As the extrinsic loss is reduced in the visible spectral region, and the techniques for reliably patterning short-pitch domain gratings are improved, applications of long periodically-poled crystals to high power visible light generation will expand. Applications beyond sources of coherent radiation are also appearing. Wavelength convertors for WDM communications systems implemented by near-degenerate difference frequency generation in QPM waveguides are characteristic of the optical signal processing functions made possible by the very high mixing efficiencies available in waveguide QPM devices.(Xu 1995) (Yoo 1996) There appear to be many other opportunities in this area, e.g., quantum optics experiments using waveguide (Anderson 1995) (Serkland 1995) and bulk (Lovering 1996) OPAs. Other materials systems are being developed that address limitations of PPLN. Periodically-poled lithium tantalate extends the UV edge to 280 nm.(Meyn 1997) Periodically-poled KTP isomorphs hold the promise of larger apertures, higher damage thresholds, and more resistance to photorefractive effects.(Reid 1997) III-V semiconductors offer large nonlinear susceptibilities and transparency beyond the 5 µm multiphonon edge characteristic of oxide ferroelectrics. Diffusion-bonded stacks of plates (Zheng 1977) and growth of films controllably twinned by a template substrate (Angell 1994) (Yoo 1996) are both progressing towards practical device implementations. The surprisingly large nonlinear susceptibilities created by electric field poling of glass fibers has been used in QPM interactions, offering interesting opportunities for long devices in convenient waveguide forms.(Kazansky 1997)

8. REFERENCES Akhmanov, S. A., Kovrygin, A. I., and Sukhorukov, A.P., 1975, Optical Harmonic Generation and Optical Frequency Multipliers, in: Quantum Electronics: A Treatise, H. Rabin and C. L. Tang, eds., Academic Press, New York. Anderson, M.E., Beck, M., Raymer, M.G., and Bierlein, J.D., 1995, Quadrature squeezing with ultrashort pulses in nonlinear-optical waveguides, Opt. Lett. 20:620. Angell, M.J., Emerson, R.M., Hoyt, J.L., Gibbons, J.F., Eyres, L.A., Bortz, M.L., and Fejer, M.M., 1994, Growth of alternating (100)/(111)-oriented II-VI regions for quasi-phase-matched nonlinear optical devices on GaAs substrates, Appl. Phys. Lett. 64:3107. Arbore, M.A., Fejer, M.M., Fermann, M.E., Hariharan, A., Galvanauskas, A., and Harter, D., 1997, Frequency doubling of femtosecond erbium-fiber soliton lasers in periodically-poled lithium niobate, Opt. Lett. 22: 13.

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Arbore, M.A., and Fejer, M.M., 1997b, Singly resonant optical parametric oscillation in periodically poled lithium niobate waveguides, Opt. Lett. 22: 151. Arbore, M.A., Galvanauskas, A., Harter, D., Chou, M.H., and Fejer, M.M., 1997c, Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate, Opt. Lett. 22: 1341. Armstrong, J.A., Bloembergen, N., Ducuing, J., and Pershan, P.S., 1962, Interactions between lightwaves in a nonlinear dielectric, Phys. Rev. 127: 1918. Batchko, R.G., Weise, D.R., Plettner, T., Miller, G.D., Fejer, M.M., and Byer, R.L., 1997, 532 nm-pumped continuous-wave singly resonant optical parametric oscillator based on periodically-poled lithium niobate, in: OSA Trends in Optics and Photonics Series. Vol.10 Advanced Solid State Lasers., Pollock, C.R., and Bosenberg, W.R., eds., p 182-4, Opt. Soc. Am., Washington, D.C. Beausoleil, R., 1992, Highly efficient second harmonic generation, Lasers and Optronics, 11: 17. Bordui, P.F., and Fejer, M.M., 1993, Inorganic crystals for nonlinear optical frequency conversion, Annu. Rev. Mat. Sci. 23:321. Bortz, M.L., Field, S.J., Fejer, M.M., Nam, D.W., Waarts, R.G., and Welch, D.W., 1994, Noncritical quasiphase-matched second harmonic generation in an annealed proton-exchanged LiNbO3 waveguide, IEEE J. Quantum Electron. 30:2953. Bortz, M.L., Arbore, M.A., and Fejer, M.M., 1995, Quasi-phasematched optical parametric amplification and oscillation in periodically-poled lithium niobate waveguides, Opt. Lett, 20:49. Bosenberg, W.R., Drobshoff, A., Alexander, J.I., Myers, L.E., and Byer, R.L.,1996,93% pump depletion, 3.5-W continuous-wave, singly resonant optical parametric oscillator, Opt. Lett. 21: 1336. Boyd, G.D., and Kleinman, D.A., 1968, Parametric interaction of focused Gaussian light beams, J. Appl. Phys. 39:3597. Brosnan S., and Byer. R.L., 1979, Optical parametric oscillator threshold and linewidth studies, IEEE J. Quantum Electron., 15:415. Butterworth, S.D., Pruneri, V., and Hanna, D.C., 1996, Optical parametric oscillation in periodically poled lithium niobate based on continuous-wave synchronous pumping at 1.047 µm, Opt. Lett. 21: 1345. Burr, K.C., Tang, C.L., Arbore, M.A., and Fejer, M.M., 1997, High-repetition-rate femtosecond optical parametric oscillator based on periodically poled lithium niobate, Appl. Phys. Lett. 70:3341. Byer, R.L., 1975, Optical parametric oscillators, in: Quantum Electronics: A Treatise, H. Rabin and C.L. Tang, eds.Academic Press, New York. Byer, R.L., 1977, Parametric oscillators and nonlinear materials, in Nonliner Optics, P.G. Harper and B.S. Wherret, eds., Academic Press, San Francisco. Byer, R.L., and Piskarskas, A., eds., 1993, Special Issue on Optical Parametric Oscillation and Amplification, J. Opt. Sci. Am. B 10(9) and 10( 11). Chemla, D.S., and Zyss. J., 1987, Nonlinear Properties of Organic Molecules and Crystals, pp. 42-43, Academic Press, Orlando. Chen, C.-T., Wu, Y.-C., and Li, R.-K., 1985, The relationship between the structural type of the anionic group and SHG effect in boron-oxygen compounds, Chinese Phys. Lett., 2: 389. Chen, Q., and Risk, W., 1994, Periodic poling of KTiOPO 4 using an applied electric field, Electron. Lett. 30:1516. Chen, Q., and Risk, W., 1996, High efficiency quasi-phasematched frequency doubling waveguides in KTiOPO 4 fabricated by electric field poling, Elect. Lett. 32: 107. Dmitriev, V.G., Gurzadyan, G.G., and Nikogosyan, D.N., Handbook of Nonliner Optical Crystals, 2nd ed., Springer-Verlag, Berlin (1997). Eimerl, D., 1987, High average power harmonic generation, IEEE J. Quantum Electron. QE-23:575. Fejer, M.M., Magel, G.A., Jundt, D.H., and Byer, R.L., 1992b, Quasi-phasematched second harmonic generation: tuning and tolerances, IEEE J. Quantum Electron. 28:2631. Fejer, M.M., 1992, Nonlinear frequency conversion in periodically-poled ferroelectric waveguides, in: Guided Wave Nonlinear Optics, D.B. Ostrowsky and R. Reinisch, eds., Kluwer Academic Publishers, Dordrecht. Fiore, A., Berger, V., Rosencher, E., Laurent, N., Theilmann, S., Vodjdani, N., Nagle, J., 1996, Huge birefringence in selectively oxidized GaAs/AlAs optical waveguides, Appl. Phys. Lett. 68: 1320. Franken, P.A., Hill, A.E., Peters, C.W., and Weinreich, G., 1961, Generation of optical harmonics, Phys. Rev. Lett. 7:118. Galvanauskas, A., Arbore, M.A., Fejer, M.M., Fermann, M.E., and Harter, D., 1996, Fiber-laser-based femtosecond parametric generator in bulk periodically poled LiNbO 3, Opt. Lett. 22:105. Gettemy, D.J., Harker, W.C., Lindholm, G., and Barnes, N.P., 1988, Some optical properties of KTP, LiIO3 , and LiNbO 3 . IEEE J, Quantum Electron. 24:2231. Giordmaine, J.A. 1962, Mixing of light beams in crystals, Phys. Rev. Lett. 8:19. Goldberg, L., McElhanon, R.W., and Burns, W.K., 1995, Blue light generation in bulk periodically field poled LiNbO3 , Electron. Lett. 31:1576. 404

Harada, A., and Nihei, Y., 1996, Bulk periodically poled MgO-LiNbO 3 by corona discharge method, Appl. Phys. Lett. 69:2629. Jundt, D. H., Magel, G.A., Fejer, M.M., and Byer, R.L., 1991, Periodically poled lithium niobate for high efficiency second-harmonic generation, Appl. Phys. Lett. 59:2657. Kazansky, P.G., Russell, P.St.J., and Takebe, H., 1997, Glass fiber poling and applications, J. Lightwave Technol. 15:1484. Kintaka, K., Fujimura, M., Suhara, T., Nishihara, H., 1996, High-efficiency LiNbO 3 waveguide secondharmonic generation devices with ferroelectric-domain-inverted gratings fabricated by applying voltage, J. Lightwave Technol. 14:462. Kitaoka, Y., Mizuuchi, K., Yamamoto, K., Kato, M., 1995, An SHG blue-light source using domain inverted LiTaO 3 , Review of Laser Engineering 23:788. Kozlovsky, W.J., Nabors, C.D., and Byer, R.L., 1988, Efficient second harmonic generation of a diode-laserpumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities, IEEE J. Quantum Electron. 26:135. Kurtz, SK., 1975, Measurement of nonlinear optical susceptibilities, in: Quantum Electronics: A Treatise, Rabin, H., and Tang, C.L., eds., Academic Press, New York (1975). Lim, E.J., Fejer, M.M., and Byer, R.L., 1989, Second harmonic generation of green light in a periodicallypoled lithium niobate waveguide, Electron. Lett. 25:174. Lines, M.E., and Glass, A. M., Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford (1997). Lovering, D.J., Levenson, J.A., Vidakovic, P., Webjorn, J., and Russell, P.St.J., 1996, Noiseless optical amplification in quasi-phase-matched bulk lithium niobate, Opt. Lett. 21:1439. Maker, P.D., Terhume, R.W., Nisenoff, M., and Savage, C.M., 1962, Effects of dispersion and focusing on the production of optical harmonics, Phys. Rev. Lett. 8:21. Meyn, J.P., and Fejer, M.M., 1997, Tunable ultraviolet radiation by second-harmonic generation in periodically-poled lithium tantalate, Opt. Lett. 22:1214. Miller, G.D., Batchko, R.G., Fejer, M.M., and Byer, R.L., 1996, Visible quasi-phasematched harmonic generation by electric-field-poled lithium niobate, in SPIE Proceedings on Nonlinear Frequency Generation and Conversion, 2700:34. Miller, G.D., Batchko, R.G., Tulloch, W.M., Weise, D.R., Fejer, M.M., and Byer, R.L., 1997, 42% efficient single-pass second harmonic generation of a continuous-wave Nd:YAG laser output in a 5.3 cm length periodically-poled lithium niobate crystal, presented at the OSA Topical Meeting on Advanced Solid State Lasers, Orlando, FL, Jan. 1997. Miller, R.C. 1964, Optical harmonic generation in BaTiO3 single crystal, Phys. Rev. 134:A1313. Mizuuchi, M., Yamamoto, K., and Taniuchi, T., 1991. Second harmonic generation of blue light in LiTaO3 waveguides, Appl. Phys. Lett. 58:2732. Mizuuchi, K., Yamamoto, K., and Kato, M., 1997, Generation of ultraviolet light by frequency doubling of a red laser diode in a first-order periodically poled bulk LiTaO 3 , Appl. Phys. Lett. 70:1201. Mizuuchi, K., Ohta, H., Yamamoto, K., and Kato, M., 1997b, Second-harmonic generation with a highindex-clad waveguide, Opt. Lett. 22:1217. Mizuuchi, K., Yamamoto, K., and Kato, M., 1997c, Harmonic blue light generation in X-cut MgO:LiNbO 3 waveguide, Electron. Lett. 33:806. Myers, L.E., Miller, G.D., Eckardt, R.C., Fejer, M.M., Byer, R.L., and Bosenberg, W.R., 1995, Quasiphasematched 1.064-µm-pumped optical parametric oscillator in bulk periodically-poled lithium niobate, Opt. Lett. 20:52. Myers, L.E., Eckardt, R.C., Fejer, M.M., Byer, R.L., Bosenberg, W.R., and Pierce, J.W., 1995b, Quasiphase-matched optical parametric oscillators in bulk periodically-poled LiNbO3, J. Opt. Sci. Am. 12:2102. Myers, L.E., Eckardt, R.C., Fejer, M.M., Byer, R.L., and Bosenberg, W.R., 1996, Multigrating quasi-phasematched optical parametric oscillator in periodically poled LiNbO3 , Opt. Lett. 214:591. Okada, M., and Ieiri, S., 1971, Influences of self-induced thermal effects on phase matching in nonlinear optical crystals, IEEE J. Quantum Electron. 7:560. Pollock, C.R., and Bosenberg, W.R. eds., OSA Trends in Optics and Photonics on Advanced Solid State Lasers. From the Topical Meeting, Opt. Soc. America, Washington (1997). Pruneri, V., Koch, R., Kazansky, P.G., Clarkson, W.A., Russell, P.S.J., and Hanna, D.C., 1995, 49 mW of CW blue light generated by first-order quasi-phase-matched frequency doubling of a diode-pumped 946-nm Nd:YAG laser, Opt. Lett. 23:2375. Pruneri, V., Webjorn, J., Russell, P.S.J., and Hanna, D.C., 1995b, 532 nm pumped optical parametric oscillator in bulk periodically poled lithium niobate, Appl. Phys. Lett. 67:2126. Pruneri, V., Butterworth, S.D., and Hanna, DC., 1996, Highly efficient green-light generation by quasiphase-matched frequency doubling of picosecond pulses from an amplified mode-locked Nd:YLF laser, Opt. Lett. 21:390. 405

Pruneri, V., Koch, R., Kazansky, P.G., Clarkson, W.A., Russell, P.S.J., and Hanna, D.C., 1996b, Highlyefficient CW blue light generation via first-order quasi-phase-matched frequency doubling of a diodepumped 946 Nd:YAG laser, OSA Trends in Optics and Photonics on Advanced Solid State Lasers. Vol.1. From the Topical Meeting, Payne, S.A. and Pollock, C.R. eds., Opt. Soc.America, Washington. Reid, D.T., Penman, Z., Ebrahimzadeh, M., Sibbett, W., Karlsson, H., and Laurell, F., 1997, Broadly tunable infrared femtosecond optical oscillator based on periodically poled RbTiOAsO4, Opt. Lett. 22:1397. Roberts, D.A., 1992, Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature, IEEE J. Quantum Electron. 28:2057. Schiller, S., Principles and applications of otpical monolithic total-internal-reflection resonators, Ph.D. dissertation, Stanford University (1993). Serkland, D.K., Fejer, M.M., Byer, R.L., and Yamamoto, Y., 1995, Squeezing in a quasi-phase-matched LiNbO 3 waveguide, Opt. Lett. 20:1649. Shen, Y.R., The Pinciples of Nonlinear Optics, John Wiley, New York (1984). Stegeman, G. and Seaton, C., 1985, Nonlinear integrated optics, J. Appl . Phys. 58: R57. Stein, A, 1974, Thermooptically perturbed resonators, IEEE J. Quantum Electron. 10:427. Sturman, B., Aguilar, M., Agullo-Lopez, E., Pruneri, V., and Kazansky, P.G., 1997, Photorefractive nonlinearity of periodically poled ferroelectrics, J. Opt. Sci. Am. B 14:2641. Tang, C.L., Bosenberger, W.L., Ukachi, T., Lane, R.J., and Cheng, L.K., Optical parametric oscillators, Proc. IEEE 80:365. Taya, M., Bashaw, M.C., and Fejer, M.M., 1996, Photorefractive effects in periodically poled ferroelectrics, Opt. Lett. 21:857. van der Poel, C.J., Bierlein, J.D., Brown, J.B., and Colak, S., 1990, Efficient type I blue second harmonic generation in periodically segmented KTiOPO4 waveguides, Appl. Phys. Lett. 57:2074. Webjorn, J., Laurell, F., and Arvidsson, G., 1989, Fabrication of periodically domain-inverted channel waveguides in lithium niobate for second harmonic generation, J. Lightwave Technol. 7:1597. Webjorn, J., Siala, S., Nam, D.W., Waarts, R.G., and Lang, R.J, 1997, Visible laser sources based on frequency doubling in nonlinear waveguides, IEEE J. Quantum Electron. 33:1673. Webjorn, J., 1997b, personal communication. Wirges, W., Yilmaz, S., Brinker, W., Bauer-Gogonea, S., Bauer, S., Jager, M., Stegeman, G.I., Ahlheim, M., Stahelin, M., Zysset, B., Lehr, F., Diemeer, M., and Flipse, M.C., 1997, Polymer waveguides with optimized overlap integral for modal dispersion phase-matching, Appl . Phys . Lett. 70:3347. Xu, C.-Q., Okayama, H., Kamijoh, T., 1995, Broadband multichannel wavelength conversions for optical communication systems using quasiphase matched difference frequency generation, Japanese J. Appl . Phys., 34:L1543. Xue, Y.H., Ming, N.B., Zhu, J.S., and Feng, D., 1984, The second harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains, Chinese Phys. 4:554. Yamada, M., Nada, N., Saitoh, M., and Watanabe, K., 1993, First order quasi-phase matched LiNbO 3 waveguide periodically-poled by applying an external field for efficient blue second harmonic generation, Appl. Phys . Lett. 62:435. Yamamoto, Y., and Mizuuchi, K., 1992, Blue-light generation by frequency doubling of a laser diode in a periodically domain-inverted LiTaO 3 waveguide, IEEE Photon. Technol . Lett. 4:435. Yariv, Amnon, Optical Waves in Crystals, Wiley, New York, (1984). Yi, S.-Y., Shin, S.-Y., Jin, Y.-S., and Son, Y.-S., 1996, Second-harmonic generation in a LiTaO3 domaininverted by proton exchange and masked heat treatment, Appl . Phys . Lett. 68:2493. Yoo, S.J.B., Caneau, C., Bhat, R., Koza, M.A., Rajhel, A., and Antoniades, N., 1996, Wavelength conversion by difference frequency generation in AlGaAs waveguides with periodic domain inversion achieved by wafer bonding, Appl. Phys. Lett. 68:2609. Zernike, F., and Midwinter, J.E., Applied Nonlinear Optics, Wiley, New York (1973). Zheng, D., Gordon, L.A., Wu, Y.S., Route, R.K., Fejer, M.M., Byer, R.L., and Feigelson, R.S., 1997, Diffusion bonding of GaAs wafers for nonlinear optics applications, J. Electrochem. Soc. 144: 1439. Zhu, S.N., Zhu, Y.Y., Zhang, Z.Y., Shu, H., Wang, H.F., Hong, J.F., Ge, C.Z., and Ming, N.B., 1995, LiTaO 3 crystal periodically poled by applying an external applied field, J. Appl. Phys . 77:5481.

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LOW-POWER SHORT WAVELENGTH COHERENT SOURCES: TECHNOLOGIES AND APPLICATIONS

D.B. Ostrowsky Laboratoire de Physique de la Matière Condensée Université de Nice-Sophia-Antipolis Parc Valrose 06108 Nice Cedex 2, France

INTRODUCTION The object of this course is to attempt to compare three techniques that currently appear to be the most promising for the realization of relatively short wavelength visible coherent light sources furnishing a few tens of milliwatts: parametric conversion via sum-frequency generation, rare-earth doped fiber upconversion lasers, and, short wavelength diode lasers. The course will essentially concentrate on the two latter types of devices since the parametric devices have been extensively described in other courses at this school. In any case, it should be emphasized that all three techniques will no doubt prove to be viable for certain applications. The comparison will not result in a “winner take all" situation. In order to emphasize this we will begin by mentionning some of the motivations for the development of such sources. This will enable us to understand the different constraints various applications will impose. We will then describe each technique in turn and determine the type of performance each can be expected to provide. We will conclude with a description of some interesting applications using the techniques and short-wavelength sources that have been described. APPLICATIONS OF SHORT-WAVELENGTH SOURCES In this course we shall define "short-wavelength" as wavelengths below those currently produced by continuous wave diode lasers, i.e. essentially wavelengths in the green to violet portions of the visible spectrum. The applications for coherent sources in this range can be, somewhat arbitrarily, grouped into three major classes: data storage and printing, display, and scientific instrumentation. Among these applications, data

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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storage is currently the most important economic motor for device development and we will, therefore, examine it in somewhat more detail than the others. It is clear that data storage will be the main consumer application for short wavelength lasers in the forseeable future. Currently, on the order of 150 million diode lasers are sold each year for this purpose - at an average price on the order of 1$ per laser! With the red diode lasers already being used in commercial Digital Video Disks (DVD's) one attains a 4 GB per face capacity, and a demonstrated possibility of using fouoverall 16 GB capacity. This improvment over the current 650 MB CD-ROM disk we know and love is attained through a reduction by 2.2 in the interline spacing - now down to around 0.8 microns, close to the finest lines used in microprocessors, and a factor of 2 in spot size due to better quality optics and a slightly shorter laser wavlength. However, only a small advantage has been gained by the passage from the 780 to 640 nanometer diode lasers, the rest has been achieved by improved coding. So, the race is on to fabricate diode lasers in the 400 nm range, with excellent beam quality, that would permit tripling or quadrupling the capacity to the 15 GB/face range. For this application it is difficult to imagine other sources being cost effective but, since the laser currently only represents a few per cent of the entire device fabrication cost, an enabling technology, such as fiber lasers, or the frequency doubling of existant high power nearIR diode lasers, could probably compete if they offer superior performance, and attain fabrication costs of under 10 $. However, whether this will be possible remains to be seen. Displays are currently an essentially non-existant market for coherent sources, but one that has enormous potential for mass dvelopment if the enabling technologies are developped. These technologies include both the three color laser sources and adequate scanning technology, or high brightness LED arrays. On a much more pedestrian level, GaN LED's are being proposed for many more immediate applications, the paradigm being traffic lights, where good energy efficiency and long life are primordial. However, this is an application field we shall not address. The other field of applications we will address is scientific instrumentation. These are often based on fluorescence techniques, which, using short wavelength sources, has always had many scientific and technological applications. In this course we shall describe three such applcations, quite arbitrarily chosen, as examples of the use of various phenomena and devices that shall be described in the paper. These are genetic identification, a fiber optic temperature sensor, and, waveguide examination. Having briefly outlined some potential applications of interest, we will now go on to describe the three most promising types of low-power short wavelength coherent sources: upconversion fiber lasers, short wavelength diode lasers, and frequency doubled near IR diode lasers. UPCONVERSION FIBER LASERS Introduction to fiber lasers Actually, the advantages of a doped fiber structure for the realization af amplifiers and lasers was recognized and demonstrated by Snitzer as early as 19611 in work that probably began before the demonstration of the first laser by Maiman in 1960. With the maturing of laser technology, resulting in the availability of a variety of "classical" laser sources for pumps, optical fibers became a remarkable medium for the fabrication of a

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new class of lasers: the laser pumped fiber lasers. The essential advantage of fiber lasers is that of "plumbing" i.e. the fiber allows the confinement of a high density of pump and laser powers over distances impossible to attain with bulk optics. The advantages of this configuration, highly schematically shown in the following figure, are numerous.

Figure 1. Grossly simplified schematic of a fiber laser.

The essential point to note is that this configuration, in contrast to the bulk laser configuration, allows an independant choice of the pump spot size and the interaction length. This allows several orders of magnitude higher pump power concentration with correspondingly lower thresholds. Furthermore, since essentially no optical power is lost over the relatively short fiber lengths ( a few centimeters to a few 10's of meters) usually used in fiber lasers, the doping concentration can be optimized for each desired phenomenon by properly adjusting the length. It is all of these degrees of freedom that have permitted the development of a variety of "standard" fiber lasers and amplifiers in both three-level and four level atomic systems 2. By standard, we mean lasers that are pumped by wavelengths that are shorter than their lasing wavelength. In the following we shall be concerned with upconversion lasers, i.e. those that are pumped with wavelengths longer than their lasing wavelength. All of the previous advantages cited continue to apply but the phenomena involved in the excitation process are somewhat more complicated. We will outline these processses in the following sections. Introduction to up conversion In the early 1960's, before the advent of visible Light Emitting Diodes (LED's) and efficient semiconductor photodetectors in the infrared, there was a major interest in the incoherent upconversion of IR light for both displays and IR detection. In part due to a suggestion by Bloembergen3 for an infrared quantum counter (IRQC) based on what we would now call Excited State Absorption (ESA), a considerable amount of work was carried out in this direction over the following decades. A number of upconversion processes identified at that time and their respective pump power density normalized efficiencies 4 are shown on the following figure.

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Figure 2 Upconversion processes and a rough estimate of their power density normalized efficiencies per pump power density following Kushida and Tamatani, and Auzel.

These normalized efficiencies were defined as

where η = Pe / Pp and P e

is the high frequency power, and S the interaction surface, were essentially estimated from experimental data on bulk samples, available around 1970 by the authors of reference 10, without taking into account the sample lengths. What was surprising to many people was the extremely high relative efficiency of the double energy transfer processes, as first pointed out by Auzel5. Being French, he called this process Addition de Photons par Transfert d'Energie and both the acronym APTE, and the name Auzel effect, were extensively used in the literature over the following years. Today, energy transfer is the commonly used name for this process. The atomic, as opposed to dielectric, phenomena most highly investigated were ESA and energy transfer processes in which several ions were involved. In both cases the active materials of prediliction were Rare Earth (RE) ions in various hosts. While the original motivations for upconversion of that ancient era have been adequately addressed by other techniques, the scientific work has proven to be invaluable with the advent of guided wave optics, leading to the the fiber amplifiers, lasers, and sensors that are of current active interest. Since the performance of many of these devices is based on the particularities of the RE ions we will summarize the essential spectroscopic aspects of these ions before outlining some pertinent examples of the actual upconversion processes . . Rare earth ion spectra The rare earths play an extremely important role as active elements in the optical region of the spectra. Since they are the essential actors in the atomic upconversion scenarios we will develop, we present here a brief outline of their key properties. The essential characteristics of these ions in various hosts consist of a multitude of relatively fine absorption and emision lines. This is due to the fact that the optically 410

active electrons in the partially filled 4f shell are inside the filled 5 s 2 and 5 p 6 shells as indicated heuristically on the following figure.

3+

Figure 3. Highly imaginative heuristic schematization of the outer orbitales of Cr

and Nd

3+

.

They are, therefore, partially fielded from the fields in the their immediate environment and their emission and absorption lines are an order of magnitude finer than those of transition metal ions, for example as can be seen on the following figure.

Figure 4. Fluorescence bands of transition metals (Ti, Cr) and rare-earth (Nd, Er) ions.

These emission and absorption lines arise from transitions between the levels in the ground states of the elements in their stable +3 ionisation state. These levels are conventionally labelled as 2 S + 1 L J where S, L, and J represent the spin quantum number, the orbital angular momentum quantum number, and, the total angular momentum quantum number respectively. The orbital angular momentum quantum number L is historically designated by the letters S,P,D,F,G,H,I,...which correspond to L= 0,1,2,3,4,5,6,7.... respectively. Within a given LSJ level the local field raises the 2J+ 1 degeneracy leading to a number of Stark levels, which depends on the symmetry properties of the field as well as J. Even more important is the fact that the field breaks the inversion symmetry of the ion’s environment which permits transitions between

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levels that would have been parity forbidden in a symmetric field. The relative weakness of the oscillator strengths for such transitions partially accounts for the metastabilty of many of the levels. In a typical rare earth the levels will consist of LSJ manifolds having a total width on the order of several hundred cm-1 sparated from one another by several thousand cm -1 , with a typical pair being shown in the following figure.

Figure 5. Energy level for two LSJ manifolds of a rare-earth ion in a crystal field. Each manifold is split into various Stark levels

It is important to note that the splitting of the individual manifolds is on the order of kT at 300°K, which corresponds to 210 cm - 1 . This means that these levels will rapidly thermalize among themselves allowing us to consider the manifold as a single homogeneous band. It is this rapid thermalization that depopulates the upper Stark levels of the 4 I 15/2 level of Er permitting the 1.48 µ m pumping of the ubiquitous 1.54 µ m amplifier. This rapid thermalization process will also serve as the basis for a temperature sensor we shall describe later. A last preliminary, but essential, subject to be treated before entering the discussion of the excitation processes themselves, are the phonon spectra associated with the possible host materials. This will be outlined in the following section. Role of the host material phonon spectra When rare earths are used to dope a given material, the excitation process, and hence the useful lasing levels, will depend on an essential material parameter: the material’s phonon spectrum. In order to demonstrate this we will begin by discussing the system that currently appears to be the most promising for the realisation of a blue fiber laser: thullium doped ZBLAN fiber. ZBLAN is an acronym for ZrF 4 - BaF 2 LaF 3 - A1F 3 - NaF, a glass almost accidentally discovered at the University of Rennes in 1974 6 . While silica fibers have led to very performant lasers in the infrared and near infrared, it is the ZBLAN fibers that have permitted the realization of most of the upconversion based visible lasers. The reason for this lies in the phonon spectrum of the two materials. ZBLAN, a so-called “soft” glass, has a much lower average phonon energy than silica. As a result, if a rare-earth LSJ manifold is less than about 4500 cm -1 above the next lower energy, manifold, in a silica host, non-radiative phonon emission processes can depopulate the upper manifold. For the ZBLAN host the the limiting energy separation neeeded to avoid this is only about 3000 cm- 1 . This is evident from the multiphoton decay rate versus energy gap figure shown in the following figure7 . . 412

Figure 6. Phonon induced decay rate as a function of the energy gap between levels in silica and ZBLAN hosts.

As a result of this, rare-earths in ZBLAN hosts exhibit many more metastable states than in silica hosts. These states will permit both lasing and the possibility of ESA or energy transfer to higher level. This is the essential reason for the widespread use of the ZBLAN host for upconversion lasers. Having introduced the necessary background material, we will go on to discuss the main excitation and de-excitation processes in the doped fiber materials. Excited State Absorption (ESA) and Energy Transfer (ET) processes We will now outline the basic excitation and de-excitation processes that will determine the population dynamics of the systems of interest for upconversion: Excited State Absorption (ESA) and Energy Transfer (ET). For ESA the basic effect is schematized on the following figure.

Figure 7. Schematic of a typical ESA upconversion process.

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As indicated on the figure, a first pump photon is absorbed leading to an excitation of level 2 which non-radiatively decays rapidly to level 1, a metastable level. The atom in level 1 then absorbs a second pump photon to attain level 4, from which it rapidly decays non-radiatively to level 3. Upconversion occurs when level 3 is radiatively coupled to the ground state.since, despite the energy losses due to the non-radiative processes, the 3-1 fluorescence has a higher frequency than the pump photons. This is the simplest possible example of ESA. Clearly the process does not have to begin or terminate on the ground state, and higher order processes are possible, but figure 7 expresses the essential idea. It is also the process which leads to green fluorescence with Er doped silica fibers when pumping at 800 nm, and which we shall describe in more detail later. Energy transfer processes can occur between identical ions, or other ions having similar energy levels, which are in close proximity to one another. In the following figure we schematize an ET process which enables an ion to non-radiatively transfer absorbed energy to a neighbouring ion.

Figure 8. In this ET neighbouring ion

process an ion absorbs a pump photon and transfers the energy to a similar level in a

In this processs a pump photon has been absorbed exciting the ET ion to the level 1. The ET ion then transfers it’s excitation to the neighbouring ion’s level 3, simultaneously relaxing to it’s ground state. The lasing ion relaxes non-radiatively to the upper lasing level, 2. If there is a second excited ET ion in the vicinity of the lasing ion, and a corresponding excited state in the lasing ion having a level whose energy is around the transfer energy above the state 2, a double energy transfer (see fig. 2a) can take place leading, eventually to upconversion. Another multiple ion phenomena which occurs is that of quenching. This process is schematized in the following figure.

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Figure 9. Quenching of an excited state due to ET to a neighbouring ion.

In this example, typical for a four-level laser ion such as Nd , the pump photon is absorbed to excite level 3 which rapidly relaxes non-radiatively to the upper lasing level, 2. In the presence of a quenching ion the excitation can be trasferred to that ion thereby reducing the population of the excited state and lowering the pumping efficiency. Such transfers do not always have deleterious effects. They can be used to advantage, for example to depopulate a relatively long lived terminal laser transition level, or, as we shall show, to realize useful technological devices such as the molecular beacon we shall describe in the application section. Having identified and described most of the essential phenomena involved in the excitation and de-excitation of rare-earth ions we are, finally, in a posi tion to describe some fiber upconversion lasers, which we shall do in the following section. Examples of upconversion lasers While upconversion lasers had been demonstrated in pulsed and CW lowternperature versions previously , the paper of Allain et al 8 on the first CW room temperature upconversion fiber laser was a major breakthrough. This laser was based on a holmium doped (1200 ppm by weight) ZBLAN monomode fiber of around 1 m length. The fiber laser was pumped by a krypton laser operating at 647 nm and had a threshold of about 150 mw. It produced up to 10 mw of power at 550 nm and exhibited a slope efficiency of approximately 20%. While this was a very impressive accomplishment, more recent work has centered on the use of thulium doped ZBLAN and has led to efficient upconversion fiber lasers operating at 480 nm and we shall describe these lasers in more detail The first CW room temperature blue upconversion laser was reported by Grubb et al 9. The fiber used was a 1000 ppm Tm 3+ doped 2 m long ZBLAN fibre with a numerical aperure of 0.21 and an LP11 cut-off wavelength of 800 nm. The pump was a Nd:YAG laser operating on three closely spaced lines at 1112, 1116, and 1123 nm. The three step ESA upconversion pumping scheme used is shown on the following figure.

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Figure 10. Three photon ESA pumping scheme for 650 and 480 nm laser action in Tm3+ doped ZBLAN fibers.

The first version of this laser had a threshold of 46 mw of coupled pump power and a 32% slope effiicency with respect to absorbed pump power. The maximum output was 56 mw. More recently, output powers of up to 230 mW of power were demonstrated with a 1.6 W Nd:YAG pump10 and 106 mw of 480 nm light has been obtained using two diode lasers as a pump for a nearly identical fiber 11 . The threshold for the latter device (an incident power threshold of 80 mw was cited) and slope efficiency were very close to those of the configuration in reference 9. We believe this to be the most interesting frequency conversion schemes realized to date with diode laser pumping and it certainly gives a glimpse of the type of devices we might see appearing commercially in the near future. Before leaving the discussion of such lasers it is worth noting that in view of the complexity of the rare-earth spectra and the various means of obtaining upconversion, it is still possible that other combinations of ESA and/or ET will lead to viable lasers. One extensively studied possible material is Pr doped and co-doped ZBLAN which has led to dual wavelength pumped lasers 12 based on ESA, and single wavelength pumping of Nd:Pr doped fiber 13 , which could also prove to be useful. Nevertheless, since the performance of these devices is not equal to the Tm doped systems we shall not discuss them any further. They are simply cited as examples of the multiplicity of posible paths to upconversion fiber lasers in general. Nevertheless, while these lasers allow quite acceptable nm tolerances for the pump wavelength they suffer from the extremely high current prices of the ZBLAN fibers. An order of magnitude reduction will probably be necessary to make such lasers commercially viable. Having concluded this discussion of upconversion fiber lasers, we shall now go on to describe the state of the art of short wavelength diode lasers.

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SHORT WAVELENGTH DIODE LASERS In view of the applications for short-wavelength lasers, it might seem surprising that the first demonstration of a blue-green injection laser came nearly 30 years after the first such lasers realised in the GaAs system. While lasing had been demonstrated in the wide-gap II-VI semiconductors using optical and e-beam pumping the report of Haase et al 14 in 1991 of injection pumped lasing at 490 nm in ZnCdSe-Zn(S)Se was practically a surprise. Research workers had clearly identified the most significant problem as the inability to realize highly p-doped materials. This was overcome through the development of nitrogen doping using an RF (radio-frequency) plasma source which permitted attaining the 10 18 cm -3 range of dopant concentration. This allowed operation of a gain guided laser having the structure shown in the following figure.

Figure 11.

Highly simplified schematic of a ZnSe injection laser structure.

The laser functionned in a pulsed mode with a threshold of around 70 mA at a temperature of 77 K. This report led to a considerable amount of research activity throughout the world, leading to a demonstration of CW operation at room temperature by a group at Sony in 1993 15 . Currently, CW operation of such lasers has attained 100 hour levels at room temperature 16 . This limitation is apparently imposed by the formation of nonradiative centers during current injection. The degradation mechanism apparently originates at pre-existing defects which are created when the first layers of ZnS nucleate on the GaAs substrate. Incorrectly aligned atoms form stacking faults which expand during subsequent crystal growth to become so-called dark-line defects which then propagate to in or near the active layer. In the active quantum well region these faults present sites for non-radiative recombination of electrons and holes which leads to a release of around 2.5 eV of energy, a level capable of creating further defects, and hence, rapid degradation. A major part of the research on this system consists of characterizing, in the hope of eventually eliminating, the defect nucleation sites. However progress at this time seems to be more rapid using another class of mateials based on GaN and it's alloys. It was only in 1994 that the extensive research on 417

the Al-Ga-In-N alloy family led to dramatic results- the commercialization by Nichia, a Japanese company, of extremely bright LED's emitting in the green to ultraviolet range. This was followed by the announcement by the same group of injection laser operation under pulsed conditions in late 1995 17 and a CW room temperature version emitting 1.5 mw per facet around 420 nm and lasting 35 hours, with operating currents between 100 to 280 mA, in 1996 18 . The type of structure used is highly schematically shown in the following figure.

Figure 12. Highly simplified schematic of a GaN injection laser.

Obvious differences with other laser structures include the fact that the substrate is an insulating crystal. This leads to the necessity of depositing the n electrode at the side of the structure and adapting the buffer layer to the sapphire structure. While these lasers still have threshold current densities on the order of 3 kA/cm2, an order of magnitude greater than GaAs lasers, the defect structures, which are proving difficult to eliminate in the II-VI lasers, appear to be far less deleterious in the nitrides. The technology is continuing to develop, with a strong push coming from the display applications for which GaN LED arrays are already commercially available. This combination of an existing market (display) as well as the promise of blue lasers seems to give a strong advantage to the GaN family in the ongoing race towards reliable short wavelength diode sources. PARAMETRIC SOURCES The parametric process that has been most developped for the practical generation of CW blue light is Second Harmonic Generation (SHG). For the power levels we are considering, on the order of 10 mW, the Quasi-Phase-Matched (QPM) waveguide configuration, realized in lithium niobate or tantalate, described in the lectures of Professors Stegeman and Fejer at this institute, appears to be the most promising. Since the underlying phenomena and theory have been developped in those courses we shall only present the briefest of outlines here before proceeding to a comparison with the other techniques. The basic QPM waveguide configuration is shown on the following figure.

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Figure 13. Basic waveguide QPM configuration for SHG.

The essential idea is that the periodically inversed domains, and hence signinverted nonlinear coefficients, correct the π dephasing developped between the nonlinear polarization and the propagating wave at the harmonic frequency. This clever idea, first proposed by the group of Bloembergen 19 was demonstrated in waveguide form in 1989 by groups at Stanford 20 and the Institute of Optical Resarch in Stockholm 21 . Extensive analyses of the tolerances on periodicity, waveguide regularity, pump wavelength spectrum, and temperature variations have been carried out by the two groups 22,23 . In reference 23 the authors presented a very useful formula giving the following first-order Taylor series expansion approximation of the full width at half maximum (FWHM) acceptance bandwith of the SHG efficiency for an arbitrary parameters variation: where ξ is the arbitrary parameter and ∆ k = k 2 – 2k 1 – K m with K m the inversion period for the m th order QPM being used For the frequency doubling of 860 nm light this indicates that the FWHM tolerances for a 1 cm long sample in lithium niobate are approximately those given in the following table. parameter

δ λ (nm) .07

δT(°K) 1.4

δυ z (mrad) 11

δυ o (mrad) 10

Table 1 FWHM acceptance bandwidth for doubling of aan 850 nm source with a 1 cm long QPM lithium niobate saample. δυ z and δυ o represent angular deviations of the sample cut around the z and ordinary axes respectively.

While such tolerances appear rather draconian, a recent development in the field of diode laser pumps should allow satisfying these conditions. The device in question is an electrically tunable, high power, single frequency diode laser fabricated by SDL24 . The stucture consists of a Distributed Bragg Reflector (DBR) configuration with separate electrodes allowing independant current injection in the gain and DBR regions. Injection of current in the DBR region locally heats the grating leading to an increase of the emittted wavelength. The device can be tuned over a 10 nm range in 0.08 nm steps, the steps being due to the longitudinal mode separation in the cavity. Such devices have 419

already demonstrated over 20,000 hour lifetimes, in ongoing tests, while emitting 200 mW continuously. That such a device, in conjunction with a lithium niobate QPM doubler, is a viable source for optical disk recording is underscored by the recent (June 9, 1997) announcement by Panasonic, of the realization of a complete recording head based on this technology. The source, using an X-cut MgO doped periodically poled lithium niobate crystal 25 with an overlay of Nb 2 O5 provides 15 mW of power at 425 nm without demonstrating photorefractive effects. The direct modulation permitted by the source allowed the demonstration of the reading and writing of optical discs corresponding to a 15 GB/face capacity, roughly a 4-fold improvement over existing technology. In conclusion, in view of the performance of the system reported by Panasonic, it looks like near IR diode laser pumped QPM parametric generators could become the first example of a nonlinear optical mass consumer application. If this should come to pass, the repercussions would be enormous for the field of nonlinear optics as the market would generate the means for an enormous expansion of related research and development. OTHER EXAMPLES OF APPLICATIONS In this section we shall describe three examples of techniques and devices based on the various excitation and de-excitation phenomena we have presented. To underline the sort of pleasant surprises science can afford us we will begin with two examples based on what are usually considered to be nefast phenomena. The first is a fiber optic temperature sensor, the second is a molecular beacon technique used for genetic identification. We shall conclude with a description of a fluorescent waveguide examination technique. An ESA based auto-referenced silica fiber optical temperature sensor In this section we will outline, as an example, a highly performant sensor prototype 26 , based on ESA in Er doped silica fiber, that will illustrate some of the particularities one can encounter in rare-earth spectroscopy. The ESA in question is the green fluorescence observed when pumping erbium doped fibers with 800 nm light, and is the essential reason that such pumps cannot be used for Er doped fiber amplifiers. The ESA phenomena used is schematically shown in fig 14.

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Figure 14. The 550 nm green fluorescence due to ESA of 800 nm light

A closer examination shows that the upper level of the green fluorescence is in fact two levels that are thermally quasi-isolated from the rest of the atomic system. These two levels, which thermalize rapidly between themselves, give rise to two overlapping components of green fluorescence as shown in the following figure.

Figure 15. Two component green fluorescence due to Er ESA

A measurement of the intensity ratio of the two lines allows determining the temperature. An important point to note is that the measurement is based on an intensity ratio eliminating the need for calibration. Another important point to note, as shown in the figure inset, is that the intensity coming from the small (3%) population of the upper excited state level at ambiant temperature is compensated by its 20 times larger emission cross section, resulting in roughly equal total fluorescence for the two components. It is this gift of nature that allows the temperature sensor to have an 421

extremely large dynamic range.A schematic of the experiment carried out to verify this, and the experimental results, are shown in the following figures.

Figure 16. Experimental schematic of the ESA based temperature sensor.

In this set-up the doped fiber, typically 50cm long, was spliced to a 20 meter length of standard telecommunication fiber. Power levels in the 50 to 200 mW range, at 850 nm were supplied by the Ti:saphire laser, which could be replaced by commercially available diode lasers.

Figure 17. Experimental results for the ESA based fiber temperature sensor.

This sensor can also be excited by an energy transfer process using Yb-Er codoped fibers and a 980 nm pump and quasi-distributed, rather than point sensors, can be realized by analyzing the ratio of 1130 and 1240 nm lines which also appear when pumping at 800 nm 27 , but a discussion of this configuration would take us too far afield from the subject of upconversion.

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Molecular beacons The molecular beacon is another application of blue light that is based on an apparently nefast phenomenon: fluorescence quenching by a neighbouring ion. This phenomenon is one in which an excited atom, capable of fluorescing, transfers it’s energy non-radiatively, to another ion, which then decays, also nonradiatively, to it’s ground state. This sort of quenching is common in the rare-earths and is the sort of process that limits the useful doping levels to a few per cent. However, such a phenomenon has been put to good use in a "device" for genetic identification: the molecular beacon 28 . The structure of such a beacon is shown in the following figure.

Figure 18. A molecular beacon consisting of a single strand of DNA folded back upon itself with a fluorescent (F) and quencher (Q) molecule attached at the extremity.

The letters A,C,G, and T represent the adenine, cytosine, guanine,and thymine molecules attached to the DNA strand. Adenine links to thymine and guanine to cytosine to form the famous double helix. In the beacon extremity the complementary molecules are used to close the structure and fluorescent and quencher molecules are attached to opposite sides of the strand. Commonly used molecules are fluorescin for the fluorophore and DABCYL for the quenching molecule. As long as the extremity of the structure is closed, the proximity of the quenching molecule will inhibit fluorescence when the beacon is illuminated with blue light. When such a structure is placed in a solution containing single strands of DNA it will move about due to thermal fluctuations, and attach itself to the normal, much longer DNA strands if it finds a complementarily coded segment. Since the longer segment is much more rigid, attachment will force the closed end of the beacon open, separating the F and Q molecules, allowing fluorescence, and hence localisation of the researched segment. Such beacons allow the discrimination between segments that differ from one another by only a single nucleotide 29 . This technique is expected to play a major role in future research on gene sequence identification and manipulation. Having concluded our two example of good uses of usually nefast phenomena, we will go on to describe our last example of blue light or upconversion processes, waveguide examination by fluorescence.

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Waveguide examination by fluorescence While calculations give beautiful graphics concerning the evolution of fields in integrated optical devices it is rather difficult to directly observe such fields. What one usually observes when looking at a waveguide are scattering centers which do not yield a reliable image of the fields. A technique that has been demonstrated for overcoming this is to coat the guide with a low index plastic film doped with a fluorescent material and observe the emitted fluorescence induced by the evanescent wave. This fluorescence, which is directly proportional to the local guided wave power, gives an excellent map of the field 30 . A schematic of the experimental setup is shown on the following figure.

Figure 19. Schematic of the waveguide observation by fluorescence set-up.

In the first experiments demonstrating this effect the guide was realized using polyphenylsiloxane (PPS), an electron resist material that allowed direct e-beam writing of guides 31 . In these experiments a structure consisted of an 800 nm thick PPS guide with an index of 1.565 at 632 nm, covered with a 300 nm polymethylmethacrylate (PMMA) film with an index of 1.49, doped with 6% by weight of Rhodamine B. This molecule exhibits an interesting anti-Stokes fluorescence line (an upconversion process we have not discussed) at 595 nm when pumped with a He-Ne laser. In the structure described approximately 10% of the guided light propagates in the evanescent field within the doped film and 3 mW of guided light leads to a fluorescence power on the order of 10 -10 W/µm 2 emitted from the guide surface which is a detectable power level. This allowed, therefore, field maps to be made in varying structures such as couplers, Y-junctions and curved guides. In order to observe the fields propagating in lithium niobate and tantalate guides, which, due to their higher indices, only have on the order of 1% of the guided light in the evanescent field, it was necessary to dope with Rhodamine 6G and pump with a blue line of an argon laser. This yielded equivalent fluorescent power and allowed the observation of fields propagating in these materials. A chosen few have even seen a film made with this technique that demonstrates the observation of switching in electrically controlled lithium niobate directional couplers 32 . Since this direct observation of switching is as close to the main thrust area of this institiute as we shall get in this contribution, we choose to draw the curtain here.

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CONCLUSION In this course we have described the current state of the art concerning three technologies being used to realize low-power coherent short wavelength sources: upconversion fiber lasers, diode lasers, and second harmonic generation. We have also gone at some length into describing three rather original applications for such sources and their technology: a fiber optic temperature sensor based on upconversion, a DNA recognition technique, and a waveguide examination technique, both of thelatter using short wavelength induced fluoresence. We hope to have shown that this is a rapidly evolving field, involving pluridisciplinary interactions, that has led to important results, and will, in the future continue to provide, as it has in the past, pleasant surprises. The author wishes to acknowledge fruitful discussions with P. Gibart concerning GaN and ZnS lasers, Jean-Pierre Lehureau, concerning compact disc technology, Albert Libchaber and Gregoire Bonnet, concerning molecular beacons, Jean-Paul Pocholle, Gerard Monnom and Bernard Dussardier concerning rare-earth doped fibers, and Arnaud Grisard, Marc De Micheli and Pascal Baldi concerning parametric sources. The author alone is totally responsable for any errors introduced as well as the opinions offered.

1 E. Snitzer, "Optical maser action of Nd 3+ in barium crown glass", Phys. Rev. Letts. 7, p 444, (1961) 2 J. R. Armitrage, Optical Fiber Lasers and Amplifiers, ed P.W. France, p14- 49, CRC Press (1991) 3 N. Bloembergen, Phys. Rev. Letts. 2, 84 (1959) 4. T. Kushida and M. Tamatani, "Conversion of infrared into visible light", supplement to J. Japan Soc. Appl. Phys. 39, pp 241-247, (1970), F. Auzel, "Materials and devices using double-pumped phosphors with energy transfer", Proc. I.E.E.E. 61, pp 758-786, (1973) 5 . F. Auzel, "Compteur quantique par transfert d'énergie enttre deux ions de terres rares dans un tungstate mixte et dans un verre", C.R. Acad. Sci.262, pp 1016-1019, (1966) 6 M. Poulai, M. Poulain, and J. Lucas, Mat. Res. Bull. 10, 243 (1975) 7 Zheng H. and Gan F. Chinese Phys. 6,978 (1986) 8 J.Y. Allain, M. Monerie, and H. Poignant, " Room temperature CW tunable green upconversion holmium fibre laser", Electron. Lett. 26,261 (1990) 9 S.G. Grubb, K.W. Bennett, R.S. Cannon and W.F. Humer, CW room-temperature blue upconversion fibre laser", Electron. Lett. 28, 1243 (1992) 10 R. Paschetta, N. Moore, W.A. Clarkson, A.C. Tropper, D.C. Hanna, G. Mazé, 230 mWof blue light from thulium:ZBLAN upconversion fiber laser, Proceedings CEO 97 paper CTuG3 p 80 (1997). 11 S. Sanders, R.G. Waarts, D.G. Mehuys, and F.D. Welch, "Laser diode pumped 106 mW blue upconversion laser", Apl. Phys. Lett. 67, p 1815, (1995). 12 R.G. Smart, D.C. Hanna, A.C. Tropper, S.T. Davey, S.F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991). 13 S.C. Goh, R. Pattie, C. Byrne, and D. Coulson, "Blue and red laser action in Nd3+: Pr3+ co-doped fluorozirconate glass", Appl. Phys. Lett., 67, 768 (1995). 14. M.A. Haase, J. Qui, J.M. Depuyt, and H. Cheng, Appl. Phys. Lett. 59, p1272, 1991 15 N. Nakayama, et al, Electron. Lett. 2, p. 2194, 1993 16 S. Tanaguchi, etal, Electron. Lett. 32, 552, 1996 17 . Nakamura, S., et al , InGaN multi-quantum-well structure laser diodes, Jpn. J. Appl. Phys. 35, L74 (1996) 18 Nakamura, S. Characteristics of InGaN multi-quantum-well structure laser diodes, Mater. Res. Sec. Soc. 449, : 1135 (1996) 19 J.A. Arrmstrong, N. Bloembergen, J. Ducuing, and P.S. Pershan, Interactins between light waves in a nonlinear dielectric, Phys. Rev. 127, 1918 (1962) 20 E.J. Lim, M.M. Fejer, R.L. Byer, and W.J. Koslovsky, Blue light generation by frequency doubling in a periodically poled lithium niobate channel waveguide, Electron. Lett.25 pp 731-732 (1989). 425

21 J. Webjorn, F. Laurell, and G. Arvidsson, Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide, IEEE Photon. Technol. Lett 1 pp 316-318 (1989) 22 S. Helmfrid and G. Arvidsson, Influence of randomly varying domain lengths and nonuniform effective index on second-harmonic-generation in quasi-phase-matched waveguides, J. Opt. Soc. Am. B. 8,797 (1991) 23 M.M. Fejer, G.A. Magel, D.H. Jundt, and R.L. Byer, Quasi-phase-matched second harmonic generation tuning and tolerances, IEEE J. Quantum Electron. 28,2631 (1992) 24 V.N. Gulgazov, H. Zhao, D. Nam, J.S. Major, and T. Koch, Tunable high-power AlGaAs distributed Bragg reflector laser diodes, Electron. Lett. 33, 58 (1997) 25 K. Mizuuchi, K. Yamamoto and M. Kato, Harmonic blue light generation in X-cut Mg0:LiNbO 3 waveguide, Electron. Lett. 33, 806, (1997) 26 E. Maurice, G. Monnom, B. Dussardier, A. Saissy, D.B. Ostrowsky, and G.W. Baxter, Erbium-doped silica fibers for intrinsic fiber-optic temperature sensors, Appl. Optics 34, p 8019, (1995) 27 E. Maurice, G. Monnom, D.B. Ostrowsky, and G.W. Baxter, 1.2-µm transitions in erbium-doped fibers: the possibility of quasi-distributed temperature sensors, Appl. Optics, 34, p 4196, ( 1995) 28 S. Tyagi and F;R; Kramer, Nature Biotechnol, 14,303 (1996) 29 G. Bonnet, S. Tyagi, F.R. Kramer, and A. Libchaber, Molecular beacons for probing information in DNA, to be published 30 D.B. Ostrowsky and A.M. Roy, Visualisation de la propagation dans un guide d'onde optique par fluorescence anti-Stokes, Revue tech. Thomson-CSF, 6,973 (1974) 31 D.B. Ostrowsky, M. Papuchon, A.M. Roy, et J. Trotel, Electron beam fabrication using an electron sensible film, App. Opt. 13 , p, 636 (1974). 32 M. Papuchon, B. Puech, C. Puech, and D.B. Ostrowsky, A movie on the visualization by fluorescence of the electrically controlled directional coupler, Paper Tu A2, Proceedings of the Topical Meeting on Integrated and Guided Wave Optics, Salt Lake City, (1978)

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ARTIFICIAL MESOSCOPIC MATERIALS FOR NONLINEAR OPTICS

C. Flytzanis Laboratoire d’Optique Quantique, Ecole Polytechnique 91128 Palaiseau cédex, France

INTRODUCTION The mesoscopic materials which have dimensions intermediate between those of the bulk material and its constituent molecular units are becoming an important and multifaceted issue in materials science and in a wide range of important technological applications. Whenever the size of such systems becomes comparable to a characteristic physical parameter with a dimension of length these mesoscopic particles and their ensembles reveal properties which can be markedly different from those of the bulk as well as of its constituent molecular units. In certain classes of composites formed with mesoscopic particles of cystalline materials like transparent dielectrics, conjugated polymers, semiconductors or metals this is strikingly revealed in their optical and spectral features and a fortiori in their nonlinear optical properties. The characteristic lengths we have in mind here are such as the optical wavelength, the electron delocalization, the Bohr radius or the electron mean free path. The observed size dependence of the optical properties of such mesoscopic systems can be traced to the change of the confinement regime that the photons or electrons undergo as their size approaches any of these characteristic lengths. This leads to a modification of the relevant density of states and the appearance of morphological resonances that profoundly affect both the magnitude and dynamics of the optical nonlinearities. It also provides a way of artificially controlling these nonlinearities by controlling the size and interface of these mesoscopic systems during the fabrication procedure. Here we shall present a succinct discussion of the optical nonlinearities in mesoscopic systems and the way they are affected by the confinement. The emphasis will be on the so called photoinduced nonlinearities in particular the optical Kerr nonlinearity both because it is bound to play an important role in all-optical beam reshaping and control and also because the confinement has its strongest impact there

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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since it is multiresonantly enhanced. After reviewing some general aspects of these nonlinearities and the mesoscopic systems in the following section we present a general discussion of the impact of the electromagnetic or photon confinement on the optical nonlinearities of the mesoscopic dielectric materials and subsequently that of the electron confinement which is manifested either as dielectric or quantum confinement and is illustrated with the metal and semiconductor nanocrystals respectively.

OPTICAL NONLINEARITIES AND MESOSCOPIC SYSTEMS. Efficiency of nonlinear optical effects The nonlinear optical effects in a medium have their origin in the induced polarization sources nonlinear in the field amplitude as they appear in the development (1,2) (1) and are classified according to the power of the field amplitudes and frequencies, wavevector and polarization vector configurations ; the above development is valid as n <Ec and χ (n) ~ 1/E c where Ec is a cohesive electric field for the outer shell long as electrons in the medium. The efficiency of the resulting nonlinear effects is certainly determined by the behavior(3) of the effective nonlinear susceptibilities χ(n) at the chosen configuration but also by geometrical and temporal features connected with the evolution of the fields amplitudes which in the simplest stationary plane wave regime in the slow varying envelope approximation are obtained by the nonlinear one-dimensional propagation equation(1.2) (2) where is the appropriate nonlinear polarization source at a frequency ω which stores and transfers energy to the field of amplitude A = ê ê is the unity polarization vector, λ the optical wavelength and n0 the refractive index all at frequency ω ; ∆k = is the wavevector mismatch (or equivalently ∆ kz is the phase mismatch) between the wavevector of the induced nonlinear polarization and the wave vector that can propagate in the medium at frequency ω for the chosen configuration namely k = ω n/c. This wavevector mismatch is a manifestation of nonlocality and introduces severe selectivity in the growth of the nonlinear process since it limits its growth over a distance = π/∆ k, the coherence length. If the phase mismatch can be compensated naturally by exploiting the linear(4) or circular (5) birefringence of the materials or artificially in the so called quasi-phase matched configuration it is quite clear from (2) that the efficiency of the nonlinear optical process will be enhanced either by increasing the optical nonlinearity of the medium as defined in (1) or by increasing the interaction length as defined by (2) when ∆ k = 0 still keeping the geometrical dimensions of the material small. A key role in all-optical beam reshaping and control is played by the optical Kerr nonlinearity which is related to the cubic nonlinear optical polarization (2,6)

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(3) stands for an effective third order susceptibility and the frequencies ω' and ω where can be different, or ω' can be equal to ω or zero. Note that this effective susceptibility besides the direct contribution also contains indirect(3,7) ones because of retardation (or cascading effects) and may consequently depend on the wavectors as well. The general case ω ≠ ω ' corresponds to the modification of the optical characteristics of a beam at frequency ω by another beam at frequency ω ' while the other two cases ω' = ω and ω ' = 0, correspond to the beam self action and static Kerr effect (or Kerr electromodulation of a beam) respectively. In all these cases one can define an effective complex refractive index (4) for the case where ω = ω ' where I = (n 0c / 2π) and n 2 = and similarly for ω ≠ ω '. The real part of n2 (or χ (3) ) affects the dispersion while the imaginary part of χ(3) can be associated with additional two photon losses or gain in the medium. Along with the magnitude and phase of n2 in the stationary regime a key role in the envisaged applications of the optical Kerr effect is also played by its temporal evolution (2,6) of n 2 which for many purposes can be modeled by the Debye equation (5) or (6) where τ is a decay time related to energy or population relaxation. In certain cases one introduces figures of merits to assess the materials as regards the optical Kerr effect and one such figure of merit is (7) where α ω is the absorption coefficient at the operating frequency ω. It is evident from its definition that this refers to the nonlinearity per photoexcited electron. Other figures of merit can also be defined according to the application one has in mind but in general one should be cautious regarding the conclusions that can be drawn from the figures of merit in general. The key issue of nonlinear materials science is clearly to improve the efficiency of optical nonlinearities taking into account several other aspects like optical losses, photochemical stability, processing, and interfacing or doping, miniaturization and other important fabrication and stability constraints and these aspects cannot be easily incorporated in such figures of merit. Mesoscopic systems. Composites The efficiency of a large class of nonlinear optical effects can be enhanced by exploiting either the photon or electron confinements. As will become shortly evident the two confinements cannot be simultaneously implemented in the same material in the optical range as they address material aspects incompatible with each other there. The

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first one is achieved by spatially confining(8,9) the interacting optical beams in guides or cavities (resonators) whose minimal dimension is of the order of the optical wavelength λ so that the electromagnetic modes can be resonantly accommodated in space and through multiple reflections effectively increase their interaction path ; this is the electromagnetic confinement and can be achieved in highly transparent nonlinear optical media where the nonlinearities involving very localized electrons, even for the most favorable cases are weak but the absorption losses can be kept minimal and the beam coherence is preserved over long propagation distances. The second one consists in enhancing the optical nonlinearity of materials with very delocalized valence electrons, like in metals, semiconductors or conjugated polymers, by artificially confining the valence electrons in regions much shorter than their natural delocalization length in the bulk, which can extend over many units cells or even infinity ; this is the case of the electron confinement (10) and its most conspicuous feature is the appearance of discrete optical resonances whose position, oscillator strength and dynamics depend on the extension of the artificial electron confinement and hence can be externally modified to meet certain requirements. Referring to our previous definition of the cohesive field E c in connection with (1) this amounts reducing E c without substantially altering the cohesion of the material. In the optical range we are interested here the electromagnetic confinement(11) concerns dimensions of the order of the optical wavelength λ namely a few hundreds nanometers (nm) up to micrometer (µm) while the electron confinement concerns(10) dimensions of the order of the electron Bohr radius or delocalization length namely a few nanometers up to several tens nanometers. These are the dimensions of the mesoscopic systems, organic or inorganic, that are increasingly being studied in modern science and technology. Besides their intrinsic features due to their finite mesoscopic size their interface with the surrounding media as well as the doping are also of crucial importance in these considerations and play a very important role in many applications. Accordingly appropriate fabrication techniques have been and are being developed to artificially grow such particles and control their size and form as well as their doping and interfacing with the surrounding medium ; the role of the later is not simply to support the fine mesoscopic particles but also to endow them with certain properties that are very important for applications. The particles that exploit the electromagnetic or photon confinement and have sizes of the order of the wavelength λ or smaller are made(11) from high quality transparent amorphous dielectrics, polymeric or inorganic ones ; they can have spherical or ellipsoidal shapes and are uniformly dispersed in a another transparent dielectric amorphous matrix, solid or liquid. Their concentration and spatial arrangement in the matrix depend on the effects one wishes to enhance. Periodic arrays of such particles, the so called colloidal crystals can also be obtained by different techniques in particular optical ones whereby the particles are displaced and positioned by interfering optical beams. The fabrication of such spherical dielectric particles of well calibrated shape and size is now well controlled and extensively used(12,13) in a wide range of applications ranging from efficient light scatterers, high quality paints and laser microcavities to name a few. In some of these applications in addition one needs to wrap these particles with an appropriate film or layer while in others to appropriately dope them with specific molecular or atomic ions. Whatever the case may be the underlying material is a high optical quality amorphous dielectric with high featureless transparency region that can extend up to the near UV to keep absorption losses as low as possible and allow the electromagnetic waves to undergo many reflections without losing their coherence.

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(14) The particles that exploit the electron confinement on the other hand and have sizes up to a few tens of nanometers are made out of crystalline materials with very delocalized electrons such as covalent semiconductors or metals. The confinement here modifies the electronic density of states distribution and by the same token that of the optical oscillator strength distribution and relaxation processes; both play an essential role in shaping the linear and nonlinear optical properties of these materials and consequently determine their potential use in microoptoelectronics and other areas. Another feature of much interest, recurrent to all composite materials obtained by interfacing materials of different chemical constitutions and more or less mobile electrons, is the interfacial charge transfer and similar processes that may take place there and can be exploited to produce new functionality(15) artificial materials. The later aspect clearly involves quite sophisticated chemical as well as physical considerations but the former, namely the confinement, can be addressed and its main consequences followed up to a large extent by physical considerations regarding the electron behavior in confined geometries. Several approaches exist for preparing such mesoscopic systems, metal(16) or semiconductor (17) nanocrystals uniformly embedded in a transparent liquid or solid matrix. For the case of semiconductor nanocrystals the most extensively studied(14,18) ones are the Cd(S,Se)-doped glasses which are used as sharp cut-off filters in the yellow to red part of the spectrum and are available from several glass makers. To make such a semiconductor-doped glass (SDG), one usually starts adding the semiconductor constituents (or their oxides) to a melt of alkali-silicate glass at very high temperature, typically 1300°C. The melt is then rapidly cooled. At this point, the glass is still almost colorless, nucleation of Cd(S,Se) particles being still at an early stage. The glass is then annealed at a temperature lower than 500°C in order to produce a stress-free optical quality glass. It then undergoes what is known as the striking process: it is further heated to a temperature of 500-700°C during which the particles grow. Using such a technique, CuCl-, CuBr-, CdS-, CdSe- and CdTe-doped glasses have been made and also GaP but in general this procedure has produced poor results for the III-V compounds because of the high volatility of the constituent ions. The growth of the particles is usually thought to be dominated by the ripening process. In the ripening stage, the volume fraction p occupied by the particles remains constant, the bigger particles growing at the expense of the smaller ones by atomic diffusion through the glass matrix. Particle growth in this ripening stage has been studied theoretically by Lifshitz and Slezov(19). Assuming spherical particles, their average radius is given by :

(8) where s is the surface tension in the interface, D is the diffusion constant, c is a constant that depends exponentially on the striking temperature and t is the duration of the striking process. Particle growth being controlled by a diffusion process, this inevitably leads to a large size dispersion. The expression of the size distribution has also been given by Lifshitz and Slezov. One can achieve better size control and eventually very narrow size dispersion in colloidal solutions where in addition one can control the particle interface by covering it with organic or inorganic layers. Metal nanocrystals, (mainly the noble ones like Cu, Ag, Au, Pt) in glasses or other transparent amorphous dielectrics are produced (16) by somewhat different approaches

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that schematically involve four steps : solution of noble metal salts, reduction, dialysis for the removal of foreign ions and solidification and eventually coating and solidification. The coating process is used in order to achieve high volume concentrations since otherwise uncoated particles tend to agregate at concentrations higher than 10- 5. Colloidal solutions are similarly obtained with very narrow size distribution. The most direct way of measuring the particle size and shape and the size distribution is electron microscopy(20), but it is a time-consuming and destructive method. Inelastic light scattering can also be used (21) to determine the average size of the nanoparticles. X-ray diffraction gives access to the crystalline structure as also does to some extent the EXAFS technique. The main conclusions drawn from these structural characterization studies is that the particles in the glass matrix behave(20) as nanocrystals with the same lattice constant and symmetry as the bulk and show facets compatible with their crystalline symmetry like large crystals do. For most purposes regarding the electron confinement however they can be regarded as almost spherical. The size may also be obtained using small angle X-ray or neutron scattering(22) which are non destructive tools. The absence of scattering at very small angles indicates that each particle is surrounded by a depletion zone. There are several indications that these nanocrystals are defect free and in particular exempt of impurities. The average crystallite size can cover a wide range of values, from several micrometers down to a few nanometers, with more or less narrow size distribution depending on the preparation technique and the quality of the interface with the surrounding dielectric that can substantially differ from case to case. The later has much relevance on the "surface" features but usually for sufficiently large nanocrystals containing thousand or more atoms corresponding to particle diameters larger than a few nanometers we may simply assume that the interface with the surrounding dielectric is sharp and of infinite potential height. The volume concentration can also vary over a wide range down to concentrations as low as 10- 6 or up as high as several percent which allows one to study single particle features as well as features that arise from interparticle interactions. In certain cases ordered arrays of such particles have been achieved by physicochemical or artificial techniques, for instance self-organization on semiconductor or polymeric substrates or artificial nanofabrication techniques involving particle or photon beams ; these are still in an experimental stage and mostly concern two dimensional arrays. Here we shall be concerned with metal or semiconductor particles uniformly and randomly dispersed in a glassy or other transparent dielectric with concentrations low enough that interparticle excitation transfer can be neglected and only the electron confinement within such a single particle embedded in the glass has to be considered.

ELECTROMAGNETIC CONFINEMENT AND NONLINEAR SCATTERING A highly transparent amorphous dielectric has very low optical Kerr nonlinearities but because of the low absorption and scattering losses there the cumulative effect of the nonlinear interactions over long distances can give rise(23) to sizable effects when the phase mismatch is not an issue. Classical examples here are those of the temporal and spatial optical solitons which are extensively discussed elsewhere in this volume. Actually it is not necessary for the light to propagate long distances to benefit from the cumulative effect of the optical interactions; indeed multiple passages over a small spatial region of few optical wavelengths λ in size can effectively provide sufficiently

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long interaction lengths for the nonlinear optical processes to fully develop. In the optical frequency range this can be achieved(12,24) by electromagnetic confinement in a few micrometer size spherical dielectric particles in air or eventually in another transparent dielectric of lower refractive index. The multiple reflections of the radiation inside such a single dielectric particle provide efficient optical feedback to enhance the nonlinear optical processes. Actually even in an assembly of randomly or periodically distributed dielectric microparticles of sizes of the order or less than the optical wavelength λ embedded in another dielectric one may expect comparable feedback and enhancement of the nonlinear optical processes this time through multiple reflections and scattering off such small nanoparticles. Such internal or external multiple reflections and nonlinear scattering which actually preserve a certain coherence can give rise to some quite striking effects that can be controlled either by the particle size, their concentration or spatial distribution and average interparticle distance. We shall briefly discuss below some aspects of these effects without going into quantitative considerations which need a detailed description of the electromagnetic propagation in confined space or in random media. Single dielectric microparticle. Electromagnetic confinement. A spherical dielectric particle or a liquid droplet with a size parameter κ = 2 πa/ λ, a being its radius, and refractive index n0 acts as a high Q optical cavity at specific wavelengths corresponding to the normal electromagnetic modes of a dielectric sphere (or morphology dependent resonances) which can be calculated (25) using the MieLorentz theory. These occur at discrete size parameters κm,l where m and l are the mode number and order respectively ; they possess a (2m+1)- degeneracy. For a spheroid with polar and equatorial radii ap and a e respectively the (2m+1)-degeneracy of the sphere is lifted and the mode frequencies are shifted with respect to those of the sphere of radius by amounts that can be calculated by perturbation theory when the distorsion e = |a e - a p| /a is small (e<<1). Their Q can be made very large and the radiation lifetime τ = Q/ω where ω is the mode frequency. The morphology dependent resonances provide(24) efficient feedback and enhance several nonlinear optical processes such as stimulated Raman and Brillouin scattering, nonlinear Mie scattering, self-phase modulation, third order harmonic generation and others ; when the particles are doped they provide very efficient stimulated emission and lasing action. Actually these resonances can be modified by processes such as electrostriction and the nonlinear optical processes being very sensitive to the resonance characteristics can be used as very efficient diagnostics to study such distorsions and other features related to the particle form and surface. In all these cases the new optical waves generated inside the dielectric microparticle through the cumulative feedback of the nonlinear optical processes adopt the spatial distribution of the morphology dependent resonances and eventually leak out from the rim of the dielectric in different directions and not in a single well defined direction as is the case in nonlinear interactions in a homogeneous dielectric. Several features affect their efficiency that are not present in the one-dimensional wave propagation (1) usually considered inside a homogeneous dielectric. Thus the processes occur only when the input laser wavelength coincides with one of the morphological resonances and similarly for the emitted frequency in the case of the stimulated processes. The spatial distribution of the modes strongly affect the overlap of the

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nonlinear optical waves and the optical feedback as also does the leakage and the Q of the resonance. The study (24) of the nonlinear optical processes in such microparticles besides providing a very elegant demonstration of such processes in a micrometer size dielectric it also provides well controlled means to efficiently enhance these processes and offers the possibility to exploit them in certain applications. Two-component dielectric composite In the previous case the multiple reflections of the radiation and its "trapping" inside the rim of a single dielectric microparticle provided efficient optical feedback to enhance the nonlinear optical processes. Under certain conditions such a feedback and efficient enhancement of nonlinear optical processes can also be achieved in the reversed configuration outside the particles namely through external multiple reflections and scattering off these particles in an orderly or disorderly distribution in a solid or liquid dielectric of different refractive index and nonlinear Kerr coefficient. Photonic band gap composites. Quasi-phase matching. In the case of ordered or periodic distribution of such particles or voids in a transparent dielectric one has a photonic band gap dielectric or semiconductor(26) and several possibilities of nonlinear processes can be envisaged. Any optical coefficient χ here can be assumed periodically modulated and written as a Fourier series (9) where R and K are vectors in the real and reciprocal space respectively R being the position vector of the nanoparticles in the periodic array with interparticle distances (lattice constant) of the order of the optical wavelength λ. Accordingly one has a situation analogous to the one prevailing(27) in X-ray scattering in a crystal or in the electron band states in a crystalline semiconductor. The optical waves can now be generated in directions fixed by Bragg von Laue type conditions in the optical range. The quasi-phase matching configuration (1,28) extensively discussed elsewhere in this volume is actually a one-dimensional realization of such a periodic array for second order processes and similar ones can be envisaged in two or three dimensions with quadratic or cubic nonlinearities. Actually in the previously considered case of multiple reflections inside a single dielectric microparticle one can unfold the nonlinear optical processes to occur in an equivalent periodic array and find several similarities between the two cases. Random composites. Here we shall concentrate our attention on the case of a random distribution of such dielectric micro or nanoparticles (component A) in a transparent dielectric (component B) of different refractive (nA ≠ nB) and optical Kerr ( n 2A ≠ n 2B ) indices. Light scattering is the dominant process here and the optical coefficients cannot be written as in (9). In the frequency preserving (elastic) scattering in the Rayleigh and Rayleigh-Gans-Debye regimes, ka << 1 and (εA/ εB - 1) ka << 1 with εA ≈ ε B respectively, where k = 2π/λ , and εA ( εB) is the dielectric constant of component A(B), the transfer of photons from the incident forward propagating mode to the scattering modes of same frequency but different directions is expressed(29) by a scattering loss coefficient

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(10) where ∆ε = εA - εB and ∆ n = nA - nB . One also introduces an elastic (coherent) photon mean free path (11) which will be assumed much larger than the inelastic mean free path la related to real absorption losses inclusive the ones involving frequency shifts during the scattering process. In the elastic (frequency preserving) scattering the relative coherence is preserved to a certain extend and this may have drastic implications on the overall scattering pattern. We shall briefly concentrate our attention in two cases where the nonlinear behavior strongly interfers with the scattering pattern because of the modification of the relative coherence in the scattering process. As the intensity of the incident beam increases the refractive indices of the components A and B, the inclusion and the embedding medium respectively, are modified and become ñA = n A + n 2A I and ñ B = n B + n2 BI where I is the beam intensity. Lacking a rigorous theory of nonlinear scattering comparable to the Lorentz-Rayleigh in the linear regime we make the ansatz that one can introduce an elastic scattering loss coefficient and mean free path or ∆ñ as in (10) and (11) or with a functional dependence on (12) respectively where (13) with ∆ n0 = n A - n B and . ∆n 2 = n 2A - n 2 B . This ansatz although intuitively acceptable is not trivial and at presently we lack a theoretical support for its validity; we shall discuss some of its implications. White self-transparency. It is clear from (12) and (13) that the scattering losses are suppressed(30,31) for a critical light intensity (14) if sign (∆ n 0 .∆ n2 ) = -1 or otherwise stated the scattering waves are recycled into the incident forward propagating mode without loss of coherence through a suppression of the refractive index fluctuation. The net result is that the incident beam propagates without attenuation in the random two-component composite for this particular intensity and independently on its frequency as long as this is in the transparency region of the composite material. This is the white self-transparency effect(31). The stability conditions for this effect are quite involved and will not be discussed here but we mention(30,32) below a few important implications that follow from such a situation:

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- for optical pulses with peak intensity exceeding the critical one (14) the center and the wings of the pulse are attenuated and one may recover pulses with rectangular temporal profile - a similar process in space allows one to reshape the spatial profile of an intense beam with peak intensity larger than Ic into rectangular spatial profile - the reflexion of an intense beam from a mirror located behind such a medium exhibits bistable behavior as its intensity is varied - one may shorten pulses - starting with a two component composite with equal linear refractive indices for the two components or ∆n 0 = 0 (index matching) but different optical Kerr coefficients ∆ n2 ≠ 0 the scattering losses grow as the intensity grows and this effect can be exploited in intensity limiting(33,34). Multiple nonlinear scattering. The previous case effectively concerned the single scattering regime which is spatially isotropic. In the multiple scattering regime(34) interferences along certain scattering paths introduces an axial asymmetry in the scattering radiation pattern along the incident propagation direction. Indeed the scattering pattern exhibits an enhanced scattering in the backward direction within a cone of solid angle θ ~ λ /lc superposed on the isotropic scattering pattern. This is the weak localization regime a precursor to the strong localization regime the latter expected to occur when the Iofe-Regel criterion(36) is satisfied, namely l s ~ λ ~ a, which qualitatively can be visualized as the establishment of stationary interference patterns in all directions following the multiple scattering processes. Within the radiative transport theory the strong localization regime appears when the effective diffusion constant vanishes and this establishes a threshold condition as also does the Iofe-Regel criterion. We see that within this context and assuming that ansatz (12) is valid in the nonlinear regime both the weak and strong localization behavior can be strongly affected. Thus in the weak localisation regime, in the case of ∆ n0 ≠ 0 (the index mismatch configuration) the backward scattered cone will become narrower as the intensity increases to the critical value I c if sign (∆ n 0 ∆n 2) = -1 and eventually disappears as this value is reached because of the suppression of the refractive index fluctuation in the two component random dielectric ; it angularly broadens in a monotonic way if sign (∆ n0 ∆ n2 ) = 1 since the refractive index mismatch cannot be now suppressed. In the case of ∆n 0 = 0 the refractive index matching configuration a cone appears and angularly broadens as the light intensity increases. In the strong localization regime when the nonlinear regime becomes operative two cases may occur ; depending whether the initial configuration is on or off the strong localization condition l s ~ λ ~ a, one may switch off switch on the strong localization and this will in particular affect both the forward as well as the backward propagating waves with respect to the incident propagation direction. These processes besides their fundamental interest can have several important applications.

ELECTRON CONFINEMENT All composite materials formed by uniformly and randomly dispersed metal or semiconductor crystallites in a transparent dielectric share in common two important

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features. First, in the metal or semiconductor nanocrystals the otherwise delocalized valence electrons in the bulk find themselves confined in regions much smaller than their natural delocalization length ; this drastically modifies their quantum motion as probed by optical beams but also their interaction with other degrees of freedom. Second, because the size of the crystallites is much smaller than the wavelength and their dielectric constant is very different from that of the surrounding transparent dielectric, the distribution of the electric field that acts on and polarizes the charges inside these crystallites can be vastly different from the macroscopic Maxwell field in the composite. These two effects, the first quantum mechanical and the second classical, go under the names of quantum and dielectric confinements respectively and are particularly conspicuous, in the optical frequency range. The first can be treated within the effective mass approximation if the particles are large enough so that surface effects can be neglected with respect to volume effects ; the second can be treated within the effective medium approach if the particle size and volume concentration are small enough. The most conspicuous signature of both confinements however is the appearance of morphological resonances related to quantum confined dipolar transitions and collective dipolar modes for the quantum and dielectric confinement respectively. They have a drastic influence on the electron dynamics and optical nonlinearities in these systems. Dielectric confinement. For very low volume fraction p of spherical crystallites in the transparent dielectric, or p << 1, and particle size d << λ, where d = 2a is the spherical particle diameter and a its radius, λ is the optical wavelength which is also larger than the interparticle distance, one can introduce an effective dielectric constant for the composite which within the effective medium approach(38) is written (15) with εd and εm(ω) being the dielectric constants of the transparent dielectric and the embedded crystallite respectively. This is a straightforward consequence of the Clausius-Mosotti approximation for the dipole induced in a spherical polarizable particle immerged in a dielectric and can also be related to the Mie theory(29) of light scattering from a diluted gas of spherical particles by imposing the vanishing of the forward scattering amplitude and neglecting all terms of higher order than the dipolar one. Taking into account the dielectric polarization effect since a << λ the field E in inside the particles is related to the Maxwell field in the composite by the relation where f l (ω) is approximately given by (16) To the extent that εd is frequency independent and real while ε m( ω) is frequency dependent and complex, ε m( ω) = ε ' m(ω ) + iε" m(ω ), the absorption coefficient of the composite is easily obtained from (15) and (16) as

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(17) where n ≈ ε d 1/2 is the refractive index of the composite. It is evident from (15) and also (16) and (17) that these quantities are resonantly enhanced close to a frequency ω s such that (18) which is the condition for the surface plasmon resonance. It arises from the restoring force of the build-up of surface charge distribution that leads(39) to resonant density fluctuations of the conduction electrons and accordingly (18) can also be viewed as the condition for a collective dipolar mode which introduces an antishielding. Its width which also measures the coherent damping of the mode is determined by the imaginary part ε"m ( ω) ; the value of ω s clearly depends on the values of ε'm and εd which can be artificially modified. An intense light beam can modify the dielectric constant through the optical Kerr nonlinearity by an amount (19) where by extending (40,41) the previous approach to the nonlinear regime one finds (20) for the effective third order susceptibility of the composite related to the optical Kerr effect, when the contribution of the surrounding dielectric is neglected with respect to that of the nanoparticles which is resonantly enhanced, being the corresponding third order susceptibility of the later. The resonances are either the surface plasmon one or the quantum confined ones and inspection of (19) and of the quantum mechanical expression of clearly shows that in the case of the optical Kerr effect because of the frequency degeneracy the resonant enhancement is multiple and the neglect of the nonresonant Kerr nonlinearity of the surrounding medium is justified. In the previous discussion we have tacitly assumed that one can replace the summation over the size distribution by an average value for the optical coefficients of the nanoparticles. In a more rigorous approach such aspects can be properly and simply incorporated and in fact lead to minor deviations from the above sketched derivation. Our field form factor fl is a macroscopic concept relating the internal field assumed to be uniform inside the particle and the field in the glass matrix. There is also a (3)

microscopic local field effect but it is built in the response coefficients, ε and χ , which describe the response of the electrons of the particles to the uniform internal field E in . The dielectric constant ε( ω) of the particle being frequency dependent with , the factor fl (ω ) may show a resonant behavior when namely at the surface plasma resonance for spherical particles. Such resonances are important in the case of metal particles but not for semiconductor ones. Indeed below the absorption edge of bulk semiconductors, the dielectric constant ε is approximately frequency indépendant, but for semiconductor nanoparticles, we will see that discrete transitions appear due to size quantization and one could then expect a

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resonant behavior of ε and fl at the corresponding frequencies. It has however been shown(42) that, for SDGs, even near such transitions, the frequency dependence of ε is weak so that it may safely be replaced by the high frequency dielectric constant ε∞ in eqn (18). The local field factor may then be considered as real and constant. For CdSedoped glasses for example, one has ε∞ ≈ 6.1 and f ≈ 0.63 which is at least an order of magnitude less than the values that prevail in noble metal doped glasses when ω ~ ω s. Quantum confinement. For metal and semiconductor nanocrystals containing a few thousand or more atoms the effective mass approximation (EMA) provides a convenient framework to discuss the modification of the electron states that results from the confinement. This approach is concerned(43) with the behavior of the delocalized electrons in a crystal perturbed by an aperiodic potential and was initially devised for impurity and electronhole pair states in crystals and subsequently extended(44) to the case of quantum confined nanostructures. In contrast to the dielectric confinement the quantum confinement has a drastic impact on the optical and spectral features of the semiconductor nanocrystals but much less so on those of the metal nanocrystals. We shall discuss to some extent the semiconductor cases. In the one-electron picture(43) of a perfect crystal the electrons occupy band states which form energy band continua separated by forbidden energy regions ; their wavefunctions are of the form (21) is the pseudo-momentum, n labels the band, has the periodicity of the where lattice and is closely connected with the atomic wave-functions that form the basis set of the bands and exp(ikr) is the enveloppe wave function. The band states are filled up to an energy level EF , the Fermi level. Within this picture the electronic transitions are viewed as the promotion of an electron from an allowed occupied state below E F to an unoccupied allowed state above EF leaving behind a positively charged hole the two interacting predominantly through Coulomb forces within the sea of all carriers. The main distinction between a metal and a semiconductor is related to the Fermi level E F being an allowed state situated within a band, a half filled band in the case of a metal, or being a forbidden state situated within a gap between a filled valence and empty conduction band in the case of a semiconductor or equivalently that the electron-hole spectrum extends down to zero or to a finite energy gap Eg respectively. We first concentrate our attention on direct-gap semiconductors, such as CdSe, in which the bottom of the conduction band and the top of the valence band occur at the same point of the Brillouin zone, most generally its center G. In the vicinity of such extrema, the k-dependence of energy may be approximately written in a free-particle form: (22) m-1 being the inverse effective mass tensor. Neglecting anisotropy, m is a scalar that we will denote m e , the effective mass of the electron, for the bottom of the conduction band

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and m h , the effective mass of the hole, for the top of the valence band. At the same time, the k-dependence of un, k is usually neglected and this lattice periodic part is denoted u c and uv for the two bands of interest; referring to the previous remark these are s- and ptype wave-functions respectively. Effective mass approximation. We assume the semiconductor to be intrinsic, the ground state of a crystallite corresponding to complete filling of the levels of the valence band, the conduction band being empty. The simplest excited states correspond to excitation of an electron from the valence to the conduction band, leaving a hole in the valence band. We assume that the wave-functions vary slowly on the unit cell scale so that, we may use (43,44) the effective mass approximation which is concerned with the behavior of electrons in a crystal perturbed by an additional aperiodic potential. We further assume that the bands are isotropic and parabolic in the vicinity of the G point, that the bands are non degenerate and that the potential barrier between the semiconductor particle and the glass matrix is infinitely high. The electron-hole wavefunction φ(re ,rh ) then obeys the equation :

(23) where ∆ is the Laplacian operator, Wc is the confining potential, assumed to be constant inside the crystallite and infinite outside. The last term in the Hamiltonian corresponds to Coulomb interaction between electron and hole and assumes this simple form only when we neglect the difference between the dielectic constant of the semiconductor and that of the glass matrix. ε is the low frequency dielectric constant of the semiconductor and r e-h the distance between electron and hole. φ (re ,rh ) is the envelope wave-function. There are two length scales that enter the problem: the Bohr radius of the Wannier exciton with µ = m emh /(me + m h ) and the radius a of the particle if we assume it to be spherical. Two different energies appear(45) in the problem: the kinetic confinement energy of the carriers of order and the Coulomb interaction energy of order e 2/ε a or e2/ εaB (whichever is larger). Since these two energies have different size dependences, it may easily be shown that, when a >> aB, the Coulomb interaction energy is much larger than the confinement energy whereas, when a << aB, the opposite is true. When a >> aB , we are in the weak confinement regime(45) , the Wc terms can be omitted as a first approximation in the Hamiltonian of eqn (23) which then reduces to the well-known hydrogenic Hamiltonian. The Wannier exciton then exists and moves freely inside the crystallite. One may then take confinement into account, the result being confinement of the exciton as a whole with its translational mass M = m e + mh and a small confinement energy of order . On the other hand, when a << aB, we are in the strong confinement regime(45) , the Coulomb interaction term can be omitted as a first approximation in the Hamiltonian of eqn (23) which is then the sum of two independent Hamiltonians. The electron and the hole are confined independently. The e electron envelope wave-function ϕ (r) for example then obeys the equation :

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(24) When the independent electron and hole problems have been solved, one may then take Coulomb interaction into account using perturbation theory. Finally, when a ~ a B , we are in the intermediate confinement regime and no simple solution exists although an adiabatic decoupling approach has been put forward. Experiment tells us that the conditions of validity of these limiting cases are not so stringent and the strong confinement regime can be used when a < a B and the particles then are denoted quantum dots. Semiconductor quantum dots are thus semiconductor particles whose size is smaller than that of the Wannier exciton in the bulk. CdSe particles, for which a B ≈ 5.6 nm, are easily obtained in the strong confinement regime. In the case of CdS x Se 1-x , aB varies between the CdS value (≈ 3.2 nm) and the CdSe one. We concentrate on the strong confinement case. As a first approximation, the electron envelope wave-function is a solution of eqn (24). This is the simple particle in a (45) box problem and for a spherical particle, ϕ e(r) assumes the form : (25) where the Ylm 's are the spherical harmonics, jl (x) is the spherical Bessel function of order l, a i n its nth zero and B In a normalization constant. r, q and j are the spherical coordinates of r. Taking the zero of energy at the top of the valence band, the corresponding eigenenergy is: (26) which shows a (2l + 1)-fold degeneracy. Instead of having a quasi-continuous distribution of allowed levels in the conduction band as in the bulk, we now have a discrete set of such levels, the first ones being 1s with l = 0, n = 1 and a01 = π , then 1p with l = 1, n=1 and a 11 = 4.49, then 1d, 2s and so on. We emphasize that the total electron wavefunction is : (27) in which uc (r e) has the periodicity of the lattice and is predominantly s- type. If we assume the valence band to be nondegenerate, the hole envelope wavefunction obeys(45) a similar equation whose eigenfunctions are identical to those obtained for the electron, the corresponding eigenenergy being given by : (28) Here again, the total hole wave-function is : (29)

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with a different lattice periodic part u v which is predominantly p-type. Several simplifications and approximations(46) are incorporated in the previous model for the quantum confinement. The spherical particle approximation is one but it should not be too drastic. For instance the main result which is size quantization of the levels is not strongly shape dependent and the spherical case correctly gives the order of magnitude of the confinement energy. Also the assumption of parabolic and isotropic bands in the vicinity of the G point is not too severe. The main consequence of non parabolicity is a slight decrease of the confinement energy. The assumption of an infinite confining potential instead of the ill-defined energy gap of the glass matrix ~ 5-6 eV has minor impact on the first levels but may become more important for higher excited levels. Finite potential barriers have been considered. The main result here is again a reduction of the confinement energy due to a larger spatial extent of the envelope wave-function. Because of the difference in dielectric constant between the semiconductor and the glass matrix, the Coulomb interaction between electron and hole is the sum of the last term in the Hamiltonian of eqn (23) and of correction terms denoted polarization(43) or solvation terms. Basically, the electron for example interacts with the hole but also with the hole image and with its own image. These additional terms have been taken into account using variational or perturbational approaches. They do not change much the value of the eigenenergy but they slightly modify the wave-functions, pushing the hole toward the semiconductor-glass interface. Far more drastic is the approximation of non-degenerate band(48). For the conduction band which originates from s atomic orbitals, this is not a problem. The conduction band is non degenerate except for the electron spin degeneracy. For the valence band, the situation is quite different. Since the valence band originates from p atomic orbitals, it shows a 3-fold orbital degeneracy and a 6-fold degeneracy when spin is taken into account and this substantially complicates the picture ; a valence band mixing results(48-50). Finally, the effective mass approximation (EMA) ceases to be valid for small semiconductor particles where the surface layer still makes an important contribution and the only way out of the EMA, which suppresses these surface states is a molecular approach (51) . In such an approach molecular orbitals are constructed, using the true atomic orbitals as the starting basis set. Projecting the MOs thus obtained on the wavefunctions of the inner or outer atoms, these preliminary results allow(52) to visualize the formation of the valence and conduction bands and show the presence of surface states. This MO approach also allows to consider particles the surface of which has been partly or totally capped or passivated. It is however very demanding in terms of numerical computation and has been limited to a total number of a few hundred atoms. Interband and intraband transitions. The selection rules for transitions between the quantized states are determined (14) by the matrix elements of the momentum operator p. We consider first interband transitions in which an electron is promoted from an (l,m,n) state in the valence band to an (l',m',n') state in the conduction band if we neglect the valence band degeneracy. Since the momentum operator is proportional to the gradient operator and since ylmn(r h ) is the product of a rapidly varying part uv and of a slowly varying one ϕh, we have : (30)

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The transition matrix element is approximately the product of what is traditionally denoted p cv and of the overlap integral of the electron and hole envelope wavefunctions; its value thus does not depend on the particle radius a. The ϕ lmn s given by eqn (25) form an orthonormal basis set and the overlap integral is simply δl l δmn' δnn' where dij is the Kronecker symbol. The only allowed transitions are then those which preserve the quantum numbers l, m and n (enveloppe conservation). We then have the 1s-1s, 1p-1p and so on transitions. They occur at frequency ω In given by .. (31) in which the last term, corresponding to Coulomb interaction, is obtained using perturbation theory. The number β is approximately 1.8 for ns-ns transitions. Taking the valence band degeneracy into account slightly modifies the selection rules. Transitions are allowed for example from all the nS3/2 levels to the 1Se one but the most important is the 1S 3/2-1S e one. Eqn (30) as well as EMA assume that ϕ varies slowly on the unit cell scale. This is the case for not too small particles when the quantum numbers 1 and n are small integers but the assumption is quickly invalidated when 1 and/or n increase. EMA is then no longer valid. The selection rules are then modified and this may be the reason why, in real spectra, only the first transitions can be identified. At higher photon energy, the spectra have the same continuum-like aspect as in the bulk. We may also consider intraband transitions in which for example an electron is promoted from an (l,m,n) state in the conduction band to another (l',m',n') state of the same conduction band. The situation is quite different here and the matrix element is approximately given by : (32) The selection rule is then ∆l = ± 1. We may have for example excitation from the 1S e state to nPe ones. The transition matrix elements (32) are now ~ a but the transition probability quickly decreases with increasing n. Above we outlined the description of the electron states in semiconductor nanoparticles within the effective mans approximation. For metal nanoparticles the situation at the outset seems to be simpler in the optical frequency range because we may concentrate our attention solely on the electrons in the half filled conduction band (s-p type). Since the low frequency dielectric constant in metals is very large (in principle infinite at ω = 0) the screening of all interchange interactions is complete and one may view the electrons as non-interacting, freely moving quasiparticles in the confining potential of the metal nanoparticle and their states are then simply described by an equation of the type (24), with m e ≈ m the free electron mass, and with wave functions of type (27). The transitions in the optical range then are either intraband ones between such states with intraband transition matrix elements given by (32) or interband ones between the filled d-band states and the quantum confined ones of the half-filled s-p band.

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METAL NANOCRYSTALS IN GLASSES Using the previous simplified discussion of the dielectric and quantum confinements we may now proceed to describe the linear and nonlinear properties of metal nanoparticles in a glass matix. Linear properties We remind (53) that in the case of bulk metals, because of the zero energy gap and is infinite for = 0 large density of states at EF , the wave vector dielectric constant resulting in a complete screening of the electron-hole interaction potential within a distance rF ≈ 1/k F, the inverse of the Fermi wavevector which is of the order of a few Angströms or roughly equal to the lattice constant. Thus the electrons and holes on either side of EF can behave and move as free noninteracting particles over any distance in a perfect metal occasionally interacting and being scattered by temporal and spatial disorders ; such interruptions of the coherent free electron motion can be expressed in terms of a free mean path for the electrons which in an ideal metal we will assume to be limited by the unklapp electron-electron and electron-phonon scattering processes lumped together : one finds there that (33) for the dielectric constant of a metal were ωp is the plasma frequency, ω2p = 4π e2n/m, n is the electron density, δεinter is the interband contribution and τ0 is a scattering time for the electrons which for an ideal defect free metal we assume to be related to the compound effect of electron-electron and to electron-phonon scattering. Within this context in a metal nanocrystal the main modification then will result from an additional restriction of the free electron motion because of the presence of the interface with the surrounding dielectric which to a good approximation can be visualized as a spherical potential well of infinite height. In addition the electronphonon coupling has to be modified because of the modification of the vibrational spectrum in the metal nanocrystal with respect to the bulk. Assuming the two processes uncorrelated this only amounts to replacing 1/τ0 in the denominator in (33) by (34) where vF is the electron velocity on the Fermi level and 1/ τ' 0 now is the electron-phonon relaxation rate in the crystallite. This classical picture which however properly incorporates the Fermi statistics can be also justified within a quantum mechanical approach(40,54) in terms of the confined electronic states in the spherical potential well set up by the interface with the surrounding dielectric. These are now discrete states which incorporate the boundary conditions because of the requirement that the wave functions must vanish at the interface ; as previously discussed more realistic models introduce only minor changes.

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Although the enveloppe equation (24) together with the boundary condition lead to a discrete spectrum with energy spacing as we move up in the energy spectrum this becomes close spaced and in fact one may again introduce a density of states (35) where V is the spherical particle volume, and a Fermi level (36) both independent on the nanocrystal size and equal to their corresponding bulk values. Calculating the transition dipole moment between such states and introducing(55) a common coherence relaxation time T2 related to electron-lattice interactions for all allowed dipole transitions one may proceed to calculate the optical properties. Taking into account also transitions from the filled d-band which is unaffected by the quantum confinement one obtains the remarkable result that the dielectric constant takes(40,54,55) the form (37) namely the same as for the bulk with the damping time τ in the Drude term given by expression (34) previously obtained by a classical argument, while the imaginary part of the interband term is given by (38) where P is the transition matrix element of the momentum operator between the d-and the conduction s-p bands, and J(ω) is the joint density of states for the two bands. Thus one recovers for the metal particles an expression similar to that of the bulk metal (33) with the the damping replaced by expression (34) ; this also justifies the use of the bulk expression for εm( ω) in équation (15) as anticipated. With this expression for εm( ω) we obtain (39) for the expression of the surface plasmon resonance in terms of the dielectric constants εd and εb = 1 + δε inter. We see that ωs can be modified by several ways by acting on the physical parameters εb, ε d or ω p in particular through photoaction. Optical nonlinearities The optical Kerr effect results from the photoinduced modification of the complex refractive index of the medium or δñ = n2 I, where I is the light beam intensity and the optical Kerr coefficient n 2 is simply related to the third order susceptibility χ (3) ( ω, – ω, ω)

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or χ (3) (ω , – ω ' ,ω ') which can be resonantly enhanced whenever ω and/or ω ' are close to a resonance of the medium. In the case of the metal or semiconductor nanocrystals in glasses the relevant resonances are the morphological ones resulting from the dielectric and quantum confinements. A simple inspection of the expressions of the third order optical Kerr susceptibilities shows that one has multiple resonant behavior there leading to strong enhancement close to these resonances and also strong dependence on the features of these resonances which also reflect those of the confinement. Broadly speaking, in such a resonant regime, the physical mechanisms contributing to χ ( 3 ) can be distinguished in two classes(48,56). The first type involves real population of excited states or excitation of charge carriers there resulting from real transitions ; it is accompanied by linear absorption and the response speed is limited by the population relaxation time. The second type of mechanisms contributing to χ(3) effectively results from light induced shifts on the electronic levels, direct or indirect ones, and can be connected with virtual transitions. The most conspicuous direct contribution here is the one related to two-photon resonances while indirect ones may result either from interpair electron-hole (biexcitonic) interactions which can be viewed as an optical Stark shift mutually induced between photoinduced e-h pairs, or through interaction with the field set up by an interface trapped photocarrier which can be viewed as a static Stark effect ; the three mechanisms can be distinguish through their different frequency dependence and different response times they imply. In the case of metal nanocrystal because of the complete screening that is operative there all contributions of the second class to the Kerr nonlinearity mentioned above are drastically reduced and only those of the first class contribute there. This has also been confirmed experimentally. In addition in metal nanocrystals the effective third order Kerr susceptibility is fourfold resonantly enhanced close to the surface plasmon resonance ωs . The optical nonlinearities of metal nanocrystals in transparent dielectrics have been studied to a certain extent over the last ten years and a good understanding of these nonlinearities and their relation to the electron dynamics has emerged that confirm the main aspects stated above. Reliable predictions regarding their behavior in noble metals (3) of the composite are indeed feasible. Thus the fourfold resonant enhancement of material close to the surface plasmon resonance (39) as predicted by (19) has been strikingly demonstrated (22) in gold and silver colloids first and subsequently confirmed (57,58) also in the case where the gold and silver particles are embedded in glass. Similar behavior has also been evidenced (59) in copper nanocrystals in glass where the surface plasmon resonance strongly overlaps with the interband transitions. Using the and and relation (18) one easily finds that experimental values of  fe ∼ has values in the range 10-20 for noble metals resulting in 3 enhancement factors in the range 10 -105 for over the surface plasmon resonance width (34). The impact of the quantum confinement is related to the behavior of which can be extracted from (19) after the factor fl (ω) has been accounted for in terms of the dielectric confinement as stated above using expression (16) and condition (18). The detailed experimental study of ref. 57 using the degenerate four wave interaction technique or optical phase conjugation where several parameters were varied, such as the size of the nanocrystals, the temperature or the polarization state of the input beam (3) and in addition the phase of χ was determined, allowed to analyse its behavior in the light of a detailed theoretical modelling and single out the origin of the nonlinearity ; it confirmed on the one hand the minor role played by the quantum confinement there and

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on the other the key role played by the photoinduced population rearrangement close the Fermi level and its dynamics. (55,57) of the quantum mechanical expression of χ m(3) (ω,– ω,ω ) Careful analysis indeed shows that the second class of contributions previously termed as photoinduced level shifts are irrelevant in metal nanocrystals and only the first class of contributions, termed photoinduced population changes are responsible for the optical nonlinearities there. These can be additionally split into three contributions : an intraband contribution involving only transitions between two quantum confined states within the conduction s-p conduction band which can be written (40) where κ1 is a constant factor, T1 et T2 are respectively the energy lifetime and dephasing time and a 0 ≈ T2(2EF /m)1/2 which is roughly 100-200Å in metal nanocrystals ; thus χ (3) intra is imaginary with Im χ(3) intra < 0 and strong size dependence characteristic of quantum confinement, an interband contribution involving transitions between a d-band state and a quantum confined s-p conduction band state which can be written (41) where κ ' is a constant, T1' and T 2 are the energy lifetime and dephasing time for the twolevel system and P and J(ω ) were previously defined in equ. (38) ; thus χ (3) is inter < 0 and size independent, imaginary with Im χ (3) inter a hot electron contribution which can be viewed as a hybrid term of the previous two and results from the photoinduced modification of the occupation of the conduction band states above and below the Fermi level accessible to the d-electrons. This modification is provoked by the absorption of photons, close to the surface plasmon resonance frequency, by the conduction electrons which are promoted to initially unoccupied states above the Fermi level and the same time liberate states below it. Assuming this leads to a quasithermal distribution in a subpicosecond to picosecond time scale at an electronic temperature Te much higher than the lattice temperature T l one gets '

(42) for this contribution where C is the conduction electrons heat capacity, τ 0 / τ e f f measures is the the number of all collisions the electrons suffer before they thermalize, ε''L imaginary part of interband term at the point L of the Brillouin zone which makes the dominant contribution close to the surface plasmon resonance frequency ; thus χ (3) he is imaginary with Imχ(3) he > 0 and size independent. The detailed experimental study in ref. 57 showed that in the size range 30-100 Å the hot electron contribution is the dominant one in χ x(3)x x x being by one and two orders of magnitude larger than the inter and intraband ones respectively. In χ (3) xyxy this hot electron contribution being incoherent does not contribute while the interband one does 3 ) / (3) and this is confirmed by the measured value and sign on the ratio χ (xxxx χxyxy . Similar conclusions were also reached in the detailled study of the optical nonlinearities in

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copper nanoparticles in glasses where the interband transitions strongly overlap with the intraband ones involved in the surface plasmon resonance. Electron dynamics The study of the optical nonlinearities in metal nanocrystals brought into focus the important role played there by the electron dynamics in a confined space. The study of such processes was initiated ( 5 3 ) quite early in electron transport in connection with the electrical conductivity in metallic wires and plates or thin films and it was associated there to the modification of the mean free path of the electrons because of their collisions with the walls. The Landau theory of Fermi liquids provides a rigorous albeit complicated framework to study such aspects in terms of the interaction of quasi particles (electrons and holes) among themselves or with other excitations and with the walls. Much insight however can also be gained with a simpler description and by proper accounting for the Fermi statistics. In the past these experimental and theoretical studies were predominantly conducted in the frequency domain under the assumption that the different interactions that cause damping were uncorrelated and the long time regime prevails. Deviations from such a behavior however were clearly noticeable but difficult to analyze in frequency domain. The recent advances in ultrashort laser pulse techniques opened the way to address these problems directly in the time domain, in particular the short time behavior, and assess the interplay of the different processes there. The most commonly used time resolved technique is the pump and time delayed probe technique and its variants : in essence in all these techniques the spatiotemporal evolution of a quickly photoinduced modification of an optical property is interrogated by a short probe pulse with variable delay and related to the dynamics of the relevant electronic transitions. The recent application (55,60,61) of time resolved nonlinear optical techniques in the femtosecond time scale opened the way to study the early stages of ultrafast dynamics of photoexcited electrons in metal nanoparticles confirmed ( 5 5 ) the importance of the interplay of the e-e and e-p collisions in the process of their thermalization. Numerical solutions with certain simplifying approximations fully confirm this effect. The case of the copper metal particles where some of these studies were performed ( 6 0) presents the inconvenience that the interband transitions in copper strongly overlap with the intraband ones of the surface plasmon resonance. The case of the gold or silver particles embedded in glass is be exempt of such complications. Recently studies were performed (55) in silver nanoparticles embedded in a glass matrix ; here the plasmon resonance and interband transitions are well separated permitting a selective probing of the surface plasmon resonance. The preliminary results obtained with 100 fs resolution indicate that thermalization does not set in before several hundred femtoseconds. The study yielded evidence of a time dependent red shift ∆ω s and broadening of the surface plasmon resonance which set in instantaneously and follow the thermalization process. The frequency shift can be related to the photoinduced modification of ε b in (39) because of the optical Kerr effect while the broadening is attributed to the increment of the collision rate of the electrons with the surface as their average velocity increases. Additional studies with shorter time resolution are needed to unveil the athermal behavior. Extensions

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The study of the optical nonlinearities and electron dynamics in metal nanocrystals embedded in glass matrix is connected with some fundamental aspects of the behavior of electrons in metals in general and in confined geometries in particular. In this respect two aspects should retain our attention : one is the crucial role played by the dielectric confinement to enhance the nonlinear optical coefficients close to the surface plasmon resonance and other morphological resonances ; the other is the restrictions imposed by the Pauli principle and Fermi statistics on the available phase space for the electrons leading to interference of the scattering processes and eventually to the breakdown of the addivity assumption of the damping rates and that of the thermal evolution of the photoexcited electronic distribution. Indeed the impact of these two aspects is such that any specifically quantum mechanical features are effectively washed out. This is in striking contrast with the situation in the semiconductor nanocrystals discussed below where specific quantum mechanical features prevail in the optical spectra and electron dynamics. At the same time the recognition of the important role played by the dielectric confinement and the Fermi statistics shows the way to some interesting future studies. Thus similar studies in transition metal nanoparticles where the d-band is also half filled or noble metal nanoparticles containing magnetic impurities should show some new features regarding electron dynamics and optical nonlinearities because of the drastic modification of the electron density distribution at the Fermi level that occur there. Another line of study is that of much higher volume concentrations for the metal nanoparticles where the Maxwell-Garnett effective medium approach breaks down and must be replaced by more sophisticated ones like the Bruggemann approach ( 6 2 ) or more complex ones where the topology and fractality are properly introduced. Such studies have already been initiated uncovering some new aspects (63) . Finally there is certainly interest (6 4 ) in the study of glasses co-doped with metal nanocrystals and rare earth ions where the dielectric confinement through the surface plasmon mode it sets up inside the nanocrystal may also show its effect on the rare earth ions outside the metal nanocrystals. In these and other cases the radiative damping which was althogether diregarded in the previous discussion may play a certain role for sufficiently large nanocrystals since its contribution grows as the cube of the particle size and must now be included. In this respect we wish to point out that the absorption losses in the far infrared of the composites formed by dispersing metal nanocrystals in dielectrics can not be accounted for even when all the refinements of the present theory are included. These and other studies will be pursued both for their fundamental interest as well as for their impact in designing artificial composite materials with specifies functions. SEMICONDUCTOR NANOCRYSTALS IN GLASSES The linear and nonlinear optical properties of semiconductor nanocrystals ( 1 4 , 1 8 ) have been extensively studied the last two decades in particular in the strong quantum confinement regime. The most conspicuous spectroscopic feature is the appearance of the quantum confinement resonances related to the morphology and size of the nanocrystals. To the extent that the interface with the surrounding dielectric contains a sizeable proportion of atoms surface states and traps play an important role and drastically effect the dynamics of the confined resonances. Below we shall first review some spectroscopic features and then discuss the nonlinear optical mechanisms in such quantum confined nanocrystals.

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From the previous considerations on interband transitions, the linear absorption spectrum of semiconductor quantum dots is expected to be structured with an absorption edge shifting to the blue when the particle size is reduced. This is indeed what we observe in low temperature absorption spectra for the extensively studied CdSe-doped glasses. The second derivative of the α(ω) spectrum clearly shows substructures which bear the signature of valence band mixing ( 4 6 ). The first two substructures correspond to the 1S3 /2 -1Se and 2S3 /2-1Se transitions. Then, usually comes the 1P3/2-1Pe one. These substructures were observed in several CdSe-doped samples grown in a pure silicate matrix. Usually however, in less pure samples, even when working at low temperature and taking the second derivative of the absorption spectrum, the 2S 3/2 -1S e feature only appears as a shoulder to the 1S 3 / 2-1S e peak. At room temperature, only one feature is observed. When discussing below the resonant non-linear optical properties of these materials, we will mainly deal with this feature and vaguely denote it 1S 3/2 -1S e . In order to better understand the linear optical properties of SDGs and the levels (18) of relevance in the vicinity of the absorption edge, luminescence measurements are also important. When SDGs are excited with a c.w. laser beam, the luminescence spectrum consists of a relatively narrow peak slightly Stokes shifted from the 1S 3 / 2 -1S e absorption peak and a usually more intense, broader luminescence band showing a much larger Stokes shift. When excited with a picosecond laser pulse, one observes that the narrow luminescence peak is a rather fast component with a lifetime of the order of 1 ns whereas the broad band is a slow component with a lifetime of the order of microseconds. In fact, in this broad band, the larger the Stokes shift, the slower the luminescence decay. The narrow peak was therefore interpreted as direct recombination of an electron in the 1Se level with a hole in the 1S 3 / 2 one. The broad luminescence band was interpreted as trapped carrier recombination. Traps. As previously stated traps play a very important role in these nanocrystals but their identification and characterization is very difficult. Until now, we assumed the semiconductor to be intrinsic and we did not pay attention to the semiconductor-glass interface. Very little is known about possible inner impurities or defects. In the same way, little is known about the role of the surface of the nanocrystals. One could imagine that dangling bonds lead to the presence of surface states. One could imagine that these dangling bonds are at least partly saturated by surface reconstruction or by bonds with the Si-O matrix. For commercial Schott filters, the broad luminescence band is much more intense than for specially made (denoted experimental) glasses grown from the same melt. This means that commercial filters contain many more traps than experimental samples. And, for experimental samples grown from a same melt, the smaller the particles are, the larger the number of traps is. The traps are therefore thought to be located at the semiconductor-glass interface. This is further supported by the observation of drastic changes in the luminescence behavior when the surface of colloidal CdSe nanoparticles is capped with organic species. Another important point was reported in ref. 65. When time resolving the rise of the luminescence signal, the authors observed that the narrow peak and the blue edge of the broad band start appearing at the same time. One does not observe a progressive rise of the broad band accompanying the temporal decay of the narrow peak as would be expected if the carriers were progressively trapped. This suggests the existence of two

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volume of the particle and directly recombine giving rise to the narrow peak and particles with traps in which the carriers very quickly trap at the interface and then slowly recombine giving rise to the broad feature. Photodarkening. When SDGs are exposed to a laser beam at frequency ω such > E g for a long time, they experience a phenomenon known as photodarkening that as first observed independently by Roussignol et al(66) and Mitsunaga et al(67) . This effect has several consequences, It leads to a drastic change of the luminescence spectrum. After darkening, the magnitude of the narrow luminescence peak is somewhat reduced whereas the broad band almost completely disappears. Darkening also reduces the magnitude of the recombination time of free carriers, which explains the reduction in the intensity of the narrow luminescence peak. It also reduces the magnitude of the optical Kerr susceptibility in the resonant case. It does not significantly change however the absorption spectrum. The disappearance of the broad luminescence feature seems to be due to the opening up of a new and more efficient decay channel in particles with traps. The degree of darkening which may be quantified in several different ways, for example by measuring the intensity of the broad luminescence feature, depends only on the integrated irradiation dose the sample has received. The photochemical process leading to darkening was elucidated by Grabovskis et al (68) using thermally stimulated luminescence: the absorbed photons create carriers which may be ejected from the nanoparticles and trapped in the glass matrix. This is a quasi permanent phenomenon. At room temperature, it may be considered as permanent but, by heating the samples to ~ 370°C for a few hours, the ejected electrons diffuse back to their original particle and the doped glass recovers its original properties. Line-broadening. The observed transitions are broad and this is mainly attributed to size dispersion. The transition frequency is a-dependent: smaller particles absorb at higher photon energy. The size distribution leads to inhomogeneous broadening. This inhomogeneous broadening hides the intrinsic or homogeneous width of the transitions as would be given by single particle spectra. If we are able to excite only one particle size, probing it will give us narrower features. This may be done using nonlinear techniques. We may fix the excitation frequency and 1) probe the change in the absorption spectrum : this is saturation or hole burning spectroscopy or 2) measure the luminescence spectrum: this is known as fluorescence line narrowing (FLN) spectroscopy and reveals spectral structures for instance in the 1S 3/2 -1S e peak due to a vibronic or phonon progression. The simplest and clearest FLN spectra are obtained by exciting only the biggest particles. Increasing the photon energy, one would excite smaller particles in the zero-phonon line and larger ones in the one-phonon line. Symmetrically to case 2) above, one may detect only the blue edge of the luminescence peak and tune the excitation frequency. One then obtains a photoluminescence excitation (PLE) spectrum. This way, we also observe the phonon progression of the 1S3/2-1S e transition. FLN or PLE spectroscopies are ideal tools to observe finer details. Using PLE, Norris and Bawendi (50) have been able to follow the size dependence of ten interband transitions and could clearly identify the first six of them. Using PLE again, Norris et al (50) could also study the very fine structure of the zero phonon line in the 1S3/2- 1S e feature. The 1S 3/2 -1S e level is eight-fold degenerate and

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the very fine structure is due to lifting of this degeneracy by deviation from sphericity, deviation from the zinc-blende structure and by the confinement-enhanced exchange interaction. Electroabsorption. A static (or very-low-frequency) electric field E leads to the static Kerr effect which modifies the optical properties. The change in the absorption spectrum δα(ω) is usually measured; it is proportional to the imaginary part of (3) (3) χ (0,0, ω). The real part of χ could be obtained using the Kramers-Kronig relationships. Such an effect may be used (69) in hybrid devices such as the SEEDs made from quantum wells. But the static Kerr effect is also interesting per se. The change in the optical properties is due to the shift of the energy levels and to admixture of neighboring wave-functions. When dealing with atoms for which the spacing between levels is large, perturbation theory is used and the effect is termed Stark effect. When dealing with a bulk semiconductor for which the levels are very closely spaced compared with eaE, the Hamiltonian: (43) where H 0 is the zero-field Hamiltonian, must be solved. The effect is then termed FranzKeldysh effect. This is one and the same effect, only the mathematical approach differs. Electroabsorption measurements were performed (70,71) on CdS 0.5 Se 0.5 samples -1

containing particles of various mean sizes. The electric field with Ein = 2.10 4 V cm is applied via indium tin oxide transparent electrodes deposited on the two sides of a thin SDG slab. The applied field oscillates sinusoidally at a frequency of 1 kHz and the change in absorption δα is measured using lock-in detection at 2 kHz. Oscillations are observed (70) in the vicinity of the 1S 3/2-1S e peak with a replica in the vicinity of the splitoff band edge. The position of the oscillations is independent of the magnitude E of the applied field and their amplitude is proportional to E². This is typical of a quantum-confined Stark effect: confinement has led to well-resolved discrete levels. Similar results have been obtained by other groups. When the particle size is larger, then Franz-Keldysh behavior is observed (71) . The amplitude of the oscillations decreases when the particle size is reduced. This is understandable since smaller particles are less polarizable. Using Rayleigh-Schrödinger perturbation theory, these experimental results could be fitted with good accuracy. Variational calculations have also been performed. We note that -12 e.s.u., a fairly large value, and that a static field leads to a decrease of (0, 0, ω) ~ 10 the absorption coefficient at the posi-tion of the 1S 3 / 2 - 1S e peak and to an increase on each side of it.

Nonlinear optical properties. The optical Kerr effect The nonlinear property we will concentrate on and the one that has the largest number of applications is the optical Kerr effect. The static Kerr effect is the modification of the susceptibility χ( ω) of a material (or of its index of refraction) by a static electric field E0 , the modification of c being proportional to the square of E0 . The

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electroabsorption previously discussed is a special case of the static Kerr effect when ω is close to a resonance. The optical Kerr effect generalizes this to the field of a laser beam of same (ω) or different (ω') frequency. The first case is a self-action, the second one is a cross-action. The real and imaginary parts of χ (3) give rise to nonlinear refraction and nonlinear absorption, respectively. As previously stated two different physical mechanisms may contribute to the third-order nonlinearity. The first type involves real population of excited states or excitation of charge carriers here. It corresponds to resonant nonlinearities when the frequency w of the field is close to that of an optical transition of the medium. It is accompanied by linear absorption losses and the response speed is limited by the carrier recombination time. The second type of mechanisms contributing to χ (3) effectively results from light induced shifts of the electronic levels, direct or indirect, and can be connected to virtual transitions. It has a femtosecond response time and is then much faster, and usually smaller, than the resonant one. The most conspicuous contribution here is the one related to two photon resonances. In a semiconductor of energy gap E g , the third-order nonlinear absorption coefficient becomes significant when the photon energy of the driving beam is larger than E g /2 and, in the transparency range, is a two-photon absorption (TPA) process. Experimental Techniques. A variety of techniques (14,18) may be used to study the Kerr nonlinearity. Nonlinear absorption, which is one of these techniques, may be implemented in different ways. One may use a single beam of frequency ω and measure the transmission of the sample as a function of the incident intensity. One may use two laser beams in a pump-probe scheme. A probe pulse of frequency ω ' measures the change in absorption coefficient δα ( ω') induced by a pump pulse of frequency ω. Delaying the probe pulse allows to study the time behavior of this nonlinear response. If a white spectrum probe pulse is used, the change in the absorption spectrum can be obtained in a single laser shot. Nonlinear absorption gives access to the imaginary part (3) (3) o f χ ( ω,– ω,ω) in the first case and χ ( ω,– ω,ω') in the second one. In optical phase conjugation (OPC), three beams at the same frequency ω are incident on the sample, two counterpropagating pump beams and a usually weaker probe beam making a small angle with the forward pump beam. A conjugate beam propagating in the direction opposite that of the probe beam is generated and detected. OPC is also known as degenerate four-wave mixing in the backward geometry. Most common configuration is on the case where the forward pump and probe beams are copolarized and when the backward pump beam is cross polarized. OPC then has a simple holographic interpretation: the forward pump and probe beams create an index grating off which the backward pump beam is diffracted. In the resonant case, a population grating is created and again, delaying the backward pump pulse allows to time resolve the nonlinear res-ponse. OPC gives access to the modulus of χ (3) ( ω,– ω,ω). In frequency mixing, two beams of frequency ω and ω' are incident on the sample and the beam generated at frequency 2ω - ω' is detected. Contrary to the other situations, we do not have automatic phase-matching in this case. This technique gives (3) (3) access to the modulus of χ (ω,ω,- ω'). When dispersion is not important, this χ i s (3) approximately equal to χ ( ω ',–ω ',ω ' ). This technique is important in the two-photon absorption regime since, when the difference ω - ω ' is much larger than the inverse carrier recombination time, modulation of the population of the excited state at

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frequency ω - ω ' is negligible: this implies that free carriers do not contribute to generation of the beam at frequency 2ω - ω '. If we consider a propagating beam with a gaussian intensity profile, the nonlinear phase follows the intensity distribution: it is maximum at the center and vanishes at the edge of the beam. The nonlinearity then produces a self-focusing (defocusing) of the (3) beam when the real part of χ is positive (negative). Scanning the sample through the focal zone of a lens and measuring the intensity transmitted through a partially closed aperture allows to measure the magnitude and, most importantly, the sign of the real part of χ(3) (ω, – ω,ω): this is the Z-scan technique extensively discussed elsewhere in this volume. An open aperture Z-scan allows to measure nonlinear absorption. The resonant regime. In the resonant case, luminescence may also give indications on the nonlinear response. When the samples are excited with an intense picosecond pulse and when concentrating on near-edge luminescence, hot luminescence peaks appear corresponding to direct recombination from the 2S 3/2 -1S e level and even higher excited states. This is indicative of state filling in the quantum dots, a behavior contrasting with the bulk-like band filling observed in larger particles. Model and theory. The first prediction regarding the resonant nonlinear response was that each particle would behave (72) as an isolated two-level system made of the ground state (level 0) and the 1S 3/2 -1S e excited state (level 1). Saturation of this twolevel system would lead to the nonlinear response. Absorption saturation had indeed (73) (74) been observed in SDGs. It was soon realized however that, after excitation of one carrier pair in level 1, a second pair may be created, leading to the presence of a twopair level 2. The Hamiltonian for such a two-pair state is the sum of the two Hamiltonians for each pair plus terms corresponding to Coulomb interaction between the two pairs. It has been shown that the interaction energy between the two pairs increases when the particle size is reduced but remains small. The transition frequency ω 21 is smaller than ω 10 but not very different from it so that, when ω ≈ 1 0 , we also have ω ≈ ω 21. Assuming the same dephasing time T2 for the 0 → 1 and 1 → 2 transitions and keeping only the triply resonant terms, the degenerate Kerr polarizability is easily obtained :

(44) where d 10 and d 21 are matrix elements of the electric dipole moment operator, proportional to those of the momentum operator, and T1 is the lifetime of level 1. g is the sum of three terms, the first one corresponding to saturation of the 0 → 1 transition, the second one to induced absorption from level 1 to level 2. The third term is a coherent term involving only off-diagonal elements of the density matrix. Since T2 << T1 , this third term is negligible. The resonant nonlinearity mainly originates from population

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of the SDG is obtained by multiplying g effects. From eqn (44), the susceptibility with the number density N of quantum dots and with the local field correction factor. In the resonant regime, the contribution χd(3) of the glass matrix is negligible. A nanoparticle should behave as a three-level system if we ignore trapped carriers. The important role of trapped carriers had been pointed out by several groups. To account for the trap levels, we add a fourth level denoted 3 whose decay is neglected on the nanosecond time scale. Carriers are excited to level 1. From there, they can relax down to level 0 with rate constant k or be trapped in level 3 with rate constant k'. k' is known to be large. The problem has been treated in two cases : one class of (or identical) particles and two classes of particles, particles with traps and particles without traps. Since the lifetime T1 of level 1 is longer than the laser pulse duration tp , the response function formalism was used. The response may also be characterized by an effective susceptibility which takes the transient nature of the response into account. The response is made of three terms: population of level 1 leads to saturation of the 0 → 1 transition and to induced absorption between levels 1 and 2, carriers in level 3 modify the optical response through the static electric field E 0 they create which is of order e /εa². Assuming rectangular pulses, the response may be calculated analytically. We only reproduce here the results pertaining to the case of two classes of particles and the OPC geometry. When t > tp , the amplitude of the nonlinear polarization giving rise to the conjugate beam is : (45) with t the delay of the backward pump pulse and: (46) (47) where -i(2a - b) is (within a numerical factor) the contribution of free carriers to the true (steady-state) :

(48) N w and N w/o are the number densities of particles with and without traps respectively. The two terms in eqn (48) correspond to the first two terms in eqn (44) and generalize them in that the dephasing time T'2 for the 1 → 2 transition is not necessarily equal to T 2 . The second factor in (46) accounts for the transient nature of the response. It is close to t p /T1 when tp << T1 . -iC f and -iC s are effective susceptibilities. The amplitude of the incident fields are defined as Ai (t) = Ai when 0 < t < t p and 0 otherwise with f, b, p standing for forward, backward and probe respectively. d' = ( ω 21 - ω)T'2 is the normalized detuning for the 1 → 2 transition. From the expression (45) of PωNL (t) , the fluence fc (t) of the conjugate pulse may be calculated.

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Regarding the induced absorption contribution, until now level 2 was supposed to be a two-pair state and we limited ourselves to interband transitions. We may extend the result to intraband transitions as well in which, for example, the electron is promoted from level 1Se to an nPe level. Close to resonance, the susceptibility is expected to be mainly imaginary. Indeed, from what has been said before, a, b and Cs are expected to be mainly real and positive. For two of the three contributions, numerical estimates can -10 be obtained. For the trapped carrier term, one gets for Cs a number of order 10 e.s.u. -8

The bleaching contribution to Cf is estimated to be of order 10 e.s.u. The induced absorption term is much more difficult to estimate. When the carrier pairs are non interacting, ω21 = ω 10 , T' 2 = T 2 and and it is is clear from eqn (48) that induced absorption cancels absorption saturation. Experimental results. Using the nonlinear absorption technique, absorption saturation as well as induced absorption have been observed (77,78) . The nonlinear response of SDGs had been observed to be usually comprised of a fast and a slow component. But the exact role of free and trapped carriers could be elucidated only recently by time resolving the nonlinear response of a CdS 0.3 Se 0.7 sample having experienced various degrees of darkening. For the fast component, the decay time was observed to decrease upon darkening whereas the magnitude of this fast component was observed to be indepen-dent of the degree of darkening. The magnitude of the slow component steadily decrea-ses upon darkening. These results support the two classes of particles hypothesis in agreement with theory (if we assume one class of particles, Cf involves a contribution from trapped carriers). Particles without traps give rise to the fast (free carrier) compo-nent, the lifetime T1 decreasing from ~ 1.4 ns to ~ 30 ps upon darkening. Particles with traps in which the carriers are very quickly trapped give rise to the slow component, darkened particles no longer contributing to the response. For a non darkened sample, the slow component usually dominates whereas, for a darkened one, the fast component dominates and the slow component eventually vanishes. But the dominant mechanism cannot be known a priori: it depends on the origin of the sample and on its past history. Using a reference sample, the magnitude of -10 Cf and Cscan be obtained. For a fresh sample, CS is found to be of order 10 e.s.u. in -10

agreement with the estimate. Cf is found to be also of order 10 -8

e.s.u. when the

numerical estimate for the bleaching term is 10 e.s.u. This indicates that, even in nanoparticles, we still have substantial cancellation between induced absorption and bleaching. If we think of applications, SDGs in the resonant regime are characterized by the figure of merit where is the absorption coefficient and τr the carrier recombination time (also denoted Tl in the previous model). The response time of the fast component can be made very small by darkening (at the same time, the slow component disappears) and a bistable device with a response time of ~ 25 ps has been demonstrated. Carrier recombination may also become faster at higher laser intensity because of Auger processes but such high intensities correspond to a saturated response and must be avoided. The size dependence of the nonlinearity and of the corresponding figure of merit are also of interest. In the resonant regime, in a working device, the SDG would rapidly be darkened. One would then be left with the fast component. It has recently been

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observed(79) that the figure of merit associated with this fast component decreases when the particle size decreases. Two-photon absorption and free carrier absorption. We now turn to the case where the laser beam is not absorbed. In the bulk, we would say ω < E g . But we point out that, for SDGs, Eg is quite often defined as the photon energy at which the -1

absorption coefficient reaches a certain value (for example 2 cm ). This defines an effective E g which takes confinement into account. Although when ω < E g we do not have one-photon absorption, when Eg /2 < ω < Eg we may have two-photon absorption (TPA). In the non resonant regime, TPA was the first property to be studied, the main point being again to pinpoint the effect of quantum confinement. When a laser beam experiences TPA, its intensity is attenuated according to: (49) where b, proportional to the imaginary part of ( ω ,– ω ,ω), is the two-photon absorption coefficient of the SDG. The second term accounts for what is known as free carrier absorption (absorption due to free carriers created by TPA). N is the number density of free carriers and is given by : (50) neglecting carrier recombination on the picosecond time scale ; σ is the absorption cross section of free carriers. The selection rules for TPA are different from those of linear absorption as has been verified (80) experimentally. Measuring the transmission of a sample as a function of the incident intensity and fitting with eqn 49, one may obtain the values of β and σ. When using picosecond laser pulses, free carrier absorption was observed(81,82) to play an -l8 important role, σ being of the order of 10 cm 2 . This role could be minimized by using femtosecond pulses which allow a direct measurement of TPA. Since the glass matrix is real and does not contribute to the imaginary part of (3) , from the measured values of β and of the volume fraction one may deduce the value of the imaginary part (3) of the nanoparticle χ . When plotted as a function of ω /E g, it was recently (80,82) observed to have the same value and behavior as in the bulk semiconductor although different results had been reported(83) earlier. The dispersive nonlinearity. There have been some studies of the real part of in the same non resonant frequency range. When TPA is present, the free carriers (5) thus created modify the index of refraction. This is a χ mechanism. When working with a single laser frequency, this free carrier refraction is usually dominant and, in optical phase conjugation measurements, the conjugate intensity was observed to scale as the fifth power of the laser intensity. To avoid the problems of free carrier refraction, frequency mixing, also known as non degenerate four wave mixing, was used. When the difference ω - ω ' is large enough, free carriers do not contribute to generation of the wave at frequency 2ω - ω '. In this way, the real part of (3) could be measured (84,85) . The

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dominant contribution was found to be due to the glass matrix. Subtracting χ(3) d and (3) taking the volume fraction p into account, the real part of the nanoparticle χ was (3) obtained and plotted as a function of ω /E g . The real part of χ was also found to be the same as in the bulk and to agree with the results of a two-band model. Here again, different experimental results have been reported. Contrary to the resonant case in which the nonlinear response is due to population changes, in the non resonant case it is due to the optical Stark effect which shifts the energy levels as was shown in . In the non resonant regime, since the absorption losses are very small and the response is very fast, a material is characterized by a figure of merit different from the one quoted above. The criterion is now that the real part of (3) be larger than its imaginary part. General remarks The optical nonlinearities of the II-VI semiconductor dots in glass matrices and in particular their optical Kerr nonlinearity are now reasonably well understood and can be related to some key spectroscopic features of these nanoparticles. This regards the magnitude, dynamics and frequency behavior of these nonlinearities. Along with the saturation mechanisms, these are affected by two-photon direct or indirect transitions and the presence of traps in the semiconductor-glass interface. The case of CdSe and its CdSx Se1-x alloys considered here is not unique and results and conclusions apply to other II-VI and III-V semiconductor quantum dots although the relative contribution of the different processes may be different. There are also cases of semiconductor quantum dots where additional studies must be performed before conclusive predictions can be made regarding the optical nonlinearities. These materials have potentials in optoelectronics pending certain improvements in the fabrication and characterization techniques. EXTENSIONS AND GENERAL REMARKS The previous discussion was concentrated on the optical Kerr nonlinearities in composite materials formed by interfacing or embedding artificial mesoscopic materials in transparent dielectrics. Besides these all optical or photoinduced modifications of the optical characteristics of these materials a whole class of other or similar effects in nonlinear optics can be envisaged there that result from the combined impact of an intense light beam and another external agent such as a static electric or magnetic field or even an acoustic wave. Here we have in mind modifications and modulations of the characteristics mediated through electro-optic, acousti-optic or magneto-optic coupling. All of them can be strongly affected by the photon or electron confinement along the lines discussed above. Of particular interest are the nonlinear magneto-optic effects as they exhibit some quite distinctly new features with respect to the other two classes. These effects and in particular the photoinduced Faraday rotation allow a very efficient photoinduced control of the polarization state of an optical field and the development of reciprocal optical devices like optical valves and others. The photoinduced Faraday effect originates(86) from the combined effect of the Faraday rotation and the optical Kerr effect. In an isotropic medium in the presence of a static magnetic field H0 the two eigenmodes of frequency ω in the direction of H 0 are the left and right circularly polarized waves with indices n- and n+ respectively ; through the

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optical Kerr effect the latter become intensity dependent for high light intensities or I. Accordingly the polarization direction of a linearly polarized input wave E of frequency ω after propagation through a length L in such a medium collinear with H 0 rotates by an angle

where θF = ω L (n- - n+) /2nc is the usual linear Faraday rotation angle and ∆θ NL = θ 2 I = ωL (n 2- - n 2+ ) / 2nc is the photoinduced change of the latter and is proportional to the difference of the optical Kerr coefficients for the left and right circular polarizations. Accordingly the previous photon and electron confinements here too will lead to an enhancement of the photoinduced Faraday rotation in mesoscopic materials when working close the morphology related resonances and certain provisions are made that the difference of the optical Kerr coefficients for left and right polarizations is large. This can be done for instance in II-VI semiconductor nanocrystals like CdS, CdSe or CdTe doped with magnetic impurities such as Mn, the so called semimagnetic or diluted magnetic semiconductors. Through the spin exchange interaction of the band and impurity electrons the Landé factor of the band electrons is enhanced (87) by almost two orders of magnitude and similarly the magneto-optical coupling and the Zeeman splitting of the electron states. Otherwise stated, the magnetic impurities act as local amplifiers of the static field. Without optimizing the interaction configuration, giant photoinduced Faraday rotations have been observed in bulk(88) semimagnetic semiconductors like Cd1-x Mn x Te and their quantum confined nanostructures(89) . In the latter case in fact these photoinduced Faraday rotations are as large as the linear Faraday rotation when the saturation regime of the quantum confined resonances is reached ; thus in a 1 µm thick multiple quantum well of ten repeat units CdTe/Cd1-x Mn xTe photoinduced Faraday rotation angles as high as 20-30 degrees are achieved(89) for moderate light and magnetic field intensities when the frequency is tuned close to the quantum confined excitonic transitions. Similar effects are expected in Cd1-x Mn xTe nanocrystals in a glass matrix and much effort is presently concentrated in the fabrication of such materials. This and other cases clearly justify the present interest and growing effort in the artificial fabrication of composite materials formed with mesoscopic particles embedded in a transparent dielectric, their interfacing and doping with other constituents as well as the control of their morphology. This is a rich field for future investigations that opens new avenues for materials research. In summary in the previous four sections we discussed the way photon and electron confinement affect the optical Kerr nonlinearities in artificial composite materials formed from mesoscopic particles in a transparent dielectric. As it was stressed in the introductory section the two confinements cannot be simultaneously implemented in the same material in the optical range. The first one leads to an enhancement of the effective interaction path through multiple reflections or scattering and can be achieved only in mesoscopic particles of highly transparent dielectrics and of dimension of the order of the optical wavelength embedded in another transparent dielectric of different linear or nonlinear index. The second one, namely the electron confinement, leads to an enhancement of the optical nonlinearities close to certain morphological resonances due to either dielectric or to quantum confinement as exemplified in metal and semiconductor nanoparticles respectively embedded in a glass

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matrix and can be achieved when the size of these nanoparticles is smaller than certain a critical length. The enhancements achieved in all these cases are substantial but the key advantage of these materials relies on the fact that the confinement is artificially implemented and accordingly their properties can be tailored to meet several criteria and requirements for their use in devices. Indeed besides the ones relevant to their intrinsic nonlinear optical properties these materials also possess most of the other properties required for device applications such as robustness, photochemical stability, miniaturisation, interfacing etc, in fact to a higher degree than bulk materials, and make them attractive and competitive with the best bulk materials. Certainly more work is needed to bring them to the level where their properties can be precisely tuned and adapted to a prescribed device application but the essential features regarding their nonlinear optical behavior are now fairly well understood and can be used as a basis for developing appropriate growth and fabrication techniques. Several new approaches are presently emerging in this direction. In addition these mesoscopic materials have several other interesting properties for physicochemical, chemical or even biological applications and the combined effort and interest in these areas has produced remarkable progress in bringing together these features and forging new interdisciplinary research topics. REFERENCES l - J.A. Armstrong, N. Bloembergen, J. Ducuing and P. Pershan, Phys. Rev. 127, 1918 (1962) 2 - Y. R. Shen, The Principles of Nonlinear Optics, John Wiley, New York, 1984 3 - C. Flytzanis, Theory of Nonlinear Optical Susceptibilities in Quantum Electronics, A Treatise, Vol. la, H. Rabin and C.L. Tang (Eds), Academic Press, New York, 1975, p.9. 4 - R.W. Terhune, P.D. Maker and C.M. Savage, Phys. Rev, Lett. 8, 404 (1962) ; J. Giordemaine, Phys. Rev. Lett. 8, 407 (1962) 5 - H. Rabin and A. Bey, Phys. Rev. 156, 1010 (1967) 6 - P.N. Butcher and D. Cotter, The Elements of Nonlinear Optics, Cambridge University Press, Cambridge, 1991 7 - E. Yablonovitch, C. Flytzanis and N. Bloembergen, Phys. Rev. Lett. 29. 865 (1972) 8 - See for instance D. Marcuse Theory of Dielectric Optical Waveguides, Academic Press, New York, 1974 9 - See for instance G.I. Stegeman, Nonlinear Guided Wave Optics in Contemporary Nonlinear Optics, G.P. Agrawal and R.W. Boyd (Eds), Academic Press, New York, 1992, p.1 10- See for instance C. Flytzanis and J, Hutter, Nonlinear Optics in Quantum Confined Structures, in Contemporary Nonlinear Optics, G.P. Agrawal and R.W. Boyd (Eds), Academic Press, New York, 1992, p.297 11 - See for instance Optical Effects Associated with Small Particles P.W. Barber and R.K. Chang (Eds), World Scientific, Singapore, 1988 ; also Optical Particle Sizing ; Theory and Practice, G. Gouesbet and G. Greham (Eds), Plenum, New York, 1988 12 - R.K. Chang and G. Chen in Nonlinear Optics and Materials, SPIE Vol. 1497, 1991, p.2

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13 - C.F. Bohren and D.F. Huffman, The Scattering of light by Small Particles, John Wiley, New York, 1983 14 - C. Flytzanis, C. Hache, M.C. Klein, D. Ricard, Ph. Roussignol, Nonlinear Optics in Composite Materials : Semiconductor and Metal Crystallites in Dielectrics in Progress in Optics, E. Wolf (Ed) North Holland, Amsterdam, Vol. XXIX, 1991, p. 321 15 - See for instance New Functionality Materials, T. Tsuruta, M. Doyanna, M. Seno (Eds), Elsevier, Amsterdam, 1993 16 - J.A.A. Perenboom, P. Wyder and F. Maier, Phys. Reports 78, 173 (1981) ; W. P. Halperin, Rev. Mod. Phys. 58, 533 (1986) 17 - See for instance Quantum Dot Materials for Nonlinear Optics Applications, R.F. Haglund (Ed), NATO/ASI Series, Kluwer Publ., Dodrecht, 1997 18 - M.C. Schanne-Klein, D. Ricard and C. Flytzanis in ref. 17 19 - I.M. Lifshitz and V.V.Slezov, Sov. Phys. JETP 35, 331 (1959) 20 - M. Allais and M. Gandais, J. Appl. Crystallogr. 23, 418 (1990) 21 - E. Duval, A. Boukenter and B. Champagnon, Phys. Rev. Lett. 56, 2052 (1986) 22 - G.P. Banfi, V. Degiogio and B. Speit, J. Appl. Phys. 74, 6925 (1993) 23 - G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 1989 24 - H.M. Tzeng, K.F. Wall, M.B. Long and R.K. Chang, Opt. Lett. 9, 499 (1984) 25 - G. Mie, Ann. Phys. (Leipzig) 25, 377 (1908) ; H.C. Van der Hulst, Light Scattering by Small Particles, Dover Publ. Inc., New York 1981 26 - E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987) ; S. John, Phys. Rev. Lett. 58, 2486 (1987) 27 - See for instance C. Kittel, Introduction to Solid State Physics, Wiley, New York 28 - J. van der Poel, J.D. Bierlein and J.B. Brown, Appl. Phys. Lett. 57, 2074 (1991), E.J. Lim, S. Matsumoto and M.M. Fejer, Appl. Phys. Lett. 57, 2294 (1991) 29 - A. Ishimaru, Wave propagation in Ramdom Media, Vols 1 and 2, Academic Press, New York (1978) 30 - G.B. Altshuler, V.S. Ermolaev, K.I. Krylov, A.A. Manenkov, Sov. Phys. Dokl. 28, 951 (1983) 31 - C. Flytzanis, Annales de Physique (Paris) Suppl. 2, 15, 49 (1990) 32 - N.C. Kothari and C. Flytzanis, Opt. Lett. 11, 806 (1986) ; 12, 492 (1987) ; N.C. Kothari, J. Opt. Soc. Am. B5, 2348 (1988) 33 - V. Jourdrier, J.C. Fabre, P. Bourdon, F. Hache and C. Flytzanis, Proceedings of the Material Research Society, San Fransisco, Spring Meeting 1997 34 - G.L. Fisher, R.W. Boyd and T.M. Moore, Opt. Lett. 21, 1643 (1996) 35 - See for instance H.C. Van der Hulst, Multiple Light Scattering, Vols 1 and 2, Academic Press, New York (1980) ; Ping Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena, Academic, San Diego, 1995 36 - A.F. Iofe and A.R. Regel, Progr. Semicond 4, 237 (1960) 37 - F. Hache and C. Flytzanis to be published 38 - J.C. Maxwell-Garnett, Philos Trans. R. Soc. (London), 203, 385 (1904) ; 205, 237 (1906) ; C. J. Böttcher, Theory of Electric Polarization, Elsevier Amsterdam, 1973 39 - A. Kawabata and R.J. Kubo, Phys. Soc. Jap. 21, 1765 (1966) 40 - K.C. Rustagi and C. Flytzanis, Opt. Lett. 9, 344 (1984) 41 - D. Ricard, Ph. Roussignol and C. Flytzanis, Opt. Lett. 10, 511 (1985) 42 - D. Ricard, M. Ghanassi and M.C. Schanne-Klein, Opt. Comm. 108, 311 (1994)

461

43 - See for instance G.G. Weinreich, Solids : Elementary Theory for Advanced Students, John Wiley, New York ; also W. Ashcroft and N.D. Mermin, Solid State Physics, Holt-Saunders, Tokyo, 1981 44 - G. Bastard, Wave Mechanics applied to Semiconductor Heterostructures, Editions de Physique, Paris, (1988) 45 - Al. L. Efros and A.L. Efros, Sov. Phys. Semic. 16, 772 (1982) 46 - L. Banyai, S.W. Koch, Semiconductor Quantum Dots, World Scientific, Singapore, 1993 47 - L.E. Brus, J. Chem. Phys. 80, 4403 48 - J.B. Xia, Phys. Rev. B40, 8500 (1989) 49 - A.I. Ekimov, F. Hache, M.C. Schanne-Klein, D. Ricard, C. Flytzanis, C. Kudryavtsev, I.A. Yazeva, T.U. Rodina and A1. L. Efros, J. Opt. Soc. Am. B10, 100 (1993) 50 - D.J. Norris and M.G. Bawendi, Phys. Rev. B53, 16338 (1996) ; D.J. Norris, A1.L Efros, A.M. Rosen and M.G. Bawendi, Phys. Rev. Lett. B53,196347 51 - P.E. Lippens and M. Lannoo, Phys. Rev. B41, 6079 (1990) ; N.A. Hill and K.B. Waley, J. Chem. Phys. 99, 3707 (1993) 52 - E. Westin, M.C. Schanne-Klein, D. Ricard and C. Flytzanis, to be published 53 - A.H. Wilson, The Theory of Metals, Cambridge University Press, Cambridge 1965 ; D. Pines and P. Nozières, The Theory of Quantum Liquids, W.A. Benjamin Inc., New York (1996) 54 - L. Genzel, T.P. Martin and U. Kreibig, Z. Phys. 132, 339 (1975) ;R. Rupin and H. Yatom, Phys. Stat. Sol. (b) 74, 14 (1975) 55 - F. Hache, D. Ricard and C. Flytzanis, J. Opt. Soc. Am. B3, 1647 (1986) ; F. Hache, Ph. D. Thesis, University of Paris, Orsay Campus (1988) 56 - F. Vallée, N. Del Fatti and C. Flytzanis in Nanostructured Materials : Clusters, Composites, and Thin Films, M. Mostovits (Ed), Am. Chem. Soc. Books, 1997 57 - F. Hache, D. Ricard, C. Flytzanis and U. Kreibig, Appl. Phys. A47, 347 (1988) 58 - M.J. Bloemer, J. W. Haus and P.R. Ashley, J. Opt. Soc. Am. B7, 790 (1990) 59 - K. Uchida, S. Kaneko, S. Omdi, C. Hata, H. Tanji, Y. Asahara, A.J. Ikushima, T. Tomizaki and A. Nakamura, J. Opt. Soc. Am. B11, 1236 (1994) 60 - T. Tomizaki, A. Nakamura, S. Kaneko, K. Uchida, S. Omi, H. Tanji and Y. Asahara, Appl. Phys. Lett. 65, 941 (1994) 61 - J.Y. Bigot, J.C. Merle, O. Chegut and A. Dannois, Phys. Rev. Lett. 75, 4702 (1995) 62 - G. Bruggemann, Annal. Phys. (Leipzig) 24, 636 (1935) 63 - El.M. Shalaev, Phys. Reports 272, 61 (1996) 64 - O.L. Malta, , P.A. Santa Cruz and F. Auzel, J. Sol. St. Chem. 68, 314 (1987) 65 - M.G. Bawendi, P.J. Carroll, W.L. Wilson and L.E. Brus, J. Chem. Phys. 96, 946 (1992) 66 - Ph. Roussignol, D. Ricard, J. Lukasik and C. Flytzanis, J. Opt. Soc. Am. B4, 5 (1987) 67 - M. Mitsunaga, M. Shinojima and K. Kubodera, J. Opt. Soc. Am. B5,, 1448 (1988) 68 - V. Grabovskis, Ya. Dzenis, A.I. Ekimov, A, Kudryavtsev, I.A. Tolstoi and U.T. Rogulis, Sov. Phys. Sol. St. 31, 149 (1989) 69 - D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood and C.A. Burrus, Appl. Phys. Lett. 45, 13 70 - F. Hache, D. Ricard and C. Flytzanis, Appl.Phys. Lett. 55, 1504 (1989) 71 - D. Cotter, H.P. Girdlestone, and K. Moulding, Appl. Phys. Lett. 58, 1455 (1991)

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85 86 -

87 88 89 -

S. Schmitt-Rink, D.A.B. Miller and D. Chemla, Phys. Rev. B35, 8113 (1987) G. Bret and F. Gires, Appl. Phys. Lett. 4, 175 (1964) L. Banyai, Y.Z. Hu, M. Lindberg and S.W. Koch, Phys. Rev. B38, 8142 (1988) Ph. Roussignol, D. Ricard, K.C. Rustagi, and C. Flytzanis Opt. Comm. 55, 143 (1985) M. Ghanassi, L. Piveteau, L. Saviot, M.C. Schanne-Klein, D. Ricard and C. Flytzanis, Appl. Phys. B61, 17 (1995) M.G. Bawendi, W.L. Wilson, L. Rothberg, P.J. Carroll, T.M. Jedin, M.L. Steigerwald and L.E. Brus, Phys. Rev. Lett. 65, 1623 (1990) N. Peyghambarian, B. Fluegel, D. Hulin, A. Migus, M. Joffre, A. Antonetti, S.W. Koch and M. Lindberg, IEEE, J. Quant. Electron. 25, 2516 (1989) M.C. Schanne-Klein, L. Piveteau, M. Ghanassi and C. Flytzanis, Appl. Phys. Lett. 67, 579 (1995) K.I. Kang, B.P. Mc Ginnis, Y.Z. Hu, S.W. Koch, N. Peyghambarian, A. Mysyrowicz, L.C. Lin and S.H. Risbud, Phys. Rev. B45, 3465 (1992) G.P. Banfi, V. Degiorgio, M. Ghigliazza, H.M. Tan and A. Tomaselli, Phys. Rev. B50, 5699 (1994) G.P. Banfi, V. Degiogio and H.M. Tan, J. Opt. Soc. Am. B12, 621 (1995) ; G.P. Banfi, V. Degiogio, D. Fortusini and H.M. Tan, Appl. Phys. Lett. 67, 13 (1995) D. Cotter, M.G. Burt and R.J. Manning, Phys. Rev. Lett. 68, 1200 (1992) H.L. Fragnito, J.M.M. Rios, A.S. Duarte, E. Palange, J.A. Medeiros Neto, C.L. Cesar, L.C. Barbosa, O.L. Alves and C.H. Brito Cruz, J. Phys. Condens. Matter 5, A179 (1993) S. Tsuda and C.H. Brito Cruz, Appl. Phys. Lett. 68, 1093 (1996) C. Buss, E. Westin, S. Wabnitz, R. Frey and C. Flytzanis in Novel Optical Materials and Applications, I.C. Khoo, F. Simoni and C. Umeton (Eds), John Wiley, New York, 1997 p. 295 J.K. Furdyna, J. Appl. Phys. 64, R29 (1988) J. Frey, R. Frey and C. Flytzanis, Phys. Rev. B45, 4056 (1992) C. Buss, R. Pankoke, P. Leisching, J. Cibert, R. Frey and C. Flytzanis, Phys. Rev. Lett. 78, 4123 (1997)

463

CONTRIBUTORS

V. M. AGRANOVICH, Institute of Spectroscopy, Troitsk, Moscow Obl., Russia G. ASSANTO, Optoelectronic Laboratory, University « Roma Tre », Rome, Italy H. BENISTY, Ecole Polytechnique, Palaiseau, France P. BONTEMPS, Dept. of Pure and Applied Physics, University of Salford, Salford, UK A; A. BOARDMAN, Dept. of Pure and Applied Physics, University of Salford, SALFORD, UK S. BRASSELET, France Telecom, Bagneux, 92220-Bagneux, France Z. CHEN, Department of Electrical Engineering, Princeton University, Princeton, New Jersey, USa B. CROSIGNANI, Dipartimento di Fisica, Universita dell’Aquila, L’Aquila, Italy P. DI PORTO, Dipartimento di Fisica, Universita dell’ Aquila, L’Aquila, Italy M. FEJER, Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA Ch. FLYTZANIS, Laboratoire d'Optique Quantique, Ecole Polytechnique, Palaiseau, France F. KAJZAR, CEA, LETI - Technologies Avancées, CE Saclay, Gif sur Yvette, France A. E. KAPLAN, Dept. of Electr. & Comp. Eng. The Johns Hopkins University, Baltimore, MD, USA K. KOUTOUPES, Dept. of Pure and Applied Physics, University of Salford, Salford, United Kingdom S. KRYSZEWSKI, Institute of Theoretical Physics and Astrophysics University of Gdansk, Gdansk, Poland M. MITCHELL. Department of Electrical Engineering, Engineering Quadrangle, Princeton University, Princeton, NJ, USA J.- M. NUNZI, CEA, LETI - Technologies Avancées, CE Saclay, Gif sur Yvette, France

465

D. OSTROWSKY Laboratoire de Physique de la Matiere Condensee, Universite de Nice - Sophia Antipolis, NICE, France R. REINISCH, LEMO-ENSERG, Grenoble, France G. SALAMO, Department of Electrical Engineering, Engineering Quadrangle, Princeton University, Princeton, NJ, USA M. SEGEV, Department of Electrical Engineering, Engineering Quadrangle, Princeton University, Princeton, NJ, USA M.-F. SHIH, Department of Electrical Engineering, Engineering Quadrangle, Princeton University, Princeton, NJ, USA P. L. SHKOLNIKOV, Dept. of Electr. & Comp. Eng, The Johns Hopkins University, Baltimore,MD, USA G. I. STEGEMAN, University of Central Florida, CREOL, Orlando, FL, USA F. STRAUB, Abteilung für Quantenoptik, Ulm University, Ulm, Germany L. TORNER, Dept. Signal Theory and Communications, Polytechnic University of Catalonia, Barcelona, Spain E. W. VAN STRYLAND, Physics and Elec.& Comp. Eng., University of Central Florida, Orlando, FL, USA C. WEISBUCH, Ecole Polytechnique, Palaiseau, France K. XIE, Dept. of Pure and Applied Physics, University of Salford, Salford, United Kingdom J. ZYSS, France Telecom, Bagneux, France

466

INDEX

Absorbance, 112, 113 Acceptance bandwith, 385–386 Active amplifier nonlinearities, 24 AgGa 1-x In xSe 2, 393 AgGaS 2 ,393 AgGaSe 2 , 393 All-optical amplifiers, 346 devices, 19 diode, 350 Mach–Zehnder switch, 19 modulators, 341, 346 poling, 31,95, 99, 104, 107, 121 processing, 341,369 switching, 46, 342,351,229 transistors, 341 Ammonium dihydrogen phosphate (ADP), 392, 394 Amplitude modulation, 341 of soliton, 229 Analog operations, 161 Angular acceptance criterion, 385 momentum, 234 steering, 365 Anharmonic oscillator, 300 potential, 309 Anomalous-dispersion regime, 193 Anticrossing region, 144 Approximation of non-degenerate band, 442 Area theorem, 140 AsGa quantum well, 139 Attenuated total reflection (ATR) technique, 119 Auger processes, 456 Auzel effect, 410,411 Axial order, 112 Azimuthal modulational instability, 248 Bandfilling, 24 Bare cavity, 141 cavity photon, 142 Basic envelope soliton solution, 223 BaTiO 3 (barium titanate), 260,270 BBO (BaB2 O 4 ), 344,356,378,393 Beam locking, 242 Biexciton two-photon transition, 133

Birefringence, 110 Birefringent media, 379 phasematching, 379, 380 walk-off, 365 Bloch 2π-soliton, 296 full equations, 295,296 optical equation, 158 Bloch–Boltzmann equations, 150, 168, 170, 171, 175, 177,297 Bohr radius, 440 Bound electron effects, 24 quantum well excitons, 137 Boussinesq-like equation, 291,299 Bragg coupling, 361 reflection, 361 Bright pulse, 266 Carrier frequency, 187, 192 heating, 27 Cascading effects, 29 phase-shift, 341 Causality, 40 Cavity model, 136, 139 cross-section of, 145 normal modes, 135 polariton modes, 142 polaritons, 141 Cd1-x Mn x Te, 459 CdS, 141,441 CdSe, 443,458 Cerenkov phase matching, 9 radiation, 320,333–340 Charge transfer molecule, 106 χ (2) grating in glass fibers, 123 χ (2) susceptibility, 29–30, 94, 97 χ (3) susceptibility’ 28–29 Chirp, 191–192 balance, 185 from dispersion, 199 from nonlinearity, 199 parameter, 190, 197 positive, 198 spatial, 195 Chirping, 190–l91, 196

467

Chromophore, 105 orientation of, 119 Cinnabar (HGS), 394 Coherence length, 344, 378 Coherent anti-Stokes Raman spectroscopy (CARS), 16 frequency degenerate effects, 15 interactions, 280 Collision integrals, 167 kernel, 159–160, 162 operator, 163–166, 168–169, 177–178 interactions, 366 in Kerr media, 280 in photorefractive nonlinear media, 261 in saturable nonlinear media, 261 Colloidal crystals, 430 Complete elliptic integral, 385 Composites, 427 Confinement, 214 energy, 440 Contact poling, 107 Continuity equation, 262 Convention factors, 103 Conversion efficiency, 378,385–386,401 assymptotic limit of, 387 Corona poling, 106–108, 126 Correlation function, 174, 175, 179 Coulomb long-range attraction, 297 Coupled-modes 319, 323, 333, 339, 340 Critically phasematched crystal, 388 Cross-phase modulation, 21 Crosslinking, 107 Crystal Violet, 82–83 Cubic nonlinearity, 343 Cut-off frequency, 313 Cyanobiphenyl, 109

Diffusion field, 263 tensor, 153, 157, 175 time, 142 Digital operations, 142 (1,5)-Dinitro,(2,6)-di(N,N)-n-butylaminobenzene (DNDAB), 82–83,94 Diode, 356 lasers, 407,408; 425 Dipolar χ (2) symmetry, 98 approximation, 101 interaction energy, 113 moment, 107 Directional coupler, 358 Disperse red #1 (DR#1), 82–83,97, 120 in PMMA, 121 Dispersion, 144 balance, 185 equation, 215 law, 141 limited propagation, 219 Dispersive nonlinearity, 457 Display, 407–408 Distributed Bragg reflector (DBR), 419 couplers, 359 feedback grating, 361 DNA, 425,425 Doppler approximation, 173 limit, 173 shift, 293 Down-conversion, 334, 342,369 DR1-MMA, 96,98 Drift velocity, 171 Drude term, 445 Duffing effect, 260–261,264,285 equation, 295, 298

DAN, 353–354 Dark intensity, 263 Dark-line defects, 417 DAST, 233 Data storage, 407 DC poling, 114 Debye length, 263 Defect nucleation, 417 Defocusing, 343 Degeneracy factors, 3,101, 103 Degenerate four-wave mixing (DFWM), 53 geometry of 18 Depletion, 344 Dielectric breakdown, 108 composite, 434 confinement, 428,437 constant, 18, 111 photonic band gap, 434 susceptibility, 141 tensor, 379 Difference frequency generation, 375 Diffraction, 202,229 length, 207 limited propagation, 219

Effective length, 390 mass approximation, 439–440 medium approach, 437 nonlinear coefficient, 380 phase mismatch, 378 Efficiency of the polar ordering, 116 Eigenmodes, 293 Eigenvalues, 353 Electric field induced second harmonic generation (EFISH), 77,80 Electroabsorption, 452 Electrode poling, 107 Electromagnetic bubbles (EMB’s), 291–318 amplitude of, 305 generation of, 302 length of, 298 non-oscillating, 296 precursor of, 303–308 propagation of, 299 pulses, 315 solution, 294,297 steady-state, 298 small-amplitude, 298 unipolar, 314

468

Electromagnetic confinement, 430, 432 solitons, 291 Electron beam poling, 107–108 confinement, 428, 430–431, 436 Electrooptic (Pockels) effect, 13, 262 coefficient, 311–312 materials, 14 tensor, 262 Energy efficiency, 386–387 flow, 233 localization 361 regulators, 70 transfer (ET), 410, 413, 414, 416, 422 Envelope conservation, 443 equation, 215,218 function, 143,215, 216 soliton, 200,213 Epitaxy, 104 Excited State Absorption (ESA), 37, 409, 410, 413–414, 416, 420 Exciton bleaching, 24 branches 134 donor-bound, 133 Exciton–exciton screening effects, 137 External self-action, 53 Extraordinary crystal directions, 350 refractive index, 379 EZ-scan technique, 53–55

Fabry–Perot cavity, 388 quasi-modes, 146 Faraday effect photoinduced, 458 Faraday rotation, 458 photoinduced, 458–459 Fermi level, 439, 442 ,447 ,449 Ferroelectrics, 399–400 Ferroelectric crystals, 350 Fibers, 354 Fiber amplifiers, 356 optic temperature sensor, 420, 425 Figure of merit, 378,429 First order hyperpolarizability, 87, 89 Fluorescence, 423 green, 414 line narrowing (FLN), 451 Focal power 390 Fock–Leontovich wave equation, 265 Focusing, 343 Four-wave mixing, 134, 349 Fourier components, 187 frequencies, 300 Franz–Keldysh behavior, 452 Fraunhofer diffraction pattern, 206 far field, 206 Free carrier absorption, 59, 457 refraction, 59 Frenkel excitons, 133 Frequency conversion, 375 Fresnel (near field) pattern, 206 Functionalized polymers 105

GaAs, 417 GaN, 408, 417–418 Gap solitons, 341, 361–362 Gauss law, 262 Gaussian beams, 311, 365 half-cycle pulses (HCP), 311 spatial profile, 311 temporal profile, 311 Gibbs–Boltzann distribution function, 112–113 Glasses, 27 Glass transition temperature, 108 Gradient velocity, 154, 157 Gratings, 359 Grating coupler, 359 Group index mismatch, 386 Group velocity, 138, 144, 188, 216, 379 dispersion, 189, 195,221 Guest–host systems, 105 Guided modes, 319–326, 336, 337, 339 powers, 353 wavenumber, 216 waves, 341,369 wave all-optical switching devices, 341 wave wavevectors, 353 Half-cycle pulses (HCP), 302–315 Half-harmonic generation, 352 Hamiltonian, 440,452 dynamical system, 233 Harmonic generation, 316 light scattering, 80–81, 82 polarized, 94, 96 Helmholtz’s equation, 264–265 Heteroepitaxy, 104 Hole burning spectroscopy, 451 Holmium, 415 Hot electron contribution, 447 Idler wave, 390 Imbalance, 351–352 Improper poles, 328, 331, 335, 336 Incoherent all-optical effects, 15 interactions, 280 Induced polarization, 428 transparency, 361 Infrared quantum counter, 409 InP:Fe, 286 Interband contribution, 447 nonlinearities, 25 transitions, 442–443 Interferometer, 357 Inverse scattering transform (IST) method, 226 Inversion symmetry, 101 Isolators, 341 K(TiO)P 1-x As x O4 , 395 KDP crystal, 97, 378 Kerr effect, 40, 259, 260, 262, 268, 286 coefficient, 342–343 limit, 278 medium, 274, 277, 280

469

Kerr effect (cont.) nonlinearity, 20, 27, 133, 143, 146, 291 polarizability degenerate, 454 regime, 272 solitons, 262, 277 suceptibility, 446 Kerr-like dark solitons, 275 manner, 203 nonlinear medium, 272 Kleinman permutation symmetry, 94 KNbO 3, 393 Kohlrausch–Williams–Watt (KWW) stretched exponential function, 128 Kortdeweg–de-Vries (KdV) differential equation, 184–185, 389 modified, 115, 300, 307 Kramers–Kronig relationship, 23, 40, 452 KTA(K(TiO)AsO 4 , 394 KTP(KTiO)PO 4 , 233, 344, 351–354, 364–368, 378, 393, 399–401 Laboratory frame time, 188–189 reference frame, 110 Landau theory of Fermi liquids, 448 Lanée factor, 459 Langmuir–Blodgett technique, 104 films, 10 Laser mode-locking, 349 pumped fiber lasers, 408 Leaky mode, 319–320, 326–328, 332–340 Legendre polynomials, 112–113 LiB 2 O 3 , 393 Light bullets, 230 Light emitting diode (LED), 408, 418 arrays of, 408 Light-induced diffusive pulling (LIDP), 156–157 drift (LID), 149, 151, 155, 171, 178 kinetic effects (LIKE), 149, 151 Lighthill criterion, 220 test, 221 LiNbO 3 (lithium niobate), 233, 260, 270, 354, 356, 358, 360, 378, 397–403, 418, 420 electrically poled, 402 waveguides, 391 Line broadening, 451 Linear Boltzmann equation, 160 cavity polaritons, 141 coupled equation, 229 polarizability, 87, 110 Liquid crystals, 116 crystalline polymer, 107 nematic, 112 smectic, 112 Liquid droplet, 433 LiTaO 3 , 270, 399–401 Local field factor, 85, 103 time, 189–189, 197 Localization regime strong, 434 weak, 434 Localized wave, 184 Lorentz–Lorenz formula, 103

470

Low conversion limit, 314, 377 frequency limit, 314 LSJ manifolds, 412 Macroscopic polarization, 102 second order NLO suceptibility χ(2) , 102 Magneto-optic coupling, 458 Main chain polymers, 105 Manakov’s solitons, 286 Manley–Rowe relation, 233 Material figure merit (FOM), 19, 387; see also Figure of merit nonlinearities, 20 systems, 20 Maxwell equation, 291, 294, 299–300 reduced, 301 Maxwell + Bloch equation, 294, 301, 306 Maxwell + full Bloch equation, 292 Maxwell–Boltzmann distribution, 83, 85 Maxwell–Garnett effective medium approach, 449 MBA-NP, 347 Mesoscopic materials, 427, 431, 444 Metal nanoparticles, 448 Metastable level, 414 Microcavities, 430 Mid-Ir OPO, 400 Mie theory, 437 Miller’s rule, 394 Modal dispersion phase matching (MDPM), 7–8 Modal fields, 391 Mode mixing, 360 Model kernel, 162 Modified Kortdeweg–de-Vries (KdV) equation, 115, 300, 307 NLS, 393 spherical Bessel functions, 113 Modulational instabilities, 236 Molecular axis, 110 beacon, 415, 420, 423 deposition techniques, 134 diode, 77 engineering, 90 epitaxy, 104 hyperpolarizabilities, 101 nonlinear optics, 99 nonlinearities, 22 reference frame, 110 re-orientation, 120 Morphological resonances, 427 Morphology dependent resonances, 433 Multiple quantum wells (MQW), 356 MSVP model, 116–117 Multi-bubble solution, 307 Multi-Electromagnetic bubbles, 303 Multiphoton absorption, 89 Multiple dark sceening solitons, 276 scattering regime, 436 Multiplicative factor, 103 Multipolar molecules, 79, 89 Multipolar symmetry, 97, 99 systems, 78

Near-resonant response, 23 Nematic liquid crystals, 112; see also Liquid crystals NLO susceptibility tensors, 102 Non-integrable equations, 186 Non-resonant response, 23 Noncentrosymmetric materials, 341 Nondegenerate nonlinearities, 62 Nonlinear absorption, 343 atoms, 291 coefficient, 377 competition, 341 drive, 377–378, 385 integrable equations, 186 integrated devices, 342, 353 length, 207 materials, 29 392–394, 398–403 Maxwell equation, 301 mechanisms, 21 medium, 219 non-integrable equations, 186 optical beam propagation, 264 partial differential equation, 184 phase shift, 341 polarization, 2 processes, 341 scattering, 432 multiple, 436 scattering cross-sections, 81 Schrödinger equation (NLS), 134, 144, 186, 191, 199, 213–214, 219–220, 231, 259, 267–268 fundamental solution of, 222, 223 spectroscopy, 16 susceptibility, 3, 375, 377 spatially varying, 396 Nonlinearity balance, 185 Nonlocality, 428 Nonresonant Kerr nonlinearity, 439 Normal modes, 137 Normalized brightness, 385 intensity, 263 Octupolar configuration, 97 χ (2) symmetry, 98 macroscopic susceptibility, 90 molecules, 93 One photon absorption, 20 contribution, 93 excitation probability, 91–93 One-dimensional diffraction, 208 Optical bistability, 34 l–342 density, 112 dielectric constant, 111, 115 engineering, 369 Kerr effect, 448, 452–453, 458–459; see also Kerr effect nonlinearity, 427–429, 438, 446 limiting, 62 parametric amplification (OPA), 375 amplifier, 9 generator (OPG), 9, 403

Optical Parametric Oscillator (OPP), 99, 390–391, 402, 403 oscillation, 9–10, 376 phase conjugation, 258, 453, 455 piston, 156 poling, 89; see also All optical poling rectification, 292, 342 signal idler, 390 Stark effect, 458 shift, 446 Optimum aspect ratio, 385 drive, 385 length, 388 Orbital angular momentum, 411 Order parameter, 111, 112 Ordinary crystal directions, 350 refractive index, 379 Orientation distribution function, 96, 112 Orientational averages, 117 Overlap integral, 7 Parametric amplifier, 390; see also Optical parametric amplifier conversion, 407 effects, 341 interplay, 368 Paraxial approximation, 311–312 Periodically poled LiNbO3 (PPLN), 344, 347, 402, 420–421 LiTaO 3 , 403 OPO, 402 Phase and polarization interferometers, 345 Phase conjugation, 260, 349; see also Optical phase conjugation conjugation, 260 matched crystals, 390 matching, 344, 378, 387 birefringent 395 condition, 5 critical, 387, 391 modal, 395 noncritical, 387, 391, 394 SHG, 5 type I, 240,380 type II, 253, 342, 347, 380 mismatch, 342, 389 shift, 369 velocity, 216 Phase-to-amplitude transducers, 346 Phase-velocity matching, 376 Phonon spectrum, 412 thermal LO, 142 Photoassisted poling, 104, 107, 119, 123 Photo-crosslinking polymers, 105, 107 Photodarkening, 451 Photogalvanic effect bulk, 270 Photoinduced χ(2) , 96 motion, 94 nonlinear response, 97 tensor, 98 processes, 78, 83, 86

471

Photoinduced χ (2) ( cont.) reorientation mechanism, 99 susceptibility, 95 Photoisomerization, 86 Photoluminescence excitation (PLE), 451 Photon confinement, 428, 430 echoes, 316 Photonic engineering, 77, 90 Photorefractive crystal, 261, 273 soliton, 274 effect, 13, 260–261, 264, 285 precursor of, 303, 306–308 in LiNbO 3 , 394 screening solitons, 271, 279, 281–282; see also Solitons spatial solitons bright and dark, 259–260; see also Solitons vector solitons, 286; see also Solitons Photothermal poling, 107–108 Photovoltaic (or photogalvanic) effect bulk, 270 current, 271 self-defocusing nonlinearity, 271 soliton, 260, 270–271 Planar waveguide, 209–210, 353 Plane wave, 2, 341 Plane waves, 341 PMMA-DR #1, 109, 121, 125; see also DR #1PMMA Pockels effect, 261, 286; see also Electrooptic effect Polar order, 112 growth of 125 orientation of chromophores, 117 spatial profile of, 125 Polarization of seeded SHG coefficient, 125 configuration, 96 driven wave function, 4 field, 108 rotation of, 351 Polaritons linear, 134 Polarized HLS, 94, 96 Poled polymer, 10, 104, 111, 129 Poling, 95, 99, 107, 123 efficiency of, 109, 129 field, 110 Polymers thermally crosslinking, 105, 107 Potassium dihydrogen phosphate (KDP), 392, 394 Potassium niobate, 344; see also KNbO 3 Poynting vector, 205 walkoff, 231, 242, 387, 390 Prism coupler, 359 Proper poles, 320, 323, 326, 328 Pulse and beam envelope soliton propagation, 226 conditioning, 349 dispersion, 346 in a linear medium, 197 in nonlinear medium, 198 propagation in linear media, 193 in nonlinear medium, 194 spreading, 189 pump-probe spectra, 421 pump-probe Z-scan, 59

472

Quasi phase matching (QPM), 6, 8, 13, 31, 233, 240, 350, 375, 377, 395–403, 418, 420 configuration of, 428,434 history of, 398 interaction, 396 materials, 400 mid-IR generation of, 400 OPO, 402 theory of, 396 Quadratic cascading, 341, 343 materials, 343 nonlinear frequency conversion, 377 Quantum confinement, 428, 446 confinement Stark effect, 452 well, 133–134, 136, 142–143, 146, 417 asymmetric, 12–13 exciton, 137 Quartz, 392 Quasi-steady-state photorefractive dark solitons, 277 solitons, 270 spatial solitons, 270 Quenching, 414, 423 Rabi frequency, 295, 301–302 phase, 301 splitting, 299 splitting dimensionless, 142 Radar signals, 192 Radiation fields, 319–321, 323, 326, 328, 334, 336 Radiative waves, 122 Random composites, 432 Rare earth (RE), 410 doped fiber, 407 Real-time holography, 260 Recombination time, 142 Refractive index nonlinear, 30 Relaxation, 128 rates, 129 times, 129 Repulsive Kerr collision, 281 Resonance, 319, 328–329, 331, 333, 335–336, 338–339 Resonant enhancement, 3 response, 23 Retardation, 429 Reverse-saturable absorber, 62 Saturable absorption, 22 plasma density, 142 Saturation spectroscopy, 451 SBN (strontium barium niobate), 260, 275–276 Scattering potential function, 227 Schott filters, 450 Schrödinger equation, 200, 222, 291 nonlinear solution of, 222 nonlinear unmodified, 235 solution of, 294, 297 Scientific instrumentation, 407 Screening solitons, 271, 279, 281–282, 272–274, 263

Second harmonic field, 377 Second harmonic generation (SHG), 4, 95, 124; 126, 230, 341, 375, 378–380, 418, 425 acceptance bandwidth, 386 asymptotic limit of efficiency, 387 Band width limitations, 387 conversion efficiency, 385–386, 390 critically phasematched, 388 effective length, 390 efficiency, 389 energy efficiency, 386–387 externally resonant, 388 figure of merit, 5, 12, 387, 388 focused, 387 growth of, 118 high average power, 389 in lossy materials, 387 noncritically phasematched, 388 normalized efficiency of, 401–402 phase matching in waveguides, 6 phase mismatch, 389 in quasi phase matched structures, 394–403 in silica, 78 single pass conversion efficiency, 389 in situ, 118 spatial walkoff effects, 387 temporal decay of, 128 thermal phase mismatch, 389 Type I, 4, 6 Type II, 6 ultrafast, 385, 387, 403 in waveguide, 12 waveguide devices, 391 Second-order hyperpolarizability, 87 soliton, 201, 234 susceptibility, 342 Seeding, 346, 361 power dependence, 127 procedure, 125, 129 Self focusing, 40, 276, 286, 202, 204–205, 207–212 229 effects, 272 Self-confined beams, 341 Self-defocusing, 276 nonlinearity of, 271 Self-guided beams, 364 Self-induced transparency, 133, 145, 291, 316 Self-induced transparency (SIT) soliton, 139 pulses, 137 Self-phase modulation, 21 Self-trapped beams, 208, 210, 212 beam propagation, 263, 265 solutions, 261 (stationary) solutions, 206, 267 Self-trapping, 221, 229 of an incoherent beam, 261, 285 Semiconducting band gap, 25 Semiconductor, 37, 47 doped glasses, 431 microcavities, 133 nanocrystals, 431, 449

Semiconductor (cont.) nonlinearities, 24 photonic band gap, 434 Semimagnetic semiconductor, 459 Shallow water, 183–184 Shock formation distance, 309 waves, 296 Shock-like wave fronts, 308 Side chain polymers, 105 Sidelobes, 345 Signal parametric gain, 390 wave, 390 Similarity rules, 235 Simultons, 362 Sine-Gordon equation, 138, 291, 301 SIT pulses, 139–141 SKS model, 116 Slowly-varying-envelope approximation, 342 Small-signal amplification, 351 Smectic liquid crystals, 112; see also Liquid crystals Solitary wave, 183–186, 190, 292, 315 waves spatial, 342, 364 Soliton, 183–286, 291, 301 (1+1), 235 (2+1), 235 amplitude of, 229, 305 annihilation of, 271 basic envelope solution, 223 beam, 274 bright, 234, 267, 268, 279 condition for, 295 envelope of, 186, 187, 199 photorefractive screening, 266 screening, 271–272 spatial, 202 vortex, 270 buried, 236 collisions, 278, 280 content, 200 critical energy, 140 dark, 186, 220–221, 267, 269, 275–276 sceening, 275 solutions, 145 vortex, 270 discrete, 250 excitation of, 239 existence conditions, 230 curves, 26 first-order, 234 fission of, 89, 271 formation, 198 fundamental, 201 fusion of, 89, 271 generation of, 183, 316 grey, 267 high-order, 200, 266–267 incoherent, 283, 285 incoherent pair, 286 induced waveguides, 261 intensity, 271

473

Soliton ( cont .) interactions, 271 like, 156 lowest-order, 198 multiple dark, 276 numbers, 201 optical Kerr, 278 fiber, 291 oscillating, 239 period, 200 photorefractive, 261, 272, 279 photovoltaic, 260, 270, 271 quadratic, 31 in quadratic nonlinear media, 230 in quasi-phase-matched samples, 270 propagation, 316 pulse nonlinear, 134 Russel’s, 185–186 solution, 144, 146, 186 spatial, 183, 206, 208, 210, 212–224, 229, 260, 269–270, 283 bright, 202, 260 dark, 260 screening, 267, 269 in a waveguide, 209 spiraling, 199, 283 temporal, 183, 187, 195, 213, 218, 221, 224–225, 260,362 velocity of, 146, 229, 243 in water, 185 width of, 272 Space charge field, 125–126, 261, 272, 286 wave, 328 Spectral hole burning, 27 Spherical Bessel functions, 112 harmonic functions, 79 Spin quantum number, 411 Split modes, 146, 364 Square pulse envelope, 191 Surface plasmon resonance, 448 Stark effect, 114 levels, 411 shifted frequency, 306 Static field poling, 107 Kerr effect, 452–453 Steady-state nonlinear susceptibility, 143 Stimulated Brillouin scattering, 16–18 cascade Raman scattering, 292 Raman scattering, 16–17 Rayleigh scattering, 16–17 Stop-gaps, 361 Strong confinement regime, 440 Sum-frequency generation, 407 Surface damage threshold (in SHG), 391 Surface plasmon resonance, 438 switching, 46, 229, 341–342 Symmetry of the induced χ (2) susceptibility, 126 Taylor expansion, 187–188 TE modes, 136 Temporal pulse, 195

474

Thermal focusing, 401 lensing, 390 mismatch, 390 phase mismatch, 389 Thermally crosslinking polymers, 105, 107 Thermo-optic effect, 24 Thermodynamic force, 154, 162 Third-order Kerr nonlinearity, 140 NLO phenomena, 15 nonlinearity, 136 soliton, 234 susceptibility, 37, 47 Three-wave mixing, 134, 347 Three-level model, 87 Three-wave interaction, 342 Threshold intensity, 391 fluence, 391 power, 199 Three photon absorption, 20 Thulium, 415 Ti:Sapphire laser, 300 Time-domain, 198 TM modes, 136 Topological charge, 270 formation of, 248 wave front dislocations, 2586 Trans-ci isomerization, 119–120 Transport coefficients, 167, 168, 171, 178–179 Transverse beam momentum, 234 radius, 312 Two-beam coupling, 37 interactions, 37 Two-color Z-scan, 56 Two-component solitons, 286 Two-Level Atom, 45, 138, 267, 291 anharmonicity, 146 medium, 137 model, 89 system, 454 Two-parabolic band model, 47 Two photon absorption (TPA), 20, 23, 91–93, 453, 457 Ultrafast nonlinear absorption, 37 refraction, 37 signal processing, 229 Unbalanced components, 348 Up-conversion, 342, 369, 409–410, 422–425 fiber lasers, 408, 416 lasers, 407, 409, 415 Vacuum Rabi splitting, 142, 146 Vakhitov–Kolokolov criterion, 237 Valence band mixing, 442 Vectorial quadratic solitary waves, 367 spatial spatial solitary waves, 364 Velocity-changing collisions (VCC), 150, 177 Visible light generation, 400 Vortex (dark), 277 solitons, 246, 260, 276

Walking solitons, 241 Walkoff angle, 379, 380 spatial, 387 Wannier exciton, 441 Wave vector mismatch, 231 Waveguide devices, 391 examination, 420, 423, 425 SHG devices overall efficiency of, 391 SHG devices, 391 Waveguides, 353 Wavelength shifters, 341, 356 Weak confinement regime, 440

Weak-wave retardation, 37, 341, 346 White self-transparency effect, 435 Whitney’s theorem, 237 Wide gap dielectric, 37

Z-scan technique, 53–75, 271, 285, 454 2-color, 56 Zakharov and Shabat problem, 226 ZBLAN, 412–417 ZnCdSe-Zn(S)Se, 417 ZnGeP2,393–394

475