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PREFACE
This volume presents four articles dealing with topics of considerable current research activities. The first article by C. Aegerter and G. Maret is concerned with localization of classical waves by strong multiple scattering, with emphasis on propagation of visible light in optically turbid media. When the multiple scattering is weak, there is a twofold enhancement of the intensity of the scattered light in the back direction, an effect known as coherent backscattering. The origin of this effect is discussed, as well as experimental investigations of multiple scattering of light in various media. The second article by Y. V. Kartashov, V. A. Vysloukh, and L. Torner gives an account of recent theoretical and experimental investigations concerning soliton manipulation of lattices. Optical lattices make it possible to control diffraction of light beams in media with periodicallymodulated optical properties to control reflection and transmission bands. This leads to a rich variety of new families of nonlinear stationary waves and solitons, offering novel opportunity for all-optical shaping, switching and transmitting of optical signals. This technique offers new possibilities of producing non-diffracting light patterns. The article which follows, by P. Gallion, F. Mendieta and S. Jiang, deals with basic quantum noise manifestations in optical amplification, optical direct detection and coherent detection systems. Applications to optical communications and quantum cryptography are also discussed. The concluding article by M. Yan, W. Yan, and M. Qiu entitled Invisibility Cloaking by Coordinate Transformation explains how recent developments of new optical materials make it possible to produce perfect invisibility cloaks. A review of recent theoretical and experimental researchers for producing such cloaks is also given. Emil Wolf Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA November 2008 v
CHAPTER
1 Coherent Backscattering and Anderson Localization of Light Christof M. Aegerter and Georg Maret Fachbereich Physik, Universit¨at Konstanz, Universit¨atstrasse 10, 78457 Konstanz, Germany
Contents
1.
Introduction Instances of Enhanced Backscattering 1.1 1.2 Coherent Backscattering 1.3 Theoretical Predictions 2. Experiments on Coherent Backscattering 2.1 Colloidal Suspensions and Turbid Powders 2.2 The Influence of a Magnetic Field 2.3 Cold Atoms 2.4 Other Types of Waves 3. The Transition to Strong Localization 3.1 Low-dimensional Systems 3.2 Static Measurements 3.3 Time-resolved Measurements 4. Conclusions and Outlook Acknowledgements References
1 3 4 5 10 11 21 24 27 30 30 34 45 56 57 57
1. INTRODUCTION Most of the time, we obtain information on an object by looking at it, that is, we exploit the light that is scattered from it. The spectral and angular distribution of the backscattered (and reflected) light gives us information about the nature of the particles making up the object. For instance, the reddish color of copper is determined by the absorption c 2009 Elsevier B.V. Progress in Optics, Volume 52 ISSN 0079-6638, DOI 10.1016/S0079-6638(08)00003-6 All rights reserved.
1
2
Coherent Backscattering and Anderson Localization of Light
properties (in the green) of the d electrons in the partially filled shell. On the other hand, the blue color of the sky is well known to originate from the scattering properties of the air molecules, which follows Rayleigh scattering with a cross-section proportional to 1/λ4 . This tells us that the molecules are much smaller than the wavelength of light. In fact, a more thorough analysis allows a characterization of the density fluctuations of the air from the scattering properties of the sky. As a final example, we mention the ‘Glory’, the halo sometimes seen around the shadow of an airplane on clouds it is flying over, which will be discussed further below. In the following, we will be concerned with instances of such enhanced backscattering in nature, where the intensity is enhanced in the direction of backscattering. As we will see below, one such effect is due to the interference of multiple scattering paths in disordered media like clouds, milk or white paint. Due to the reciprocity of light propagation, such paths will always have a counterpart of exactly the same length, which implies that they will always interfere constructively in the backward direction. We will also discuss how this effect can lead to a marked change in the transport behaviour of the light waves in a disordered system, where diffuse transport comes to a halt completely. This transition is known as Anderson localization, and has been of great influence in the development of the theory of electrons in metals and condensed-matter physics. However, as will be seen in the discussion of backscattering enhancement below, the effect is also present in classical waves such as light, and there have been great efforts to try and experimentally observe the transition to Anderson localization of light. In the rest of the introduction, we will discuss the different instances of enhanced backscattering in nature and their possible connection to coherent backscattering. Then we will discuss the connection of coherent backscattering to Anderson localization in more detail, before discussing the main predictions of Anderson localization in order to guide the experimental search for the effect. Section 2 will return to coherent backscattering and will discuss in detail the different experimental observations connected to recurrent scattering, the influence of absorption and finite size of the medium, as well as the problem of energy conservation. In that section we will also discuss other instances of coherent backscattering, that is, with light scattered by cold atoms as well as with waves other than light. In Section 3 we will discuss the quest for Anderson localization of light, describing the different experimental approaches used in the past, as well as their advantages and disadvantages. Finally, we will concentrate on our studies of time-resolved transmission and the corresponding determination of critical exponents of Anderson localization of light.
Introduction
3
1.1 Instances of Enhanced Backscattering As first realized by Descartes (1637), the rainbow is an enhancement in intensity (different for different colors due to dispersion) due to refraction of light in the rain drops, which, due to the dispersion of water, is highest at different angles for different colors. However, this is a purely geometric effect, which does not yield information on the size of the rain drops reflecting the light. Something akin to a rainbow can be seen when flying in an airplane over an overcast sky. When the sun is low and the cloud cover not too thick, one can see a beautiful halo around the shadow of the plane on the clouds. The effect is also well-known to alpinists who can observe this halo around their own shadow on a day that is hazy in the valley. In contrast to what one might think, this ‘Glory’ as it is called, is not in fact a rainbow. One can see this for instance by considering the angle of this colorful enhancement, which is usually only a few degrees and hence much smaller than the 42◦ corresponding to a rainbow. Therefore another mechanism has to be at work. It has been shown that the size of the scattering droplets influences the angle of the glory (Bryant and Jarmie, 1974). It turns out that this is due to the Mie-scattering properties (Mie, 1908) of the droplets. With a typical size of 10 µm, the droplets in a cloud are large compared to the wavelength of light. Furthermore, as illustrated by experiments on a levitating droplet of water, Glory is the property of a single drop (Lenke, Mack and Maret, 2002). Enhanced backscattering is also commonly observed in forests, where the leaves of dew-covered trees, or the blades of dew-covered grass, have a halo. This effect is called sylvanshine (see e.g. Fraser (1994)) and is due to the focusing action of the droplet on the reflecting surface of the leaf. By the same principle, the diffuse reflection from the leaf is channeled back through the lens (i.e. the drop) which decreases the angle of reflection. Hence the leaves or the grass blades are brighter than the background. The grass does not even need to be dew-covered to observe a halo, as there is an additional effect increasing the intensity in the direct backscattering direction. Exactly opposite to the incidence, any ensemble of rough objects will be brightest. This is because in this direction, we see the reflected light directly and none is lost due to shadows of other objects (Fraser, 1994). This is known as the corn-field effect. As a final instance of enhanced backscattering, let us mention observation of the intensity of objects in the solar system, such as the moon or other satellites of planets, when the earth and the sun are in opposition to the moon. In that case, it was observed by Gehrels (1956) for the moon and subsequently for many other satellites (Oetking, 1966) that the intensity of the satellite is in fact increased over its usual value. Due to the arrangement of sun and satellite when the effect is observed, this was called the ’opposition effect’. In this effect, coherent backscattering
4
Coherent Backscattering and Anderson Localization of Light
as we will discuss below, works in concert with analogues of the effects described above, such as the corn-field effect. The presence of coherent backscattering in the opposition effect was discovered (Hapke, Nelson and Smythe, 1993). With this knowledge it was then possible to actually study the surface properties (e.g. the granularity) of these satellites from remote observations.
1.2 Coherent Backscattering Among instances of enhanced backscattering, here we will be concerned mostly with coherent backscattering, an interference effect that survives all averages in a random medium. Fundamentally, the enhancement is due to the fact that, because of time-reversal symmetry, every path through a random medium has a counterpropagating partner. Light elastically scattered on these two paths interferes constructively, because the pathlengths are necessarily the same. This leads to an enhancement of exactly a factor of two in the direction directly opposite to the incidence. In contrast to Glory or other effects discussed above (Lenke, Mack and Maret, 2002), coherent backscattering is not an interference due to the properties of a single scatterer, but relies fundamentally on multiple scattering. In fact, in the single-scattering regime there cannot be a coherent backscattering cone as there cannot be a counterpropagating light path. The entry- and exit-points of a multiple-scattering path can then be seen as the two points of a double slit, which, due to the coherence of the time-reversed paths, necessarily interfere with each other. The different interference patterns corresponding to different light paths in the disordered medium have to be averaged over, which will lead to the shape of the backscattering cone discussed in Section 1.3 below. What can be seen from this picture is that in the exact backscattering direction, the averaging will always lead to an enhancement factor of two. These principles behind the origin of the backscattering cone will strongly influence the transport through a random system. Taking the end points of the counterpropagating paths to coincide somewhere inside the sample, there will be a two-fold enhancement at this point on such a closed loop. This in turn leads to a decreased probability of transport through the system. This effect is what causes Anderson localization of light (Anderson, 1958), i.e. the loss of diffuse transport due to increasing disorder. As disorder increases, the probability of forming closed loops on which intensity is enhanced increases. At a certain critical amount of disorder, these closed loops start to be macroscopically populated, which leads to a loss of diffuse transport. This critical amount of disorder has been estimated using dimensional arguments by Ioffe and Regel (1960) to be when the mean free path roughly equals the inverse wavenumber, i.e. when kl ∗ ∼ 1. Such a mechanism was first proposed for the transport of
Introduction
5
electrons in metals, where it was found that an increase in disorder can turn a metal into an insulator (see e.g. Bergmann (1984)). Historically, the first instances of localization were discussed in the context of electron transport in metals, and thus localization was thought to be a quantum effect. Moreover, due to the fact that localization should always be present in two dimensions (see scaling theory below) and is not influenced too much by the presence of correlations, these studies were carried out in thin films. A review of these experiments can be found in Altshuler and Lee (1988) and Bergmann (1984) and these studies of localization in lower dimensions have had a big influence on the study of other quantum effects in low-dimensional electron systems, such as the quantum Hall effect (Klitzing, Dorda and Pepper, 1980; Laughlin, 1983). Eventually however, it was realized that the quantum nature of electrons is not a necessary ingredient for the occurrence of Anderson localization as, in fact, this is purely a wave effect. Thus, it should also be possible to observe localization effects with classical waves, such as light, as was proposed by John (1984) and Anderson (1985). As we will see below, coherent backscattering, that is, weak localization, was observed with light shortly thereafter; subsequently, there was a vigorous programme to also observe signs of strong localization of light, because the study of photon transport in disordered media has many advantages over the study of electrons in metals. This is because in the latter case there are alternatives that may also lead to localization: in the case of electrons, a random potential can lead to a trapping of particles, which also strongly affects transport, while not being connected to localization. On the other hand, electrons also interact with each other via Coulomb interaction, so that correlations in electron transport are again not necessarily due to localization effects, but may more likely be explained by electron–electron interactions. In fact, it can be shown that in the presence of particle interactions, the effects of localization vanish (Lee and Ramakrishnan, 1985). However, as we will discuss below, the photonic system is not completely free either of possible artifacts masking as localization. For instance, light will be absorbed by materials to a certain extent, which leads to a loss of energy transport similar to localization. Furthermore, resonant scattering can lead to a time delay in the scattering process, which leads to a slowing down of transport, which again may be mistaken for localization. In Section 3 we will discuss in detail how these possible artifacts can be circumvented and localization can in fact be observed.
1.3 Theoretical Predictions As discussed above, the enhanced backscattering from turbid samples, known as coherent backscattering, is a manifestation of weak localization
6
Coherent Backscattering and Anderson Localization of Light
of light. Localization has been studied intensely in electronic systems, and many of the predictions found there can be applied also to optics. Here we will discuss the most important predictions, which will also serve as a guiding line in the quest to observe Anderson localization of light. Most prominent in these are the predictions of the change in static transmission (Anderson, 1985; John, 1984) which turned out to be difficult to observe experimentally due to the presence of absorption in real samples. The critical prediction for Anderson localization concerns the fact that there should be a phase transition to a state where diffusion comes to a halt. This is described by scaling theory (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979), which can also be investigated by ¨ a self-consistent diagrammatic theory (Vollhardt and Wolfle, 1980). This version of the theory can also be extended to describe open systems with absorption, a situation much more suitable for experiments (Skipetrov and van Tiggelen, 2004, 2006). First of all, however, we will describe the shape of the backscattering cone as calculated by Akkermans, Wolf and Maynard (1986) and van der Mark, van Albada and Lagendijk (1988). 1.3.1 The Cone Shape Given the nature of the backscattering cone due to interference of photons on time-reversed paths, one can explicitly calculate the shape of the enhancement as a function of angle. In order to do this, the interference patterns, corresponding to two counterpropagating paths with end-to-end distance ρ, need to be averaged weighted by the probability distribution of such an end-to-end distance occurring. Like in a double-slit experiment with slit separation ρ, each of these interference patterns will contribute a factor 1 + cos(qρ), such that the enhancement above the incoherent background is simply given by the real part of the Fourier transform of the end-to-end distance distribution: α(q) =
Z p(ρ) · cos(qρ) dρ.
(1)
In the diffusion approximation, this probability distribution can be calculated (Akkermans, Wolf, Maynard and Maret, p 1988; van der Mark, van Albada and Lagendijk, 1988) to be 1/a(1 − ρ/ ρ 2 + a 2 ) in the case of a semi-infinite planar half-space. Here, the length scale a = 4γ l ∗ describes how the diffuse intensity penetrates the sample as described by the Milne parameter γ and the transport mean free path l ∗ . The parameter γ can be calculated from the radiative transfer equation to be ∼0.71 and in the diffusion approximation is exactly γ = 2/3. In the following, we will always use the value of γ = 2/3. This leads to the following expression
Introduction
7
for the backscattering enhancement: α(q) =
1− p
!
ρ
Z
ρ2 + a2
· cos(qρ) dρ,
(2)
which can be solved to give (Akkermans, Wolf and Maynard, 1986; Akkermans, Wolf, Maynard and Maret, 1988; van der Mark, van Albada and Lagendijk, 1988): α(q) =
3/7 (1 + ql ∗ )2
1+
1 − exp(−4/3ql ∗ ) . ql ∗
(3)
This gives a cone shape in very good agreement with the experiments that will be discussed in Section 2. As can be seen from an investigation of the angle dependence, the cone tip is triangular with an enhancement of 1 in the exact backscattering direction. The enhancement then falls off on an angular scale proportional to 1/(kl ∗ ); in fact the full width at half maximum of the curve is given by 0.75/(kl ∗ ). Thus the investigation of the backscattering cone is a very efficient method of determining the turbidity of a sample as given by 1/l ∗ . A similar description following diagrammatic theory, where the most crossed diagrams have to be added up, was given by Tsang and Ishimaru (1984). The main features of the curve remain the same, however the different theories use different approximations for the Milne parameter. 1.3.2 Static Transmission One of the main predictions of Anderson localization in electronic systems is the transition from a conducting to an insulating state. This of course has strong implications for the transmission properties of localized and nonlocalized samples. For a conducting sample, the transmission is described by Ohm’s law, which describes diffusive transport of particles and hence a decrease of transmission with sample thickness as 1/L. This is also the case in turbid optical samples, where the transmission in the diffuse regime is simply given by T (L) = T0l ∗ /L (see e.g. Akkermans and Montambaux (2006)). In the presence of absorption, this thickness dependence of the total transmission will change to an exponential decay for thick samples according to e.g. Genack (1987) T (L) = T0
l ∗ /L a , sinh(L/L a )
(4)
8
Coherent Backscattering and Anderson Localization of Light
√ where L a = l ∗la /3 is the sample absorption length corresponding to an attenuation length la of the material, which describes the absorption of the light intensity along a random scattering path. The localization of photons will similarly affect the transmission properties of a sample. As the diffusion coefficient of light becomes scale dependent close to the transition to localization, the total transmission will decrease. Scaling theory of localization, to be discussed below, predicts that the diffusion coefficient at the transition will decrease as 1/L (John, 1984, 1985, 1987). This should then be inserted into the expression for the diffuse transmission of the sample, resulting in a different thickness dependence T (L) ∝ 1/L 2 . Again, this ignores the effects of absorption, and Berkovits and Kaveh (1987) have calculated the effects of absorption in the presence of a renormalized diffusion coefficient, finding T (L) = T0 exp(−1.5L/L a ).
(5)
Again, this leads to an exponential decrease of the transmitted intensity for very thick samples, where, however, the length scale of the exponential decrease has changed. When photons are fully localized, the transport is exponentially suppressed, as only the tails of the localized intensity can leave the sample. Thus Anderson (1985) has predicted the transmission in the localized case to be given by T (L) = T0 exp(−L/L loc ), where L loc describes the length scale of localization. As was the case above, this derivation again does not take into account absorption, and a fuller description would be given by T (L) = T0
l ∗ /L a exp(−L/L loc ). sinh(L/L a )
(6)
Again, this gives an exponential decrease of the transmitted intensity for thick samples with an adjusted length scale not solely given by the absorption length L a . In an experimental investigation of Anderson localization therefore, static transmission measurements will have to find an exponential decrease of the transmission that is faster than that given by absorption alone. This implies that the absorption length must be determined independently for such an investigation to be able to indicate localization of light. 1.3.3 Scaling Theory When studying the thickness dependence of the conductance (i.e. the transmission), its dependence on disorder has to be taken into account. Abrahams, Anderson, Licciardello and Ramakrishnan (1979) produced the first version of such a theory, where they introduce the ‘dimensionless
Introduction
9
conductance’ g as the relevant parameter to study. In electronic systems, this simply is the measured conductance normalized by the quantum of conductance, e2 / h. In optics, the conductance is naturally dimensionless and can be defined simply via the transmission properties of the sample. In fact, g can be calculated in three dimensions from the ratio of the sample volume to that occupied by a multiple scattering path. This volume of the multiple scattering path is given by λ2 s, where s is the length of the path, which in the case of diffusion is s ∝ L 2 /l ∗ . Thus one obtains g ≈ (W/L)(kW )(kl ∗ ), where W is the width of the illumination, which could also be obtained from the static transmission discussed above. In the case of a localized sample, the transmission decreases exponentially with L, which has to be reflected in a renormalization of the path-lengths in order to give an exponentially decreasing g. The main ansatz of Abrahams, Anderson, Licciardello and Ramakrishnan (1979), in treating the problem of the localization transition in the following, is to suppose that the logarithmic derivative β = d(ln g)/d(ln L) can be expressed as a function of g only. The transition to a localized state is then given by the criterion that β changes from a positive value to a negative one. Ohm’s law as a function of dimensionality states that the conductance scales as g ∝ L d−2 . Therefore, making a sample larger and larger in low-dimensional systems will in fact lead to a reduction of the conductance and hence be associated with localization. Actually Ohm’s law straightforwardly implies that β = d − 2 for large L (and thus g), such that d = 2 is the lower critical dimension for a transition to localization to occur. In fact, for low-dimensional systems the waves are always localized (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979). Where there is a transition to localization (i.e. in d > 2), more details about that transition can be obtained by assuming the dependence of β on g to be linear at the crossing of the null-line. In this case, the scaling function β describes how one arrives from a diffuse conductance to one which is exponentially suppressed in the localization length. This transition is a function of the disorder in the system, such that one can describe it in terms of a diverging length scale of localization at the transition. This would be given by an exponent ν, such that L loc ∝ |(g−gc )/gc |−ν . With the assumption that close to the transition, β can be approximated by a linear function in ln g, this exponent is simply given by the inverse slope of β at the transition. In the framework of scaling theory, no exact value can be given for this exponent, however extrapolating β from its known dependencies at large and small disorder, Abrahams, Anderson, Licciardello and Ramakrishnan (1979) obtain an upper bound of ν < 1. As a matter of fact, John (1984) has shown that expanding the treatment around the lower critical dimension, the exponent should be given by ν = 1/2 in d = 2 +
10
Coherent Backscattering and Anderson Localization of Light
dimensions. Such a value for the critical exponent would also be expected for d > 4, where it should simply be given by the mean field value of a critical exponent of an order parameter (Schuster, 1978). At the transition, the loss of transmission can be described by a scale dependence of the diffusion coefficient, such that D becomes smaller as the sample size L increases. As discussed above, this results in D ∝ 1/L. Such a scale-dependent diffusion coefficient can however also be described in the time domain, where the scale dependence corresponds to a decrease of D with increasing path-length. To quantify this, one has to insert the scale-dependent D into the diffuse spread of the photon cloud: hr 2 i = Dt. Since D depends on the length scale as 1/L, we obtain that D ∝ t −1/3 at the transition (Berkovits and Kaveh, 1990). For states which are localized, i.e., with an exponential decrease of the transmission, the spread of the photon cloud has to be limited to the length scale of L loc , so that in this case we obtain D ∝ 1/t. Such a timedependent diffusion coefficient will constitute the hallmark of Anderson localization, and can also be described by self-consistent theories, which ¨ 1980). These predict the temporal scaling of D (Vollhardt and Wolfle, theories have been adapted to a semi-infinite, open medium in order to describe the influence of localization on the coherent backscattering cone by van Tiggelen, Lagendijk and Wiersma (2000). They obtain a rounding of the cone, which experimentally is difficult to distinguish from absorption. Subsequently, Skipetrov and van Tiggelen (2004) and Skipetrov and van Tiggelen (2006) have applied self-consistent theory to open slabs, which are comparable to an experimental situation. Here they indeed find that in time-resolved experiments, a measure of D(t) could be found that can be studied experimentally. We will describe this in detail below.
2. EXPERIMENTS ON COHERENT BACKSCATTERING As we have seen above, the enhancement of backscattered light is due to the wave nature of light and the time-reversal symmetry (or reciprocity) of wave propagation. As such it is an illustration of the principle behind Anderson localization. Since light does not interact with itself and thus correlation effects can be ruled out, numerous experiments on coherent backscattering of light – and other waves – have studied directly the influence of disorder, polarization and the scattering process on Anderson localization. In this section we will discuss these experiments, starting with the discovery of coherent backscattering and continuing with other influential factors, such as sample thickness and absorption. Then we will discuss the effects of increased disorder on the backscattering cone before discussing experiments on multiple scattering in clouds of cold atoms. There, the nature of the scattering process is of paramount importance
Experiments on Coherent Backscattering
11
FIGURE 1 The different setups used by van Albada and Lagendijk (1985) (left) and Wolf and Maret (1985) (right) to measure backscattering cones (see text). Reproduced with permission from van Albada and Lagendijk (1985) and Wolf and Maret (1985) c 1985, American Physical Society
and the symmetries responsible for backscattering can be broken due to internal degrees of freedom of the atom involved in the scattering process. Finally we describe some experiments on coherent backscattering using waves other than light, such as acoustic and matter waves.
2.1 Colloidal Suspensions and Turbid Powders Soon after the prediction by John (1984) and Anderson (1985) that Anderson localization may be observed with light waves, weak localization was observed in the backscattering from colloidal suspensions by van Albada and Lagendijk (1985) as well as by Wolf and Maret (1985). These two groups used slightly different setups to study suspensions of polystyrene particles, see Figure 1. van Albada and Lagendijk (1985) illuminated their sample using a beamsplitter, such that the backscattered light can be observed directly using a photomultiplier on a translation stage. In present comparable setups, a CCD camera is used to capture the backscattered light. Wolf and Maret (1985), on the other hand, illuminated the sample using a glass slide as a beamsplitter and placed the detector on a goniometer. As can be seen from inspection of Figure 2, the two setups obtain very similar results. When the volume fraction of polystyrene particles is increased (thus decreasing l ∗ ), the observed backscattering cone gets wider. Sizeable enhancement factors are found in both cases, but they are still far from the ideal theoretical value of 1. This is due to the fact that the setups lack angular resolution very close to the centre, as well as to a residual effect of direct reflection which is not suppressed completely. These problems were later solved in the setups discussed below.
12
Coherent Backscattering and Anderson Localization of Light
FIGURE 2 The dependence of the backscattering cones on the density of suspended particles (i.e. the mean free path). On the left, the data of van Albada and Lagendijk (1985) are shown with densities varying from 1.4 × 1017 to 1.4 × 1016 m−3 and beads of diameter 1.09 µm. On the right, the corresponding data of Wolf and Maret (1985) are shown, where the volume fractions change from 0.004 to 0.11 and beads have diameter 0.49 µm. Due to limited angular resolution, stray light and single backscattering contributions, the enhancements are between 0.4 and 0.6. Reproduced with permission from van Albada and Lagendijk (1985) and Wolf and Maret (1985) c 1985, American Physical Society
An enhancement of backscattered light was also found by Tsang and Ishimaru (1984) and Kuga and Ishimaru (1984), however, found the enhancements there were much smaller than those discussed above. Furthermore, both van Albada and Lagendijk (1985), as well as Wolf and Maret (1985) have discussed their findings in the context of weak localization, which was not the case in Kuga and Ishimaru (1984). Random interference of photons on multiply scattered paths can lead to very large fluctuations in the intensity. These fluctuations are called a speckle pattern. In order to observe a coherent backscattering cone at all, the fluctuations due to the speckle pattern need to be averaged. Using a colloidal suspension, as carried out by van Albada and Lagendijk (1985) and Wolf and Maret (1985), this averaging is achieved by the motion of the scatterers, which leads to a redistribution of the pathlengths. In fact, studying the decrease of the time autocorrelation of a speckle spot directly gives information of the motion of the scatterers. This was developed into the technique known as diffusing wave spectroscopy by Maret and Wolf (1987) and Pine, Weitz, Chaikin and Herbolzheimer (1998) to extract information on particle size, flow rates and relaxation dynamics in complex turbid fluids. In turbid powders, the averaging over the speckle pattern is usually done by rotating the sample, which leads to a ¨ configurational average (see, for instance, Gross, Storzer, Fiebig, Clausen, Maret and Aegerter (2007)).
Experiments on Coherent Backscattering
13
2.1.1 Experimental Setups for Large Angles In order to be able to characterize highly turbid samples, in addition to the relatively dilute suspensions discussed above, an apparatus capable of resolving rather large angles is needed. A rough estimate of the angles needed for samples close to the Ioffe–Regel criterion (Ioffe and Regel, 1960) results in a cone width of up to 40◦ . Even taking into account the narrowing of the cone due to internal reflections at the surface (see below) this means that in order to properly determine the level of the incoherent background, angles up to at least 40◦ need to be measured. However, at the same time the setups should be able to resolve the cone tip with great accuracy in order to observe deviations from the ideal tip shape (see below). These two requirements pose a big challenge, which has been solved to some extent (angles up to 25◦ ) by Wiersma, van Albada and ¨ Lagendijk (1995) and Gross, Storzer, Fiebig, Clausen, Maret and Aegerter (2007) (angles up to 85◦ ) in two very different ways. The setup of Wiersma, van Albada and Lagendijk (1995) combines a movable detector with the method of using a beamsplitter to be able to observe the most central angles to high accuracy. Instead of just moving the arm of the detector, an ingenious scheme is used whereby the sample, detector and beamsplitter are moved in concert to ensure that the detected light is always perpendicular to the detector and the incident light is always perpendicular to the sample surface. This is to ensure that the polarizer (P in Figure 3) is always arranged such that direct reflections are extinguished completely. The angular range covered by the setup is, however, limited by the presence of the beamsplitter to below 45◦ , such that the incoherent background in extremely wide cones cannot be assessed. On the other hand, a single setup can cover all angles up to 25◦ at almost unlimited resolution with an extinction rate for singly reflected light of nearly 100 per cent (Wiersma, 1995). ¨ A radically different approach was chosen by Gross, Storzer, Fiebig, Clausen, Maret and Aegerter (2007). Here, moveable parts are completely absent and the backscattered intensity is measured at all angles simultaneously (Figure 4). This is done via a set of 256 highly sensitive photodiodes placed around an arc of a diameter of 1.2 m. At the very centre of the arc, photodiode-arrays are used that yield a limiting angular resolution of 0.14◦ . At higher angles the diodes are increasingly far apart, such that at angles >60◦ the resolution is 3◦ . In addition, the central 3◦ of the cone are measured using a beamsplitter and a CCD camera similar to those described above. This gives good overlap with the central part of the wide-angle apparatus, such that effectively the whole angular range is covered, while still measuring the tip of the cone with a resolution of 0.02◦ . The problems of perpendicular incidence onto the circular polarizers discussed above is solved by using a flexible polarizer
14
Coherent Backscattering and Anderson Localization of Light
FIGURE 3 The wide-angle setup of Wiersma, van Albada and Lagendijk (1995). The top and bottom panels show the setup at two different angles and illustrate the rotation of the sample, the beamsplitter and the detector are in concert in order to always keep the incident light perpendicular on the sample and the detected light perpendicular on the detector. This is to reduce aberrations in the polarizer when the light is incident at an angle, such that enhancement factors of unity can actually be measured. Reproduced with permission from Wiersma, van Albada and Lagendijk c 1995, American Physical Society (1995)
foil placed in front of the whole arc. Such a polymer-based polarizer has the disadvantage that only about 96 per cent of the reflected light is extinct, so that enhancements of 2 as obtained by Wiersma, van Albada and Lagendijk (1995) are not possible with this setup. On the other hand, such a polarizer is much cheaper to obtain and can be used in a larger window of wavelengths than a linear polarizer and quarter-wave plate. For wavelength-dependent studies this is a great advantage. Furthermore the renunciation of movable parts makes it possible to measure the small intensities at very large angles with reasonable accuracy. In addition, very broad backscattering cones pose a problem in that they would seriously breech conservation of energy. As the total reflectivity of an infinitely thick, non-absorbing sample should be R = 1, the photon energy within the backscattering cone must be obtained from destructive interference at other scattering angles. In order to be able to
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FIGURE 4 The wide-angle setup of Gross (2005). The sample is at the centre of an arc of 1.2 m diameter, which holds 256 sensitive photodiodes. At the centre the diodes are minimally spaced, yielding a resolution of 0.14◦ ; outwards they are increasingly farther apart up to 85◦ . To shield ambient light, the whole setup is placed inside a black box. Direct reflections are suppressed by the use of circular polarization, which is achieved using a flexible polarization foil placed in front of the whole arc. With this, enhancement factors up to 0.96 are possible. The different diodes are calibrated using a teflon sample, which in this angular range gives a purely incoherent signal. ¨ Fiebig, Clausen, Maret and Aegerter Reproduced with permission from Gross, Storzer, c 2007, American Physical Society (2007)
tackle this problem experimentally, a calibrated energy scale would be ¨ needed. A simple extension of the setup of Gross, Storzer, Fiebig, Clausen, Maret and Aegerter (2007) is capable of this, as will be described below. 2.1.2 Absorption and Finite Thickness In all of the above, we have assumed that the sample can be treated as in infinite half-space, such that all incident photons are eventually backscattered at the sample surface. In reality, this is not always a good approximation and photons may either be absorbed or leave the sample at the other end or the sides. This implies that the photon pathlength distribution needs to be adjusted by suppressing such long paths. This can be done, for instance, by introducing an exponential cut-off to the probability distribution p(s) discussed above, and we expect that the tip of the cone, which corresponds to these long paths, is altered. van der Mark, van Albada and Lagendijk (1988), Akkermans, Wolf, Maynard and Maret (1988) and Ishimaru and Tsang (1988) have studied this problem quantitatively and find that, indeed, the tip of the cone is rounded. For the simple case of absorption, the cone shape can be p 2 obtained by replacing q in equation by 1/L a + q 2 (Akkermans, Wolf, Maynard and Maret, 1988), where √ L a is the absorption length of the multiple scattering sample, i.e. l ∗la /3, with la the absorption length of the material. This leads to a rounding on the angular scale of 1θ = 1/k L a .
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Coherent Backscattering and Anderson Localization of Light
FIGURE 5 Absorption, finite thickness, but also localization of light would lead to a rounding of the cone tip, which ideally would be linear as discussed above. This is because in all of these cases, photons on long paths are not reflected from the sample and therefore do not contribute to the backscattering cone. In these data from Schuurmans, Megens, Vanmaekelbergh and Lagendijk (1999), this rounding can be clearly seen for a sample of photoanodically etched GaP. Due to the lack of an independent determination of the absorption length it is difficult to associate this rounding unambiguously with absorption or localization. Reproduced with permission from Schuurmans, Megens, Vanmaekelbergh and Lagendijk (1999) c 1999, American Physical Society
The situation is somewhat more complicated for finite samples, but van der Mark, van Albada and Lagendijk (1988) have derived that, in that case, the rounding is on an angular scale of 1θ = coth(L/L a )/k L a . This rounding of the cone has been observed by Wolf, Maret, Akkermans and Maynard (1988) and Schuurmans, Megens, Vanmaekelbergh and Lagendijk (1999), see e.g. Figure 5. Similarly, the scaling of the width of the rounding with sample thickness L and absorption length L a has been tested experimentally (see Figure 24 below). However, from the discussion above, it is also plausible that localization would lead to a rounding of the backscattering cone since localization too leads to a redistribution of the path-lengths for very long paths. This has been suggested by Berkovits and Kaveh (1987) and calculated using self-consistent theory by van Tiggelen, Lagendijk and Wiersma (2000). We will discuss these issues further in the context of strong localization below. 2.1.3 Surface Reflections In the above discussion of the shape of the backscattering cone, we have assumed that the cone is directly given by the diffuse path-length distribution of photons at the free sample surface. However, since samples
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usually have an effective refractive index higher than that of air, this distribution can be influenced by internal reflections of the light as it exits the sample. Such reflections will effectively broaden the path-length distribution, which will lead to a narrowing of the cone. This fact is illustrated in Figure 6. When the path-length distribution broadens, the average distance between the end points of time-reversed paths increases. As is evident within the picture treating the time-reversed paths as double slits, this leads to an increased distance and hence a narrowing of the resulting interference pattern. An averaging over all end-to-end distances then leads to a narrowing of the cone shape. This correction has been calculated quantitatively by Zhu, Pine and Weitz (1991) and Lagendijk, Vreeker and de Vries (1989), who found a strong dependence of the resulting value of kl ∗ estimated from the full width at half maximum. Instead of FWHM = 0.75/(kl ∗ ) for the scaling of the width as obtained from Equation (3), Zhu, Pine and Weitz (1991) find a scaling as: FWHM−1 =
2 2(1 + R) + kl ∗ , 3 3(1 − R)
(7)
where R is the angular averaged reflectivity due to the index mismatch. Thus the values of kl ∗ obtained from a fit to Equation (3) need to be corrected by a factor of 1/(1 − R). This correction becomes important in the quest for Anderson localization as in that case the particles are very strongly scattering and the samples therefore have relatively high refractive indices. Thus they show increased internal reflections, which would lead to an overestimation of the value of kl ∗ solely from the width of the backscattering cone. In order to perform the above correction, the refractive-index mismatch at the surface of the sample needs to be known, i.e. the effective refractive index of the sample needs to be calculated. To a first approximation, this can be done following Garnett (1904), but this approach is strictly valid only for particles with a small refractive index. In order to take into account the strong scattering cross-sections of the particles, more elaborate schemes are necessary. Such calculations have been pioneered by Soukoulis and Datta (1994) and Busch and Soukoulis (1996). 2.1.4 Recurrent Scattering As the turbidity of the samples increases, there is an increased probability for multiple-scattering paths to return upon themselves. In the extreme case, this will lead to Anderson localization, when such paths become macroscopically populated. Wiersma, van Albada, van Tiggelen and Lagendijk (1995) have studied the backscattering cone for increasingly turbid samples and have found that with decreasing kl ∗ , the enhancement
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Coherent Backscattering and Anderson Localization of Light
FIGURE 6 If the effective refractive index of the scattering medium is high, internal reflections at the sample boundary may occur. These internal reflections in turn lead to an overpopulation of longer end-to-end distances of photon paths. Since the backscattering cone originates from the interference of time-reversed photon paths, this overpopulation then artificially narrows the measured cones, such that the determination of kl ∗ directly via the width leads to an overestimation of its inherent value
factor of the backscattering cone is reduced. When the first and last scatterer of a multiple-scattering path are the same, the contribution of the interference with the time-reversed path will be the same as that of the background. This implies that the background will be overestimated, leading to a reduction of the enhancement factor. This is illustrated in Figure 7 for two different samples with values of kl ∗ of 22 and 6, respectively. Due to the high resolution and wide angular range of their setup described above (Wiersma, van Albada and Lagendijk, 1995), the enhancement factor is claimed to be determined to roughly 1 per cent. Thus the reduction shown in Figure 7 should be significant. It should be noted however that in these measurements of the backscattering cone there is no absolute determination of the intensity scale. The level of the incoherent background is simply determined from a cosineshaped fit in addition to the backscattering cone described by Equation (3). As such, the backscattering cone would violate the conservation of energy, so that in such strongly scattering samples the absolute intensity needs to be known. This will be discussed in more detail below, where the enhancement is determined over the full angular range with an absolute intensity scale. In fact, the incoherent background can differ by a few per cent as the turbidity changes. For instance, as the turbidity increases so does the effective size of the sample, such that the albedos of the different samples may no longer be comparable due to losses at the sample boundary. Similarly, the absorption lengths of the different samples will be different, such that the intensity scale may need to be adjusted. This might be
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FIGURE 7 For very turbid samples, the enhancement in the backscattering direction is reduced, as can be seen from a close-up at the cone tip of different samples. This is argued to be due to an underpopulation of time-reversed paths because for very turbid samples, there is an increased probability of visiting the same scatterer twice in a multiple-scattering path. Therefore such paths do not contribute fully to the interference pattern resulting in the backscattering cone. Reproduced with permission c 1995, American from Wiersma, van Albada, van Tiggelen and Lagendijk (1995) Physical Society
the case for the data in Figure 7, where the broad cone is more consistent with a rounded tip, and thus seems to have a somewhat higher absorption than the sample with a perfect two-fold enhancement. As it stands, in the absence of an absolute determination of the incoherent background, the enhancement factor cannot be determined with an accuracy of one per cent. Thus the observed decrease may not be significant. 2.1.5 Energy Conservation From the discussion so far it would seem that coherent backscattering violates the conservation of energy: In all of the theoretical calculations discussed above (e.g. Akkermans, Wolf and Maynard (1986) and van der Mark, van Albada and Lagendijk (1988)), the enhancement of the cone is always positive irrespective of angle and polarization channel. Thus for a non-absorbing sample covering an infinite half-space (i.e. with a reflectivity of 1), more intensity would be scattered back from the sample than is incident. Obviously this cannot be, and there has to be a correction to the angular intensity distribution at high angles, which compensates for the enhancement in the back-direction. However, this correction is small, and in order to observe it one needs to determine the incoherent background absolutely. This was not done so far (e.g., Wiersma, van ¨ Albada, van Tiggelen and Lagendijk (1995) and Storzer, Gross, Aegerter and Maret (2006)) and the backscattering cones thus obtained were well described by Equation (3). Figure 8 shows the result of such a correction.
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Coherent Backscattering and Anderson Localization of Light
FIGURE 8 A backscattering cone taking into account the absolute intensity scale. Note that there is a negative contribution at high angles balancing the intensity in the cone. This negative part of the enhancement can be described by a correction based on the finite width of the time-reversed paths. Such a corrected theory is shown by ¨ ¨ the dashed line. Data from Fiebig, Aegerter, Buhrer, Storzer, Montambaux, Akkermans and Maret (2008)
Here the incoherent reference was a teflon sample, where the absorption was determined using a time-of-flight measurement (Fiebig, Aegerter, ¨ ¨ Buhrer, Storzer, Montambaux, Akkermans and Maret, 2008). Also shown in the figure is a corrected theory, taking into account the fact that timereversed paths start to overlap when the mean free path gets smaller than the wavelength of light. This leads to an underpopulation on these paths and hence to a reduction of the backscattering enhancement. This is indicated in the sketch in Figure 9, where the overlap of two Gaussian distributed lightpaths is shown. Since this reduction takes place on a length scale of λ, the corresponding reduction of the backscattering enhancement is at high angles. To a first approximation, this correction ¨ ¨ Storzer, Montambaux, can be calculated as (Fiebig, Aegerter, Buhrer, Akkermans and Maret, 2008): 1 − exp(−4/3ql ∗ ) 3/7 α(q) = 1+ ql (1 + ql ∗ )2 9π cos θ − . 7(kl ∗ )2 cos θ + 1
(8)
The dashed line in Figure 8 is a fit of this equation to the backscattering data with kl ∗ as the only fit parameter; it is in very good agreement with the data. Furthermore, the integrated intensity over the backscatter-
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FIGURE 9 Illustration of the physics behind the reduction of the backscattering enhancement. When the mean free path gets shorter, the end points of the multiple scattering paths start to overlap. Describing these as Gaussian distributions with a width λ, one obtains the correction of Equation (8)
ing half-space of this expression (and of the corresponding data) is nearly zero for all values of kl ∗ , showing that, by including this correction, energy is indeed conserved for coherent backscattering. This result can also be obtained from diagrammatic theory (Akkermans and Montambaux, 2006; van Tiggelen, Wiersma and Lagendijk, 1995). Here it can be shown that the Hikami-box (Hikami, 1981) describing coherent backscattering contains not only the most-crossed diagrams, but also those dressed with an impurity. These diagrams give a contribution of the same order, but negative. It can then be shown exactly that the integral over the whole Hikami-box is exactly zero, which corresponds to the conservation of energy.
2.2 The Influence of a Magnetic Field As discussed in detail above, coherent backscattering is fundamentally an interference effect due to the wave nature of light. In order to show this experimentally, one needs to change the phase of the light on counterpropagating paths, such that time-reversal symmetry is broken. A possible mechanism for the breaking of time-reversal symmetry is the application of a magnetic field. As shown by Faraday (1846), an applied magnetic field will change the polarization angle of light passing through a material. This is very pronounced for materials containing paramagnetic rare-earth elements, as they possess absorption bands that lead to a very strong Faraday effect. The magneto-optical rotation of a material is quantified by the Verdet constant, which is the constant of proportionality between the change in phase angle and the applied magnetic field times the length of the light path. Given the importance of time-reversal symmetry to coherent backscattering and the possibility of influencing the phase of light inside a multiple scattering medium via the Faraday effect, Golubentsev (1984)
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Coherent Backscattering and Anderson Localization of Light
and MacKintosh and John (1988) have theoretically studied the effect of a medium with a high Verdet constant on the coherent backscattering cone. Due to the fact that we are dealing with a multiple-scattering medium, things are not so simple that it would suffice to project the multiplescattering path on to the applied field in order to obtain the mean angle of rotation of the phase. In fact, because multiple scattering leads to a depolarization of the light, the average rotation of the phase will be exactly zero irrespective of the applied field and the path-length through the material. However, MacKintosh and John (1988), in a model where every scattering event is assumed to randomly change the polarization of the light, found that the mean square displacement of the phase rotation does follow the Faraday effect. They found that h1α 2 i =
4 2 2 ∗ V B sl f , 3
(9)
where s is the length of the path and l ∗f is a length scale describing the depolarization of the photons. This length scale will be of the order of the mean free path l ∗ , but will depend on the depolarization properties (and hence sizes) of the scattering particles. We will discuss a numerical investigation of these issues in more detail below. From this result it can be concluded that on path-lengths exceeding (l ∗f (2V B)2 )−1 , photons on counterpropagating paths can no longer interfere with each other, such that localization is destroyed. Other effects of magnetic fields on light transport in random media have been discussed as well; these include the existence of the analogue of the Hall effect for photons (Rikken and van Tiggelen, 1996; Sparenberg, Rikken and van Tiggelen, 1997), as well as that of transverse diffusion of light (van Tiggelen, 1995). 2.2.1 Destruction of the Backscattering Cone As discussed above, for sufficiently strong magnetic fields, Verdet constants and path-lengths, the Faraday effect will lead to a suppression of interference of counterpropagating photons. As we have seen above, the cone tip is due to the longest paths, so that to a first approximation, one could describe the cone in the presence of a magnetic field by introducing the length scale (lf∗ (2V B)2 )−1 as an absorption length in the expression for the cone. With increasing field, this length scale decreases, such that eventually the cone should disappear completely. The field strength at which the cone would be reduced to half its size can be estimated by noting that the width corresponds to a length scale of l ∗ , so that (taking l ∗f = 2l ∗ for simplicity) B = 1/V l ∗ . For a molten Faraday active glass, with a Verdet constant of roughly 103 1/Tm and a mean free path of
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FIGURE 10 Destruction of the backscattering cone by a magnetic field. The left-hand panel illustrates the destruction of the cone in both angular dimensions, while the right-hand panel describes the reduction of the enhancement as a function of different applied fields. The data are from Lenke and Maret (2000)
roughly 100 µm, one obtains a field of roughly 10 T. A corresponding experiment was carried out by Erbacher, Lenke and Maret (1993), who studied the field dependence of the backscattering cone in a Faradayactive glass powder in fields up to 23 T. As can be seen in Figure 10, the application of 23 T to the material leads to an almost complete destruction of the backscattering cone, in accordance with the theoretical prediction. For the theoretical curves, q 2 was replaced by q 2 + q F2 , where q F = 2V B describes the depolarization due to the magnetic field. 2.2.2 Polarization Effects The simple helicity-flip model of MacKintosh and John (1988) provides a satisfactory description of the data when the incident light is parallel to the applied field. However, if the field is perpendicular to the illumination, the cone shape can no longer be described by a modified version of Equation (3), as was shown by Lenke, Lehner and Maret (2000). In fact, as can be seen in the left-hand panel of Figure 11, the backscattering cone may even split into two peaks, which then diminish in intensity. In order to describe these data, the polarization dependence of the scattering process has to be taken into account, which goes beyond the helicity-flip model and has to be done numerically. Such an investigation was carried out by Lenke and Maret (2000). In their treatment, Faraday rotation takes place only between scattering events, as is the case in the helicity-flip model of MacKintosh and John (1988), but at each point in a simulation of a
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Coherent Backscattering and Anderson Localization of Light
FIGURE 11 The influence of the field direction and the incident polarization on the backscattering cone in a magnetic field. Experimental data are on the left, simulation results on the right. If the field is not applied parallel to the incident light, the destruction of the cone cannot be described by a modified version of Equation (3). However, a simulation taking into account the full scattering matrix for all scattering events on a multiple scattering path can describe the data. Adapted from Lenke and Maret (2000) and Lenke, Lehner and Maret (2000)
random walk, the full scattering matrix of Rayleigh–Debye–Gans theory is applied to the polarization. The result of such a simulation is shown in the right-hand panel of Figure 11. As can be seen by comparing both parts of the figure, the simulation can qualitatively describe the data. Physically, this splitting of the cone peak is due to the fact that in this transverse setup there is a net, magnetic-field dependent phase change on the timereversed paths given by the end-to-end distance. This phase difference needs to be compensated for by the phase change due to the path-length difference at different angles. For circular polarization, this leads to a shift of the peak, whereas in linear polarization, the different angular directions are equivalent, such that a splitting of the cone peak is observed (Lenke and Maret, 2000). In this description, the magnetic-field effects on the backscattering cone are fundamentally determined by the length scale l ∗f , which describes the polarization. For Rayleigh scattering, it can be shown that l ∗f = 2l ∗ (Lenke and Maret, 2000). However, as the scattering particles increase in size, Mie theory has to be used to describe the polarization effects of each scattering event. This has been studied by Lenke, Eisenmann, Reinke and Maret (2002) for different-sized particles of the order of the wavelength of light, where good agreement is found with the predictions from Mie theory.
2.3 Cold Atoms With the advent of laser cooling and the corresponding successes in cooling atomic gases to very low temperatures, a new field of investigation
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FIGURE 12 Backscattering cone from a cloud of cold Rb atoms (left panel shows data ¨ Miniatura and Kaiser (1999), right panel from Labeyrie, de Tomasi, Bernard, Muller, shows data from Kupriyanov, Sokolov, Kulatunga, Sukenik and Havey (2003)). Note that the enhancement factor is very low compared to that seen for colloidal suspensions or powders. This is connected to the internal degrees of freedom of the atoms in question as will be discussed in the text. Reproduced with permission from ¨ Labeyrie, de Tomasi, Bernard, Muller, Miniatura and Kaiser (1999) and Kupriyanov, c 1999, and 2003 American Physical Sokolov, Kulatunga, Sukenik and Havey (2003) Society
of multiple scattering has been opened. Due to the fact that in a cold cloud of atoms all scatterers are identical, one can exploit the properties of resonant scattering in order to increase the scattering cross-section manyfold. In the future this may allow a reduction of kl ∗ for these samples to such values that the Ioffe–Regel criterion is fulfilled and Anderson localization can be observed. So far however, only the backscattering cone has been observed and the situation has proved to be somewhat more complicated than was hoped at first. This is because of the importance of microscopic degrees of freedom in atomic scattering, which can greatly influence, e.g., time-reversal symmetry. This will be discussed in detail below, and can lead to the observation that the backscattering cone is not destroyed by the presence of a magnetic field as we have seen above, but rather is recovered due to a magnetic field. At present, investigations of multiple scattering of light in cold atomic gases are limited to Rb and Sr, which show vastly different behaviours due to their different ground-state degeneracies. 2.3.1 Rb Atoms Due to the fact that the cooling of Rb atoms is well known and understood, the first backscattering cones from cold atomic gases were scattered ¨ by Rb atoms (Labeyrie, de Tomasi, Bernard, Muller, Miniatura and
26
Coherent Backscattering and Anderson Localization of Light
Kaiser, 1999; Kupriyanov, Sokolov, Kulatunga, Sukenik and Havey, 2003). However, as can be seen in Figure 12, the observed enhancement is only between 1.1 and 1.15, much smaller that that observed in colloidal suspensions and powders. Due to the internal structure of the Rb atoms, especially the fact that the ground state is degenerate, time-reversal symmetry is partially broken. This is similar to the Faraday rotation effects discussed above for colloidal powders. The degeneracy of the ground state may lead to a change in helicity of the photon during a scattering event, by changing the ground state of the atom (Jonckheere, ¨ Muller, Kaiser, Miniatura and Delande, 2000). This could be treated by a model similar to the helicity-flip model (MacKintosh and John, 1988) devised to take into account the effect of Faraday rotation inside a material ¨ with high Verdet constant. Muller, Jonckheere, Miniatura and Delande (2001) calculated this explicitly and found good agreement with the experimental reduction of the cone enhancement (Labeyrie, de Tomasi, ¨ Bernard, Muller, Miniatura and Kaiser, 1999). They also found that different orders of scattering contribute differently to the effect. In fact, if only double scattering were taken into account, the reduction effect would be much less pronounced, with enhancement factors of up to 1.7 being ¨ possible (Jonckheere, Muller, Kaiser, Miniatura and Delande, 2000). By lifting this degeneracy using an applied magnetic field, the enhancement of the backscattering cone could be recovered. 2.3.2 Sr Atoms In order to be able to study a system with a good enhancement factor in the absence of a magnetic field, one needs to use a cloud of atoms with a non-degenerate ground state. This is much more difficult as the cooling transitions are harder to excite in this case. However, using Sr atoms it was possible to cool a cloud sufficiently to observe a coherent backscattering cone (Bidel, Klappauf, Bernard, Delande, Labeyrie, Miniatura, Wilkowski and Kaiser, 2002). The resulting cone is shown in Figure 13 and has an enhancement factor of nearly two in accordance with expectation. Thus the study of Sr clouds may hold the promise of increased coherence length, such that multiple-scattering samples with very long coherent light paths can be studied. This may then lead to the observation of Anderson localization in such samples. In this context it should be noted, however, that due to the exploitation of resonance scattering to reduce the mean free path, the propagation ¨ speed of photons is slowed down remarkably (Labeyrie, Vaujour, Muller, Delande, Miniatura, Wilkowski and Kaiser, 2003). This means that the increased dwell time in the scattering cavity may also lead to a loss of coherence due to the motion of scatterers on this time scale. This was ¨ investigated using Monte Carlo simulations by Labeyrie, Delande, Muller,
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FIGURE 13 Backscattering cone from a cloud of cold Sr atoms (Bidel, Klappauf, Bernard, Delande, Labeyrie, Miniatura, Wilkowski and Kaiser, 2002). Here, almost perfect enhancement is observed due to the fact that the magnetic moment of Sr does not allow for internal degrees of freedom to be scattered from. Reproduced with permission from Bidel, Klappauf, Bernard, Delande, Labeyrie, Miniatura, Wilkowski c 2002, American Physical Society and Kaiser (2002)
Miniatura and Kaiser (2003), who showed that only a few scattering events are taking place with coherent light, such that the long multiplescattering paths necessary for Anderson localization to occur are out of reach. We will get back to the point of reduced transport velocity due to resonant scattering in the discussion of colloidal powders below.
2.4 Other Types of Waves In addition to localization of light waves and electronic waves, localization has been searched for in many other types of waves. Given the difficulties faced by electron localization due to the presence of interactions, these studies have focused on non-interacting waves, such as acoustic, seismic and matter waves. Due to the fact that strong scattering cross-sections are difficult to obtain in these waves, most studies have concentrated on the observation of weak localization. 2.4.1 Seismic Waves Multiple scattering of seismic waves has become a very interesting subject, leading, for instance, to an increased precision in the determination of the earth’s structure from the noise in seismographs (Snieder, Grˆet, Douma and Scales, 2002; Campillo and Paul, 2003; Shapiro, Campillo, Stehly and Ritzwoller, 2005). In the context of interest here, Larose, Margerin, van Tiggelen and Campillo (2004) have studied the reflection of a stimulus from a sledge hammer that was repeated fifty times for each measurement as a function of distance from the source. The results are shown in Figure 14. The different lines correspond to different delay times between
28
Coherent Backscattering and Anderson Localization of Light
FIGURE 14 Backscattering cone of seismic waves produced by Larose, Margerin, van Tiggelen and Campillo (2004). The different lines show the signal of buried seismographs at a certain distance from the stimulus as a function of delay time. After prolonged times, the multiply scattered paths in the back-direction show a coherent backscattering cone with an enhancement factor of nearly 2. Reproduced with permission from Larose, Margerin, van Tiggelen and Campillo (2004) c 2004, American Physical Society
the stimulus and the reflected signal. As expected from the theoretical description above, the backscattering cone arises from long multiplescattering paths, such that the signal only appears at long delay times. In particular, as can be seen in the figure, an enhancement factor of two can be observed from the long paths observed at late times. In order to be able to observe this enhancement, i.e. to suppress any incoherent background, the experiment was carried out at night in a sparsely populated region as well as under anticyclonic conditions. 2.4.2 Acoustic Waves Well before the study of seismic waves propagating in disordered media, the behaviour of multiple scattered ultrasonic waves in the MHz range was investigated. Kirkpatrick (1985) has theoretically calculated the properties of localized waves in a random medium, and Bayer and Niederdr¨ank (1993) have experimentally studied the backscattering cone from, e.g., gravel, using ultrasonic waves. This was done in both two- and three-dimensional systems and good agreement with theory was found, as illustrated in Figure 15. There have also been experiments studying time-resolved transmission of acoustic waves through samples of aluminium beads by Page (Page, J.H. (2007) private communication). In these experiments, a nonexponential decay of the time-resolved transmission was found. As will be discussed below in the context of light, this is a strong indication of non-classical transport and localization. In addition, Page (Page, J.H. (2007) private communication) studied the statistics of speckles for these
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FIGURE 15 Backscattering cone of ultrasonic waves through (left) gravel stones and ¨ (right) brass rods. These data were obtained by Bayer and Niederdrank (1993). The thin lines show the theoretical expectation (Kirkpatrick, 1985), whereas the thick lines show the experimental results. The data correspond to signals at a certain time delay, similar to those of the seismic waves in Figure 14 of 22 µs and 40 µs, respectively. c 1993, American ¨ (1993) Reproduced with permission from Bayer and Niederdrank Physical Society
samples, and again found strong deviations from the behaviour of diffuse waves. 2.4.3 Matter Waves Two different types of matter waves are presently the subject of localization efforts. In the first instance, the advent of laser cooling and optical traps has led to the proposal of studying Anderson localization of cold atoms in disordered optical traps. These optical traps are usually provided by a speckle pattern from a laser source passed through a disordered medium. It has been found however that in that case, the average spacing of the speckle spots is difficult to reduce to a scale where the Ioffe–Regel criterion can be reached (Lye, Fallani, Modugno, ´ Wiersma, Fort and Inguscio, 2005; Cl´ement, Varon, Hugbart, Retter, Bouyer, Sanchez-Palencia, Gangardt, Shlyapnikov and Aspect, 2005; ¨ Kuhn, Miniatura, Delande, Sigwarth and Muller, 2005). In addition, dense clouds of cold atoms are troubled by strong interactions, such that a Mott insulator can be observed, but Anderson localization is still out of reach in present experiments. In fact, no observation of a backscattering cone of cold atoms in disordered optical lattices has been reported to date. A second type of proposal is to localize ultracold neutrons in the presence of disorder (Igarashi, 1987). Here, progress has been made in cooling the neutrons sufficiently to be able to observe their multiple scattering. The angular resolution of neutron detectors is, however, not sufficient at present to observe the corresponding, narrow backscattering
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Coherent Backscattering and Anderson Localization of Light
cone (Stellmach, Abele, Boucher, Dubbers, Schmidt and Geltenbort, 2000; Stellmach, 1998).
3. THE TRANSITION TO STRONG LOCALIZATION In the experiments described above, the critical parameter for localization, the disorder as measured by (kl ∗ )−1 , was small compared to unity. However, the effects of weak localization could still be observed as there is a counterpropagating path to every path in the back-direction. In order to see the effects of strong localization, one needs to strongly increase the probability of formation of closed loops, so that the renormalization discussed in the theory part becomes important. This can be done in principle by increasing the disorder so as to reach the limit proposed by Ioffe and Regel (1960). In practice however, this turns out to be difficult as one needs to have samples with both strong scattering and low absorption, two conditions which are usually mutually exclusive. However, it can be accomplished by making use of the properties of Mie-resonances in the scattering cross-section (Mie, 1908), as we will see below. On the other hand, spatially restricting the propagation will lead to a strong increase in the probability of observing crossings of the paths. This is exploited in the study of quasi-one-dimensional systems where localization of microwaves has been observed (see, e.g., Chabanov, Stoytchev and Genack (2000)).
3.1 Low-dimensional Systems Low-dimensional systems present several advantages for studying localization. First of all, scaling theory predicts that in less than two dimensions, localization will always be present (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979). The occurrence of localization will in this case go together with the increase in system size, and localization effects can be observed for large enough samples. The spatial restriction in quasi-one-dimensional samples leads to a natural reduction of the dimensionless conductance, thus leading to the presence of localization even far above the turbidity demanded by Ioffe and Regel (1960). This, in turn, also implies that the transition to localization cannot be studied in low-dimensional systems, and a proper study of the scaling theory of localization requires three-dimensional systems. The most successful experimental low-dimensional system so far constitutes a quasi-one-dimensional case, where alumina spheres with a diameter of roughly a centimeter are placed inside a copper tube with a diameter of 7 cm and a length of roughly one meter (Chabanov and Genack, 2001). The alumina particles then scatter microwaves, which are contained in the copper tube as in a wave guide, thus producing the quasione-dimensional character of the system.
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FIGURE 16 Measurements of the fluctuations of photons in a quasi-one-dimensional sample of alumina spheres obtained by Chabanov, Zhang and Genack (2003). (a) The intensity as a function of input frequency for a certain realization of the disorder in the tube. (b) Averaging over many realizations of the disorder, the intensity probability distribution is obtained. This shows clear deviations from the classical exponential distribution and instead shows a stretched exponential with an exponent of 1/3 (Chabanov, Zhang and Genack, 2003). Reproduced with permission from Chabanov, c 2001, American Physical Society Zhang and Genack (2003)
3.1.1 Statistical Features The transmitted microwave intensity through the cavity shows strong fluctuations as a function of input frequency (see Figure 16a). Such fluctuations always arise in the case of a multiple-scattering sample (Genack and Garcia, 1991). They arise from interferences of the randomly distributed field and are known as speckle. Due to the fact that speckle originates from a random distribution of fields it is easy to show that a diffusive speckle shows an exponential intensity distribution. As can be seen in Figure 16b, in the case of a quasi-one-dimensional sample longer than the localization length, the intensity distribution of the fluctuations is much wider than the exponential distribution expected for a diffusive speckle indicated by the dashed line (Chabanov, Stoytchev and Genack, 2000). The intensity distribution is instead given by a stretched exponential with an exponent of 1/3 (Chabanov, Zhang and Genack, 2003), in agreement with the prediction of Kogan, Kaveh, Baumgartner and Berkovits (1993), whereas Nieuwenhuizen and van Rossum (1995) have predicted a stretched exponential with an exponent of 1/2. This result was later confirmed by Kogan and Kaveh (1995). The advantage of studying the fluctuations of the intensity rather than the static intensity is that this measure is not affected by the presence of absorption. As we will see in the discussion of static transmission measurements in three-dimensional samples, the presence of absorption can be a great problem in identifying localization from transmission
32
Coherent Backscattering and Anderson Localization of Light
FIGURE 17 Measurements of the path-length distribution of photons in a quasi-one-dimensional sample of alumina spheres of different lengths as obtained by Chabanov and Genack (2001). As can be seen from the lower part of the figure, the diffusion coefficient in these samples shows a time dependence indicating a breakdown of diffusion due to the presence of pre-localized states. Reproduced with c 2001, American Physical Society permission from Chabanov and Genack (2001)
measurements alone. In addition to the probability distribution of the speckle intensities, Sebbah, Hu, Klosner and Genack (2006) have measured the spatial distribution of the localized modes. 3.1.2 The Path-Length Distribution Another strategy for avoiding problems with absorption influencing the interpretation of experimental results is to study time-resolved transmission. This will be discussed in more detail below in the context of three dimensional systems, but time-resolved measurements were also carried out in the quasi-one-dimensional system described above by Chabanov and Genack (2001). The results of such measurements are shown for four different samples in Figure 17. The samples differ in tube length, ranging from 61 cm (sample A) to 183 cm (sample C). Samples B and D are both 90 cm long but in sample D the absorption was artificially enhanced by adding a titanium foil to the tube.
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FIGURE 18 The number of diffusive modes is inverse to the probability of crossings of paths inside the sample. For very constrained samples, such as in the case of the quasi-one-dimensional microwave experiments or the measurements on universal conductance fluctuations, the probability of crossings is high and thus the number of modes, i.e. the control parameter for localization, is low. Adapted with permission c 1999, American Physical Society from Scheffold and Maret (1998)
The data in Figure 17 are shown on a semi-logarithmic scale and show a slight deviation from the purely exponential decrease at long times expected from diffusion. This is shown more clearly in the lower part of the figure, which shows the derivative of the logarithm of the intensity with respect to time. At long times, this should be a constant given by a combination of the absorption length and the diffusion coefficient. As can be seen, the diffusion coefficient instead decreases with time, which is most prominent for sample A. In addition, comparison of samples B and D shows that indeed a change in absorption only leads to a constant shift in the decay rate and thus does not influence the results of the time dependence shown in the figure. In contrast to the results in three dimensions to be discussed below, the diffusion coefficient here decreases linearly with time. This decrease is obtained from weak localization corrections in the quasi-one-dimensional case as discussed by Cheung, Zhang, Zhang, Chabanov and Genack (2004) and Skipetrov and van Tiggelen (2004). 3.1.3 The Connection to Bulk Experiments As mentioned above, experiments in quasi-one-dimensional systems exploit the increased probability of paths crossing due to the constricted geometry. As a matter of fact, a similar approach was used by Scheffold, H¨artl, Maret and Matijevi´c (1997) and Scheffold and Maret (1998) to study the universal conductance fluctuations of light, which are suppressed by a factor of 1/g 2 compared to the usual fluctuations. These experiments were carried out in a colloidal suspension of TiO2 particles with values of kl ∗ of the order of 20. This shows that a geometric confinement gives rise to mesoscopic effects (Figure 18); these are similar to the bulk effects of Anderson localization in that they depend on interference between different paths, but they are due solely to geometric effects and thus should not be confused with bulk Anderson localization.
34
Coherent Backscattering and Anderson Localization of Light
In fact, when estimating the dimensionless conductance (Scheffold and Maret, 1998) for a true bulk sample with dimensions of roughly 104 l ∗ , one finds that g is very large. For the samples with very low values of kl ∗ ¨ ¨ discussed below, one obtains g ≈ 104 (Aegerter, Storzer, Fiebig, Buhrer and Maret, 2007). This demonstrates that in bulk samples, the critical parameter is indeed kl ∗ as opposed to g, and one cannot think of the problem in terms of separated modes. This difference also clearly shows up in the self-consistent theory of Skipetrov and van Tiggelen (2004) adjusted for finite systems. In the quasi-one-dimensional case (Skipetrov and van Tiggelen, 2004) there are pre-localized states and a cross-over to localization, whereas in the three-dimensional version of the theory (Skipetrov and van Tiggelen, 2006) there are no pre-localized states and diffusion only breaks down at later times.
3.2 Static Measurements Static measurements have the strong advantage experimentally of being reasonably simple to set up. However, in terms of observing localization, there is a great problem with most static measurements insofar as a photon that is lost due to absorption cannot be distinguished from one lost due to localization. Thus, pure measurements of numbers of photons either in transmission or reflection are difficult to interpret in the context of localization. One way around this will be discussed at the end of this chapter; it consists of studying the fluctuations of the static intensity (i.e. the speckle). There, the interference terms are of great importance, so that one does not simply look at numbers of photons, and the effects of localization and absorption can be distinguished (Chabanov, Stoytchev and Genack, 2000). Another possibility is offered by timeresolved measurements, which will be discussed in the next subsection. 3.2.1 Decrease in Transmission As discussed above, localization will lead to a strong decrease in the transmission of photons through the sample. In fact, due to the renormalization of the diffusion coefficient, the dependence of the transmission on the sample thickness can be predicted. In the critical regime, where there is no length scale in the diffusion left, scaling theory predicts that the diffusion coefficient will decrease as 1/L (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979). This leads to a decrease in transmission proportional to 1/L 2 (Anderson, 1985; John, 1984). Deep in the localized regime, where the diffusion coefficient vanishes, the transmission will be suppressed exponentially, as only the tails of the probability distribution are capable of leaving the sample at the boundary (Anderson, 1985; John, 1984). These predictions have been at the basis of an experimental search for localization using static
The Transition to Strong Localization
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FIGURE 19 Static transmission through samples of GaAs of average particle size ∼ 1 µm (Wiersma, Bartolini, Lagendijk and Righini, 1997). As can be seen, the thickness dependence of the transmission does not follow a 1/L dependence. However, the dashed line in the figure indicating classical diffusion is inconsistent with a physical interpretation of the data. Any physical effect, be it localization or absorption, would lead to decreased transmission as compared to the classical expectation, whereas the dashed line in fact indicates an increased transmission with respect to the diffusive expectation. This is most probably due to the fact that the value of kl ∗ is underestimated due to neglect of absorption in the sample (see text and Figure 20)
transmission measurements of strongly scattering samples (Wiersma, ´ Bartolini, Lagendijk and Righini, 1997; Wiersma, Gomez Rivas, Bartolini, Lagendijk and Righini, 1999). In this work, the transmission properties of ground samples of GaAs were studied in the near infrared, at a wavelength of 1064 nm. For different degrees of grounding and hence different average particle sizes, marked differences in the thickness dependence of transmission were observed. The scattering properties of the samples were characterized using the initial slope of the coherent backscattering cone, yielding a value of kl ∗ . The results for a sample consisting of particles with an average diameter of 1 µm are reproduced in Figure 19. As can be seen in the figure, there are deviations from the expected 1/L behaviour corresponding to diffusion. The theoretical prediction shown by the dashed line however, is in strong contradiction with a simple understanding of localization. As shown in the figure, the deviations from classical transmission due to localization lead to an increase in static transmission, which is physically impossible. Moreover, the deviations increase with decreasing sample thickness, again in contradiction with a physical understanding of the situation. The theoretical prediction of classical diffusion was obtained from the measurement of kl ∗ due to the initial slope of the backscattering cone. This yields a mean free path of 0.17 µm and a corresponding value of kl ∗ = 1. The transmission measurements from thin samples are,
36
Coherent Backscattering and Anderson Localization of Light
FIGURE 20 The data of Wiersma, Bartolini, Lagendijk and Righini (1997) (see Figure 19) as re-analysed by Scheffold, Lenke, Tweer and Maret (1999). In this analysis, the influence of absorption has been taken into account as well. Thus, the tip of the cone is rounded and the slope at that point cannot be used for a reliable estimate of kl ∗ . The analysis of the cone shape (right-hand side) yields a value of kl ∗ ≈ 5 and an absorption length of L a ≈ 8 µm, which fits the data very well. Using these parameters, the static transmission measurements on the left-hand side can be described without additional parameters. The inclusion of absorption yields the solid line, whereas the classical expectation is given by the dashed line. Note that in contrast to Figure 19, the expectation for pure diffusion is above the data, as it should be
however, more consistent with a value of kl ∗ ≈ 5. In this case the deviations in thicker samples are such that the number of transmitted photons decreases compared to the classical expectation. This implies that the determination of kl ∗ from the initial slope of the backscattering cone is systematically flawed and underestimates the value of kl ∗ . As we have seen above, absorption can lead to a rounding of the cone tip. Such a rounding strongly influences the initial slope of the cone, while leaving the width more or less unchanged. Thus the presence of absorption may very well lead to an underestimation of kl ∗ from the initial slope of the backscattering cone, whereas an estimate from the width of the cone is less susceptible to absorption. This is corroborated by the fact that the value of kl ∗ estimated above is in good agreement with a re-analysis of the cone shape using the width of the cone to determine kl ∗ and including absorption, shown in Figure 20 (Scheffold, Lenke, Tweer and Maret, 1999). This re-analysis also leads to an estimation of the absorption length of L a ≈ 8 µm, which is consistent with the transmission data, again shown in Figure 20. This means that the deviations from diffusive behaviour are most probably due to an increased absorption induced by the longer grinding. Such absorption may for instance be due to the increased appearance of surface states. For even smaller particle sizes (average diameter 300 nm), Wiersma, Bartolini, Lagendijk and Righini (1997) obtain an exponential decrease of transmission with a typical length scale of L ≈ 5 µm. Supposing,
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FIGURE 21 Even for a classical sample (with kl ∗ ≈ 20) that appears white and which does not show significant rounding of the backscattering cone, absorption may be high enough to produce an exponential decrease in transmitted intensity, which ¨ appears incompatible with classical diffusion of light (dotted line) (Aegerter, Storzer, ¨ Fiebig, Buhrer and Maret, 2007). When determining the absorption length using time-resolved measurements (see below), the resulting exponential decrease (dashed line) fits very well with the static measurements
in the absence of absorption, that with decreasing particle size the mean free path also decreases, this would be in accordance with the prediction of Anderson localization in the localized regime (Anderson, 1985; John, 1984). Unfortunately however, the scattering properties of this sample were not characterized by any measurement, so we do not know what the value of kl ∗ for this sample should be. This also makes it impossible to estimate the absorption length of this sample. Based on the above arguments, absorption will be present also in this sample and the corresponding absorption length would not be inconsistent with the length scale of the exponential decrease in transmission. Furthermore, the decrease in kl ∗ with particle size is certainly not linear and will certainly show a minimum as the scattering cross-section decreases when the particle size is much smaller than the wavelength of light (Fraden and Maret, 1990). Therefore it is questionable whether or not the scattering strength of this sample will be strong enough to be beyond the Ioffe–Regel criterion. In addition, in the absence of an independent determination of the absorption length, an exponential decrease of transmission cannot be claimed to be due to localization, as absorption is a much more likely candidate. This is illustrated in Figure 21, which shows the static transmission ¨ ¨ through a sample with kl ∗ ≈ 20 (Aegerter, Storzer, Fiebig, Buhrer and Maret, 2007). For very thick samples, absorption will always dominate
38
Coherent Backscattering and Anderson Localization of Light
and a simple comparison with the expectation from diffusion (dotted line) will always overestimate the transmission. However, in this case, the absorption length was determined directly as well, using time-resolved measurements (see below). Adding this to the description yields the dashed line, which perfectly describes the data without a single adjustable parameter. Here the shaded area between the dashed lines indicates the error bar in the experimental determination of the absorption length. For samples with much lower values of kl ∗ ≈ 2.5, which also show effects of localization in time-resolved measurements (i.e. a spatially dependent diffusion coefficient, see below) the situation is markedly ¨ different (Aegerter, Storzer and Maret, 2006). This is shown in Figure 22. Again the dotted line, corresponding to diffusion in the absence of absorption, strongly overestimates the transmission through the samples. However, the description including the experimentally determined absorption (dashed line) is in contradiction with the data as well. Thus in this case the reduced transmission is most probably due to localization of photons. This conclusion is strongly supported by the fact that the timeresolved measurements also allow a determination of the localization length (see below). Including this in the description of the transmission measurements yields the solid line, which describes the data perfectly over twelve orders of magnitude and without any adjustable parameters. This shows that in static transmission measurements, the problem of absorption may be circumvented by an independent determination of the absorption length. This is most conveniently done in time-resolved measurements as we will discuss below. 3.2.2 Influence on the Cone Shape As discussed above, the renormalization of the diffusion coefficient arising from localization can also be treated as a path-length dependence of D (van Tiggelen, Lagendijk and Wiersma, 2000). Since the tip of the backscattering cone consists mostly of photons from long paths, such a path-length dependence of the diffusion coefficient should also be visible in the tip of the cone, as a decrease of the slope as the tip is approached. This effect has been calculated explicitly in the framework of self-consistent theory by van Tiggelen, Lagendijk and Wiersma (2000). Their main result is shown in Figure 23, where the decrease of the diffusion coefficient inside the sample is shown together with the corresponding rounding of the cone tip. As discussed above, however, absorption also leads to a rounding of the cone at small angles due to the lack of photons coming from very long paths. Unfortunately, van Tiggelen, Lagendijk and Wiersma (2000) obtain a diffusion coefficient which decreases exponentially with sample thickness (analogous to the exponentially decreasing transmission), so that the effect of localization again has the same shape as that of absorption. This
The Transition to Strong Localization
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FIGURE 22 Static transmission measurements through a localizing sample as shown ¨ and Maret, 2006). by a long-time tail in time-resolved transmission (Aegerter, Storzer As will be shown below, in this case it is possible to determine not only the absorption length, but also the localization length. Again, the dotted line represents the expectation from pure diffusion and the dashed line that of diffusion including absorption. Both curves are incompatible with the measurements, and satisfactory description of the data becomes possible only upon incorporating the experimentally determined localization length. In fact, there is good agreement between theory and experiment over twelve orders of magnitude without any adjustable parameters
means that, as in the case of static transmission measurements discussed above, measurements of cone-rounding can only be used as arguments for the observation of localization in the presence of data on the absorption properties of the samples. Indeed, as a function of sample thickness, Schuurmans, Megens, Vanmaekelbergh and Lagendijk (1999) find an increasing rounding of the cone when the thickness is decreased, as demanded by theory (van der Mark, van Albada and Lagendijk, 1988). Samples with reasonably high kl ∗ (shown as solid triangles and open squares in Figure 24) are well described by the linear increase of the cone-rounding with 1/k L. For samples with smaller values of kl ∗ however, there are marked deviations for thicker samples. In order to determine the influence of absorption, Schuurmans, Megens, Vanmaekelbergh and Lagendijk (1999) filled the photoanodically etched GaAs sample with dodecanol, showing increased cone-rounding (see Figure 5). This leads to an increase in kl ∗ as can be seen from a decrease in the width of the backscattering cone (see Figure 25). Hence the √ absorption length is also increased according to L a ∝ l ∗la . A description of the dependence of the cone-rounding on sample thickness of the nonfilled sample due to absorption (dashed line in Figure 24) is not compatible
40
Coherent Backscattering and Anderson Localization of Light
FIGURE 23 The influence of localization on the tip of the backscattering cone. Due to the fact that photons on long paths are localized and therefore no longer contribute to the backscattered light, the cone shape is rounded close to the backscattering directon. Plotted here are results from a self-consistent theory assuming a spatially dependent diffusion coefficient calculated by van Tiggelen, Lagendijk and Wiersma (2000). The left-hand panel shows the spatial dependence of the diffusion coefficient, while the right-hand panel gives the corresponding tip of the backscattering cone. The dashed line on the right is the classical cone shape in the absence of localization effects. Reproduced with permission from van Tiggelen, Lagendijk and Wiersma c 2000, American Physical Society (2000)
with the thickness dependence of the cone-rounding of the filled sample. However, one has to note that in this calculation, Schuurmans, Megens, Vanmaekelbergh and Lagendijk (1999) did not take into account the narrowing of the cone due to internal reflections. The filling of the etched holes will lead to a change in the effective refractive index and hence to a change in the value of kl ∗ determined from the cone width. Thus the filling of the voids may well lead to a decrease in the refractive index and hence to an underestimation in the increase in kl ∗ . Due to these uncertainties a direct determination of the absorption length in the lowkl ∗ samples would have been very useful in order to check whether effects of absorption can be ruled out. In addition, subsequent time-resolved experiments on the same samples by Rivas, Sprik, Lagendijk, Noordam and Rella (2000, 2001) and Johnson, Imhof, Bret, Rivas and Lagendijk (2003) did not show effects of localization in the time domain. While this could be due to the fact that the time-resolved measurements were done on thinner samples (see also below), we note that the transmission data of
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FIGURE 24 The rounding of the cone shown in Figure 5 as a function of sample thickness for different samples (Schuurmans, Megens, Vanmaekelbergh and Lagendijk, 1999). For low values of kl ∗ (open circles), there are deviations from the expectation of a finite sample. This is in contrast to samples with a higher kl ∗ (solid triangles and open squares). The open squares are from a similar sample to the open circles, where the pores have been filled with dodecanol. The solid line is a description of the data with absorption. Assuming an unchanged absorption with pore-filling, the dashed line should then correspond to the open squares. Reproduced with permission from Schuurmans, Megens, Vanmaekelbergh and Lagendijk (1999) c 1999, American Physical Society
Johnson, Imhof, Bret, Rivas and Lagendijk (2003) can be described with an absorption length corresponding to the solid line in Figure 24. It therefore would seem that the increased cone-rounding observed in these samples cannot be used as an indication of the onset of Anderson localization as long as absorption is not quantified. 3.2.3 Transport Speed Another quantity that can be determined from static measurements is the transport speed of photons in multiple scattering. Since the strong scatterers employed in the search for Anderson localization are roughly of the same size as the wavelength to increase the scattering crosssection, these particles also show resonant scattering (Wigner, 1955). This was first discussed by van Albada, van Tiggelen, Lagendijk and Tip (1991) in the context of multiple scattering. Contrary to what might be thought intuitively, the resonant scattering properties are still present after averaging over the random distribution of scatterers in multiple scattering (van Tiggelen, Lagendijk, Tip and Reiter, 1991). This leads to a strong decrease in the transport speed of photons, as can be seen in Figure 26,
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Coherent Backscattering and Anderson Localization of Light
FIGURE 25 The backscattering cones for photoanodically etched GaP both as produced and filled with dodecanol (Schuurmans, Megens, Vanmaekelbergh and Lagendijk, 1999). As can be seen, the cone of the filled material is narrower, indicating an increase in kl ∗ . Figure 24 has shown that for the filled samples, significantly less cone-rounding has been observed. Reproduced with permission from Schuurmans, c 1999, American Physical Society Megens, Vanmaekelbergh and Lagendijk (1999)
where the speed of light is shown as a function of particle size. These results were obtained from a calculation using the properties of TiO2 with a filling fraction of 36 per cent. It shows that earlier measurements of anomalously low values of the diffusion coefficient in TiO2 samples by Drake and Genack (1989) were most probably due to resonant scattering, reducing the transport speed and hence the diffusion coefficient, and not to the onset of Anderson localization. Subsequently, these calculations were improved by Busch and Soukoulis (1996) and Soukoulis and Datta (1994) to also be valid for higher filling fractions more appropriate to describe experiments. Using a combination of time-resolved transmission measurements (see below) and coherent backscattering measurements, ¨ Storzer, Aegerter and Maret (2006) measured the transport speed directly for a number of samples with different sizes. These measurements clearly show resonant reductions in the transport speed. The results are shown in Figure 27 and compared with the theoretical descriptions in the inset (upper line Soukoulis and Datta (1994), lower line van Albada, van Tiggelen, Lagendijk and Tip (1991)). There is reasonable agreement with the appropriate theory for higher filling fractions. In addition, these measurements show that the reduction in transport speed can be well separated from signatures of localization. Some of the samples studied here do show a non-exponential tail in the time-resolved transmission intensity as discussed below. However, these samples do not necessarily
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FIGURE 26 Calculation of the transport velocity as a function of the size parameter van Albada, van Tiggelen, Lagendijk and Tip (1991). The calculations were done for particles with a refractive index of 2.72, corresponding to that of TiO2 in the rutile structure, and for a filling fraction of 3 per cent. Due to the fact that correlations between different scatterers are not taken into account in the calculation, the theory does not fully apply at high filling fractions. Reproduced with permission from van c 1991, American Physical Society Albada, van Tiggelen, Lagendijk and Tip (1991)
show a decrease in transport speed. This is because the Mie-resonances responsible for the increase in scattering cross-section (Mie, 1908) as well as resonant scattering are complemented by resonances in the structure factor, which influences only the scattering cross-section. Thus a suitable particle size and packing fraction can lead to a separation of the effects of ¨ Aegerter and Maret, 2006). localization and resonance scattering (Storzer, In addition, resonant scattering was also observed in multiple ¨ scattering measurements on cold atoms (Labeyrie, Vaujour, Muller, Delande, Miniatura, Wilkowski and Kaiser, 2003), where a decrease in transport speed up to a factor of many orders of magnitude has been observed. 3.2.4 Statistical Features As was discussed in the context of microwave experiments, the statistics of transmitted or reflected photons can also give valuable information about the samples and their possible localization properties. Due to its wave nature, multiply scattered light shows a characteristic interference pattern known as speckle. The intensity of each speckle spot will be determined by the differing phase lags between photon paths. For a diffusive sample, the phase delay at differents point will be given by a Gaussian distribution, such that the corresponding intensity distribution is given by an exponential. This intensity distribution of the speckle pattern has been characterized by Wolf, Maret, Akkermans and Maynard (1988), where good agreement with the exponential decay of
44
Coherent Backscattering and Anderson Localization of Light
FIGURE 27 The transport velocity as measured from a combination of time-of-flight ¨ Aegerter and Maret (2006). and backscattering measurements as obtained by Storzer, The main figure shows the velocity relative to its expectation using the Garnett approach. This strongly overestimates the transport speed at particle sizes corresponding to integers of half the wavelength inside the scatterer. The inset shows the transport speed as a function of size parameter compared to theoretical expectations such as that shown in Figure 26 (lower line), and to calculations based on the model of Busch and Soukoulis (1996) that takes into account correlations between scatterers and thus should be applicable for high filling fractions (upper ¨ Aegerter and Maret (2006) line). Reproduced with permission from Storzer, c 2006, American Physical Society
the probability has been found. Vellekoop, Lodahl and Lagendijk (2005) have measured the phase delay directly using interferometric methods (see Figure 28). This allows a study not only of the intensity distribution, but also of the phase distribution. From the width of this distribution, an independent measure of the diffusion coefficient can be found, which Vellekoop, Lodahl and Lagendijk (2005) find in good agreement with several other ways of determining D. When photons are localized within a sample, the phase-delay distribution changes accordingly. Due to the presence of closed loops, there will be an increase in constructive interference of the different paths. In turn this leads to an intensity distribution with a non-exponential tail at high intensities (as well as a suppression at small intensities due to conservation of energy). This has been calculated in one-dimensional and quasi-one-dimensional systems (Nieuwenhuizen and van Rossum, 1995; Sebbah, Hu, Klosner and Genack, 2006). These calculations are in agreement with the results obtained from microwaves discussed above (Garcia and Genack, 1989), however, the situation in three-dimensional
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FIGURE 28 Distribution of the phase delay in a random sample. Left: delay time; right: delay time weighted by intensity. For classical samples, the width of these distributions yields a measure of the diffusion coefficient of light. In Vellekoop, Lodahl and Lagendijk (2005), this was done for TiO2 particles of several sizes and a typical result is shown here. The determination of the diffusion coefficient in this way agrees very well with that from time-of-flight measurements. Reproduced with permission c 2005, American Physical Society from Vellekoop, Lodahl and Lagendijk (2005)
systems is less clear. There have, as yet, been no experimental findings of changed phase statistics close to Anderson localization. In addition, there are no explicit calculations for the phase distribution in a threedimensional localizing sample.
3.3 Time-resolved Measurements As we have seen above, static measurements of transmission or reflection are not readily suited for observing effects of strong localization. This is due to the fact that a simple loss of the number of photons in transmission in thick samples cannot distinguish localization from absorption. Therefore, one has to determine the phase of the photons as well. This can be done either via a quantification of the fluctuations, as discussed above, or via the time-resolved measurements we will discuss below. Since localization acts differently on photons that have spent different amounts of time inside the sample, localization and absorption can be separated in this case, as can be seen by the different functional dependencies implied by the effects. Absorption invariably leads to an exponential decrease also of the time-resolved intensity, while localization and its corresponding renormalization of the diffusion coefficient lead to a decay that is slower than exponential. 3.3.1 The Diffusion Coefficient In a typical time-resolved measurement, the path-length distribution of photons inside a sample of finite thickness is obtained. This can be done either in transmission or in reflection. Due to the much faster time scale of
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Coherent Backscattering and Anderson Localization of Light
the signal in reflection (most of the intensity is only delayed a time l ∗ /v), an experiment in transmission is much more feasible, although there have also been experiments in reflection (Johnson, Imhof, Bret, Rivas and Lagendijk, 2003). In the diffusion approximation, the path-length distributions can be calculated analytically (for a derivation see, for instance, Lenke and Maret (2000)). In transmission, one obtains: ! # " X 1 n2π 2 D n+1 + t , T (t) ∝ (−1) exp − τabs L2 n
(10)
where τabs is the absorption length. Thus, for a sample of given length the time-resolved intensity is determined solely by D and τabs . These two parameters have very little covariance, as the time delay, until sizeable transmission through the sample is achieved, is solely determined by D, while the long-time behaviour is given by an exponential decay with a slope of π 2 D/L 2 + 1/τabs . In order to measure the time-resolved transmission, several types of setups have been used. Usually, a pulsed-laser system capable of producing picosecond pulses shines light on the sample. Behind the sample, a photodetector starts a clock that is subsequently stopped by a delayed reference pulse. For a more detailed description of such setups ¨ see, for instance, Watson, Fleury and McCall (1987) and Storzer, Gross, Aegerter and Maret (2006). For very thin samples, pulses on the scale of a few fs are needed, so that interferometric methods are needed for detection. This was done by Johnson, Imhof, Bret, Rivas and Lagendijk (2003) using samples of etched GaP (this will be discussed in more detail below in the context of time-resolved reflection measurements). Figure 29 ¨ shows the result of a measurement using a ps system (Storzer, Gross, Aegerter and Maret, 2006), in the case of a sample of TiO2 particles of average diameter 540 nm at a wavelength of 590 nm. This sample has a value of kl ∗ = 6.3(3) and thus shows purely diffusive behaviour as can be seen from the fit to Equation (10) shown as a thick solid line, which perfectly describes the data. Due to the fact that time-resolved transmission thus allows a direct determination of the diffusion coefficient, many early experiments have looked for anomalously low values of D, or a thickness dependence of D (see e.g. Watson, Fleury and McCall (1987) and Drake and Genack (1989)). In these experiments, Drake and Genack (1989) found very low values of D and interpreted them as indications of the onset of localization (see Figure 30). However, due to resonance scattering, as discussed above and pointed out by van Albada, van Tiggelen, Lagendijk and Tip (1991), a low value of D does not necessarily imply a low value of l ∗ nor the onset of
The Transition to Strong Localization
47
FIGURE 29 Time-resolved transmission for a classical sample with kl ∗ = 6.3 (data ¨ from Storzer, Gross, Aegerter and Maret (2006)). The thick solid line is a fit to classical diffusion theory through a slab of length L, which allows determination of the diffusion coefficient and the absorption length. There is little covariance between the two quantities as D determines the time lag before any photons are transmitted through the sample, while the absorption length only influences the slope of the exponential long-time tail.
localization. This is because the reduction in transport velocity induced by the increased dwell time in resonance scattering will reduce the value of D obtained from time-of-flight measurements. Similar information can also be gathered from a time-resolved measurement in reflection geometry. This setup presents additional experimental difficulties due to the much shorter time scales of the reflection signal. In reflection, most photons exit the sample after very few scattering events, therefore the peak in the time-resolved intensity is of the order of l/v, where l is the scattering mean free path. For samples close to the localization transition, i.e. with a mean free path comparable to the wavelength, this time is of the order of a few fs and thus very difficult to measure. After this peak, it can again be calculated in the diffusion approximation (see Johnson, Imhof, Bret, Rivas and Lagendijk (2003)) where the intensity decreases as "
R(t) ∝
X n
n2π 2 D 1 n exp − + τabs L2 2
! # t .
(11)
For short times this corresponds to a power-law decay with an exponent of 3/2, whereas at long times (the time scale of transmission) there is an exponential decay on the same time scale as in transmission measurements. Due to the time scale of the resulting intensity,
48
Coherent Backscattering and Anderson Localization of Light
FIGURE 30 The diffusion coefficient and the absorption length of light through powders of TiO2 as determined by time-of-flight measurements (Drake and Genack, 1989). The decrease of the diffusion coefficient with incident wavelength was interpreted as the onset of the localization transition, which predicts vanishing of the diffusion coefficient at a phase transition with kl ∗ , i.e. the wavelength. However, it has later been shown van Albada, van Tiggelen, Lagendijk and Tip (1991) that such a decrease is more likely to be due to an increased dwell time caused by resonant scattering of particles having roughly the same size as the wavelength of light. c 1989, American Reproduced with permission from Drake and Genack (1989) Physical Society
measurements of time-resolved reflection need to be done with a fs-pulsed laser and the signal needs to be recorded interferometrically. In addition, the fact that most signals will be from photons which have only gone through a few scattering events, the signal-to-noise ratio will limit the time resolution to which reflection measurements can be performed. In spite of these differences, Johnson, Imhof, Bret, Rivas and Lagendijk (2003) have carried out measurements of time-resolved reflection on porous GaP samples with very small values of kl ∗ . Figure 31 clearly shows the initial power-law decay and the subsequent exponential decrease due to the finite sample and possible absorption. Measurements of the diffusion coefficient from such time-resolved measurements, both in transmission and reflection, do show good
The Transition to Strong Localization
49
FIGURE 31 Time-resolved measurements of reflection (data from Johnson, Imhof, Bret, Rivas and Lagendijk (2003)). The left-hand panel presents the data on a logarithmic scale showing the exponential decrease at long times due to finite thickness and absorption. On the right the same data are shown on a double-logarithmic scale after deconvolution with the pulse shape. This demonstrates that at shorter times, the data can be described by a power-law decay with an exponent of 3/2 (straight line) in agreement with diffusion theory. Reproduced with c 2003, American permission from Johnson, Imhof, Bret, Rivas and Lagendijk (2003) Physical Society
agreement with determinations from the phase fluctuations as found by Vellekoop, Lodahl and Lagendijk (2005). 3.3.2 Spatially Dependent Diffusion Coefficient Since time-resolved measurements of transmission or reflection are capable of determining the diffusion coefficient very accurately, it is also possible to employ such measurements in the search for a scale dependence of the diffusion coefficient. One of the hallmarks of localization, as discussed above, is that the diffusion coefficient becomes renormalized (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979). This renormalization with the scale of the sample can be translated into a path-length dependence of the diffusion coefficient, as was first calculated by Berkovits and Kaveh (1987), at the critical point. Subsequently, they inserted this path-length dependence into the diffusion theory of time-resolved transmission (Berkovits and Kaveh, 1990). This changes the classical expectation (Equation (10)) to " ! # 2 X n 2 π 2 D(t) 1 n+1 D(t) T (t) ∝ (−1) exp − + t . D0 τabs L2 n
(12)
50
Coherent Backscattering and Anderson Localization of Light
Similarly, a path-length dependence of the diffusion coefficient was used to calculate the influence of localization on the cone shape discussed above (van Tiggelen, Lagendijk and Wiersma, 2000). Later investigations by some of these authors (Skipetrov and van Tiggelen, 2004, 2006) on self-consistent theory in open systems have explicitly calculated the effect of localization on time-resolved measurements. In reflection geometry, they find a change of the exponent of the power-law decay from 3/2 to 2 as the localization transition is crossed (Skipetrov and van Tiggelen, 2004). This will be extremely difficult to observe experimentally however. As discussed above, reflection measurements have to be done on short time scales and are limited by the signal-to-noise ratio due to high intensities at very short times. In addition, absorption and a finite sample will also lead to a decrease in intensity from the t −3/2 power law, which will be exceedingly difficult to distinguish from the t −2 predicted by localization theory. In transmission, Skipetrov and van Tiggelen (2004) find a similar result as Berkovits and Kaveh (1990) in that the pathlength dependence of the diffusion coefficient leads to a non-exponential tail in time-resolved transmission with a decreasing slope. In addition to Berkovits and Kaveh (1990) and Skipetrov and van Tiggelen (2004) also explicitly studied the effect of the dimensionality. Consistent with the scaling theory of Abrahams, Anderson, Licciardello and Ramakrishnan (1979), they find that in the quasi-one-dimensional case, effects of localization can already be observed above the transition (Skipetrov and van Tiggelen, 2004). With this it is possible, for instance, to describe the results on micro-wave transmission (Chabanov, Zhang and Genack, 2003) discussed above. In three-dimensional systems however, no signs of localization are observed above the transition at all (Skipetrov and van Tiggelen, 2006). In Figure 32, we show time-resolved transmission measurements on ¨ a TiO2 sample with a value of kl ∗ ≈ 2.5 (Storzer, Gross, Aegerter and Maret, 2006), the particles having a diameter of 250 nm. As can be seen, the transmission in this sample cannot be described by classical diffusion (Equation (10), shown by the dashed line) alone. There is a nonexponential decay with a decreasing slope as predicted by localization theory. This can be quantified in the same way as was done by Chabanov, Zhang and Genack (2003) as well as Skipetrov and van Tiggelen (2004), by taking the negative derivative of the log of the transmission data. This is shown in Figure 33 for several samples with varying values of kl ∗ . As can be seen, with decreasing kl ∗ there are increasing deviations from classical diffusion theory (solid line). This is, however, only a qualitative measure of possible signs of localization, and a more quantitative description is still needed. Unfortunately, the predictions of Skipetrov and van Tiggelen (2006) cannot be directly compared to the data, as the samples are much
The Transition to Strong Localization
51
FIGURE 32 Time-resolved transmission of a localizing sample with kl ∗ = 2.5 (data ¨ Gross, Aegerter and Maret (2006)). As can be seen, the transmitted from Storzer, intensity at long times shows a non-exponential tail indicative of a renormalized diffusion coefficient (see Figure 33). In fact, the solid line is a fit to diffusion theory ¨ including a scale-dependent diffusion coefficient as done in Aegerter, Storzer and Maret (2006). For comparison, a fit to classical diffusion including absorption is shown by the dashed line
FIGURE 33 The long-time behaviour of time-of-flight measurements allows a direct determination of the spatial dependence of the diffusion coefficient. Taking the negative time-derivative of the logarithm of the transmitted intensity one obtains an effective diffusion coefficient, which should be constant at long times. This is shown here for three different samples, where the sample with kl ∗ = 6.3 agrees perfectly with diffusion theory and a constant diffusion coefficient. The sample with kl ∗ = 2.5 however shows a decrease of the diffusion coefficient at long times
52
Coherent Backscattering and Anderson Localization of Light
thicker than can be described theoretically. However, using the analytic description of Berkovits and Kaveh (1990) (Equation (12)), it is possible to obtain a path-length dependence of the diffusion coefficient from the data by way of a fit with D(t). This is shown by the solid line in Figure 32, which describes the data reasonably well. The time dependence of D(t) obtained from this fit is consistent with earlier simulation results by Lenke, Tweer and Maret (2002), where a self-attracting random walk was simulated and effective diffusion coefficients were determined. The result of these simulations in three dimensions shows that the diffusion coefficient is constant for some time, after which it decreases as 1/t. This behaviour, the same as used to obtain the fit in Figure 32, can be physically explained from the fact that up to the localization length, roughly given by the size of the closed loops, the diffusion must be classical, as interference effects appear only after a closed loop has been traversed. At later times, the photons are localized to a specific region in space, such that hr 2 i tends towards a constant. Describing this behaviour with an effective D(t) immediately leads to a dependence of D(t) ∝ 1/t. This allows a quantitative discussion, not only of the time-resolved transmission experiments and the determination of the localization length discussed below, but also of the static transmission measurements discussed above. The fact that Johnson, Imhof, Bret, Rivas and Lagendijk (2003) did not find a non-exponential decay in their time-resolved measurements while their samples had similar values of kl ∗ is probably due to the small thicknesses used in that study. As can be seen from Figure 32, in the TiO2 samples the localization effects only start to appear after a few ns. This corresponds to roughly a million scattering events. Comparing this to the transmission data of Johnson, Imhof, Bret, Rivas and Lagendijk (2003), this is almost an order of magnitude bigger than their maximum time of flight (their maximum sample thickness is 20 µm). More quantitatively, this is connected to the size of the localization length discussed below. 3.3.3 The Localization Length Given the phenomenological description of the diffusion coefficient based on the simulations of the self-attracting random walk discussed above (Lenke, Tweer and Maret, 2002), the deviations from the diffusion picture in time-resolved measurements can be quantified. In the localized state, the effective diffusion coefficient will decrease ∝ 1/t, which corresponds to the limited extent of the photon cloud. On the other hand, above the transition the diffusion coefficient should be constant. This implies that a systematic study of the deviations from classical diffusion with decreasing kl ∗ should show a transition between these two asymptotic behaviours. In this case, the localization length would simply be given
The Transition to Strong Localization
53
FIGURE 34 The scale dependence of the diffusion coefficient can be quantified by the exponent, a, with which D decreases as a function of time. Its value is plotted here ¨ and Maret (2006)). As can be seen, as a function of kl ∗ (data from Aegerter, Storzer above the transition to localization, the diffusion coefficient is constant as indicated by an exponent a = 0, whereas below kl ∗ ≈ 4 it increases to a = 1, which corresponds to a localized state
√ by D0 τloc , which however can only be determined as long as this length scale is smaller than the sample thickness. Considering the predictions of one-parameter scaling theory however, the situation is somewhat more complicated. The fact that D is renormalized to be dependent on the sample thickness as 1/L, has been translated to a path-length dependence by Berkovits and Kaveh (1990) to imply a path-length dependence as D(t) ∝ t −1/3 . To take this into account, the time-of-flight measurements have been fitted with a power¨ law dependence of D(t) ∝ t −a at long times (Aegerter, Storzer and Maret, 2006). In the approach to localization, this exponent increases from its classical value of zero to its localized value of unity. At the transition, even the exponent of 1/3 can be observed, showing the critical point renormalization of the diffusion coefficient, see Figure 34. This plot also shows that the exponent is given by unity at low values of kl ∗ , corresponding to a localized state, while it is zero above the transition. This transition can be determined, from the dependence of the localization exponent in the figure, to be klc∗ ≈ 4. The algebraic decay of the diffusion coefficient in the critical regime does somewhat complicate the determination of the localization length. Due to the fact that a classical behaviour can be obtained also from a change in the localization length now has to be determined √the exponent, a via L 1−a D0 τloc . In the limiting cases of a = 1, a = 0 this gives the same values as above, while giving an interpolation in the critical regime. The
54
Coherent Backscattering and Anderson Localization of Light
FIGURE 35 The dependence of the inverse localization length on the critical ¨ parameter kl ∗ (adapted from Aegerter, Storzer and Maret (2006)). Below a critical √ value of klc∗ ≈ 4, the localization length, as given by D0 τl , becomes smaller than the sample thickness, indicating the transition to a macroscopic population of localized states
inverse of the localization length is the order parameter in the transition to localization. Therefore a systematic study of the localization length as a function of kl ∗ gives a description of the transition including the critical point and the critical exponent. In Figure 35, this dependence is plotted with the localization length normalized to the sample thickness. For a finite sample, localization can only be observed if the localization length is smaller than the sample thickness. Therefore, very thin samples having values of kl ∗ below the transition will not show effects of localization, and only very thick samples (with L far exceeding 100 l ∗ ) can show the underlying transition. This is probably why the time-of-flight measurements of Johnson, Imhof, Bret, Rivas and Lagendijk (2003) are well described by classical diffusion in spite of the fact that their values of kl ∗ are close to or beyond the transition. Their samples, which consist of photoanodically etched GaP as already discussed above in the context of cone-tip measurements, are rather thin (L ≈ 40l ∗ ). This implies that the paths of the light traversed inside the sample are not sufficiently long to form enough closed loops and thus show localization. 3.3.4 Determination of the Critical Exponent The systematic determination of the localization length for different samples around the localization transition also allows an experimental ¨ investigation of the critical exponent (Aegerter, Storzer and Maret, 2006). In this context, scaling theory (Abrahams, Anderson, Licciardello
The Transition to Strong Localization
55
FIGURE 36 The inverse localization length as a function of the critical parameter |kl ∗ − klc∗ |. As can be seen, the localization length diverges in the approach to a critical point at klc∗ = 4.4(2) with an exponent ν consistent with a value of 0.65(15) (solid line). This is in accordance with a mean-field argument for the value of the critical exponent but in contradiction to numerical simulations (MacKinnon and Kramer, 1981)
and Ramakrishnan, 1979) predicts a critical exponent ν < 1, without specifying a precise value. As discussed above, an epsilon expansion in the dimension starting from the lower critical dimension (dl = 2) yields a value of 1/2 for the critical exponent (John, 1984), however due to the fact that the system considered here is fully three-dimensional, such a comparison cannot be considered precise. On the other hand, the fact, that above the upper critical dimension (du = 4) the mean-field value ν = 1/2 is always obtained for the exponent of the order parameter in a secondorder phase transition (Schuster, 1978), would indicate that the value for a three-dimensional system should not be too far from these two limiting cases. In contrast, numerical evaluations of the Green–Kubo formalism consistently obtain a value of ν = 1.5 (MacKinnon and Kramer, 1981; Lambrianides and Shore, 1994; Rieth and Schreiber, 1997), which is not only inconsistent with the experimental data shown below, but also with one-parameter scaling theory (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979). It should be noted however that these numerical investigations are carried out on quasi-periodic lattices. As such they thus do not necessarily conform to the nature of Anderson localization, which is fundamentally based on a completely disordered structure. This may explain the discrepancy with the analytic as well as the experimental results. When plotting the inverse localization length against the critical ¨ parameter |kl ∗ − klc∗ |/kl ∗ , Aegerter, Storzer and Maret (2006) obtained a divergence as shown in Figure 36, with a critical exponent of ν = 0.45(10).
56
Coherent Backscattering and Anderson Localization of Light
The above expression for the critical parameter was defined by Berkovits and Kaveh (1990), while John (1984) and others used the expression |kl ∗ − klc∗ |. Using this latter critical parameter, the data yield a different exponent, namely ν = 0.65(15). The critical value of kl ∗ is not affected by the choice of critical parameter. Such an experimental determination can be used to test the different kinds of theoretical predictions discussed above. One notes that the result is consistent with the rather unspecific prediction of one-parameter scaling theory. Furthermore, it is in striking agreement with the result of the epsilon expansion, as well as the meanfield prediction. This is somewhat surprising given that the experiments are carried out in a system of intermediate dimensionality, where both the epsilon expansion and the mean-field result should not explicitly hold. The numerical results finally are strongly inconsistent with the data, which might be due to the fact that the numerical results are obtained from quasi-periodic systems.
4. CONCLUSIONS AND OUTLOOK Watching the light scattered back from an object can not only give a wealth of information on the scattering object, but also on some properties of light itself. As long as the scatterers are sufficiently random – and the samples thus opaque – the photonic analogue of the metal–insulator transition can be observed. Due to the fact that in this case there is no interaction between the diffusing particles (the photons, in contrast to the electrons in a metal), a theoretical treatment of photon localization is closer than that of electrons. As we have seen however, great care has to be taken in the experimental investigation of photon localization. Absorption, resonant scattering or other external effects may well pose as localization in that they also produce an exponential decrease in transmission or a slowing down of transport, respectively. Therefore, investigations of localization have to concentrate on measures that are unaffected by absorption or transport speed, such as the speckle intensity distribution or timeresolved measurements. On the other hand, static measures are still useful, however one then needs an independent quantification of the absorption and the transport speed. ¨ Gross, Aegerter and Maret Using time-resolved measurements, Storzer, (2006) have found clear indications of non-classical diffusion, which show all the hallmarks of localization and cannot be explained by the above ¨ artefacts. In fact, a quantitative description by Aegerter, Storzer and Maret (2006) of these data with qualitative localization theory not only finds localized states as given by a constant hr 2 i, but can also describe the thickness dependence of the static transmission over twelve orders
References
57
of magnitude without a single adjustable parameter. However, these measurements cannot provide evidence for the interference nature of the effect. For this purpose, measurements affecting the phase of the propagating photons would be necessary. In this context, it is useful to remember the work of Erbacher, Lenke and Maret (1993) and Golubentsev (1984), showing that weak localization can be destroyed by applying a strong magnetic field to a Faraday-active multiple-scattering medium. Using the same approach, it might be possible to add a Faraday-active material to a sample showing localization and apply a strong magnetic field. A destruction of the non-exponential tail in this case would clearly show the interference nature of the effect and thus Anderson localization. Work to this effect is under way.
ACKNOWLEDGEMENTS We would like to thank all the current and previous members of the localization team in Konstanz/Strasbourg/Grenoble for their efforts over the years in studying multiple scattering. Without their work many results presented here would not have been possible. In particular, we thank ¨ ¨ M. Storzer, S. Fiebig, W. Buhrer, P. Gross, R. Tweer, R. Lenke, R. Lehner, C. Eisenmann, D. Reinke, U. Mack, F. Scheffold, F. Erbacher, and P.E. Wolf. In addition we would like to thank many colleagues for valuable discussions on pertinent questions as well as for making available some of their data to be presented here. These are E. Akkermans, N. Borisov, M. Fuchs, A.Z. Genack, M.D. Havey, R. Kaiser, A. Lagendijk, R. Maynard, G. Montambaux, M. Noginov, J. Pendry, P. Sebbah, P. Sheng, S.E. Skipetrov, H. Stark, B. van Tiggelen, and D.S. Wiersma. Finally, this work is funded (2008) by the International Research Training Group “Soft Condensed Matter Physics of Model Systems” by the DFG and the Centre of Applied Photonics jointly financed by Ministry ¨ of Science, Research and Arts of Baden-Wurttemberg as well as the University of Konstanz. We are very grateful for their contributions.
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CHAPTER
2 Soliton Shape and Mobility Control in Optical Lattices Yaroslav V. Kartashov a , Victor A. Vysloukh b and Lluis Torner a a ICFO-Institut de Ciencies Fotoniques, and Universitat
Politecnica de Catalunya, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain b Departamento de Fisica y Matematicas, Universidad de las Americas – Puebla, Santa Catarina Martir, 72820, Puebla, Mexico
Contents
1. 2. 3.
4.
5.
Introduction Nonlinear Materials Diffraction Control in Optical Lattices 3.1 Bloch Waves 3.2 One- and Two-dimensional Waveguide Arrays 3.3 AM and FM Transversely Modulated Lattices and Arrays 3.4 Longitudinally Modulated Lattices and Waveguide Arrays Optically-induced Lattices 4.1 Optical Lattices Induced by Interference Patterns 4.2 Nondiffracting Linear Beams 4.3 Mathematical Models of Wave Propagation One-dimensional Lattice Solitons 5.1 Fundamental Solitons 5.2 Beam Shaping and Mobility Control 5.3 Multipole Solitons and Soliton Trains 5.4 Gap Solitons 5.5 Vector Solitons 5.6 Soliton Steering in Dynamical Lattices 5.7 Lattice Solitons in Nonlocal Nonlinear Media
64 66 69 69 71 72 74 76 76 78 79 83 84 86 88 88 90 91 92
E-mail address:
[email protected] (Yaroslav V. Kartashov). c 2009 Elsevier B.V. Progress in Optics, Volume 52 ISSN 0079-6638, DOI 10.1016/S0079-6638(08)00004-8 All rights reserved.
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6.
Two-dimensional Lattice Solitons 6.1 Fundamental Solitons 6.2 Multipole Solitons 6.3 Gap Solitons 6.4 Vector Solitons 6.5 Vortex Solitons 6.6 Topological Soliton Dragging 7. Solitons in Bessel Lattices 7.1 Rotary and Vortex Solitons in Radially Symmetric Bessel Lattices Multipole and Vortex Solitons in Azimuthally 7.2 Modulated Bessel Lattices 7.3 Soliton Wires and Networks Mathieu and Parabolic Optical Lattices 7.4 8. Three-dimensional Lattice Solitons 9. Nonlinear Lattices and Soliton Arrays 9.1 Soliton Arrays and Pixels 9.2 Nonlinear Periodic Lattices 10. Defect Modes and Random Lattices 10.1 Defect Modes in Waveguide Arrays and Optically-induced Lattices 10.2 Anderson Localization 10.3 Soliton Percolation 11. Concluding Remarks Acknowledgements References
94 95 98 100 101 102 105 108 108 113 115 115 116 119 120 121 125 125 128 129 130 132 132
1. INTRODUCTION Nonlinear wave excitations are ubiquitous in Nature. They play an important role in our description of a vast variety of natural phenomena and their applications in many branches of science and technology, from optics to fluid dynamics and oceanography, solid-state physics, Bose–Einstein condensates (BECs), cosmology, etc. Examples are widespread (see, e.g., Bloembergen (1965), Boyd (1992), Infeld and Rowlands (1990), Whitham (1999), Moloney and Newell (2004), Hasegawa and Matsumoto (1989), Akhmediev and Ankiewicz (1997), Kivshar and Agrawal (2003) and Stegeman and Segev (1999)). Until recently, research in this area was commonly divided into two distinct, separated categories: Excitations in systems which can be described by continuous mathematical models and those in systems that can be effectively described by discrete equations. The difference between them is not merely quantitative, but qualitative, and often drastic. The paradigmatic example is the case of systems described by the ¨ nonlinear Schrodinger equation (NLSE). This is one of the few canonical
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equations that are encountered in many branches of nonlinear science when phenomena are studied at the proper scale length. The continuous version of NLSE belongs to that very special class of equations that in onedimensional geometries are referred to as completely-integrable and thus have rigorous single and multiple soliton solutions, which describe the stable and robust propagation of wave packets with an infinite number of conserved quantities. In contrast, systems modeled by the discrete NLSE may not conserve some of the fundamental quantities, such us the paraxial wave linear momentum, and thus feature a correspondingly restricted mobility. From a practical point of view, such qualitative differences are not necessarily advantageous or disadvantageous. For example, in the case of propagating light beams, the restricted mobility inherent in the discrete NLSE suggests new strategies for the routing or switching of light. By and large, the point is that the key differences existing between continuous and strictly-discrete mathematical models manifest themselves in the generation of correspondingly different physical phenomena. However, there is a whole world in between systems modeled by totally continuous and totally discrete evolution equations. This world has been made accessible to experimental exploration in its full scope with the advent of optically-induced lattices in optics (see Fleischer, Segev, Efremidis and Christodoulides (2003)) and in Bose–Einstein condensates (see Morsch and Oberthaler (2006)). Tuning the strength of the lattices causes the system behavior to vary between a predominantly continuous model and a predominantly discrete one. The corresponding powerful concept which may be termed “tunable discreteness” has direct important applications, e.g., for all-optical routing and shaping of light in optics, or for the generation and manipulation of quantum correlated matter waves in Bose–Einstein condensates. In addition to such applications, from a broad perspective, the ability to tune the strength of the genuinely discrete features of nonlinear systems affords important applications in other areas of nonlinear science where the concept can be implemented. Optical lattices provide a unique laboratory to undertake such exploration. Here we present a progress overview focused on soliton control. We focus our attention on the salient possibilities afforded by the concept of tunable discreteness for soliton manipulation at large, and we refer the readers to other reviews for in-depth experimental advances (Lederer, Stegeman, Christodoulides, Assanto, Segev and Silberberg, 2008). Thus, we start by addressing the stationary properties of the different types of lattice solitons, and focus our attention on the dynamical properties by highlighting the main advances achieved to date. We address optical solitons, but most of the results hold as well for BECs.
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The paper is organized as follows. In Sections 2 and 3 we discuss briefly the available nonlinear materials and structures that are currently used in most experiments with optical lattice solitons. In Section 4, we describe the experimental techniques available for optical lattice induction and the mathematical models used to describe nonlinear light excitations in lattices. In Section 5 we address the salient properties of scalar and vector solitons supported by one-dimensional lattices. We discuss the impact on the soliton shape and soliton mobility caused by tunable static and dynamic optical lattices. We also address the potential stabilization of complex lattice soliton structures. In Section 6, the properties of solitons supported by two-dimensional optical lattices are considered. Addition of the second transverse coordinate remarkably affects properties of steadystate nonlinear waves. Optical lattices may have a strong stabilizing action for two-dimensional solitons and allow the existence of new types of solitons such as vortex solitons, possibilities that we discuss explicitly. In Section 7 lattices imprinted with different types of nondiffracting beams (including Bessel, Mathieu and parabolic ones) are discussed. Such lattices offer new opportunities for soliton managing and switching, and constitute one of the key points to be highlighted in this area. In the absence of refractive index modulation, three-dimensional solitons in selffocusing Kerr-type nonlinear media experience rapid collapse, but under appropriate conditions optical lattices may stabilize not only groundstate solitons but also higher-order three-dimensional solitons, even if the dimensionality of the lattice is lower than that of the solitons, as briefly reviewed in Section 8. The properties of nonlinear optical lattices (including soliton arrays), together with the techniques that can be used to cause their stabilization, are addressed in Section 9. In Section 10 we review briefly the specific properties introduced by defect modes and by random lattices. Finally, in Section 11 the salient conclusions are given.
2. NONLINEAR MATERIALS The advent and availability of suitable materials and fabrication techniques for the generation of optical lattices has been a key ingredient for the advancement of the field. In this section we briefly introduce the properties of only a few optical materials which are routinely used for the experimental excitation of lattice solitons. Fused silica is known to be a weakly nonlinear material (n 2 ' 2.7 × 10−16 cm2 /W). It has found applications in experiments with lattice solitons due to its unique ability to write two-dimensional optical waveguides by tightly focusing femtosecond laser pulses generated by amplified Ti:Sa lasers. Permanent refractive index changes up to 1.3 × 10−3 have been reached, with a typical spacing between 20 mm long
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waveguides of the order of ∼20 µm (Pertsch, Peschel, Lederer, Burghoff, ¨ Will, Nolte and Tunnermann, 2004; Szameit, Blomer, Burghoff, Schreiber, ¨ Pertsch, Nolte and Tunnermann, 2005). Formation of two-dimensional solitons in such waveguiding arrays was observed by Szameit, Burghoff, ¨ Pertsch, Nolte, Tunnermann and Lederer (2006). Evolution of linear light beams in arbitrary waveguide arrays, including laser-written ones, was ¨ addressed by Szameit, Pertsch, Dreisow, Nolte, Tunnermann, Peschel and Lederer (2007). Nonresonant nonlinearities in semiconductor materials may be orders of magnitude stronger than in silica. For instance, in AlGaAs at the wavelength 1.53 µm the nonlinearity coefficient amounts to n 2 ∼ 10−13 cm2 /W, and self-action effects are observable already for light intensities ∼10 GW/cm2 . Such light intensities can be readily achieved with tightly focused pulsed radiation. Owing to highly advanced semiconductor processing technologies it is possible to fabricate individual semiconductor waveguides as well as waveguide arrays with the effective core areas ∼20 µm2 . Such arrays were used to demonstrate discrete spatial solitons at power levels ∼500 W (Eisenberg, Silberberg, Morandotti, Boyd and Aitchison, 1998). Waveguiding arrays fabricated from polymer inorganic-organic materials also have found important applications in the field of diffraction management (Pertsch, Zentgraf, Peschel, Brauer and Lederer, 2002a,b), and in the experimental observation of the optical analog of Zener tunneling (Trompeter, Pertsch, Lederer, Michaelis, Streppel and Brauer, 2006). Metal vapors may feature resonant enhanced nonlinearities. Under proper conditions, e.g., in the absence of saturation, the nonlinear coefficient of sodium vapors for large frequency detuning is given by n 2 = 8π 2 N µ4 /[n 20 c h¯ 3 (ω − ω R )3 ], where N is the atomic concentration, µ is the dipole moment, and ω R is the resonance frequency (Grischkowsky, 1970). In rubidium vapor, resonant enhancement is achieved close to the D2 line at λ = 780 nm. The accessible values of the nonlinear coefficient range from n 2 = 10−10 cm2 /W to 10−9 cm2 /W for typical concentrations N ∼ 1013 cm−3 . Self-trapping of a CW laser beam in sodium vapor was observed as early as 1974 (Bjorkholm and Ashkin, 1974). Saturation of the nonlinearity is typical for rubidium and sodium vapors. Then, the nonlinear contribution to the refractive index can be approximated as δn = n 2 I /(1 + I /I S ), where I S is the saturation intensity. The nonlinearity sign, its strength, and saturation degree may be varied by changing the vapor concentration and by changing the detuning from resonance. Solitons were observed in metal vapors at power levels ∼100 mW (Tikhonenko, Kivshar, Steblina and Zozulya, 1998). Nematic liquid crystals have emerged as suitable materials for experimentation with lattice solitons, thanks to their strong reorientational non-
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linearity that may exceed the nonlinearity of standard semiconductors by several orders of magnitude (Simoni, 1997). Such crystals might be used for fabrication of periodic voltage-controlled waveguide arrays (Fratalocchi, Assanto, Brzdakiewicz and Karpierz, 2004). In such arrays the top and bottom cell interfaces provide planar anchoring of the liquid crystal molecules along the direction of light propagation. A set of periodically spaced electrodes (typical spacing ∼6 µm) allows a bias to be applied across a ∼5 µm thick crystal cell, thereby modulating the refractive index distribution through molecular reorientation. The typical power required for spatial soliton formation in such arrays is ∼35 mW at biasing voltage 1.2 V (see, e.g., reviews on solitons in nematic liquid crystal arrays by Brzdakiewicz, Karpierz, Fratalocchi, Assanto and Nowinowski-Kruszelnick (2005); Assanto, Fratalocchi and Peccianti (2007), and Section 5.7 of the present review). The photovoltaic nonlinearity of photorefractive LiNbO crystals (see Bian, Frejlich and Ringhofer (1997)), sets a rich playground for experiments with spatial solitons. For instance, self-trapping of an optical vortex in such crystals was reported by Chen, Segev, Wilson, Muller and Maker (1997). Two-dimensional bright photovoltaic spatial solitons have been observed in Cu:KNSBN crystal featuring focusing nonlinearity (She, Lee and Lee, 1999). Photorefractive materials also can be used for fabrication of waveguide arrays. Such arrays might be created by titanium indiffusion in a copper-doped LiNbO crystal where the optical nonlinearity (defocusing and saturable) arises from the bulk photovoltaic effect. Such waveguides combine high nonlinearity with adjustable contrast of linear refractive index. A typical LiNbO waveguide array consists of 4 µm-wide titanium doped stripes separated by the same distance. Each channel forms a single-mode waveguide at 514.5 nm wavelength for refractive index modulation depth ∼ 3 × 10−3 , while the coupling constant ∼ 1 mm−1 . The experimental observation of spatial gap solitons in such ¨ arrays at the power levels ∼µW was reported by Chen, Stepic, Ruter, Runde, Kip, Shandarov, Manela and Segev (2005). An excellent material for experimentation with different types of nonlinear lattice waves is a doped photorefractive SBN crystal biased externally with a DC electric field. Due to a strong anisotropy of the electrooptic effect, the ordinary-polarized lattice-forming beams, propagating almost linearly along the crystal, are capable of inducing relatively strong refractive index modulation for the extraordinary polarized probe beam, as was suggested by Efremidis, Sears, Christodoulides, Fleischer and Segev (2002). It was demonstrated that nonlinear lattice waves in photorefractive SBN crystals might be observed at extremely low power levels of a hundred nanowatts (Tr¨ager, Fisher, Neshev, Sukhorukov, Denz, Krolikowski and Kivshar, 2006).
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Summarizing, there are a variety of materials suitable for the exploration of nonlinear wave propagation in periodic media. However, optical induction in suitable materials constitutes a real landmark advance for the purpose that we explore in this review, as it affords control of the shape, strength and properties of the lattice with an unprecedented flexibility (see Section 4 below for a detailed discussion).
3. DIFFRACTION CONTROL IN OPTICAL LATTICES The ability to engineer and control diffraction opens broad prospects for light beam manipulation. In this section we discuss briefly a few milestones achieved in this area in lattices and waveguide arrays. In a linear bulk medium the spatial Fourier modes (plane waves) form an adequate set of eigenfunctions for the analysis of diffraction and refraction phenomena. Thus, diffraction of a collimated wave beam might be interpreted as dephasing of spatial harmonics that leads to a convex wave front. Inside an optical lattice standard Fourier analysis becomes inefficient because of the energy exchange between an incident and Braggscattered waves. The adequate functional basis for description of wave propagation in such a periodic environment is provided by the so-called Floquet–Bloch set of basis functions. Figure 1 illustrates the dependencies of the propagation constants of Bloch waves from the different bands on transverse wavenumber and the corresponding profiles of Bloch waves. In regions of convex band curvature the central mode propagates faster than its neighbors and the beam acquires a convex wave front during propagation. There are regions of normal diffraction where the wave behavior is fully analogous to that in the homogeneous media. By contrast, a group of modes in concave regions of band curvature evolve anomalously, producing a concave wave front. Thus, modification of the input angle of the beam entering the periodic structure results in different rates and even signs of diffraction. Therefore, the effect of a periodic lattice on the propagation of a laser beam depends, to a great extent, on the overall size of the input beam, relative to the lattice period, and on the depth of refractive index modulation.
3.1 Bloch Waves The linear interference of optical Bloch waves in periodically stratified materials was studied both theoretically and experimentally almost two decades ago (see Russell (1986) and references therein). Eisenberg, Silberberg, Morandotti and Aitchison (2000) proposed a scheme to design structures with controllable diffraction properties based on arrays of evanescently coupled waveguides. The scheme proposed is suitable for exploiting the negative curvature of the diffraction curve of light in
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FIGURE 1 (a) Dependencies of propagation constants of Bloch waves from different bands on transverse wavenumber k at = 4 and p = 3. (b) Band-gap lattice structure at = 4. Bands are marked with gray color, gaps are shown white. Profiles of Bloch waves from first (c),(d) and second (e),(f) bands, corresponding to k = 0 (panels (c) and (e)) and k = 2 (panels (d) and (f)) at = 4 and p = 3
waveguide arrays in order to control diffraction. Arrays with reduced, canceled, and even reversed diffraction were technologically fabricated. Excitation of linear and nonlinear modes belonging to high-order bands of the Floquet–Bloch spectrum of periodic array was demonstrated by Mandelik, Eisenberg, Silberberg, Morandotti and Aitchison (2003a). To excite a single Bloch mode in a waveguide array the light beam was coupled at a grazing angle to the array from a region of a planar waveguide. Changing the tilt angle of the input beam allows selective excitation of periodic waves from different bands. A prism-coupling method for excitation of Floquet–Bloch modes and direct measurement of band structure of one-dimensional waveguide arrays was introduced ¨ Wisniewski and Kip (2006). Solitary waves bifurcating from by Ruter, Bloch-band edges in two-dimensional periodic media have been studied theoretically by Shi and Yang (2007). In optically-induced photonic lattices, the Bragg scattering of light was studied experimentally in a biased photorefractive SBN:60 crystal (Sukhorukov, Neshev, Krolikowski and Kivshar, 2004). This experiment showed the splitting of an input slant laser beam into several Bloch modes due to Bragg scattering. A novel, powerful experimental technique was presented for linear and nonlinear Brillouin zone spectroscopy of optical
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lattices (Bartal, Cohen, Buljan, Fleischer, Manela and Segev, 2005). This method relies on excitation of different Bloch modes with random-phase input beams and far-field visualization of the emerging spatial spectrum. This technique facilitates mapping the borders of extended Brillouin zones and the areas of normal and anomalous dispersion. The possibility of acquiring the full bandgap spectrum of a photonic lattice of arbitrary profile was discussed by Fratalocchi and Assanto (2006c). Nonlinear adiabatic evolution and emission of coherent Bloch waves in optical lattices was analyzed by Fratalocchi and Assanto (2007a). Modulational instability of Bloch waves in one-dimensional saturable waveguide arrays ¨ was studied both theoretically and experimentally (Stepic, Ruter, Kip, ¨ Maluckov and Hadzievski, 2006; Ruter, Wisniewski, Stepic and Kip, 2007). Various types of two-dimensional Bloch waves were generated in a square photonic lattice by employing the phase imprinting technique (Tr¨ager, Fisher, Neshev, Sukhorukov, Denz, Krolikowski and Kivshar, 2006). The unique anisotropic properties of lattice dispersion resulting from the different curvatures of the dispersion surfaces of the first and second spectral bands were demonstrated experimentally. Nonlinear interactions between extended waves in optical lattices may lead also to new phenomena. For example, two Bloch modes launched into a nonlinear photonic lattice evolve into a comb or a supercontinuum of spatial frequencies, exhibiting a sensitive dependence on the difference between the quasi-momenta of the two initially excited modes (see, Manela, Bartal, Segev and Buljan (2006)). Finally, it is also worth mentioning that spatial four-wave mixing with Bloch waves was considered by Bartal, Manela and Segev (2006).
3.2 One- and Two-dimensional Waveguide Arrays The fabrication of simple GaAs waveguide arrays was achieved and analyzed more than three decades ago (see Somekh, Garmire, Yariv, Garvin and Hunsperger (1973)). Channel waveguides were formed by proton bombardment through a gold mask that allowed creation of channels with width 2.5 µm separated by 3.9 µm and featuring relatively deep refractive index modulation ∼0.006. Optical coupling accompanied by complete light transfer from the incident channel into adjacent ones was observed. Since diffraction properties of light beams in lattices strongly depend on the propagation angle, tandems of different short segments of slant waveguide arrays (zigzag structure) might be used in order to achieve a desired average diffraction. This approach for diminishing the power requirement for lattice soliton formation was suggested by Eisenberg, Silberberg, Morandotti and Aitchison (2000), and it is closely linked to the idea of dispersion management from fiber optics technology. It was also
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shown that at higher powers, normal diffraction may lead to self-focusing and formation of bright solitons, but slight modification of the input tilt, causing change of diffraction sign, results in defocusing. Self-focusing and self-defocusing were achieved with the same medium, structure, and wavelength (Morandotti, Eisenberg, Silberberg, Sorel and Aitchison, 2001). Detailed experimental studies of anomalous refraction and diffraction of cw radiation in a one-dimensional array imprinted in inorganic-organic polymer was reported by Pertsch, Zentgraf, Peschel, Brauer and Lederer (2002b). Discrete Talbot effects in a one-dimensional waveguide array was observed by Iwanow, May-Arrioja, Christodoulides, Stegeman, Min and Sohler (2005). It was shown that in discrete configurations a recurrence process is only possible for a specific set of periodicities of the input patterns. Experimental demonstration of the discrete Talbot effect was carried out in an array consisting of 101 waveguides imprinted in a 70 mm long Z-cut LiNbO3 wafer with lithography and titanium in-diffusion techniques. Linear diffraction as well as nonlinear dynamics in two-dimensional waveguide arrays can have more complicated character and can be much richer than those in one-dimensional arrays (Hudock, Efremidis and Christodoulides, 2004; Pertsch, Peschel, Lederer, Burghoff, Will, Nolte ¨ and Tunnermann, 2004). For instance, diffraction in these arrays can be effectively altered by changing the beam’s transverse Bloch vector orientation in the first Brillouin zone. In general, diffraction in twodimensional arrays can be made highly anisotropic and therefore permits the existence of elliptic solitons when nonlinearity comes into play. Under appropriate conditions not only the strength but also the sign of diffraction can differ for different directions (e.g., a beam can experience normal diffraction in one direction and anomalous diffraction in the other one).
3.3 AM and FM Transversely Modulated Lattices and Arrays In this subsection we discuss the new physical effects afforded by rather smooth (at the beam width scale) transverse modulation of the optical lattice parameters. As was shown in the context of Bose–Einstein condensates, matter-wave soliton motion can be effectively managed by means of smooth variations of the parameters of the optical lattice (Brazhnyi, Konotop and Kuzmiak, 2004). Thus, linear, parabolic, and spatially-localized modulations of lattice amplitude and frequency were considered. For instance solitons could be accelerated, decelerated, or undergo reflection depending on the modulation function profile. The energy, effective mass, and soliton width can also be effectively controlled in such lattices. In this case the effective particle approximation provides
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a qualitative explanation of the main features of soliton dynamics, if the soliton width substantially exceeds the lattice period and the lattice modulation is smooth enough. Grating-mediated waveguiding was proposed by Cohen, Freedman, Fleischer, Segev and Christodoulides (2004). Such waveguiding is driven by a shallow one-dimensional lattice with bell- or trough-shaped amplitude that slowly varies in the direction normal to the lattice wavevector (lattice with amplitude modulation). Optical lattices with linear amplitude or frequency modulation were considered specifically for soliton control (Kartashov, Vysloukh and Torner, 2005d). It was revealed, that the soliton trajectory can be controlled by varying the lattice depth and amplitude or frequency modulation rate. Also, it was discovered that the effective diffraction of a light beam launched into the central channel of a lattice with a quadratic frequency modulation can be turned in strength and sign (Kartashov, Torner and Vysloukh, 2005). Finally, complete suppression of linear diffraction in the broad band of spatial frequencies is accessible in FM lattices. A direct visual observation of Bloch oscillations and Zener tunneling was achieved in two-dimensional lattices photoinduced by four interfering plane waves in biased photorefractive crystals (Trompeter, Krolikowski, Neshev, Desyatnikov, Sukhorukov, Kivshar, Pertsch, Peschel and Lederer, 2006). In this experiment, a coordinate-dependent background illumination was used to create a refractive index distribution in the form of a periodic modulation superimposed on to the linearly increasing background. Previously, it had been shown theoretically that Bloch oscillations can also emerge in discrete waveguide arrays with propagation constants linearly varying across the array (Peschel, Pertsch and Lederer, 1998). The occurrence of Bloch oscillations was demonstrated experimentally in an array of 25 AlGaAs waveguides with transversally varied refractive index and spacing (Morandotti, Peschel, Aitchison, Eisenberg and Silberberg, 1999b), as well as in a thermo-optic polymer waveguide array with applied temperature gradient (Pertsch, Dannberg, Elflein, Brauer and Lederer, 1999). Photonic Zener tunneling between the bands of the polymer waveguide array was observed and investigated experimentally by Trompeter, Pertsch, Lederer, Michaelis, Streppel and Brauer (2006). Such a one-dimensional polymer waveguide array was fabricated by ultra-violet lithography from an inorganic-organic polymer, while creating a temperature gradient in the sample caused a linear refractive index increase due to the thermo-optic effect. Zener tunneling in onedimensional liquid crystal arrays was demonstrated by Fratallocchi, Assanto, Brzdakiewich and Karpierz (2006) and Fratalocchi and Assanto (2006a). A model allowing the description of one-dimensional and more
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general two-dimensional Zener tunneling in two-dimensional periodic photonic structures and calculation of the corresponding tunneling probabilities was derived by Shchesnovich, Cavalcanti, Hickmann and Kivshar (2006). Fratalocchi and Assanto (2007b) investigated nonlinear energy propagation in an optical lattice with slowly varying transverse perturbations. Symmetry breaking in such a system can provide a wealth of new phenomena, including nonreciprocal oscillations between Bloch bands and macroscopic self-trapping effects. A binary waveguide array composed of alternating thick and thin waveguides was introduced by Sukhorukov and Kivshar (2002a). In such a structure the effective refractive index experiences an additional transverse modulation. As a result, the existence of discrete gap solitons that possess the properties of both conventional discrete and Bragg grating solitons becomes possible. Such solitons and the effect of interband momentum exchange on soliton steering were observed experimentally in binary arrays fabricated in AlGaAs (Morandotti, Mandelik, Silberberg, Aitchison, Sorel, Christodoulides, Sukhorukov and Kivshar, 2004). The possibility of controlling the magnitude of dispersion experienced by a BEC wave-packet at the edges of spectral bands by modifying the shape of a double-periodic optical super-lattice was explored by Louis, Ostrovskaya and Kivshar (2005). It was shown that an extra periodicity opens up additional narrow stopgaps in the band-gap spectrum, while the effective dispersion at the edges of these mini-gaps can be varied within a much greater range than that accessible with a single-period lattice. Nonlinear light propagation and linear diffraction in disordered fiber arrays (or arrays with random spacing between waveguides) was investigated experimentally by Pertsch, Peschel, Kobelke, Schuster, ¨ Bartlet, Nolte, Tunnermann and Lederer (2004). It was shown that for high excitation power, diffusive spreading of a light beam in such arrays is arrested by the focusing nonlinearity, and formation of a two-dimensional discrete soliton is possible. The linear transmission and transport of discrete solitons in quasiperiodic waveguide arrays was studied by Sukhorukov (2006b) and it was shown that, under proper conditions, dramatic enhancement of soliton mobility in such arrays is possible.
3.4 Longitudinally Modulated Lattices and Waveguide Arrays Longitudinal modulation of the diffractive/nonlinear properties of the transmitting medium is a powerful tool for controlling the light beam parameters. This technique can be extended to the case of optical lattices and waveguide arrays. A model governing the propagation of an optical beam in a diffraction managed nonlinear waveguide array with a steplike diffraction map was introduced by Ablowitz and Musslimani (2001).
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This model allows the existence of discrete solitons whose width and peak amplitude evolve periodically. Its generalization to the case of vectorial interactions of two polarization modes propagating in a diffraction managed array was developed later by Ablowitz and Musslimani (2002). Interactions of diffraction managed solitons have been studied in detail. Pertsch, Peschel and Lederer (2003) considered soliton properties in inhomogeneous one-dimensional waveguide arrays in the framework of a discrete model. It was shown that longitudinal periodic modulations of the coupling strength may lead to soliton oscillations and decay. Different types of optically induced dynamical lattices were considered, including lattices with the simplest longitudinal amplitude modulation (Kartashov, Vysloukh and Torner, 2004), where amplification of soliton centre swinging is possible, or more complex three-wave lattices (Kartashov, Torner and Christodoulides, 2005) that are capable of dragging stationary solitons along predetermined paths (see also Section 6.6). Two-dimensional lattices evolving in a longitudinal direction can be produced with several nondiffracting Bessel beams and it has been shown that two-dimensional solitons may undergo spiraling motion with a controllable rotation rate in such lattices (Kartashov, Vysloukh and Torner, 2005c). Linear and nonlinear light propagation in a one-dimensional waveguide array with a periodically bent axis was studied by Longhi (2005). In the linear regime the analog of the Talbot self-imagining effect was predicted. The dynamical localization of light in linear periodically curved waveguide arrays was observed experimentally (Longhi, Marangoni, Lobino, Ramponi, Laporta, Cianci and Foglietti, 2006; Iyer, Aitchison, Wan, Dignam and Sterke, 2007). A suitable periodic waveguide bending is capable of suppressing discrete modulational instability of nonlinear Bloch waves. The dynamics of light in the presence of longitudinal defects of arbitrary extent in nonlinear waveguide arrays and uniform nonlinear media was studied by Fratalocchi and Assanto (2006b,d). Dynamical super-lattices also may be used for manipulation with solitons, as was suggested by Porter, Kevrekidis, Carretero-Gonzales and Frantzeskakis (2006), for the case of one-dimensional BECs. Some soliton modes supported by optical lattices with localized defects can be driven across the lattice by means of the transverse lattice shift, provided that this shift is performed with small enough steps (Brazhnyi, Konotop and Perez-Garcia, 2006a). It was predicted that Rabi-like oscillations and stimulated mode transitions occur with linear and nonlinear wave states in properly modulated waveguides and lattices (Kartashov, Vysloukh and Torner, 2007a). In particular, the possibilities of cascade stimulated transitions in systems supporting up to three modes of the same parity were shown. Such phenomena occur also in the nonlinear regime and
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in multimode systems, where the mode oscillation might be viewed as analogous to interband transitions.
4. OPTICALLY-INDUCED LATTICES The method of optical lattice induction is especially attractive because it allows creation of reconfigurable refractive index landscapes that can be fine-tuned by lattice-creating waves and easily erased, in contrast to permanent, technologically fabricated waveguiding structures. Periodic, optically induced lattices may operate on both weakly and strongly coupled regimes between neighboring lattice guides, depending on the intensities of waves inducing the lattice. This affords tunability of soliton behavior in such lattices from predominantly continuous to predominantly discrete, which is the core idea that we address in this review. The idea of optical lattice induction was put forward by Efremidis, Sears, Christodoulides, Fleischer and Segev (2002). For completeness, in this section we discuss typical parameters of optical lattices that can be made with this method. One or two-dimensional photoinduced lattices have to remain invariable along the propagation distance (typically, up to several centimeters). Experiments with a photorefractive SBN crystal take advantage of its strong electro-optic anisotropy. The lattice-writing beams are polarized in the ordinary direction, while the probe was polarized extraordinarily along the crystalline c-axis. By applying a static electric field across this axis, the probe would experience photorefractive screening nonlinearity, while the lattice beams propagate in linear regime (Fleischer, Carmon, Segev, Efremidis and Christodoulides, 2003), which enables an invariable lattice profile in the longitudinal direction to be obtained. A further benefit of this system is that the sign of the nonlinearity (focusing/defocusing) can be altered by changing the polarity of the externally applied voltage.
4.1 Optical Lattices Induced by Interference Patterns In the pioneering experiments, the periodic refractive index profile was photoinduced by interfering two ordinary polarized plane waves in a biased photorefractive SBN crystal (Fleischer, Carmon, Segev, Efremidis and Christodoulides, 2003). An external static electric field applied to the crystal creates periodic changes of the refractive index through the electro-optic effect. In SBN crystals orthogonally polarized waves feature dramatically different electro-optic coefficients (r33 ' 1340 pm/V, r13 ' 67 pm/V), so that ordinary-polarized interfering plane waves propagate almost linearly and create the stable one-dimensional periodic lattice, whereas the extraordinary polarized probe beam experiences strong
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nonlinear self-action, which can be described through the nonlinear change of the refractive index δn ∼ −n 3e r33 E 0 /(Idark + I0 cos2 (π x/d) + I ), where Idark characterizes dark irradiance level, I0 is the peak intensity of the lattice with the spatial period d, I is the intensity of the probe beam, and E 0 is the static electric field. The intersection angle between plane waves determines the lattice periodicity (d ∼ 10 µm is a representative value for experiments with optically induced lattices), while experimentally achievable refractive index modulation depth in such a lattice is ∼10−3 . The propagation direction of the lattice-creating waves and probe beam may not necessarily coincide. Laser beams launched into tilted lattices may experience anomalous refraction and experience transverse displacement, as demonstrated by Rosberg, Neshev, Sukhorukov, Kivshar and Krolikowski (2005). In such an experiment a periodic lattice was induced in a SBN:60 photorefractive crystal by interfering two ordinary polarized beams from a frequency-doubled Nd:YVO4 laser at 532 nm. Variation in bias voltage alters refractive index modulation depth and bandgap lattice structure, which, in turn, modifies the diffraction properties of Bloch waves and allows us to observe both positive or negative refraction of probe beams that selectively excite first or second spectral bands. Optical lattices might be also made partially incoherent and created by amplitude modulation rather than by coherent interference of multiple plane waves (Chen, Martin, Eugenieva, Xu and Bezryadina, 2004). Such lattices feature enhanced stability, and might propagate even in weakly nonlinear regimes, due to suppression of incoherent modulation instability. Optical induction also allows us to produce square (Fleischer, Segev, Efremidis and Christodoulides, 2003), hexagonal (Efremidis, Sears, Christodoulides, Fleischer and Segev, 2002), or triangular (Rosberg, Neshev, Sukhorukov, Krolikowski and Kivshar, 2007) two-dimensional photonic lattices by interfering four or three plane waves. A representative example of a lattice generated by four waves is shown in Figure 2. A second dimension brings fundamentally new features into light propagation dynamics in periodic environments and allows a wealth of new lattice geometries. Spatial light modulators allow us to generate a rich variety of optical potentials, including nondiffracting ones, that might be used, for example, for guiding or trapping of atoms, as was suggested by McGloin, Spalding, Melville, Sibbett and Dholakia (2003). Such modulators can be used as reconfigurable, dynamically controllable holograms and under proper conditions replace prefabricated micro-optical devices. For instance, current spatial light modulators operating in a phase-modulation regime offer 1024 × 1024 pixels with high diffraction efficiencies, which are
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FIGURE 2 Typical profile of a two-dimensional photonic lattice induced by the interference of four plane waves in highly anisotropic SBN:75 crystal. Each waveguide is approximately 7 µm in diameter, with an 11 µm spacing between nearest neighbors (Fleischer, Segev, Efremidis and Christodoulides, 2003)
comparable with etched holograms produced lithographically. The technique can be extended for optical lattice induction in photorefractive materials. A programmable phase modulator was used for engineering of specific Bloch states from the second band of a two-dimensional optical lattice (Tr¨ager, Fisher, Neshev, Sukhorukov, Denz, Krolikowski and Kivshar, 2006).
4.2 Nondiffracting Linear Beams Nondiffractive linear light beams offer very attractive opportunities for photoinduction of steady-state photonic lattices of diverse topologies. The properties of solitons supported by photonic lattices imprinted by complex nondiffracting Bessel and Mathieu beams will be discussed in Section 7. Here, we briefly describe the most representative features of nondiffracting beams in linear homogeneous and periodic media. Propagation of linear beams in uniform media is described by the three-dimensional Helmholtz equation that admits separation into the transverse and longitudinal parts only in Cartesian, circular cylindrical, elliptic cylindrical, and parabolic cylindrical coordinates. Each of these coordinate systems then gives rise to a certain type of nondiffracting beam associated with a fundamental solution of the Helmholtz equation in these coordinates. Details of experimental observation of parabolic, Bessel, and Mathieu beams can be found in papers by Durnin, Miceli and Eberly (1987), Gutierrez-Vega, Iturbe-Castillo, Ramirez, Tepichin, Rodriguez-Dagnino, Chavez-Cerda and New (2001) and Lopez-Mariscal, Bandres, Gutierrez-Vega and Chavez-Cerda (2005).
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A powerful and universal representation of the amplitude of any nondiffracting beam is provided by the reduced Whittaker integral: Q(ζ, η) =
Z
π
G(ϕ) exp [−ik (ζ cos ϕ + η sin ϕ)] dϕ,
(1)
−π
where G(ϕ) is the angular spectrum defined on the ring of radius k in frequency space. In the simplest case of Cartesian coordinates, PM superposition of M plane waves G(ϕ) = m=1 δ(ϕ − ϕm ) produces kaleidoscopic patterns (rectangular, honeycomb, etc). The angular spectrum G(ϕ) = exp(imϕ) produces m th order Bessel beams that represent fundamental nondiffracting solutions in a circular cylindrical coordinate system, while G(ϕ) = cem (ϕ; a)+isem (ϕ; a), with cem (ϕ; a) and sem (ϕ; a), being angular Mathieu functions, produces m th order Mathieu beams in an elliptic coordinate system. Parabolic nondiffracting optical fields were discussed by Bandres, Gutierrez-Vega and Chavez-Cerda (2004). Some representative examples of transverse intensity distributions in most known types of nondiffracting beams are shown in Figure 3. Using such beams in the technique of optical induction opens broad prospects for creation of the refractive index landscapes with novel types of symmetry. Besides this, nondiffracting beams find applications in diverse areas of physics, such as optical manipulation of small particles (McGloin and Dholakia, 2005), frequency doubling, or atom trapping. Nondiffracting linear beams may also form in periodic media. Such beams were considered in two-dimensional periodic lattices (Manela, Segev and Christodoulides, 2005). The links between the symmetry properties, phase structure of such beams and the number of spectral bands that gives rise to the beam have been established. The possibility of diffraction-free propagation of localized light beams in materials with both transverse and longitudinal modulation of refractive index was predicted (Staliunas and Herrero, 2006; Staliunas, Herrero and de Valcarcel, 2006; Staliunas and Masoller, 2006). In such materials the dominating (second) order of diffraction may vanish and thus the overall diffraction broadening of the beam is determined by the fourth-order dephasing of spatial Fourier modes.
4.3 Mathematical Models of Wave Propagation Several mathematical models are commonly accepted for description of soliton evolution in optical lattices. In the case of optical solitons all of them are based on the parabolic propagation equation for the slowly varying amplitude of light fields coupled to the material equation describing the corrections to the refractive index. Thus, in the simplest
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FIGURE 3 Lattices induced by the nondiffracting beams of different type: (a) zero-order radially symmetric Bessel lattice; (b) first-order radially symmetric Bessel lattice; (c) lattice produced by the azimuthally modulated Bessel beam of third order; odd Mathieu lattices with small (d) and large (e) interfocal parameters that closely resemble azimuthally modulated Bessel and quasi-one-dimensional periodic lattices, correspondingly; (f) lattice produced by the combination of odd and even parabolic beams
case of cubic nonlinear media their combination results in the canonical ¨ nonlinear Schrodinger equation describing the evolution of optical wave packets in the presence of the lattice, which in the case of one transverse dimension reads: i
1 ∂ 2q ∂q =− + σ |q|2 q − p R(η)q. ∂ξ 2 ∂η2 −1/2
(2)
Here q = (L dif /L nl )1/2 AI0 is the dimensionless complex amplitude of the light field; A is the slowly varying envelope; I0 is the input intensity; the transverse η and longitudinal ξ coordinates are normalized to the beam width r0 and the diffraction length L dif = n 0 ωr02 /c, respectively; ω is the carrying frequency; n 0 is the unperturbed refractive index; L nl = 2c/ωn 2 I0 ; σ = −1 for focusing nonlinearity and σ = +1 in the case of defocusing nonlinearity; p = L dif /L ref is the guiding parameter; L ref = c/δnω; δn is the refractive index modulation depth, while the function
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R(η) describes the transverse refractive index profile. In the particular case of optically induced or technologically fabricated harmonic refractive modulation one can set R(η) = cos(η), where is the lattice frequency. The generalization of the evolution equation (2) to the case of two transverse dimensions requires replacing ∂ 2 /∂η2 by the two-dimensional transverse Laplacian ∂ 2 /∂η2 + ∂ 2 /∂ζ 2 . Note the analogy between the equations describing propagation of optical wave-packets and matter waves. Thus, Equation (2) also describes dynamics of one-dimensional Bose–Einstein condensate confined in an optical lattice generated by a standing light wave of wavelength λ. In this case, q stands for the dimensionless mean-field wave function, the variable ξ stands for time in units of τ = 2mλ2 /π h, with m being the mass of the atoms and h the Planck’s constant, η is the coordinate along the axis of the quasi-one-dimensional condensate expressed in units of λπ −1 . The parameter p is proportional to the lattice depth E 0 expressed in units of recoil energy E rec = h 2 /2mλ2 . In quasi-one-dimensional condensates one has σ = 2λas Na /π `2 , where as is the s-wave scattering length, Na is the number of atoms, and ` is the harmonic oscillator length. The evolution equations typically admit of several conserved quantities. Thus, Equation (2) conserves the total energy flow U and the Hamiltonian H : ∞
Z
|q|2 dη,
U= −∞
1 H= 2
Z
(3)
∞
(|∂q/∂η| − 2 p R |q| + σ |q| )dη. 2
2
4
−∞
The method of creation of periodic lattices (especially two-dimensional ones) in nonlinear crystals generally relies on an optical induction technique (see Section 3). With this technique an optical lattice can be imprinted in a photosensitive crystal, for example, by interfering two or more plane waves. The interference pattern of several plane waves is a propagation invariant pattern and should not be affected by the nonlinearity of the crystal. At the same time the soliton beam released into the lattice should experience strong nonlinearity. Such conditions can be achieved in materials with a strong anisotropy of nonlinear response, e.g., in suitable photorefractive crystals. In this case, the lattice-creating waves and the soliton beam are polarized in orthogonal directions, so that nonlinearity affects only the soliton beam. In this configuration the lattice itself is not affected by the soliton beam, while the soliton beam does feel the periodic potential.
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Since nonlinearity saturation is inherent for the photorefractive materials, the model equation i
∂q 1 ∂ 2q Eq =− − [S |q|2 + R(η)] 2 ∂ξ 2 ∂η 1 + S |q|2 + R(η)
(4)
is also frequently used for description of solitons in photorefractive optical −1/2 lattices. Here q = AI0 , I0 is the input intensity; the normalization for transverse and longitudinal coordinate is similar to that for Equation (2); S = I0 /(Idark + Ibg ) is the saturation parameter; Idark and Ibg are dark and background radiation intensities; R = Ilatt /(Idark + Ibg ), where Ilatt represents the intensity distribution in the lattice-creating wave; E = (1/2)(ωr0 n 20 /c)2reff E 0 is the dimensionless static biasing field applied to the crystal; reff is the effective electro-optic coefficient corresponding to polarization of the soliton beam. Under the assumptions S |q|2 , R(η, ξ ) 1, E 1, the Equation (4) can be transformed into the cubic nonlinear ¨ Schrodinger equation (2). It is important to stress that the depth of optically induced lattices can be tuned by changing the intensities of the lattice-creating waves. The model (4), generalized to the case of two transverse dimensions by replacing the transverse Laplacian with its two-dimensional counterpart ∂ 2 /∂η2 + ∂ 2 /∂ζ 2 , is also frequently used in the literature, though it describes thestrongly anisotropic and nonlocal response of such crystals only approximately (see Section 9 where fully anisotropic model of photorefractive response is discussed). Trivial-phase stationary solutions of Equation (2) or (4) can be obtained in the form q(η, ξ ) = w(η) exp(ibξ ), where w(η) is a real function describing the transverse profile, and b is a real propagation constant. Thus, mathematically, families of lattice solitons are defined by the propagation constant b, lattice depth p, and particular lattice shape given by the function R(η). Various families of soliton solutions can be obtained from a known family by using scaling transformations, which, in the case of Equation (2), are q(η, ξ, p) → χq(χ η, χ 2 ξ, χ 2 p), where χ is the arbitrary scaling factor. The equation for soliton profiles obtained upon substitution of the light field in such form into the evolution equation can be solved numerically with standard relaxation or spectral methods. Analogously, a split-step fast-Fourier method may be used to solve the very evolution equations with a variety of input conditions. Stability of solitons can be tested by searching for perturbed solutions in the form q = (w + u + iv) exp(ibξ ), where u(η, ξ ) and v(η, ξ ) are real and imaginary parts of perturbation, that can grow with a rate δ upon propagation. Substitution of the light field in such form into Equation (2), linearization around a stationary solution w and derivation of real and
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imaginary parts yields the following linear eigenvalue problem 1 ∂ 2v + bv + σ w2 v − p Rv, 2 ∂η2 1 ∂ 2u −δv = − + bu + 3σ w 2 u − p Ru, 2 ∂η2
δu = −
(5)
that can be solved numerically with standard linear eigenvalue solvers. The presence of perturbations corresponding to Re δ 6= 0 indicates an instability of the soliton solutions, while Re δ ≡ 0 indicates linear stability. The results of linear stability analysis may additionally be tested by direct numerical propagation of the soliton solutions perturbed with broadband input noise. The generalization of these techniques to the case of two transverse dimensions is straightforward.
5. ONE-DIMENSIONAL LATTICE SOLITONS In this section we describe the properties of scalar and vector solitons supported by one-dimensional lattices and briefly discuss the possibilities for control and manipulation of the soliton shape and width, its internal structure, and transverse mobility afforded by static and dynamical optical lattices with tunable parameters, imprinted in the materials with local and nonlocal nonlinear responses. We discuss the role of the main factors that can lead to stabilization of complex lattice soliton structures. Notice that the state of art in the field of spatial optical solitons in uniform nonlinear materials has been summarized in several books (Akhmediev and Ankiewicz, 1997; Trillo and Torruellas, 2001; Kivshar and Agrawal, 2003) and reviews (Stegeman and Segev, 1999; Kivshar and Pelinovsky, 2000; Torner, 1998; Etrich, Lederer, Malomed, Peschel and Peschel, 2000; Buryak, Di Trapani, Skryabin and Trillo, 2002; Kivshar and Luther-Davies, 1998; Malomed, Mihalache, Wise and Torner, 2005; Conti and Assanto, 2004). Lattice solitons are continuous counterparts of discrete solitons existing in waveguide arrays (for a recent reviews on discrete solitons see Aubry (1997), Flach and Willis (1998), Kevrekidis, Rasmussen and Bishop (2001), Christodoulides, Lederer and Silberberg (2003) and Aubry (2006)). The discrete NLSE that is used to describe the evolution of nonlinear excitations in such systems has a rather universal character and can be used to describe light propagation even in tensorial systems and in the presence of nonparaxial and vectorial effects (Fratalocchi and Assanto, 2007c). As mentioned above, mathematically, the equations describing propagation of laser radiation in periodic media are analogous to those describing evolution of Bose–Einstein condensates
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in optical lattices, hence, many soliton phenomena predicted in nonlinear optics were encountered in BECs and vice versa (Dalfovo, Giorgini, Pitaevskii and Stringari, 1999; Pitaevskii and Stringari, 2003; Abdullaev, Gammal, Kamchatnov and Tomio, 2005; Morsch and Oberthaler, 2006).
5.1 Fundamental Solitons Lattice solitons form due to the balance of diffraction, refraction in the periodic lattice, and nonlinear self-phase-modulation. The existence of such solitons is closely linked to the band-gap structure of a onedimensional periodic lattice spectrum as depicted in Figure 1. Since the spectrum is composed of bands where only Bloch waves can propagate, and gaps where propagation of Bloch waves is forbidden, lattice solitons emerge as defect modes in the gaps of the lattice spectrum. The band-gap structure depends crucially on the lattice depth p, but in one-dimensional it always includes a single semi-infinite gap and an unlimited number of finite gaps. The internal soliton structure is determined by the position of the soliton propagation constant inside the corresponding gap. In focusing media, the simplest fundamental (odd) solitons are formed in a semi-infinite gap (see Figure 4(a) for an illustrative example of a soliton profile in the lattice R(η) = cos(η) with = 4 and focusing Kerr nonlinearity); the position of the intensity maximum for such a soliton coincides with one of local lattice maximums. The properties of odd and other types of lattice solitons have been analyzed in different physical settings, including photorefractive optical lattices and BECs (Efremidis, Sears, Christodoulides, Fleischer and Segev, 2002; Louis, Ostrovskaya, Savage and Kivshar, 2003; Efremidis and Christodoulides, 2003; Kartashov, Vysloukh and Torner, 2004a). The monotonic increase of energy flow with propagation constant indicates stability of odd lattice solitons in accordance with the VakhitovKolokolov stability criterion (Vakhitov and Kolokolov, 1973). Solitons existing near the gap edges transform into Bloch waves with the same symmetry. Besides odd solitons, optical lattices also support even solitons centred between the neighboring maxima of R(η) (Figure 4(b)). In Kerr nonlinear media even solitons are unstable, while strong nonlinearity saturation may result in stabilization of even solitons accompanied by destabilization of odd ones (Kartashov, Vysloukh and Torner, 2004a). Experimentally odd solitons in photorefractive lattices were observed by Fleischer, Carmon, Segev, Efremidis and Christodoulides (2003) and by Neshev, Ostrovskaya, Kivshar and Krolikowski (2003). Figure 5 illustrates the experimental generation of odd and even lattice solitons. Increasing the optical lattice depth inhibits localization of light in the vicinity of local lattice maxima, a process accompanied by the development of strong modulation of soliton profiles. Under such
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FIGURE 4 Odd (a), even (b), and twisted (c) solitons originating from semi-infinite gaps of optical lattice with focusing nonlinearity at b = 1.4. Odd (d), even (e), and twisted (f) gap solitons originating from the first finite gap of an optical lattice with defocusing nonlinearity at b = −0.5. In all cases lattice depth p = 4
conditions, one can employ the so-called tight-binding approximation to reduce Equation (2) to a discrete NLSE obtained by Christodoulides and Joseph (1988) for arrays of evanescently coupled waveguides, where discrete solitons were observed by Eisenberg, Silberberg, Morandotti, Boyd and Aitchison (1998). Discrete matter-wave solitons were introduced by Trombettoni and Smerzi (2001). Dark discrete solitons in one-dimensional arrays with defocusing ¨ Stepic, Kip and nonlinearity have been also investigated (Smirnov, Ruter, Shandarov, 2006). Localized nonlinear dark modes displaying a phase jump in the centre located either on-channel or in-between channels were observed, and the ability of the induced refractive index structures to guide light of a low-power probe beam was demonstrated. The interaction of a probe beam with dark and bright blocker solitons was studied ¨ Stepic, Shandarov in counter-propagating geometry by Smirnov, Ruter, and Kip (2006). Hadzievski, Maluckov and Stepic (2007) presented a numerical analysis of the dynamics of dark breathers in lattices with saturable nonlinearity. Comparison of discrete dark solitons properties in lattices with saturable and Kerr nonlinearities was performed by Fitrakis, Kevrekidis, Susanto and Frantzeskakis (2007). Dark matter-wave solitons in optical lattices were discussed by Louis, Ostrovskaya and Kivshar (2004). The properties of dark solitons in dynamical lattices with cubic-
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FIGURE 5 Experimental demonstration of odd and even spatial solitons. (a) Input beam and optical lattice (power 23 µW). (b) Output probe beam at low power (2 × 10−3 µW). (c) Localized states (87 × 10−3 µW). Left, propagation in the absence of the grating; middle, even excitation; right, odd excitation. Biasing electric field 3600 V/cm (Neshev, Ostrovskaya, Kivshar and Krolikowski, 2003)
quintic nonlinearity have been addressed by Maluckov, Hadzievski and Malomed (2007).
5.2 Beam Shaping and Mobility Control Optical lattices offer rich opportunities for soliton control, by varying the lattice depth and period. In a lattice, at a given energy flow, the field amplitude necessary for soliton-like propagation amounts to lower values than that in homogeneous media. The width of the lattice soliton also changes with increase of the lattice depth. More importantly,
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the transverse refractive index modulation profoundly affects soliton mobility. The simple effective particle approach (Scharf and Bishop, 1993; Kartashov, Zelenina, Torner and Vysloukh, 2004), based on the equation d2 p hηi = U dξ 2
Z
∞
−∞
|q|2
dR dη dη
(6)
R∞ for the integral soliton centre hηi = U −1 −∞ |q|2 ηdη, might be used to identify regimes of soliton propagation in the lattice. This approach requires substitution of a trial function for the light field, that can be chosen in the form q = q0 sech[χ (η −hηi)] exp[iα(η −hηi)], with q0 , χ, and α being amplitude, inverse width, and incident angle, respectively. At small incident angles Equation (6) predicts periodic oscillations of the soliton centre inside the input channel. When the input angle exceeds a critical value αcr = 2[ p(π/2χ ) sinh−1 (π/2χ )]1/2 (associated with the height of the Peierls–Nabarro barrier created by the periodic potential (Kivshar and Campbell, 1993)) the soliton leaves the input channel and starts traveling across the lattice, a process accompanied by radiative losses (Yulin, Skryabin and Russell, 2003). The radiation may cause soliton trapping in one of the lattice channels. The controllable trapping in different lattice locations may find practical applications and has been studied in a variety of periodic systems, including discrete waveguide arrays (Aceves, De Angelis, Trillo and Wabnitz, 1994; Krolikowski, Trutschel, CroninGolomb and Schmidt-Hattenberger, 1994; Aceves, De Angelis, Peschel, Muschall, Lederer, Trillo and Wabnitz, 1996; Krolikowski and Kivshar, 1996; Bang and Miller, 1996; Morandotti, Peschel, Aitchison, Eisenberg and Silberberg, 1999a; Pertsch, Zentgraf, Peschel, Brauer and Lederer, 2002a; Vicencio, Molina and Kivshar, 2003). The transverse mobility of lattice solitons can be substantially enhanced in saturable media in the regime of strong saturation (Hadzievski, Maluckov, Stepic and Kip, 2004; Stepic, Kip, Hadzievski and Maluckov, 2004; Melvin, Champneys, Kevrekidis and Cuevas, 2006; Oxtoby and Barashenkov, 2007), an effect related to the stability exchange between even and odd solitons. Due to radiation emission by the moving solitons, soliton collisions in such systems are strongly inelastic (Aceves, De Angelis, Peschel, Muschall, Lederer, Trillo and Wabnitz, 1996; Meier, Stegeman, Silberberg, Morandotti and Aitchison, 2004; Meier, Stegeman, Christodoulides, Silberberg, Morandotti, Yang, Salamo, Sorel and Aitchison, 2005; Cuevas and Eilbeck, 2006).
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5.3 Multipole Solitons and Soliton Trains Optical lattices can support scalar multipole solitons composed of several out-of-phase bright spots, when repulsive “forces” acting between spots in a bulk medium are compensated by the immobilizing action of the lattice (Figure 4(c)). In discrete waveguide arrays such solitons are known as twisted strongly localized modes (Darmanyan, Kobyakov and Lederer, 1998; Kevrekidis, Bishop and Rasmussen, 2001). The important feature of multipole solitons in continuous lattices is that they become completely stable when their energy flow exceeds a certain threshold (Efremidis, Sears, Christodoulides, Fleischer and Segev, 2002; Louis, Ostrovskaya, Savage and Kivshar, 2003; Efremidis and Christodoulides, 2003; Kartashov, Vysloukh and Torner, 2004a). Such stabilization takes place for multipole solitons of arbitrary higher order (soliton trains), containing multiple spots, provided that the phase alternates by π (in the case of focusing medium) between each two neighboring spots. This immediately suggests the important possibility of constructing and manipulating multi-peaked “soliton packets” beyond single “soliton bits”, a feature that might open a new door in all optical-switching schemes. Notice that individual solitons can then be extracted or added into such trains. The simplest dipole solitons in optical lattices were observed by Neshev, Ostrovskaya, Kivshar and Krolikowski (2003). He and Wang (2006) addressed specific dipole solitons residing completely in a single channel of a low-frequency optical lattice. Individual solitons and soliton trains have been also studied in lattices with competing cubic-quintic nonlinearities, where bistability was encountered for several soliton families (Merhasin, Gisin, Driben and Malomed, 2005; Wang, Ye, Dong, Cai and Li, 2005), as well as in the pure quintic case (Alfimov, Konotop and Pacciani, 2007). Periodic lattices also support truly infinite periodic waves that were studied in the context of BEC (for an overview, see Deconinck, Frigyik and Kutz (2002)).
5.4 Gap Solitons The finite gaps of the Floquet–Bloch spectrum of the periodic lattice may also give rise to solitons that are usually termed gap solitons. In contrast to solitons emerging from the semi-infinite gap (or total internal reflection gap), solitons from finite gaps are possible due to Bragg scattering from the periodic structure and exist in conditions of strong coupling between modes having opposite transverse wavevector components. Gap solitons exhibit multiple amplitude oscillations (Figure 4(d)), while their internal symmetry depends on the particular gap that supports them. Gap-type excitations in discrete waveguide arrays were analyzed by Kivshar (1993). One-dimensional gap solitons in continuous optically
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induced lattices were observed in photorefractive crystals by Fleischer, Carmon, Segev, Efremidis and Christodoulides (2003). Controlled generation and steering of gap solitons in optically induced lattices was reported by Neshev, Sukhorukov, Hanna, Krolikowski and Kivshar (2004). Matter-wave gap solitons in Bose–Einstein condensates with repulsive inter-atomic interactions were observed by Eiermann, Anker, Albiez, Taglieber, Treutlein, Marzlin and Oberthaler (2004). Gap solitons may also form in arrays of evanescently coupled waveguides fabricated in AlGaAs (Mandelik, Morandotti, Aitchison and Silberberg, 2004) and ¨ Runde, Kip, Shandarov, Manela and Segev, LiNbO (Chen, Stepic, Ruter, 2005; Matuszewski, Rosberg, Neshev, Sukhorukov, Mitchell, Trippenbach, Austin, Krolikowski and Kivshar, 2005). The steering properties of gap solitons strongly depend on the sign and magnitude of the spatial group-velocity dispersion near the corresponding gap edge and may be anomalous, i.e. solitons may move in the direction opposite to the input tilt. Similarly to solitons forming in a semi-infinite gap, solitons from finite gaps may form trains. In defocusing media, trains of in-phase gap solitons may be stable (Figure 4(e)), while twisted modes (Figure 4(f)) undergo strong exponential instabilities. Trains of gap solitons have been predicted in one-, two-, and threedimensional optical lattices (Kartashov, Vysloukh and Torner, 2004a; Alexander, Ostrovskaya and Kivshar, 2006). Experimental observation of higher-order one-dimensional gap solitons in defocusing waveguide ¨ arrays was presented by Smirnov, Ruter, Kip, Kartashov and Torner (2007). Gap soliton collisions are determined mostly by soliton localization. While broad low-energy solitons emerging close to the upper gap edge may interact almost elastically, the collisions of high-power gap solitons is inelastic and may lead to soliton fusion (Dabrowska, Osrovskaya and Kivshar, 2004; Malomed, Mayteevarunyoo, Ostrovskaya and Kivshar, 2005). The interaction between two well-localized parallel solitons in one-dimensional discrete saturable systems has been investigated using defocusing lithium niobate nonlinear waveguide arrays by Stepic, ¨ ¨ Smirnov, Ruter, Pronneke, Kip and Shandarov (2006). Gap solitons have been studied in more complicated but still periodic systems such as binary waveguide arrays (Sukhorukov and Kivshar, 2002a) or quasiperiodic optical lattices (Sakaguchi and Malomed, 2006). Gap solitons are typically stable in the middle of the gap, but near the band edge they feature specific oscillatory instabilities similar to those observed in fiber Bragg gratings (Barashenkov, Pelinovsky and Zemlyanaya, 1998). Weak instabilities of gap solitons arise due to resonant energy redistributions between different gaps, while
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perturbation eigenmodes associated with such instability are poorly localized (Pelinovsky, Sukhorukov and Kivshar, 2004). Motzek, Sukhorukov and Kivshar (2006) studied the dynamical reshaping of polychromatic beams in periodic and semi-infinite photonic lattices and the formation of polychromatic gap solitons. The propagation of polychromatic light in nonlinear photonic lattices has been reviewed by Sukhorukov, Neshev and Kivshar (2007). In particular, the observation of localization of supercontinuum radiation in waveguide arrays imprinted in LiNbO3 crystals and the possibilities for dynamical control of the output spectrum were reported by Neshev, Sukhorukov, Dreischuh, Fischer, Ha, Bolger, Bui, Krolikowski, Eggleton, Mitchell, Austin and Kivshar (2007).
5.5 Vector Solitons Vectorial interactions (cross-phase modulation) between several light fields may significantly enrich the dynamics of soliton propagation in the periodic structure. The propagation of two mutually incoherent beams in the lattice with focusing (σ = −1) or defocusing (σ = +1) Kerr nonlinearity can be described by the system of equations: i
∂q1,2 1 ∂ 2 q1,2 =− + σ q1,2 (|q1 |2 + |q2 |2 ) − p R(η)q1,2 . ∂ξ 2 ∂η2
(7)
The model (7) can be enlarged to take into account coherent interaction between orthogonally polarized beams in birefringent media, when fourwave-mixing is taken into account. Vectorial coupling between strongly localized modes in discrete waveguide arrays generates new soliton families having no counterparts in the scalar case (Darmanyan, Kobyakov, Schmidt and Lederer, 1998; Ablowitz and Musslimani, 2002). The simplest vector solitons were observed in AlGaAs arrays in the presence of fourwave-mixing (Meier, Hudock, Christodoulides, Stegeman, Silberberg, Morandotti and Aitchison, 2003). In lattices, vectorial interaction between the solitons emerging from the same gap stabilizes beams that are unstable when propagating alone. Thus, even solitons in focusing media are stabilized when coupled with twisted mode, and vice-versa, in defocusing media in-phase trains stabilize their twisted counterparts (Kartashov, Zelenina, Vysloukh and Torner, 2004). Localization of two-component Bose–Einstein condensates and formation of bright-bright and dark-bright matter-wave solitons in optical lattices was considered by Ostrovskaya and Kivshar (2004a). Coupling between solitons emerging from the neighboring gaps of a lattice spectrum may also lead to formation of composite vector states as predicted by Cohen, Schwartz, Fleischer, Segev and Christodoulides (2003) and Sukhorukov and Kivshar (2003). Importantly,
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one of the components in such vector states is always unstable alone due to its symmetry dictated by the gap number. Transient interband mutual focusing and defocusing of waves exhibiting diffraction of different magnitude and sign was observed by Rosberg, Hanna, Neshev, Sukhorukov, Krolikowski and Kivshar (2005) in optical lattices. The coherent interactions between two beams closely resembling profiles of Bloch waves from two neighboring bands also results in formation of nonlinear localized excitations or breathers that experience beating upon propagation in the array (Mandelik, Eisenberg, Silberberg, Morandotti and Aitchison, 2003b). The impact of nonlinearity saturation on the properties of vector solitons in discrete waveguide arrays was studied by Fitrakis, Kevrekidis, Malomed and Frantzeskakis (2006), while experimental observation of vector solitons in saturable discrete ¨ waveguide arrays was performed by Vicencio, Smirnov, Ruter, Kip and Stepic (2007).
5.6 Soliton Steering in Dynamical Lattices All lattices considered so far were invariant along the direction of propagation. Variation of the lattice shape in the longitudinal direction opens a wealth of new opportunities for soliton control. Several types of dynamical lattices were considered in the literature, including lattices with monotonic or periodic variation of depth and width of guiding channels as well as lattices with periodically curved channels. In particular, a longitudinal variation of coupling strength in a modulated waveguide array can resonantly excite internal modes of discrete solitons, which may lead to soliton breathing, splitting, or transverse motion, depending on the symmetry of the excited mode (Peschel and Lederer, 2002; Pertsch, Peschel and Lederer, 2003). Self-imaging at periodic planes may occur in arrays of periodically curved waveguides, while modulational instability is inhibited under conditions of self-imaging (Longhi, 2005). Periodic waveguide arrays built of several tilted segments support breathing discrete diffraction managed scalar and vector solitons, reproducing their shapes on each period of the structure (Ablowitz and Musslimani, 2001, 2002). A longitudinal modulation causes parametric amplification of soliton swinging inside the lattice channel that may be used to detect submicron soliton displacements (Kartashov, Vysloukh and Torner, 2004). Modulated lattices may be used for stopping and trapping moving solitons, while initially stationary solitons can be transferred to any prescribed position by a moving lattice (Kevrekidis, Frantzeskakis, Carretero-Gonzalez, Malomed, Herring and Bishop, 2005; Dabrowska, Osrovskaya and Kivshar, 2006). Gap solitons undergo abrupt delocalization and compression in an optical lattice with varying strength (Baizakov and Salerno, 2004).
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An interesting opportunity to control output soliton position was encountered in dynamical optical lattices that exhibit a finite momentum in the transverse plane. Such lattices can be induced by three plane waves A exp(±iαη) exp(−iα 2 ξ/2), B exp(iβη) exp(−iβ 2 ξ/2), where a, b are wave amplitudes and ±α, β are propagation angles, so that the lattice profile is given by R(η, ξ ) = 4A2 cos2 (αη) + B 2 + 4AB cos(αη) cos[βη + (α 2 − β 2 )ξ/2]. In this case the transverse momentum can be transferred from the lattice to the soliton beam, thus leading to a controllable drift (Kartashov, Torner and Christodoulides, 2005; Garanovich, Sukhorukov and Kivshar, 2005). Perturbation theory, based on the inverse scattering transform, might be used to calculate the tilt φ (or instantaneous propagation angle) of the soliton beam q(η, ξ ) = χ sech[χ (η −φξ )] exp[iφη −i(χ 2 −φ 2 )ξ/2] in a threewave lattice. One gets φ=
β +α β −α − (β + α) sinh[π(β − α)/2χ] (β − α) sinh[π(β + α)/2χ] ! β 2 − α2 1 − cos ξ , (8) 2
2π AB χ ×
Thus, the drift grows linearly with the amplitude B of the third plane wave. All-optical beam steering in modulated photonic lattices induced by three-wave interference was observed experimentally by Rosberg, Garanovich, Sukhorukov, Neshev, Krolikowski and Kivshar (2006) and is illustrated in Figure 6. Soliton steering and fission in optical lattices that fade away exponentially along the propagation direction was considered by Kartashov, Vysloukh and Torner (2006b). Garanovich, Sukhorukov and Kivshar (2007) discussed the phenomenon of nonlinear light diffusion in periodically curved arrays of optical waveguides. Soliton steering by a single continuous wave that dynamically induces a photonic lattice, as well as interactions of several solitons in the presence of such waves were addressed by Kominis and Hizanidis (2004) and Tsopelas, Kominis and Hizanidis (2006).
5.7 Lattice Solitons in Nonlocal Nonlinear Media Under appropriate conditions the nonlinear response of materials can be highly nonlocal, a phenomenon that drastically affects the propagation of light. New effects attributed to nonlocality were encountered, e.g., in photorefractive media, liquid crystals and thermo-optical media (see Krolikowski, Bang, Nikolov, Neshev, Wyller, Rasmussen and Edmundson (2004) for a review). Nonlocality of the nonlinear response also strongly affects properties of lattice solitons. Propagation of a laser beam in nonlocal media with an imprinted transverse refractive index modulation
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FIGURE 6 (a) Experimental and theoretical (inset) linear output in a straight lattice. (b) Shift of the nonlinear probe beam output versus the modulating beam power for k3x = 1.18k12x and ϕ/2π = 0.22, where k3x and k12x stand for the transverse wavenumbers of the third and first two waves, ϕ is the phase difference between the third wave and the other two waves. (c),(d) Experimental and theoretical (inset) nonlinear output in straight and modulated lattice for intensity of the third wave I3 = 0 and I3 = 4I12 , respectively, and k3x = 1.18k12x . (e),(f) Numerical simulations of the longitudinal propagation (Rosberg, Garanovich, Sukhorukov, Neshev, Krolikowski and Kivshar, 2006)
can be described by the equation: i
∂q 1 ∂ 2q =− −q ∂ξ 2 ∂η2
Z
∞
G(η − λ) |q(λ)|2 dλ − p R(η)q,
(9)
−∞
where G(η) is the response function of the nonlocal medium. In the limit G(η) → δ(η) one recovers the case of local response, while a
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strongly nonlocal medium is described by slowly decaying kernels G(η). Symmetric strongly nonlocal response reduces symmetry-breaking instabilities of lattice solitons, including even solitons in focusing and twisted gap modes in defocusing media. The Peierls–Nabarro potential barrier for a soliton moving across the lattice is reduced, due to nonlocality, an effect that results in the corresponding enhancement of the transverse soliton mobility (Xu, Kartashov and Torner, 2005a). By tuning the lattice depth one can control the mobility of lattice solitons in materials with asymmetric nonlocal response (Kartashov, Vysloukh and Torner, 2004b; Xu, Kartashov and Torner, 2006). Liquid crystals belong to the rare class of nonlinear materials, simultaneously featuring intrinsically nonlocal nonlinearity and allowing for formation of tunable refractive index landscapes (Peccianti, Conti, Assanto, De Luca and Umeton, 2004; Fratalocchi, Assanto, Brzdakiewicz and Karpierz, 2004; Fratalocchi and Assanto, 2005). Figure 7 shows the setup and experimentally observed lattice solitons in a periodically biased nematic liquid crystal. The lattice depth is tuned by varying the voltage applied to periodically spaced electrodes on the liquid crystal cell. A similar technique was used for demonstration of multiband vector breathers (Fratalocchi, Assanto, Brzdakiewicz and Karpierz, 2005a). The experimental observation of power-dependent beam steering over angles of several degrees in a voltage-controlled array of channel waveguides defined in a nematic liquid crystal via electrooptic reorientational response was observed by Fratalocchi, Assanto, Brzdakiewicz, and Karpierz (2005b). Such steering was demonstrated for light beams at mW power levels. An overview of the latest experimental and theoretical advances in investigation of discrete light propagation and self-trapping in nematic liquid crystals is given in Fratalocchi, Brzdakiewicz, Karpierz, and Assanto (2005).
6. TWO-DIMENSIONAL LATTICE SOLITONS In this section we describe the properties of solitons supported by twodimensional optical lattices. By and large, adding the second transverse coordinate profoundly affects properties of self-sustained nonlinear excitations. In particular, two dimensional solitons in uniform Kerr nonlinear media may experience catastrophic collapse in contrast to their one-dimensional counterparts (Berge, 1998). Optical lattices typically play a strong stabilizing action on two-dimensional solitons and allow for the existence of new types of solitons with unusual internal structures. Here, we describe the conditions required for existence of stable twodimensional lattice solitons. Two-dimensional settings also give rise to vortex solitons that are impossible in lower-dimensional settings. The
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FIGURE 7 (Top) Cell for liquid-crystal waveguide array: 3, array period, d, cell thickness, V , applied voltage, ITO, indium tin oxide electrodes. (Bottom) Experimental results with light propagation at 1064 nm wavelength: (a) one-dimensional diffraction at V = 0 V, (b) low-power discrete diffraction at V = 1.2 V and 20 mW input power, (c) discrete nematicon at V = 1.2 V and 35 mW input power. The residual coupling and transverse-longitudinal power fluctuations shown are due partly to the limits of the camera and partly to nonuniformities in the sample (Fratalocchi, Assanto, Brzdakiewicz and Karpierz, 2004)
properties of optical and matter-wave vortices are described in detail in a review by Desyatnikov, Torner and Kivshar (2005). In this section we address the existence and specific features of vortex solitons in periodic media. Various types of two-dimensional soliton have been studied in discrete nonlinear systems (for reviews see Aubry (1997), Flach and Willis (1998), Hennig and Tsironis (1999), Kevrekidis, Rasmussen and Bishop (2001), Christodoulides, Lederer and Silberberg (2003), Aubry (2006); as well as the focus issues Physica D 119, 1 (1998) and Physica D 216, 1 (2006) devoted to discrete solitons in nonlinear lattices). Recent developments in the field of two-dimensional matter-wave lattice solitons are summarized by Kevrekidis, Carretero-Gonzalez, Frantzeskakis and Kevrekidis (2004), Morsch and Oberthaler (2006).
6.1 Fundamental Solitons As in the case of one-dimensional solitons addressed in Section 5, the properties of two-dimensional solitons in optical lattices are dictated
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by the band-gap structure of the lattice spectrum. However, in clear contrast with the one-dimensional case, bands in the spectrum of a twodimensional lattice can strongly overlap, thereby considerably restricting the number of gaps in the spectrum or even completely eliminating all finite gaps. This is the main property that distinguishes two-dimensional lattices from their one-dimensional counterparts possessing an infinite number of finite gaps. The number of finite gaps in the spectrum of a twodimensional lattice is dictated by the particular lattice shape, depth, and period. The simplest two-dimensional lattice solitons in focusing media emerge from the semi-infinite gap (Efremidis, Sears, Christodoulides, Fleischer and Segev, 2002; Yang and Musslimani, 2003; Efremidis, Hudock, Christodoulides, Fleischer, Cohen and Segev, 2003; Baizakov, Malomed and Salerno, 2003). The absolute intensity maximum for such solitons coincides with one of the local lattice maxima. In Kerr media, an increase of the soliton amplitude at a fixed lattice depth is accompanied by a gradual energy flow concentration within the single lattice channel, while the soliton profile approaches that of an unstable Townes soliton. When b approaches the lower edge of the semi-infinite gap, the soliton amplitude decreases, a process that is accompanied by soliton expansion over many lattice periods (transition into the corresponding Bloch wave). In Kerr nonlinear media periodic refractive index modulation results in collapse suppression and stabilization of the fundamental soliton, almost in the entire domain of its existence, except for the narrow region near the lower edge of a semi-infinite gap, where the energy flow of two-dimensional solitons increases with decrease of b, in contrast to the energy flow of their one-dimensional counterparts (Musslimani and Yang, 2004). Fundamental two-dimensional lattice solitons were observed in a landmark experiment conducted in an optically induced lattice in a photorefractive SBN crystal (Fleischer, Segev, Efremidis and Christodoulides, 2003). Photorefractive crystals possess strong electrooptic anisotropy, so that optical lattices produced by four ordinarily polarized interfering plane waves propagate in the linear regime, while extraordinarily polarized probe beams experience a significant screening nonlinearity (see Section 3). Since the nonlinear correction to the refractive index is determined by the total intensity distribution, the lattice creates the periodic environment, invariable in propagation direction, where the probe beam may experience discrete diffraction or form a lattice soliton when nonlinearity is strong enough (see Figure 8). Photonic lattices might be induced with partially incoherent light (Martin, Eugenieva, Chen and Christodoulides, 2004; Chen, Martin, Eugenieva, Xu and Bezryadina, 2004). In this case the lattice is created by amplitude modulation rather than by coherent interference of multiple
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FIGURE 8 Experimental results presenting the propagation of a probe beam launched into a single waveguide at normal incidence. (a) Intensity structure of the probe beam at the exit face of the crystal, displaying discrete diffraction at low nonlinearity (200 V). (b) Intensity structure of a probe beam at the exit face of the crystal, displaying an on-axis lattice soliton at high nonlinearity (800 V). (c) Interferogram, showing constructive interference of the peak and lobes between the soliton and a plane wave. (d) Reduction of probe intensity by a factor of 8, at the same voltage as (b) and (c), results in recovery of discrete diffraction pattern (Fleischer, Segev, Efremidis and Christodoulides, 2003)
waves and might be operated in either linear or nonlinear regimes (the latter regime is achieved when the lattice is extraordinarily polarized) due to suppression of incoherent modulational instabilities and allow for observation of two-dimensional fundamental solitons and soliton trains. Quasi-one-dimensional (i.e., uniform in one transverse direction and periodic in the other direction) optical lattices created in a bulk medium with a pair of interfering plane waves also support quasi-one-dimensional or two-dimensional solitons. In focusing media, quasi-one-dimensional lattice solitons are transversally unstable, developing a snake-like shape at low powers or experiencing neck-like break-up at high powers, as observed by Neshev, Sukhorukov, Kivshar and Krolikowski (2004). However, the periodic refractive index modulation inhibits the development of transverse instability. Note that the modulational instability of discrete solitons in coupled waveguides with group velocity dispersion may lead to formation of mixed spatio-temporal quasi-solitons (Yulin, Skryabin and Vladimirov, 2006). Fully localized two-dimensional solitons in quasi-one-dimensional lattices can be completely stable in most of their existence domain in both Kerr and saturable media. Such solitons can be set into radiationless
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motion in the direction where the lattice is uniform, which opens a wealth of opportunities for studying tangential soliton collisions (Baizakov, Malomed and Salerno, 2004; Mayteevarunyoo and Malomed, 2006). Weakly transversally modulated quasi-one-dimensional lattices support localized 2D solitons even in defocusing media (Ablowitz, Julien, Musslimani and Weinstein, 2005). Two-dimensional solitons have also been thoroughly studied in various discrete systems (see for example Pouget, Remoissenet and Tamga (1993), Mezentsev, Musher, Ryzhenkova and Turitsyn (1994), Kevrekidis, Rasmussen and Bishop (2000) and reviews mentioned above). Mobility of two-dimensional solitons in both continuous and discrete systems can be strongly enhanced in the regime of nonlinearity saturation (Vicencio and Johansson, 2006). Specific optical discrete X-waves are possible in normally dispersive nonlinear waveguide arrays with focusing nonlinearity. Such waves can be excited for a wide range of input conditions while their properties strongly depart from the properties of X-waves in bulk or waveguide configurations (Droulias, Hizanidis, Meier and Christodoulides, 2005). Discrete X-waves were observed in AlGaAs arrays by Lahini, Frumker, Silberberg, Droulias, Hizanidis, Morandotti and Christodoulides (2007).
6.2 Multipole Solitons Multipole solitons in two-dimensional optical lattices may have much richer shapes than their one-dimensional counterparts. The basic properties of such solitons were investigated by Musslimani and Yang (2004) and Kartashov, Egorov, Torner and Christodoulides (2004). Multipole solitons in focusing media are characterized by a π phase jump between neighboring spots (two-dimensional lattices also support even solitons comprising two in-phase spots, but such solitons are typically unstable). Multipole solitons exist above some minimal energy flow determined by the lattice parameters. In contrast to fundamental or even solitons, multipole solitons do not transform into delocalized Bloch waves at the cutoff for existence and their existence domain does not occupy the whole semi-infinite gap. Lattices stabilize multipole solitons even in Kerr nonlinear media when the energy flow (or b) exceeds a certain critical value. In principle, the number of solitons that can be incorporated into the multipoles is not limited, but the energy threshold for stabilization grows monotonically with the number of poles. This suggests the possibility of packing a number of individual solitons with properly engineered phases into one stable matrix to encode complex soliton images and letters. Experimental observation of dipole solitons was conducted by Yang, Makasyuk, Bezryadina and Chen (2004a,b) in an optical lattice induced with partially coherent light beams in photorefractive crystals.
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FIGURE 9 Experimental results of charge-4 vortex propagating without (top) and with (bottom) the two-dimensional lattice for different applied electric fields. (a) input, (b) linear diffraction, (c) output at a bias field of 90 V/mm, (d) output at 240 V/mm, (e) discrete diffraction at 90 V/mm, (f)–(h) discrete trapping at 240 V/mm. From (f) to (h), the vortex was launched into the lattice from off-site, to on-site, and then back to off-site configurations. Distance of propagation through the crystal is 20 mm; average distance between adjacent spots in the necklace is 48 µm (Yang, Makasyuk, Kevrekidis, Martin, Malomed, Frantzeskakis and Chen, 2005)
Higher-order stationary necklace solitons were experimentally generated by launching vortex beams into a partially coherent lattice (Yang, Makasyuk, Kevrekidis, Martin, Malomed, Frantzeskakis and Chen, 2005). This effect results in the generation of octagonal necklace solitons featuring π phase difference between adjacent spots (see Figure 9). Notice that for the observation of formation of necklace-like structures in optically induced lattices, Yang, Makasyuk, Kevrekidis, Martin, Malomed, Frantzeskakis and Chen (2005) fixed the intensity of the input beam and increased the voltage applied to the crystal, something that results in a simultaneous increase of the lattice and nonlinearity strengths, in contrast to conventional method of observation of transition between linear discrete diffraction and nonlinear self-trapping upon increase of the intensity of the input beam at fixed applied voltage (see, e.g. Fleischer, Segev, Efremidis and Christodoulides, 2003). It should be pointed out that even more complex soliton trains can be generated when a relatively broad stripe beam is launched into a two-dimensional lattice (Chen, Martin, Eugenieva, Xu and Bezryadina, 2004). In this case, due to the highly anisotropic character of the nonlinear response of the crystal, increasing the nonlinearity facilitates soliton train formation or, vice versa, enhances discrete diffraction for orthogonal orientations of the input stripe.
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An experimental study of the dynamics of off-site excitations conducted by Lou, Xu, Tang, Chen and Kevrekidis (2006) has shown that a single beam launched between two sites of a periodic photonic lattice excites an asymmetric lattice soliton when nonlinearity reaches a threshold. This is an indication that the branch corresponding to stable asymmetric lattice solitons may bifurcate from the unstable even soliton branch at high powers. Multipole solitons in two-dimensional discrete systems have been also thoroughly studied (see, for example, paper by Kevrekidis, Malomed and Bishop (2001) and references therein). Extended structures in nonlinear discrete systems occupying multiple channels were addressed by Kevrekidis, Gagnon, Frantzeskakis and Malomed (2007).
6.3 Gap Solitons If the parameters of a two-dimensional lattice are chosen in such a way that the lattice spectrum possesses at least a single finite gap, conditions are set to form two-dimensional gap solitons. Such solitons were observed in photorefractive optically induced lattices by Fleischer, Segev, Efremidis and Christodoulides (2003). A probe beam was launched at an angle with respect to the lattice plane to ensure that the beam experiences anomalous diffraction, while the polarity of the biasing field was chosen to produce defocusing nonlinearity. As a result, a self-trapped wave packet was observed with a staggered phase structure, where the phase changed by π between neighboring lattice channels. A theoretical analysis of the properties of two-dimensional gap solitons shows that they always exhibit power thresholds for their existence, and they might be stable provided that the propagation constant is not too close to the gap edges (Ostrovskaya and Kivshar, 2003; Efremidis, Hudock, Christodoulides, Fleischer, Cohen and Segev, 2003). Such solitons can form bound states, whose stability properties are defined by the nonlinearity sign. For example, in defocusing media the in-phase combinations of two-dimensional gap solitons are stable, while out-of-phase combinations (or twisted solitons) may be prone to symmetry-breaking instabilities. Stability of gap solitons was addressed numerically (Richter, Motzek and Kaiser, 2007). Experimental observation of gap soliton trains in a twodimensional defocusing photonic lattices was presented by Lou, Wang, Xu, Chen and Yang (2007). Peleg, Bartal, Freedman, Manela, Segev and Christodoulides (2007) reported on specific conical diffraction and gap soliton formation in honeycomb photorefractive lattices. Dipole-like twodimensional gap solitons in defocusing optically induced lattices were observed by Tang, Lou, Wang, Song, Chen, Xu, Chen, Susanto, Law and Kevrekidis (2007).
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Arrays of two-dimensional gap solitons might be generated via Bloch wave modulational instabilities (Baizakov, Konotop and Salerno, 2002; Brazhnyi, Konotop and Kuzmiak, 2006). Novel types of localized beams supported by the combined effects of total internal and Bragg reflection in two-dimensional lattices were observed by Fischer, Tr¨ager, Neshev, Sukhorukov, Krolikowski, Denz and Kivshar (2006). Such gap states originate from the X-symmetry point of the lattice spectrum and possess a reduced symmetry and highly anisotropic diffraction and mobility properties. Hybrid semi-gap spatio-temporal solitons may exist in quasione-dimensional lattices, where localization in space is achieved due to interplay of lattice, diffraction, and defocusing nonlinearity, while localization in time is due to defocusing nonlinearity and normal group velocity dispersion (Baizakov, Malomed and Salerno, 2006a). Higherorder solitons in two-dimensional discrete lattices with defocusing nonlinearity were analyzed by Kevrekidis, Susanto and Chen (2006). Inband (or embedded) solitons that appear in trains and that bifurcate from Bloch modes at the interior high-symmetry X points within the first band of lattice spectrum were observed in two-dimensional photonic lattices with defocusing nonlinearity (Wang, Chen, Wang and Yang, 2007).
6.4 Vector Solitons In contrast to their one-dimensional counterparts, two-dimensional vector solitons in periodic lattices have been studied only for the case of incoherent interactions between soliton components. Experimental observation of the simplest two-dimensional vector soliton in an optically induced partially coherent photonic lattice was performed by Chen, Bezryadina, Makasyuk and Yang (2004), Chen, Martin, Eugenieva, Xu and Yang (2005). It was demonstrated that two mutually incoherent beams can lock into a fundamental vector soliton while propagating along the same lattice site, although each beam alone would experience discrete diffraction under similar conditions. The components w1,2 of such solitons feature similar functional shapes and can be considered as φ-projections of the profile of scalar lattice solitons w(η, ζ ), i.e., w1 = w cos φ and w2 = w sin φ. When two mutually incoherent beams are launched into neighboring lattice sites, they form dipole-like vector solitons featuring strong mutual coupling between components in both poles. Lattices also support vector solitons formed by two dipole field components in the case of orthogonal orientation of dipoles in different components. The total intensity distribution in such solitons is reminiscent of that for quadrupole scalar solitons, but stability of vector states is enhanced in comparison with their scalar counterparts (Rodas-Verde, Michinel and Kivshar, 2006). Lattices can also support two-dimensional vector solitons whose components
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emerge from the same or different finite gaps of the lattice spectrum. Such solitons were studied by Gubeskys, Malomed and Merhasin (2006) for the case of repulsive interspecies and zero intraspecies interactions in the context of binary BECs. It was found that intragap solitons, that are typically bound states of tightly and loosely bound components originating from first and second gaps, can be found in lattices that are deep enough and are stable in a wide region of their existence. Vectorial interactions in photonic lattices, optically induced in photorefractive crystals, result in the formation of counter-propagating mutually incoherent self-trapped beams analyzed by Belic, Jovic, Prvanovich, Arsenovic and Petrovic (2006). Optical lattices may play a strong stabilizing role for such counter-propagating solitons (Koke, Tr¨ager, Jander, Chen, Neshev, Krolikowski Kivshar and Denz, 2007). The time-dependent rotation of counter-propagating mutually incoherent self-trapped beams in optically induced lattices was discussed by Jovic, Prvanovic, Jovanovic and Petrovic (2007). Interactions of counterpropagating discrete solitons in a photorefractive waveguide array were ¨ Shandarov and Kip studied experimentally by Smirnov, Stepic, Ruter, (2007). Finally, the existence and stability of strongly localized twodimensional vectorial modes in discrete arrays were analyzed by Hudock, Kevrekidis, Malomed and Christodoulides (2003).
6.5 Vortex Solitons Vortex solitons are characterized by a special beam shape, in amplitude and in phase, that carries a nonzero orbital angular momentum, which is related to energy circulation inside the vortex. Thus, vortex solitons realize higher-order, excited states of the corresponding nonlinear systems. Vortex solitons carry screw phase dislocations located at the points where the intensity vanishes. In uniform focusing media, vortex solitons feature localized ring-like intensity distribution. However, they are highly prone to azimuthal modulational instabilities that result in their spontaneous self-destruction into ground-state solitons (for a review on vortex solitons see Desyatnikov, Torner and Kivshar (2005)). The properties of vortex solitons in periodic media differ dramatically from the properties of their radially symmetric counterparts in uniform materials. The profiles of lattice vortex solitons have the form q(η, ζ, ξ ) = [wr (η, ζ )+iwi (η, ζ )] exp(ibξ ), where wr and wi are real and imaginary parts of the complex field q. The topological winding number m (or dislocation strength) of such complex scalar field can be defined by the circulation of the field phase arctan(wi /wr ) around the phase singularity. The result is an integer. Substitution of the vortex field in such a form into the nonlinear
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103
¨ Schrodinger equation in the simplest case of Kerr nonlinearity yields 1 − 2
∂2 ∂2 + 2 2 ∂η ∂ζ
!
1 − 2
∂2 ∂2 + ∂η2 ∂ζ 2
!
wr + bwr + σ (wr2 + wi2 )wr − p R(η, ζ )wr = 0, (10) wi + bwi + σ (wr2
+ wi2 )wi
− p R(η, ζ )wi = 0,
where σ = −1 corresponds to focusing nonlinearity and σ = 1 corresponds to defocusing nonlinearity, while R(η, ζ ) stands for the profile of the lattice. Linear stability analysis for lattice vortex solitons can be conducted by considering perturbed solutions in the form q = (wr +U + iwi + iV ) exp(ibξ ), where U = u(η, ζ ) exp(δξ ), V = v(η, ζ ) exp(δξ ) are real and imaginary parts of perturbation that can grow with a complex rate δ upon propagation. The linearized equations for perturbation components then become 1 δu = − 2 1 −δv = − 2
∂2 ∂2 + ∂η2 ∂ζ 2
!
∂2 ∂2 + ∂η2 ∂ζ 2
v + bv + 2σ wi wr u + σ (3wi2 + wr2 )v − p Rv, (11)
! u + bu
+ σ (3wr2
+ wi2 )u
+ 2σ wi wr v − p Ru.
Absence of perturbations satisfying Equation (11) for Re δ > 0 indicates vortex stability. Vortex lattice solitons were initially introduced in discrete systems (Aubry, 1997; Johansson, Aubry, Gaididei, Christiansen and Rasmussen, 1998; Malomed and Kevrekidis, 2001; Kevrekidis, Malomed, Bishop and Frantzeskakis, 2001; Kevrekidis, Malomed, Chen and Frantzeskakis, 2004). Detailed theoretical analysis of properties and stability of vortex solitons in continuous lattices induced in focusing Kerr media was performed by Yang and Musslimani (2003). It was shown that even the simplest lattice vortices exhibit strongly modulated intensity distributions with four pronounced intensity maxima whose positions almost coincide with the positions of the lattice maxima (Baizakov, Malomed and Salerno, 2003; Yang and Musslimani, 2003). In a focusing medium such vortices originate from the semi-infinite gap of the lattice spectrum. Phase changes by π/2 between neighboring vortex spots. Two types of charge-1 vortices were found: off-site and on-site. Contrary to naive expectations, the intensity distribution for high-amplitude vortices exhibits four very well localized bright spots concentrated in the vicinity of the lattice maxima. Both on-site and off-site vortex solitons that are oscillatory unstable near
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the lower cutoff become completely stable above a certain critical value bcr . Note that in the isotropic photorefractive model widely used in the literature, vortex solitons feature a limited stability domain since strong nonlinearity saturation destabilizes them (Yang, 2004). Experimental observation of off-site vortices in photorefractive optical lattices was performed by Neshev, Alexander, Ostrovskaya, Kivshar, Martin, Makasyuk and Chen (2004), while Fleischer, Bartal, Cohen, Manela, Segev, Hudock and Christodoulides (2004) have reported the generation of both on-site and off-site vortex configurations. Typically, vortex lattice solitons can be excited with input ring-like beams carrying a screw phase dislocation of unit charge. The interaction of a vortex soliton with the surrounding photonic lattice can modify the vortex structure. In particular, nontrivial topological transformations such as flipping of vortex charge and inversion of its orbital angular momentum are possible (Bezryadina, Neshev, Desyatnikov, Young, Chen and Kivshar, 2006). Periodic media impose important restrictions on the available charges of vortex solitons. Using general group theory arguments, Ferrando, Zacares and Garcia-March (2005); Ferrando, Zacares, Garcia-March, Monsoriu and Fernandez de Cordoba (2005) demonstrated that, unlike in homogeneous media, no symmetric vortices of arbitrary high order (or charge) can be generated in two-dimensional nonlinear systems possessing a discrete-point rotational symmetry. In particular, in the case of square optical lattices produced by interference of four plane waves, the realization of discrete symmetry forbids the existence of symmetric vortex solitons with charges higher than two. Still, dynamical rotating quasivortex double-charged solitons, reversing the topological charges and the direction of rotation through a quadrupole-like transition state, can exist in two-dimensional photonic lattices (Bezryadina, Eugenieva and Chen, 2006). Stable stationary vortex solitons with topological charge 2, introduced by Oster and Johansson (2006), incorporate eight main excited lattice sites and feature a rhomboid-like intensity distribution. Vortex solitons in discrete-symmetry systems are shown to behave as angular Bloch modes, characterized by an angular Bloch momentum (Ferrando, 2005). Ideally, symmetric periodic lattices can support asymmetric vortex solitons, including rhomboid, rectangular, and triangular vortices (Alexander, Sukhorukov and Kivshar, 2004). Such nonlinear localized structures describing elementary circular flows can be analyzed using energy-balance relations. Higher-order asymmetric vortex solitons and multipoles containing more than four lobes on a square lattice were constructed by Sakaguchi and Malomed (2005). Finite gaps of the lattice spectrum may give rise to specific gap vortex solitons carrying vortex-like phase dislocation and existing due to Bragg scattering. Vortices originating from the first gap were studied
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105
in defocusing media by Ostrovskaya and Kivshar (2004b). Such vortices experience delocalizing transition akin to flat-phase gap solitons when their energy flow becomes too high. Due to their gap nature, such vortex solitons develop very unusual chessboard-like phase structure. The process of dynamical generation of spatially localized gap vortices was studied by Ostrovskaya, Alexander and Kivshar (2006). The anisotropic nature of the photorefractive nonlinearity may substantially affect the properties of gap vortex solitons propagating in such media (see Richter and Kaiser (2007)). In a focusing media, vortex lattice solitons arising from the X symmetry points and residing in the first finite gap of the lattice spectrum were suggested by Manela, Cohen, Bartal, Fleischer and Segev (2004). Such solitons have the phase structure of a counter-rotating vortex array and are stable for moderate lattice depths and soliton intensities. Figure 10 illustrates such a gap vortex that was experimentally observed by Bartal, Manela, Cohen, Fleischer and Segev (2005). Lattices with more complicated geometries support vortex solitons featuring unusual symmetries, such as multivortex solitons in triangular lattices (Alexander, Desyatnikov and Kivshar, 2007). Kartashov, Ferrando and Garc´ıa-March (2007) predicted the existence of a new type of discrete-symmetry lattice vortex soliton that can be considered as a coherent state of dipole solitons carrying a nonzero topological charge. Vectorial interactions between lattice vortices substantially enriches their propagation dynamics. For example composite states of vortex solitons originating from different gaps can be stable in certain parameter regions, while the decay of unstable representatives of such families may be accompanied by exchange of angular momentum between constituents and transformation into other types of stable composite solitons (Manela, Cohen, Bartal, Fleischer and Segev, 2004). Some novel scenarios of chargeflipping instability of incoherently coupled on-site and off-site vortices were encountered by Rodas-Verde, Michinel and Kivshar (2006). Counterpropagating vortices in optical lattices are also the subject of current studies (Petrovic, 2006; Petrovic, Jovic, Belic and Prvanovic, 2007).
6.6 Topological Soliton Dragging Two-dimensional settings give rise to a new class of lattices possessing higher topological complexity, e.g., regular lattices containing dislocations. In contrast to lattices that feature regular shapes, the guiding properties of lattices with dislocations can drastically change in the vicinity of a dislocation, a possibility that leads to new phenomena. Such lattices may be produced, e.g., by interference of a plane wave and a wave carrying one or several nested vortices (Kartashov, Vysloukh and Torner, 2005b). In the simplest case of a single-vortex wave, the lattice profile is given by R(η, ζ ) = |exp(iαη) + exp(imφ + iφ0 )|2 , where α is the plane wave
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Soliton Shape and Mobility Control in Optical Lattices
FIGURE 10 Experimental observation of a gap vortex lattice soliton. (a) Intensity distribution of the input vortex ring-beam photographed (for size comparison) on the background of the optically induced lattice. (b) Output intensity distribution of a low intensity ring-beam experiencing linear diffraction in the lattice. (c) Output intensity distribution of a high intensity ring-beam, which has evolved into a gap vortex soliton in the same lattice as in (b). (d)–(f) Phase structure of a gap vortex lattice soliton. The phase information is obtained by interference with a weakly diverging Gaussian beam. (d) Phase distribution of the input vortex ring-beam. (e) Output phase distribution of a high intensity ring-beam that has evolved into a gap vortex soliton (interference pattern between the soliton whose intensity is presented in (c) and a weakly diverging Gaussian beam). (f) Numerical validation of the phase information in (e) (Bartal, Manela, Cohen, Fleischer and Segev, 2005)
propagation angle, φ is the azimuthal angle, m is the vortex charge, and φ0 defines the orientation of the vortex origin. Such lattices are not stationary and distort upon propagation, but they can be implemented in two-dimensional condensates that are strongly confined in the direction of light propagation. Representative profiles of lattices produced by interference of a plane wave and a wave with nested vortices are shown in Figure 11. They feature clearly pronounced channels where solitons can travel. The screw phase dislocation results in appearance of the characteristic fork, so that some channels may fuse or disappear in the dislocation. Notice that lattice periodicity is broken only locally, in close proximity to the dislocation. The higher the vortex charge, the stronger the lattice topological complexity and the more channels that fuse or disappear in the dislocation. Lattice
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FIGURE 11 (a) and (b) show lattices with single dislocations produced by the interference pattern of a plane wave and a wave with nested charge-1 vortex for various vortex orientations. Arrows show the direction of motion for solitons placed into lattice channels in the vicinity of a dislocation. Positive (c) and negative (d) topological traps created with two vortices with charges m = ±3. Map of Fζ force for solitons launched in the vicinity of positive (e) and negative (f) topological traps produced by two vortices with charges m = ±1. Arrows show the direction of soliton motion. In (c)–(f) trap length δζ = π (Kartashov, Vysloukh and Torner, 2005b)
intensity gradient in the vicinity of a dislocation results in appearance of forces acting on the soliton, that lead to the soliton’s fast transverse displacement when it is launched into one of the lattice channels in the vicinity of the dislocation (Figure 11(a),(b)). The dependence of sign and magnitude of forces acting on solitons on the lattice topology makes it possible to use combinations of several dislocations (produced by spatially separated oppositely charged vortices) for creation of double lattice defects (Figure 11(c),(d)), or soliton traps, which feature equilibrium regions that are capable of supporting stationary solitons. Positive traps form when several lattice rows fuse to form the region where R(η, ζ ) is locally increased, while negative traps feature breakup of several lattice channels. The effective particle approach (see Section 5.2) yields the equations d2 p hηi = U dξ 2
Z Z
∞
−∞
|q|2
dR dη dζ, dη
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Soliton Shape and Mobility Control in Optical Lattices
d2 p hζ i = 2 U dξ
Z Z
∞
|q|2
−∞
dR dη dζ, dζ
(12)
for soliton centre coordinates hηi, hζ i, which allow one to construct the map of forces Fη ∼ d2 hηi /dξ 2 and Fζ ∼ d2 hζ i /dξ 2 acting on the solitons. Such maps built for the simplest Gaussian ansatz for the soliton profile are shown in Figure 11(e),(f) for both positive and negative traps. Both traps repel all outermost solitons (some of the possible soliton positions are indicated by circles in this figure) but support stationary solitons located inside traps, which are stable for positive traps and unstable for negative traps. This result confirms the crucial importance of lattice topology for properties and stability of solitons that it supports. Notice that solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures were also investigated in details by Ablowitz, Ilan, Schonbrun and Piestun (2006).
7. SOLITONS IN BESSEL LATTICES The simplest optical lattices exhibit a square or honeycomb shape. Lattices with different types of symmetry also exist, and offer new opportunities for soliton existence, management, and control. Such lattices can be imprinted with different types of nondiffracting beams, including Bessel beams, parabolic beams or Mathieu beams. All these beams are directly associated with the fundamental nondiffracting solutions of the wave equation in circular, parabolic, or elliptical cylindrical coordinates and thus exhibit a different symmetry. The techniques to generate various types of nondiffracting beams can be found in Durnin, Miceli and Eberly (1987), Gutierrez-Vega, Iturbe-Castillo, Ramirez, Tepichin, Rodriguez-Dagnino, Chavez-Cerda and New (2001), Lopez-Mariscal, Bandres, Gutierrez-Vega and Chavez-Cerda (2005). In this section we describe the properties of solitons supported by lattices imprinted by radially symmetric and azimuthally modulated Bessel beams, and by Mathieu beams.
7.1 Rotary and Vortex Solitons in Radially Symmetric Bessel Lattices Radially symmetric Bessel lattices can be imprinted optically with zeroorder or higher order Bessel beams, whose field distribution can be written as q(η, ζ, ξ ) = Jn [(2blin )1/2r ] exp(inφ) exp(−iblin ξ ), where r = (η2 + ζ 2 )1/2 , φ is the azimuthal angle, n is the charge of the beam, and the parameter blin sets the transverse extent of the beam core. Such beams preserve their transverse intensity distribution upon propagation in a linear medium. Rigorously speaking, such beams are infinitely extended
Solitons in Bessel Lattices
109
and thus carry infinite energy. However, accurate approximations can be generated experimentally in a number of ways. Bessel lattices can be imprinted in photorefractive crystals with the technique used for the generation of harmonic patterns by incoherent vectorial interactions (Efremidis, Sears, Christodoulides, Fleischer and Segev, 2002) and in Bose–Einstein condensates. The simplest Bessel lattices, that exhibit an intensity given by the zero-order Bessel beam, contain a central guiding core surrounded by multiple bright rings. Lattices produced by higherorder beams have zero intensity at r = 0. The field of the radially symmetric soliton in such lattices can be written as q(η, ζ, ξ ) = w(r ) exp(imφ) exp(ibξ ), where m is the topological charge. This yields the following equation for the soliton profile in the case of Kerr nonlinear response: 1 2
d2 w 1 dw m 2 w − 2 + r dr dr 2 r
! − bw − σ w 3 + p R(r )w = 0.
(13)
To analyze the linear dynamical stability of the stationary soliton families, one has to search for perturbed solutions in the form q(η, ζ, ξ ) = [w(r ) + u(r, ξ ) exp(inφ) + v ∗ (r, ξ ) exp(−inφ)] exp(ibξ + imφ), where the perturbation components u, v could grow with complex rate δ on propagation, n is the azimuthal perturbation index. Linearization of the ¨ nonlinear Schrodinger equation around a stationary solution yields the eigenvalue problem: 1 iδu = − 2 1 −iδv = − 2
d2 (m + n)2 1 d + − r dr dr 2 r2
!
d2 (m − n)2 1 d − + r dr dr 2 r2
u + bu + σ w2 (v + 2u) − p Ru, (14)
! v + bv + σ w (u + 2v) − p Rv. 2
In the case of focusing nonlinear media (σ = −1), the simplest solitons are located in the central core of the Bessel lattice (Kartashov, Vysloukh and Torner, 2004c). At low energy flows such solitons are wide and extend over several lattice rings, while at high energies they are narrow and concentrate mostly within the core of the lattice. Solitons in lattices produced by higher-order Bessel beams exhibit an intensity deep at r = 0. Importantly, a Bessel lattice imprinted in a Kerr nonlinear medium can suppress collapse and thus stabilize radially symmetric solitons in most of their existence domain. A fascinating example of localized structure supported by Bessel lattices is given by solitons trapped in different rings of the lattice. Such
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Soliton Shape and Mobility Control in Optical Lattices
solitons suffer from exponential instabilities close to low-amplitude cutoff where they are broad, but they are completely stable above a threshold b value. Solitons trapped inside the rings can be set into rotary motion by launching them tangentially to the lattice rings. The important point is that such solitons do not radiate upon rotation. A rich variety of interaction scenarios is encountered for collective motion of solitons inside the lattice rings (Figure 12). Interactions between in-phase solitons located in the same ring may cause their fusion into a high-amplitude slowly rotating soliton, while collision of out-of-phase solitons results in formation of steadily rotating soliton pairs with a fixed separation between components. Collisions of out-of-phase solitons in different rings may be accompanied by reversal of their rotation direction after each act of collision. Notice that radially symmetric Bessel lattices may also support multipole solitons composed of out-of-phase spots residing in different lattice rings. Experimental demonstration of soliton formation in optically induced ring-shaped radially symmetric photonic lattices was performed by Wang, Chen and Kevrekidis (2006). The transition from diffraction (symmetric in the case of excitation of the central guiding core and asymmetric in the case of excitation of one of the outer lattice rings) to guidance of the extraordinarily polarized probe beam by the lattice was observed by tuning the lattice and nonlinearity strengths. In addition to immobile solitons trapped in the central lattice core and in different lattice rings, Wang et al observed controlled rotation of solitons launched at different angles into lattice rings (see Figure 13). The polarity of the biasing electric field applied to the photorefractive crystal can be selected in such a way that Bessel-like beams with a bright central core induce photonic lattices featuring a decreased refractive index in the centre. Such a ring lattice with a low-refractive-index core surrounded by concentric rings (akin to a photonic band-gap fiber) may guide linear beams whose localization is achieved due to reflection on the outer rings of the lattice. Guidance of linear beams in a Bessel-like lowrefractive-index core lattice was demonstrated experimentally by Wang, Chen and Yang (2006). Solitons supported by ring-shaped radially-periodic lattices were studied by Hoq, Kevrekidis, Frantzeskakis and Malomed (2005) and by Baizakov, Malomed and Salerno (2006b). In contrast to Bessel lattices, whose intensity slowly decreases at r → ∞, radially-periodic lattices do not decay at infinity, a property that substantially affects soliton behavior and stability domains. In particular, no localized states exist in such lattices at the linear limit and all localized states are truly nonlinear. In focusing media, radially-periodic lattices support fundamental and higher-order (i.e., concentrated in outer rings) axially uniform solitons.
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FIGURE 12 (a) Interaction of out-of-phase solitons in the first ring of a zero-order Bessel lattice. One of solitons is set into motion at the entrance of the medium by imposing the phase twist ν = 0.1. (b) The same as in column (a) but for in-phase solitons. (c) Interaction of out-of-phase solitons located in first and second lattice rings. The soliton located in the first ring is set into motion by imposing the phase twist ν = 0.2. All solitons correspond to the propagation constant b = 5 at p = 5. Soliton angular rotation directions are depicted by arrows (Kartashov, Vysloukh and Torner, 2004c)
In such settings all higher-order solitons are azimuthally unstable. In contrast, solitons of the same type may be stable in defocusing media. Lattices with defocusing nonlinearity also give rise to radial gap solitons.
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FIGURE 13 Soliton rotation in ring lattice. From (a) to (c), the probe soliton beam was aimed at the far right side of the first, second, and third lattice ring with the same tilting angle (about 0.4◦ ) in the y direction. From (d) to (f), the probe soliton beam was aimed at the same second lattice ring but with different tilting angles of 0◦ , 0.4◦ , and 0.6◦ , respectively (Wang, Chen and Kevrekidis, 2006)
Surface waves localized at the edge of radially-periodic guiding structures, consisting of several concentric rings, are also possible (Kartashov, Vysloukh and Torner, 2007b). Such surface waves rotate steadily upon propagation and, in contrast to non-rotating waves, for high rotation frequencies they do not exhibit power thresholds for their existence. There exists an upper limit for the surface wave rotation frequency, which depends on the radius of the outer guiding ring and on its depth. Bessel optical lattices imprinted in defocusing media with higher-order radially symmetric nondiffracting Bessel beams can also support stable ring-profile vortex solitons (Kartashov, Vysloukh and Torner, 2005a). Such solitons exist because a Bessel lattice compensates defocusing and diffraction, thereby affording confinement of light that is impossible in the uniform defocusing medium. In the low-energy limit, vortex solitons transform into linear guided modes of the Bessel lattice, while high-energy vortices greatly expand across the lattice and acquire multi-ring structure. Since modulational instability is suppressed in defocusing media, ringshaped vortices in Bessel lattices may be stable. The higher the soliton topological charge the deeper the lattice modulation necessary for vortex stabilization. Rotating quasi-one-dimensional and two-dimensional guiding structures are also interesting. In the limit of fast rotation, light beams propagating in such structures feel the average refractive index distribution
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that may be equivalent, for example, to the refractive index distribution optically-induced by the Bessel beams. Solitons supported by twodimensional rotating lattices with focusing nonlinearity were introduced by Sakaguchi and Malomed (2007). Rotating lattices, even quasi-onedimensional, were found to support localized ground-state and vortex solitons in defocusing media, provided that the rotation frequency exceeds the critical value (Kartashov, Malomed and Torner, 2007). He, Malomed and Wang (2007) showed that rotary motion of a two-dimensional soliton trapped in a Bessel lattice can be controlled by applying a finitetime push to the lattice, due to the transfer of linear momentum of the lattice to the orbital soliton momentum. Solitons in discrete rotating lattices were considered by Cuevas, Malomed and Kevrekidis (2007).
7.2 Multipole and Vortex Solitons in Azimuthally Modulated Bessel Lattices Holographic techniques might be also used to produce higherorder azimuthally modulated beams and lattices. Lattices induced by azimuthally modulated Bessel beams of order n have a functional shape: R(η, ζ ) = Jn2 [(2blin )1/2r ] cos2 (nφ). The local lattice maxima situated closer to the lattice centre are more pronounced than others, and form a ring of 2n guiding channels where solitons can be located. Such lattices support both individual solitons in any of the guiding channels and multipole solitons arranged into necklaces (Kartashov, Egorov, Vysloukh and Torner, 2004). Complex soliton configurations in focusing media may be stable when the field changes its sign between neighboring channels. Thus, necklaces consisting of 2n out-of-phase spots may form in the guiding ring of nth order lattice. They are completely stable for high enough energy flow levels (see Figure 14 for representative examples of necklace profiles). Another intriguing opportunity afforded by azimuthally modulated Bessel lattices is that a single soliton, initially located in one of the guiding sites of a lattice ring and launched tangentially to the guiding ring, starts traveling along the consecutive guiding sites, so that it can even return to the input site. Since solitons have to overcome a potential barrier when passing between sites, they radiate a small fraction of energy and can be trapped in different sites, thus realizing azimuthal switching that can be controlled by the input soliton angle and energy flow (Figure 14). Localization of light in modulated Bessel optical lattices was observed experimentally by Fisher, Neshev, Lopez-Aguayo, Desyatnikov, Sukhorukov, Krolikowski and Kivshar (2006). In the experiment, higherorder lattices were generated in photorefractive crystals by employing a phase-imprinting technique. The result of linear diffraction of low power beams was shown to depend crucially on the specific location of the input excitation. Thus, diffraction was observed to be strongest for excitation in
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FIGURE 14 Stable soliton complex supported by a third-order Bessel lattice at b = 3 and p = 20 (a) and by a sixth-order Bessel lattice at b = 5 and p = 40 (b). Azimuthal switching of single tilted soliton to second (c) and fourth (d) channels of a third-order Bessel lattice for tilt angles αζ = 0.49 and αζ = 0.626 at p = 2 and input energy flow Uin = 8.26. Panels (e) and (f) show switching to third and sixth channels of a sixth-order lattice for αζ = 0.8 and αζ = 0.93 at p = 5 and input energy flow Uin = 8.61. Input and output intensity distributions are superimposed. Arrows show direction of soliton motion and Sin , Sout denote input and output soliton positions (Kartashov, Egorov, Vysloukh and Torner, 2004)
the origin of the lattice and almost suppressed for excitation of a single lattice site. Nonlinearity resulted in strong light confinement in one of the sites of a guiding lattice ring. As discussed above, vortex solitons may be stabilized by imprinting optical lattices in the focusing medium. However, the very refractive index modulation causing stabilization of vortex solitons, simultaneously imposes certain restrictions on the possible topological charges of vortices dictated by the finite order of allowed discrete rotations (Ferrando, Zacares and Garcia-March, 2005). A corollary of such a result is that the maximum charge of a stable symmetric vortex in a two-dimensional square lattice is equal to 1. Azimuthally modulated Bessel lattices offer a wealth of new opportunities because the order of rotational symmetry in such lattices may be higher than 4, in contrast to square lattices, a property that has direct implications in the possible topological charges of symmetric vortex solitons in Bessel lattices. By using general group theory arguments Kartashov, Ferrando, Egorov and Torner (2005) have shown
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that Bessel optical lattices may support vortex solitons with charges higher than one and that the allowed value of vortex charge m in a lattice of order n is dictated by the charge rule 0 < m 6 n − 1. A detailed stability analysis has shown that vortex solitons in Bessel lattices may be stable only if the topological charge satisfies the condition n/2 < m 6 n − 1, with the exception that when n = 2, the vortex with m = 1 may be stable.
7.3 Soliton Wires and Networks Arrays of lowest order Bessel beams might be used to produce reconfigurable two-dimensional networks in Kerr-type nonlinear media. The possibility of blocking and routing solitons in two-dimensional networks was suggested by Christodoulides and Eugenieva (2001a,b), Eugenieva, Efremidis and Christodoulides (2001). It was predicted that solitons can be navigated in two-dimensional networks and that this can be accomplished via vectorial interactions between two classes of solitons: broad and highly mobile signal solitons and powerful blockers. Interactions with signal and blocker beams combined with the possibility for solitons to travel only along network wires might be used for implementation of a rich variety of blocking, routing, and logic operations. Solitons in such networks may propagate around sharp bends with angles exceeding 90 degrees, while engineering of the corner site of the bend allows minimization of bending losses. While fabrication of such networks remains technically challenging, they could be induced optically with arrays of Bessel beams in suitable nonlinear material. Solitons launched into such networks can travel along the complex predetermined paths (such as lines, corners, circular chains) almost without radiation losses (Xu, Kartashov and Torner, 2005b), while the whole network can be easily reconfigured just by blocking selected Bessel beams in a network-inducing array. Several mutually incoherent Bessel beams can induce reconfigurable directional couplers and Xjunctions (Xu, Kartashov, Torner and Vysloukh, 2005).
7.4 Mathieu and Parabolic Optical Lattices As mentioned above, Bessel beams are associated with the fundamental nondiffracting solutions of the wave equation in circular cylindrical coordinates. Another important class of nondiffracting beams, so-called Mathieu beams, stem from the elliptical cylindrical coordinate system (Gutierrez-Vega, Iturbe-Castillo, Ramirez, Tepichin, Rodriguez-Dagnino, Chavez-Cerda and New, 2001). The lattices induced by Mathieu beams are particularly interesting because they afford smooth topological transformation of a radially symmetric Bessel lattice into a quasi-onedimensional periodic lattice with modification of the so-called interfocal parameter, thereby connecting two classes of optical lattices widely
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studied in the literature. The transformation of lattice topology finds its manifestation in an important change in the shape and properties of ground-state and dipole-mode solitons. In particular, solitons are strongly pinned by almost radially symmetric lattices with small interfocal parameters and can hardly jump into neighbouring lattice rings, while in quasi-one-dimensional lattices with large interfocal parameters even small input tilts cause considerable transverse displacements of groundstate and dipole-mode solitons (Kartashov, Egorov, Vysloukh and Torner, 2006a). The stability of dipole-mode solitons is determined by the very shape of the Mathieu lattice. Transformation of the lattice into a quasi-onedimensional periodic structure destabilizes dipole solitons in the entire domain of their existence. The unique properties exhibited by solitons in lattices produced by parabolic nondiffracting beams were introduced by Kartashov, Vysloukh and Torner (2008). Parabolic lattices exhibit a nonzero curvature of their channels that results in asymmetric shapes of higher-order solitons. It was predicted that, despite such symmetry breaking, complex higherorder states can be stable, and it was shown that the specific topology of parabolic lattices affords oscillatory-type soliton motion.
8. THREE-DIMENSIONAL LATTICE SOLITONS In this section we briefly refer to the recent advances in the search for three-dimensional lattice solitons. Such solitons may form in 3D Bose–Einstein condensates loaded in optical lattices (for a review see Morsch and Oberthaler (2006)) as well as in potentially suitable nonlinear optical materials with a spatial refractive index modulation, when the full spatiotemporal dynamics is considered (for a review on spatiotemporal solitons, or light bullets, see Malomed, Mihalache, Wise and Torner (2005)). Notice that in the absence of guiding structures, 3D solitons in pure Kerr media experience strong collapse (for a review, see Berge (1998)). Under appropriate conditions optical lattices may completely stabilize ground-state and higher order 3D solitons, even in the case when dimensionality of the lattice is lower than that of the soliton. The dynamics of 3D excitations in 3D or lower-dimensional optical ¨ lattices is described by the nonlinear Schrodinger equation that, in the case of a focusing Kerr nonlinearity, takes the form: ∂q 1 i =− ∂ξ 2
∂ 2q ∂ 2q ∂ 2q + 2+ 2 2 ∂η ∂ζ ∂τ
! − q |q|2 − p R(η, ζ, τ )q,
(15)
where for matter waves τ stands for the third spatial coordinate and ξ stands for the evolution time, while for the spatiotemporal wave-
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packets τ is the time (we assume anomalous group-velocity dispersion) and ξ is the propagation distance. The function R(η, ζ, τ ) describes the profile of the lattice, that can be of any dimensionality (up to 3D) in the case of matter waves and two-dimensional or one-dimensional for spatiotemporal solitons. Three-dimensional periodic lattices with focusing nonlinearity can stabilize fundamental 3D solitons (Baizakov, Malomed and Salerno, 2003; Mihalache, Mazilu, Lederer, Malomed, Crasovan, Kartashov and Torner, 2005). The energy of three-dimensional solitons in optical lattices always diverges at the cutoff and monotonically decreases as b → ∞. However, a limited stability interval, where dU/db > 0, appears when the lattice depth exceeds a critical value. The width of the stability domain as a function of b and U increases with growing lattice depth. Illustrative profiles of three-dimensional solitons in the three-dimensional lattice R = cos(4η) + cos(4ζ ) + cos(4τ ) are depicted in Figure 15(a)–(c), where transformation of a broad unstable soliton covering many lattice sites into a stable localized soliton with increase of peak amplitude is clearly visible. Importantly, the Hamiltonian-versus-energy diagram for threedimensional lattice solitons features two cuspidal points, resulting in a typical swallowtail loop. Although this loop is one of the generic patterns known in catastrophe theory, it rarely occurs in physical models. Repulsive Bose–Einstein condensate confined in a three-dimensional periodic optical lattice supports gap solitons and gap vortex states which are spatially localized in all three dimensions and possess nontrivial particle flow. Both on-site and off-site planar vortices with unit charge that are confined in the third dimension by Bragg reflection were encountered in the three-dimensional lattice (Alexander, Ostrovskaya, Sukhorukov and Kivshar, 2005). Such planar states can be stable over a wide range of their existence region. Besides planar vortices with axes corresponding to the X symmetry point in the Brillouin zone, solitons with other symmetry axes become possible in three-dimensional lattices. Three-dimensional vortices in discrete systems were studied by Kevrekidis, Malomed, Frantzeskakis and Carretero-Gonzalez (2004), Carretero-Gonzalez, Kevrekidis, Malomed and Frantzeskakis (2005). In such systems stable vortices with m = 1 and 3 were encountered, while vortices with m = 2 are unstable and may spontaneously rearrange themselves into charge-3 vortices. Multicomponent discrete systems allow for existence of complex stable composite states, such as states consisting of two vortices whose orientations are perpendicular to each other. Low-dimensional lattices may also support a variety of stable soliton states. Thus, quasi-two-dimensional lattices stabilize three-dimensional solitons, while quasi-one-dimensional lattices may support stable twodimensional solitons in focusing cubic media (Baizakov, Malomed and
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FIGURE 15 Isosurface plots of (a) unstable and (b), (c) stable 3D solitons in a 3D periodic lattice. In (a) soliton energy U = 2.4 and its peak amplitude A = 1.8, in (b) U = 2.04, A = 2.2, in (c) U = 2.4, A = 3. In (a)–(c) depth of periodic lattice p = 3. (d)–(f) Isosurface plots of stable 3D solitons supported by two-dimensional Bessel lattice at (d) p = 5, b = 0.11, (e) p = 5.5, b = 0.85, and (f) p = 6, b = 2 (Mihalache, Mazilu, Lederer, Malomed, Crasovan, Kartashov and Torner, 2005; Mihalache, Mazilu, Lederer, Malomed, Kartashov, Crasovan and Torner, 2005)
Salerno, 2004; Mihalache, Mazilu, Lederer, Kartashov, Crasovan and Torner, 2004). Since a lattice is uniform in one direction the solitons can freely move in that direction and experience head-on or tangential collisions, which may be almost elastic (as in the case of head-on collision of out-of-phase solitons) or may result in soliton fusion (for slow tangential collisions). Solitons in such lattices typically acquire strongly asymmetric shapes since they are better localized in the dimension where the lattice is imprinted than in a uniform dimension. Notice that two cuspidal points in the Hamiltonian-versus-energy diagram, resulting in a swallowtail loop, also appear for three-dimensional solitons in quasi-twodimensional lattices. Such solitons feature a limited stability interval in b or U when the lattice depth exceeds the critical level, which is much higher than the critical depth for soliton stabilization in a fully-three-dimensional lattice. Cylindrical Bessel lattices imprinted in Kerr self-focusing media also support stable three-dimensional solitons (Mihalache, Mazilu, Lederer, Malomed, Kartashov, Crasovan and Torner, 2005). In J0 Bessel lattices, the fundamental solitons might be stable within one or even two intervals of their energy, depending on the depth of the lattice. In the latter case
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the Hamiltonian-versus-energy diagram has a swallowtail shape with three cuspidal points. Stability properties for solitons in Bessel lattices are dictated to a large extent by the actual lattice shape and its asymptotical behavior as r → ∞. Thus, J02 lattices allow for a single soliton stability domain only. While high-amplitude solitons are mostly confined in the central core of Bessel lattices, their low-amplitude counterparts acquire clearly pronounced multi-ring structure (see Figure 15(d)–(f)). The basic properties of three-dimensional vortex solitons in quasi-two dimensional lattices have been analyzed by Leblond, Malomed and Mihalache (2007). Quasi-one-dimensional lattices cannot stabilize three-dimensional solitons in media with focusing Kerr nonlinearity (Baizakov, Malomed and Salerno, 2004). However, it was predicted that a combination of quasi-one-dimensional lattices with Feshbach resonance management of scattering length in three-dimensional Bose–Einstein condensates results in the existence of stable three-dimensional breather solitons (Matuszewski, Infeld, Malomed and Trippenbach, 2005). Such stable solitons may exist only if the average value of the nonlinear coefficient and the lattice strength exceed certain minimum values. Note that both onedimensional and two-dimensional solitons in fully-dimensional optical lattices with nonlinearity management have been studied too (Gubeskys, Malomed and Merhasin, 2005). It was found, for example, that the energy thresholds required for existence of all types of two-dimensional solitons in dynamical systems is essentially higher than those in static systems. A detailed description of the soliton properties in systems with nonlinearity management is beyond the scope of this review. We refer to Malomed (2005) for a comprehensive description of the relevant background and methods.
9. NONLINEAR LATTICES AND SOLITON ARRAYS One of the key ingredients required for the successful generation of lattice solitons is the robustness of the periodic optical pattern that effectively modulates the refractive index of the nonlinear medium, thereby creating a shallow grating whose depth is comparable with the nonlinear contribution to the refractive index produced by the soliton beam. In all previous sections the robustness of the optically induced lattice is achieved because the lattice propagates almost in the linear regime, so that self-action of the lattice and cross-action from the copropagating soliton are negligible. In photorefractive media this can be achieved by selecting proper (ordinary) polarization for lattice waves (Fleischer, Segev, Efremidis and Christodoulides, 2003). At the same time, extraordinarily polarized lattices would experience strong self-action and may interact with soliton beams. The elucidation of conditions of robust
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propagation of such patterns is of interest, since it allows extension of the concept of optically induced gratings beyond the limit of weak material nonlinearities. In this section we briefly describe the properties of the nonlinear optical lattices and the techniques that can be used for their stabilization.
9.1 Soliton Arrays and Pixels A simple example of a periodic nonlinear pattern is provided by an array of well-localized solitons. Such arrays can be created, for instance, by launching a spatially modulated incoherent beam in a focusing medium (Chen and McCarthy, 2002). The array of solitons generated in such a way can be robust, provided that the coherence of the beam and the nonlinearity strength are not too high. If the coherence is too high, the array tends to break up into a disordered pattern rather than into ordered soliton structures. Thus, the incoherence plays a crucial role for stabilization of periodic patterns since it reduces the interaction forces between neighboring soliton pixels in the array. Soliton arrays imprinted in photorefractive crystals with partially incoherent light might be used to guide probe beams at other wavelengths or for transmission of images. Such arrays can be created dynamically by seeding spatial noise on to a uniform partially spatially incoherent beam due to the development of the induced modulational instability. In strongly anisotropic photorefractive media, the dimensionality of the emerging pattern depends on the strength of the nonlinearity (Klinger, Martin and Chen, 2001). Large soliton arrays were also created with coherent light (Petter, Schroder, Tr¨ager and Denz, 2003). It was demonstrated that the attractive forces acting between in-phase solitons in such arrays might be exploited for controlled fusion of several solitons in an array caused by additional beams launched between array channels. However, such attractive forces simultaneously put restrictions on the minimal separation between in-phase solitons in the array. Decreasing the separation results in increase of the interaction forces between neighboring spots and enhances the probability of fast amplification of asymmetries, introduced by input noise or slight deviations of the shapes of separate spots, that could lead to decay of the whole array. The stability of a soliton array can be substantially improved by utilizing phase-dependent interactions between soliton rows (Petrovic, Tr¨ager, Strinic, Belic, Schroder and Denz, 2003). In particular, changing the phase difference between neighboring rows of an array by π makes the interactions between them repulsive, prevents solitons in different rows from fusion, and greatly enhances the stability of the entire array. The technique of phase engineering allows achievement of a much denser soliton packing rate.
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9.2 Nonlinear Periodic Lattices The properties and stability of truly infinite two-dimensional stationary periodic waves in nonlinear media were analyzed by Kartashov, Vysloukh and Torner (2003). Such patterns might be termed cnoidal waves, by analogy with their one-dimensional counterparts. In saturable nonlinear media such waves transform into harmonic patterns (akin to linear periodic lattices discussed in Sections 5 and 6) in low- and high-power limits, while at intermediate power levels they describe arrays of localized bright spots in focusing media and arrays of dark solitons in defocusing media. Different cnoidal waves were found that can be divided into classes (three in focusing media and one in defocusing media) depending on their shape (see Figure 16). A stability analysis of such periodic patterns has shown that all of them are unstable from a rigorous mathematical point of view, but practically, the instability growth rates can be very small in the two limiting cases of relatively low and high energy flows, indicating that nonlinearity saturation has a strong stabilizing action on periodic patterns. Importantly, stability of patterns comprising out-of-phase spots and having chessboard phase structure (like cn-cn or sn-sn waves of Figure 16) is substantially enhanced in comparison with that for patterns built of in-phase spots. Two-dimensional nonlinear lattices with chessboard phase structure were demonstrated in highly anisotropic photorefractive media (Desyatnikov, Neshev, Kivshar, Sagemerten, Tr¨ager, Jagers, Denz and Kartashov, 2005). The propagation of radiation in such media under steady-state conditions is described by the system of equations for dimensionless light field amplitude q and electrostatic potential φ of the optically induced space-charge field Esc = ∇Usc : ! ∂ 2q ∂ 2q ∂φ + 2 + σq , 2 ∂η ∂η ∂ζ (16) 2 2 ∂ φ ∂φ ∂ ∂ φ ∂φ ∂ 2 2 + 2 + E+ ln(1 + S |q| ) + ln(1 + S |q| ) = 0, ∂η ∂η ∂ζ ∂ζ ∂η2 ∂ζ
1 ∂q =− i ∂ξ 2
−1/2
where q = AI0 , φ = (1/2)k 2 n 2r0 |reff | Usc , I0 is the input intensity, σ = sgn(reff ), reff is the effective electro-optic coefficient involved, E = (1/2)k 2 n 2r02 E 0 |reff | is the dimensionless static electric field applied to the crystal, k is the wavenumber, r0 is the beam width, S = I0 /(Idark + Ibg ) is the saturation parameter. It was found that, due to anisotropy of photorefractive response, refractive index modulation induced by the periodic lattice is also highly anisotropic and nonlocal, and it depends on the lattice orientation relative to the crystal axis. Moreover, stability
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FIGURE 16 Cn-cn (a) and sn-sn (b) periodic waves in saturable medium with S = 0.1. Plots in the first row show energy flow concentrated within one wave period T = 2π versus propagation constant. Surface plots in each column show evolution of wave shape with increase of energy flow U . Column (a) corresponds to focusing nonlinearity, column (b) corresponds to defocusing nonlinearity (Kartashov, Vysloukh and Torner, 2003)
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properties of stationary periodic waves in such media are also strongly affected by the pattern orientation: square patterns parallel to the crystal’s c-axis are less robust than diamond patterns oriented diagonally. As well as lattices with the simplest chessboard phase structure, a variety of self-trapped periodic patterns was demonstrated by Desyatnikov, Sagemerten, Fisher, Terhalle, Tr¨ager, Neshev, Dreischuh, Denz, Krolikowski and Kivshar (2006), including triangular lattices produced by interference of six plane waves (Figure 17) and vortex lattices produced by waves with nested arrays of vortex-type phase dislocations. A detailed analysis of two-dimensional nonlinear self-trapped photonic lattices in anisotropic photorefractive media was carried out by Terhalle, Tr¨ager, Tang, Imbrock and Denz (2006). Nonlinear periodic structures may interact with localized soliton beams. Localized beams cause strong deformations of periodic patterns and under appropriate conditions they can form composite states (Desyatnikov, Ostrovskaya, Kivshar and Denz, 2003). Thus, a periodic nonlinear lattice propagating in a defocusing medium localizes other component in the form of a stable gap soliton. A variety of stationary ¨ composite lattice-soliton states of the vector Kerr nonlinear Schrodinger equation were found via the Darboux transformation technique by Shin (2004). In focusing media all such composite states were found to be unstable due to the instability of periodic components, while the only stable state was found in the defocusing medium. Experimentally, one-dimensional composite gap solitons supported by nonlinear lattices were observed in LiNbO3 crystals possessing saturable defocusing nonlinearity. Such states form when a Gaussian beam is launched at a Bragg angle into a nonlinear lattice (Song, Liu, Guo, Liu, Zhu and Gao, 2006). In contrast to the case of fixed lattices, transverse soliton mobility can be greatly enhanced in nonlinear lattices that experience deformations at soliton locations due to cross-modulation coupling (Sukhorukov, 2006a). Interaction of two-dimensional localized solitons with partially coherent nonlinear lattices was studied experimentally by Martin, Eugenieva, Chen and Christodoulides (2004). Such lattices might be produced with amplitude masks that modulate otherwise uniform incoherent beams, while changing the degree of incoherence allows control of the stability of the periodic pattern. When the intensity of the soliton is comparable with that of the lattice and it is launched at a small angle relative to the lattice propagation direction, the lattice dislocation is created due to soliton dragging. In this case the transverse soliton shift is much smaller in the presence of the lattice due to the strong soliton-lattice interaction.
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FIGURE 17 Theoretical (color) and experimental (grayscale) results for self-trapped triangular lattices created by interference of six plane wave (see far field images in the top panel). Two distinct orientations of the triangular lattices with respect to the crystallographic c axis are compared (top and bottom panels) as well as low (left) and high (right) saturation regimes. LW, lattice field (color) and intensity (grayscale) of the lattice wave. GW, calculated profiles of the refractive index (color) and measured intensity of the probe linear wave guided by the lattice (grayscale) (Desyatnikov, ¨ Sagemerten, Fisher, Terhalle, Trager, Neshev, Dreischuh, Denz, Krolikowski and Kivshar, 2006). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Polaron-like structures may form in a partially coherent nonlinear lattice with small enough spacing between its spots. In such structures the soliton drags toward it some of the lattice sites while pushing away the others. Interaction between vortex beams and nonlinear partially coherent lattices may result in lattice twisting driven by the angular momentum carried by the vortex beam, with the direction of twisting being opposite for vortices with opposite charges (Chen, Martin, Bezryadina, Neshev, Kivshar and Christodoulides, 2005). The formation of two-dimensional
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composite solitons upon interaction of mutually incoherent lattice and stripe beams was observed by Neshev, Kivshar, Martin and Chen (2004). Such interaction results in decrease of the spacing between the lattice rows in the interaction region while moving stripes cause lattice compression and deformation.
10. DEFECT MODES AND RANDOM LATTICES In this section we summarize extensive theoretical and experimental activity of different research groups in the field of so called defect modes. Some structural imperfections are inevitable during fabrication of waveguide arrays as well as upon optical lattice induction, so their influence on the properties of solitons should be carefully studied. Moreover, specially designed defects could even be applied for controllable filtering, switching, and steering of optical beams in lattices.
10.1 Defect Modes in Waveguide Arrays and Optically-induced Lattices The propagation of discrete solitons in a waveguide array in the presence of localized coupling constant perturbations was studied by Krolikowski and Kivshar (1996). Some potential schemes for controllable soliton switching and steering, utilizing array defects were discussed. Later, suppression of defect-induced guiding due to focusing nonlinearity in an AlGaAs waveguide array was demonstrated experimentally (Peschel, Morandotti, Aitchison, Eisenberg and Silberberg, 1999). The existence and stability of nonlinear localized waves in a one-dimensional periodic medium described by the Kronig-Penney model with a nonlinear defect was reported by Sukhorukov and Kivshar (2001). A novel type of stable nonlinear band-gap localized state has been found. The existence and stability of bright, dark, and twisted spatially localized modes in arrays of thin-film nonlinear waveguides that can be viewed as multiple point-like nonlinear defects in an otherwise linear medium (Dirac-comb nonlinear lattice) was presented by the same authors (Sukhorukov and Kivshar, 2002b). An overview of the properties of nonlinear guided waves in such structures can be found in the paper by Sukhorukov and Kivshar (2002c), where the authors also discuss the similarities and differences between discrete and continuous models. Kominis (2006) and Kominis and Hizanidis (2006) applied phasespace methods for construction of analytical bright and dark soliton solutions of the nonlinear Kronig-Penney model, describing the periodic photonic structure with alternating linear and nonlinear layers. The existence of stable nonlinear localized defect modes near the band edge of a two-dimensional reduced-symmetry photonic crystal with Kerr
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nonlinearity was predicted by Mingaleev and Kivshar (2001). The physical mechanism resulting in the mode stabilization was revealed. A new type of waveguide lattice that relies on the effect of bandgap guidance in the regions between waveguide defects was proposed by Efremidis and Hizanidis (2005). It was shown that spatial solitons that emerge from different spectral gaps of a periodic binary waveguide array can be selectively reflected or transmitted through an engineered defect, which acts as a low- or high-pass filter (Sukhorukov and Kivshar, 2005). Defect modes in a one-dimensional optical lattice were discussed by Brazhnyi, Konotop and Perez-Garcia (2006a) in the context of matter waves transport in optical lattices. Such modes supported by the localized defects in otherwise periodic lattices can be accurately described by an expansion over Wannier functions, whose envelope is governed by the ¨ coupled nonlinear Schrodinger equations with a δ-impurity (Brazhnyi, Konotop and Perez-Garcia, 2006b). The interaction of moving discrete solitons with defects in onedimensional fabricated arrays of semiconductor waveguides was investigated experimentally by Morandotti, Eisenberg, Mandelik, Silberberg, Modotto, Sorel, Stanley and Aitchison (2003). Important changes of the output soliton positions (a sudden switch across a narrow repulsive defect) were observed for relatively small changes of input conditions. The formation of defect modes in polymer waveguide arrays was investigated both theoretically and experimentally by Trompeter, Peschel, Pertsch, Lederer, Streppel, Michaelis and Brauer (2003). It was shown that variation in the defect strength (or effective index) is accompanied by the modification of the character and number of modes bound to the defect. Although the symmetric defect waveguide becomes multimode with the increase of effective index it does not support antisymmetric modes. The dynamical properties of one-dimensional discrete NLSE with arbitrary distribution of on-site defects were considered theoretically by Trombettoni, Smerzi and Bishop (2003). Several possible regimes of beam propagation, including its complete reflection, oscillations around fixed points and formation of self-trapped states, have been predicted. The trapping of moving discrete solitons by linear and nonlinear impurities in discrete arrays was discussed by Morales-Molina and Vicencio (2006). Linear defect modes can also exist in photoinduced lattices with both positive (locally increased refractive index) and negative (locally decreased refractive index) defects. Guided modes supported by relatively extended positive and negative defects in hexagonal optical lattices and far-field power spectra of probe beams propagating in such lattices that clearly reveal the physical mechanisms leading to localization of linear modes (total internal or Bragg reflection) were presented by Bartal, Cohen, Buljan, Fleischer, Manela and Segev (2005). The band-gap
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FIGURE 18 Top panel: Intensity pattern of a two-dimensional optically induced lattice with a single-site negative defect at crystal input (a) and output (b), (c). The defect disappears in the linear regime (b) but can survive with weak nonlinearity (c) after propagating through a 20-mm-long crystal. Bottom panel: Input (a) and output (b), (c) of a probe beam showing band-gap guidance by the defect (c) and normal diffraction without the lattice (b) under the same bias condition (Makasyuk, Chen and Yang, 2006)
guidance of light in optically induced 2D lattice with single-site negative defect was observed by Makasyuk, Chen and Yang (2006). Defect modes supported by such lattices feature long tails in the directions of the lattice principal axes as shown in Figure 18. In linear case the strongest defect mode confinement appears when the lattice intensity at the defect site is nonzero rather than zero (Fedele, Yang and Chen, 2005; Wang, Yang and Chen, 2007). It was shown that in nonlinear regime defect solitons bifurcate from every infinitesimal linear defect mode (Yang and Chen, 2006). At high powers in the medium with focusing nonlinearity defect soliton modes become unstable in the case of positive defects, but may remain stable in the case of negative defect. Defect solitons in optically induced one-dimensional photonic lattices in LiNbO3 :Fe crystals were observed experimentally by Qi, Liu, Guo, Lu, Liu, Zhou and Li (2007). Wave and defect dynamics in twodimensional nonlinear photonic quasicrystals was studied experimentally by Freedman, Bartal, Segev, Lifshitz, Christodoulides and Fleischer (2006). The main properties of solitons supported by defects embedded in superlattices have been analyzed by Chen, He and Wang (2007).
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10.2 Anderson Localization The concept of Anderson localization was originally introduced in the field of condensed matter physics for the phenomenon of disorderinduced metal–insulator transition in linear electronic systems. Anderson localization refers to the situation where electrons, when released inside a random medium, may stay close to the initial point (Anderson, 1978). The mechanism behind this phenomenon has been attributed to multiple scattering of electrons by random potential, a feature of the wave nature of electrons. This concept may also be applied to classical linear wave systems (Maynard, 2001). Different linear regimes of light localization in disordered photonic crystals were considered theoretically and experimentally by Vlasov, Kaliteevski and Nikolaev (1999). In particular, it was shown that Bloch states are disrupted only when local fluctuations of the band-edge frequency (caused by randomization of the refractive index profile) become as large as the band-gap width. Band theory of light localization in one-dimensional linear disordered systems was presented by Vinogradov and Merzlikin (2004). It was detected that Bragg reflection is responsible not only for the appearance of band-gaps but also for Anderson localization of light in the one-dimensional case. Schwartz, Bartal, Fishman and Segev (2007) reported on the experimental observation of Anderson localization in perturbed periodic potentials. In the setup used by authors the transverse localization of light was caused by random fluctuations on a two-dimensional photonic lattice imprinted in a photorefractive crystal. The authors demonstrated how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at high levels of disorder. In this experiment the controllable level of disorder was achieved by combining random and regular lattices, while optical induction techniques allowed easy creation of the required ensemble of lattice realizations. Anderson localization in one-dimensional disordered waveguide arrays has also been observed by Lahini, Avidan, Pozzi, Sorel, Morandotti, Christodoulides and Silberberg (2008). From the viewpoint of fundamental nonlinear physics the challenging problem is the exploration of nonlinear analogues of Anderson localization. The interplay between disorder and nonlinearity was studied ¨ in systems described by the one-dimensional nonlinear Schrodinger equation with random-point impurities (see, e.g., Kivshar, Gredeskul, Sanchez and Vazquez (1990), Hopkins, Keat, Meegan, Zhang and Maynard (1996)) as well as in one-dimensional (Kevrekidis, Kivshar and Kovalev, 2003) and two-dimensional (Pertsch, Peschel, Kobelke, Schuster, ¨ Bartlet, Nolte, Tunnermann and Lederer, 2004) discrete waveguide arrays. Anderson-type localization of moving spatial solitons in optical lattices
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with random frequency modulation was studied by Kartashov and Vysloukh (2005). In this case dramatic enhancement of soliton trapping probability on lattice inhomogeneities was detected with increase of frequency fluctuation level. The localization process is strongly sensitive to lattice depth, since in shallow lattices moving solitons experience random refraction and/or multiple scattering, in contrast to relatively deep lattices, where solitons are typically immobilized in the vicinity of modulation frequency local minima. It is worth noticing that analogous phenomena were encountered in trapped BECs (Greiner, Mandel, Esslinger, H¨ansch and Bloch, 2002; Abdullaev and Garnier, 2005; Kuhn, Miniatura, Delande, Sigwarth and Muller, 2005; Schulte, Drenkelforth, Kruse, Ertmer, Arlt, Sacha, Zakrzewski and Lewenstein, 2005; Gavish and Castin, 2005).
10.3 Soliton Percolation The combined effect of periodic and random potentials on the transmission of moving and formation of stable stationary excitations has been addressed in a number of studies and reviews (Bishop, Jimenez, and Vazquez, 1995; Abdullaev, Bishop, Pnevmatikos and Economou, 1992; Abdullaev, 1994). A universal feature of wave packet and particle dynamics in disordered media in different areas of physics is percolation (Grimmett (1999); Shklovskii and Efros (1984)). Percolation occurs in different types of physical settings, including high-mobility electron systems, Josephson-junction arrays, two-dimensional GaAs structures near the metal–insulator transition, or charge transfer between a superconductor and a hoping insulator, to mention a few. The nonlinear optical analog of biased percolation was introduced by Kartashov, Vysloukh and Torner (2007c). The phenomenon analyzed is the disorder-induced soliton transport in randomly modulated optical lattices with Kerr-type focusing nonlinearity in the presence of a linear variation of the refractive index in the transverse plane, thus generating a constant deflecting force for light beams entering the medium. When such a force is small enough, solitons in perfectly periodic lattices are trapped in the vicinity of the launching point due to the Peierls–Nabarro potential barriers, provided that the launching angle is smaller than a critical value. Under such conditions soliton transport is suppressed, and thus the lattice acts as a soliton insulator. However, random modulations of the lattice parameters makes soliton mobility possible again, with the key parameter determining the soliton current being the standard deviation of the phase/amplitude fluctuations. Kartashov, Vysloukh and Torner (2007c) discovered that the soliton current in lattices with amplitude and phase fluctuations reaches its maximal value at intermediate disorder levels and that it drastically reduces in both, almost regular and strongly
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disordered lattices. This suggests the possibility of a disorder-induced transition between soliton insulator and soliton conductor lattice states.
11. CONCLUDING REMARKS To conclude, we would like to add some comments on two important fields emerging from theoretical and experimental research in the area of lattice solitons, which fall out of the main stream of this review. The first topic is the generalization of the lattice soliton concept to partially coherent or incoherent (such as white-light) optical fields. From a physical point of view, the diffraction spreading of a partially coherent light beam is determined by the transverse coherence length rather than by the beam radius. In shallow quasicontinuous lattices, this means that self-trapping and soliton formation would normally require higher power levels for incoherent radiation. In this case, the ratio between the transverse coherence length and the lattice period becomes crucial, since the soliton formation process is sensitive to the phase relations between several neighboring lattice channels. Thus, the complex interplay between coherence, nonlinearity, and waveguiding becomes central. A powerful theoretical approach to describe the propagation and self-focusing of partially spatially incoherent light beams in bulk photorefractive media was developed by Christodoulides, Coskun, Mitchell and Segev (1997), Mitchell, Segev, Coskun and Christodoulides (1997). It was followed by the experimental observation of self-trapping of white light in bulk media (Mitchell and Segev, 1997). Further theoretical progress was reported by Buljan, Segev, Soljacic, Efremidis and Christodoulides (2003). The existence of random phase solitons in nonlinear periodic lattices was predicted by Buljan, Cohen, Fleischer, Schwartz, Segev, Musslimani, Efremidis and Christodoulides (2004). Importantly, such solitons exist when the characteristic response time of the medium is much longer than the coherence time. The prediction mentioned above was confirmed experimentally (Cohen, Bartal, Buljan, Carmon, Fleischer, Segev and Christodoulides, 2005). Lately the existence of lattice solitons made of incoherent white light, originating from an ordinary incandescent light bulb, was analyzed by Pezer, Buljan, Bartal, Segev and Fleisher (2006). Prospects of incoherent gap optical soliton formation in a self-defocusing nonlinear periodic medium was analyzed theoretically (Motzek, Sukhorukov, Kaiser and Kivshar, 2005; Pezer, Buljan, Fleischer, Bartal, Cohen and Segev, 2005), and further supported by experimental observation (Bartal, Cohen, Manela, Segev, Fleischer, Pezer and Buljan, 2006). The phenomenon of modulational instability of extended incoherent nonlinear waves in photonic lattices was addressed by Jablan, Buljan, Manela, Bartal, and Segev (2007).
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A second intriguing and practically important topic directly linked with lattice solitons is the physics of nonlinear surface wave formation at the interface of periodic structures. Surface waves were originally introduced by Igor E. Tamm in 1932 in the context of condensedmatter physics and have been intensively studied in different areas of science. However the progress in experimental investigation of nonlinear optical surface waves was severely limited by the high power thresholds required for their formation. It has been predicted theoretically (Makris, Suntsov, Christodoulides, Stegeman and Hache, 2005) and confirmed experimentally (Suntsov, Makris, Christodoulides, Stegeman, Hache, Morandotti, Yang, Salamo and Sorel, 2006) that surface waves formation at the very edge of a one-dimensional waveguiding array is accessible for moderate power levels. Formation of gap surface solitons at the edge of a defocusing lattice is also possible as it was shown theoretically (Kartashov, Vysloukh and Torner, 2006a) and confirmed experimentally in defocusing LiNbO3 waveguiding arrays (Rosberg, Neshev, Krolikowski, Mitchell, ¨ Vicencio, Molina and Kivshar, 2006; Smirnov, Stepic, Ruter, Kip and Shandarov, 2006). Siviloglou, Makris, Iwanow, Schiek, Christodoulides, Stegeman, Min and Sohler (2006) reported on experimental observation of discrete quadratic surface solitons in self-focusing and defocusing periodically poled lithium niobate waveguide arrays. Lattice interfaces imprinted in nonlocal nonlinear media support interesting families of multipole surface solitons (Kartashov, Torner and Vysloukh, 2006). Two-dimensional geometries offers even richer possibilities for surface wave formation (Kartashov and Torner, 2006; Kartashov, Vysloukh, Mihalache and Torner, 2006; Kartashov, Egorov, Vysloukh, and Torner, 2006b; Makris, Hudock, Christodoulides, Stegeman, Manela and Segev, 2006). Solitons residing at the lattice edges, in the corners, and on specially designed defects are illustrative examples. Experimental observations of two-dimensional surface waves were realized at the boundaries of a finite optically induced lattice (Wang, Bezryadina, Chen, Makris, Christodoulides and Stegeman, 2007), and in laser-written waveguide ¨ and arrays (Szameit, Kartashov, Dreisow, Pertsch, Nolte, Tunnermann Torner, 2007). Many problems remain open to explore the combination of surface solitons and optical lattices. We thus conclude by stressing that we have presented a concise review of the advances in optical soliton formation, manipulation, shaping, and control in optical lattices, with special emphasis on optically-induced lattices. Many concepts investigated in the context of optical lattices have implications and applications in several branches of physics, such as nonlinear optics, condensed-matter theory, and quantum mechanics. Optical soliton control in optical lattices offers a unique laboratory
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to investigate universal phenomena, and as such provides a unique exploration tool.
ACKNOWLEDGEMENTS We thank all our colleagues and collaborators for invaluable discussions in conducting joint research reported here and in completing this review. Most especially we thank the members of our group, the co-authors of joint papers described here, and G. Assanto, R. Carretero-Gonzalez, Z. Chen, D. Christodoulides, L. Crasovan, C. Denz, A. Desyatnikov, A. Ferrando, D. Kip, Y. Kivshar, F. Lederer, M. Lewenstein, B. Malomed, D. Mazilu, D. Mihalache, M. Molina, D. Neshev, T. Pertsch, M. Segev, G. Stegeman, A. Szameit, Z. Xu, F. Ye for contributing their knowledge and figures included in this review. This work was produced with financial support by the Generalitat de Catalunya and by the Government of Spain through the Ramon-y-Cajal program and grant TEC2005-07815.
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Yang, J. and Musslimani, Z. H. (2003). Fundamental and vortex solitons in a two-dimensional optical lattice. Opt. Lett., 28, 2094–2096. Yulin, A. V., Skryabin, D. V. and Russell, P. St. J. (2003). Transition radiation by matter-wave solitons in optical lattices. Phys. Rev. Lett., 91, 260402. Yulin, A. V., Skryabin, D. V. and Vladimirov, A. G. (2006). Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion. Opt. Express, 14, 12347–12352.
CHAPTER
3 Signal and Quantum Noise in Optical Communications and Cryptography Philippe Gallion a,1 , Francisco Mendieta b,2 and Shifeng Jiang a,1 a Ecole Nationale Sup´erieure des T´el´ecommunications,
TELECOM ParisTech, and CNRS LTCI, Paris b CICESE, Carretera Tijuana-Ensenada km. 107, Ensenada, Baja California, 22860, Mexico Contents
1.
2.
3.
Introduction Quantum and Thermal Noises 1.1 1.2 Limit of the Semi-classical Approach for Quantum Noise 1.3 New Trends in Optical Communication Engineering 1.4 Chapter Objective and Organization Basic Concepts of Quantum Optics 2.1 Quantum Description of Optical Fields 2.2 Quantum States of Optical Fields 2.3 Quantum Probability Distributions 2.4 Engineering Description of Quantum Noise Non-commutating Quadrature Measurements and Quantum State Distinguishability 3.1 Non Commutating Quadrature Measurements 3.2 Quantum Detection 3.3 Symmetrical Coherent States 3.4 Open Loop Quantum Receivers 3.5 Feedback Quantum Receivers
150 150 151 152 153 155 156 161 165 170 177 178 184 185 187 189
1 Tel.: +33 (0)1 45 81 77 02. 2 Tel.: +52 646 1750500.
E-mail addresses:
[email protected] (Philippe Gallion),
[email protected] (Francisco Mendieta),
[email protected] (Shifeng Jiang). URLs: http://www.telecom-paristech.fr/ (P. Gallion,S. Jiang), http://www.cicese.mx (F. Mendieta). c 2009 Elsevier B.V. Progress in Optics, Volume 52 ISSN 0079-6638, DOI 10.1016/S0079-6638(08)00005-X All rights reserved.
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4.
Optical Amplification 4.1 Minimum Output Additive Noise 4.2 Attenuation and Amplification Noises 4.3 Quantum Langevin Approach 4.4 Equivalent Lumped Amplifier Noise Factor 4.5 Pre-amplified Optical System Applications 4.6 Technical Noise and Impairment in Optical Amplification 4.7 Linear and Phase-Sensitive Amplification 4.8 Optical Amplification Sensitivity 5. Single Quadrature Homodyne Detection 5.1 Quantum Theory of Homodyne Detection 5.2 Second Order Momentum and Signal to Noise Ratio 5.3 Comparison with the Classical Approach Result 5.4 Coherent Optical System Applications 5.5 Application to Cryptography 5.6 Technical Noise and Impairment in Homodyne Detection 6. In-phase and Quadrature Measurements 6.1 Phase Estimation with Classical Signals 6.2 Phase Estimation with Quantum Signals 6.3 Heterodyne Detection 6.4 I–Q Measurements 6.5 Suppressed Carrier Phase Estimation 6.6 In-phase and Quadrature Measurements Applications 7. Conclusion Acknowledgments Appendix A.1 General Quantum Field Input–Output Relations A.2 Abbreviation Index A.3 Notation Index References
190 191 194 196 199 199 202 202 203 204 207 210 211 212 217 220 226 227 229 231 233 234 236 238 244 245 245 247 249 251
1. INTRODUCTION 1.1 Quantum and Thermal Noises In addition to the thermal noise corresponding to black body radiation in a single mode of propagation, the quantum noise, corresponding to the uncertainty principle, is one of the two fundamental limitations for any electrical or optical engineering system designed to handle and deal with a weak signal. As the quantum noise is unavoidable, the thermal noise influence may theoretically be reduced at high frequency ranges and at low temperature. However, even though the photon energy hν of the
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order of 10−19 J for a wavelength in the 1500 nm range, for instance, is roughly one hundred times larger than the thermal energy kT at room temperature, the thermal noise influence, up to now, remains important in optical engineering, mainly when optical-to-electronic (OE) downconversion is concerned. Moreover, high levels of technical and thermal noise and drifts are usually present. Furthermore, because of the high number of photons involved during each observation time, for instance the bit duration in an optical communication system, the major aspects of the quantum nature of light are smoothed out by ensemble averaging.
1.2 Limit of the Semi-classical Approach for Quantum Noise Until recently quantum noise was of little interest to optical and electronic engineers since for major application fields the influence of intrinsic light fluctuations was described in a nearly satisfactory way by using coarse models involving, at the same time and in an uncorrelated manner, a classical corpuscular description of the light and a classical wave one. For direct detection optical systems as far as photo detection is concerned, only a single quadrature of the optical field is observed, and the classical Poissonian shot noise, sometimes thought of as only occurring in photo detection, provides an appropriate description of optical quantum noise (Personick, 1973a,b). However, for the same systems, the wave nature of optical signals is independently invoked for descriptions of propagation, dispersion, multi-path interference, signal mixing and beating. Using the wave and corpuscular descriptions of light in the same experiment is not in agreement with Bohr’s principle of complementarity (Winzer and Leeb, 1998), making such approaches unable to provide an accurate description for future generations of systems with performances closer to the quantum limits. For direct detection of an optically pre-amplified signal, a classical wave description in terms of the well-known “signal-against-noise” and “noise-against-noise” beating (Kingston, 1978; Yamamoto, 1980) is usually preferred. This approach correctly gives the dominant noise terms but fails to give an exact and complete description since, for instance, the heuristic addition of an output shot noise must be included afterwards, as pointed out by Donati and Giuliani (1997). In such an approach, the averaged amplified spontaneous emission (ASE) noise power is calculated first, and its stochastic nature only appears in the square-law process of detection by beatings of random phase under implicit Gaussian assumption, but also under the quite strange assumption of a constant amplitude noise signal. Since, in this case, the post detection noise is not additive, it has been pointed out by Haus (1998) that the definition of an electrical signal-to-noise ratio (ESNR) and its corresponding noise figure (NF) are both tedious and inaccurate. For instance, the ENSR at the amplifier
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input refers to a hypothetical photo detection before the amplifier (Baney, Gallion and Tucker, 2000), the independence of noise in the incoming signal is an approximation and the standard noise figure cascading formula does not hold. For coherent detection, the usual description of homodyne or heterodyne optical detection, sometimes referred to as “dyne” measurement, is usually done in terms of beating as far as the signal is concerned, and is therefore closely related to the wave description of light. However, where the output noise is concerned, it is common to switch to a corpuscular description of light, leading to the usual conclusion that local-oscillator shot noise, sometimes thought of as only occurring in photo detection, is the fundamental noise. Quantum phase noise and the subsequent phase diffusion, which is a wave phenomenon, are usually reintroduced afterwards independently in terms of laser line width, and usually discussed by using the corpuscular representation of spontaneous emission proposed by Henry (1982). Using the two descriptions of light in the same experiment is again not in agreement with Bohr’s complementarity principle. Such an approach assumes that signal and local beams entering the mixer are noise free. Thanks to work on nonlinear propagation of optical signals and device-oriented research, the development of optical communication systems now includes more advanced information along with digital communication theory concepts. In standard digital communication theory, a classical two-quadrature component description of a signal is widely used, in which any signal is the vector sum of the informative envelope and an additive two-dimensional band-limited stationary Gaussian noise with a flat spectrum within a pass-band bandwidth (Rice, 1948; Rowe, 1965; Davenport and Root, 1985).
1.3 New Trends in Optical Communication Engineering The coherent technique was first widely studied for free space optical communications (Gagliardi and Karp, 1976). After the preliminary work on fiber coherent communication in the 1980s, less interest was paid to this technique because of the development of Erbium-doped amplifiers (EDFA) and, more recently, of Raman distributed amplifiers (DRA). During the same period, the quantum theory of coherent, i.e. homodyne or heterodyne, detection, was developed by Schumaker (1982), Yuen and Chan (1983), Machida and Yamamoto (1986) and Shapiro, Yuen, and Machado Mata (1979) and Shapiro (1985). A strong interest has now been revived in coherent techniques, which allow simultaneously, a noise-free mixing gain, a constant envelope phase modulation robust to fiber nonlinearities, a high spectral efficiency modulation more robust to optical fiber dispersion, an intermediate frequency electrical processing
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for propagation impairment corrections to deal with frequency and phase mismatch effects. In addition to optical communication systems, the coherent techniques are also of wide interest for future radio-overfiber (RoF) systems, free space optics, coherent lidar and optical sensors, coherent optical instrumentation and quantum tomography. Quantum cryptography is now moving from the promise of physics to the hard reality of electrical engineering and it obviously requires the full quantum nature of light. It is also dealing with coherent detection techniques for the continuous variable protocol (Timothy, 1999; Grosshans and Grangier, 2001, 2002), for the protocol proposed by Yuen (2001) and Yuen (2003) and referred to as “Yuen 2000”, and also for the Bennett and Brassard (1984) (referred to as the BB84 protocol) for which Hirano, Yamanaka, Ashikaga, Konishi and Namiki (2003) have recently proposed to use a homodyne detection in association with more system efficient phase modulation schemes. Nowadays a more accurate optical signal representation including fundamental quantum optical fluctuation is called for to permit the future development of optical engineering systems.
1.4 Chapter Objective and Organization The original works on quantum mechanics and its application to optical engineering already include a substantial and exciting part of the physics literature. The first aim of this chapter is to provide a selfconsistent and easily understandable treatment of quantum noise, in which light is described by a simplified quantum theory which includes at the same time phase and amplitude fluctuations. Such a treatment is useful, and some times mandatory, for applied physics, scientific experimentation and associated optical and electrical technologies as well, to deal with new applications in which the quantum nature of light is more and more determinant. The second aim is to review the major theoretical and experimental exploration of the ways of approaching the so-called standard quantum limit (SQL) in weak optical signal detections. The last aim is to improve on the very weak bridge that exists between the basic concepts of physics and the world of applications, experimentation and engineering which, thanks to recent development of devices, techniques and applications, needs to incorporate an enlarged background of quantum physics concepts. Some illustrative examples of applications belonging to the field of optical communications and quantum cryptography, are reported by using standard electrical engineering concepts, notations and semantics. Such an approach would be Promethean without the very valuable contributions in already published papers (Oliver, Haus and Townes,
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1962; Haus, 1995; Nilsson, 1994; Berglind and Gillner, 1994; Leonhardt, 1997; Yamamoto and Nilsson, 1997; Gordon, 1999; Shapiro, 2003). After this introduction, the second section of this chapter is devoted to reviewing the minimum basics of quantum electrodynamics now required for optical system design. It includes a short introduction to the photon creation and annihilation operators and to the commutation relations. Much attention is paid to coherent states which are the more classical ones and up to now the more useful, since they are easily produced by well-stabilized conventional, reliable and inexpensive, light sources and, therefore, cover the larger part of today’s application fields. The coherent states are also a good representation of strongly fainted laser pulses used in cryptography application. Furthermore, the excess noise of a nearly coherent state source disappears though attenuation, turning it to a coherent state. By using a symmetrized noise energy operator, a pragmatic additive Gaussian white noise (AGWN) description is introduced, before a brief discussion of the quantum noise limited optical channel capacity. The third section of this chapter addresses the problem of distinguishing quantum states and of their discrimination, which is the key issue in major applications. We start with a simple analysis of the quantum limits of the non-commutating quadrature measurements of optical fields. A more general discussion of the input-to-output relation is proposed in the appendix. Despite their wide pedagogical and application-oriented interest, coherent states are not orthogonal in the Hermitian product sense, making their error-free distinction impossible. Minimum error theoretical bounds, optimum structures for the binary channel, nulling and homodyne receivers are discussed and compared. The fourth part is devoted to the quantum noise in optical amplification. The phase insensitive linear optical amplifier minimum added noise is first derived from the commutations relationship, independent of the amplifier structure. Performances of lumped and distributed amplifiers are compared in terms of noise figure and equivalent noise figure. The performances of pre-amplified direct detection optical systems applications are addressed. Technical noises and their relative importance are finally discussed, as well as noise generation in phase sensitive and nonlinear amplification. Homodyne detection is the subject of the fifth section. Homodyne detection is sometimes referred to as degenerate homodyne detection since the usual two mixed fields of heterodyne detection have the same frequency in this case. Yamamoto and Haus (1986) have pointed out that homodyne detection is, with degenerate parametric amplification and cavity four-wave mixing (FWM), one of the noise-free single quadrature measurement techniques available today. Homodyne detection is obviously the easier to be implemented. A simplified
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quantum theory of homodyne detection will be proposed and compared to the classical one. The bit error ratio (BER) induced by the signal, quantum noise, optimal performance and the usual technical noise and impairments are presented. Single threshold homodyne detection in application to coherent optical systems, and double threshold application to quantum cryptography, first proposed by Namiki and Hirano (2003), will be reviewed. The sixth section is devoted to in-phase and in-quadrature (IQ) measurements. Heterodyne detection and phase estimation for the classical and quantum channel, tracking structures based on IQ measurements, particularly for suppressed carrier modulations, are reviewed in relation to the number-phase uncertainty relationship. Finally applications of IQ detection in telecommunications and in cryptography are mentioned. The chapter ends with a short conclusion summarizing the fundamental limits and the future perspective to reach them.
2. BASIC CONCEPTS OF QUANTUM OPTICS Quantum theory differs from linear circuit theory fundamentally, both in objectives and formalism. Quantum theory is a description of the time evolution of representing observables. This is performed in the ¨ Heisenberg representation, in opposition to the Schrodinger one, with invariant states and time evolution of operators. On the other hand, linear circuit theory is the expression of a circuit output as a function of a spatially separated input. As pointed out by Haus (1995), these two points of view at first seem very different but they meet when the time evolution of narrow-band wave packets is addressed, because the impulse response and the time delay between input and output are equivalent, by the Fourier transform, to a transfer function. In agreement with Ehrenfest’s correspondence principle, Rice’s representation, in which a classical narrow-band signal can be considered as the sum of two quadrature components, is the classical counterpart of the phase and quadrature components of the non Hermitian (non self-adjoint) photon creation operator. In this section, we introduce some fundamental theoretical aspects of quantum optics that will be used in this chapter. First, we will briefly present the quantum description of quasi-monochromatic optical fields, which includes the field operators and quantum states of the field. The quantum probability distributions will then be introduced, as well as an engineering description of quantum noise.
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2.1 Quantum Description of Optical Fields 2.1.1 Quasi-monochromatic Optical Field Operators In the classical wave theory, a quasi-monochromatic optical field E (t) in a single spatial mode and with an angular central carrier frequency ω0 = 2π ν, is described by its complex slowly time-varying envelope a (t) written as r E (t) =
hν a (t) exp jω0 t + c.c. T
(1)
where c.c. signifies the complex conjugate, and where the field is assumed to refer to a single polarization mode and therefore to be scalar, for the sake of simplicity. The time duration T will refer in practice to the observation time which is simultaneously assumed as far longer than the optical period and as far smaller than the coherence duration of the optical source. The field envelope is so normalized that the short-time-average optical power P is given by hPi = hνh|a(t)|2 i = h|E(t)|2 i, where hν = h¯ ω is the photon energy, h = 6.63.10−34 J.s is Planck’s constant, h¯ = h/2π being the reduced Planck’s constant. E (t) is therefore expressed as (Joules per second)1/2 . Over the finite time interval T , one can further perform the standard Fourier expansion of the slowly time-varying complex envelope as a (t) =
+∞ X
2kπ t . ak exp j T k=−∞
(2)
The total time-dependent envelope as well as each of its time-independent Fourier components can be decomposed into a real in-phase component (denoted by the subscipt I ) and an imaginary quadrature component (denoted by the subscipt Q) such that a (t) = a I (t) + ja Q (t)
and ak = ak,I + jak,Q .
(3)
The fluctuations of the field are expressed either by the temporal fluctuations of the envelope a (t) or by the uncertainties of the Fourier spectral component coefficients ak . In the Heisenberg picture of the quantum description, the classical field E (t) is replaced by a time varying Hermitian operator Eˆ (t) analogously
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written as r
i hν h aˆ (t) exp jω0 t + aˆ Ď (t) exp −jω0 t T r hν =2 aˆ I (t) cos ω0 t − aˆ Q (t) sin ω0 t T
Eˆ (t) =
(4)
where aˆ is the non Hermitian photon annihilation quantum operator and its adjoint aˆ + is the photon creation operator. aˆ I and aˆ Q are, respectively, the in-phase and quadrature Hermitian components of a. ˆ In the same way as for the classical field, we can perform the Fourier expansion of the normalized slowly varying operator as aˆ (t) =
+∞ X
2kπ aˆ k exp j t T k=−∞
(5)
where aˆ k is the non Hermitian photon annihilation quantum operator and its adjoint aˆ k+ is the photon creation operator in the corresponding Fourier mode. For the sake of simplification, as far as a single Fourier mode is concerned, we will omit the corresponding subscript. For the timedependent envelope multimode field, we will always write out explicitly the time-argument of the photon number operator Nˆ and photon annihilation operator aˆ so as to avoid any possible ambiguity. Thus, for a single-mode field the average optical power Hermitian operator Pˆ and the photon number Hermitian operator Nˆ are, respectively, given by hν ˆ N Pˆ = T
and
Nˆ = aˆ Ď a. ˆ
(6)
The optical power P and the average photon number N during the time T are given by D E hν P = Pˆ = N T
and
D E D E N = Nˆ = aˆ Ď aˆ
(7)
where h Xˆ i stands for the average value over the quantum state |ψi of the observable associated to the operator Xˆ , as long as the result does not
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depend explicitly on |ψi D E Xˆ = hψ| Xˆ |ψi .
(8)
2.1.2 Commutation Relationships Quantum operators transform any quantum state into one of its eigenstates. Excepted for their own eigenstates, they modify the state remaining available for another operator action whose eigenstates may be different. Therefore the order in which two operators act affects the obtained result. Therefore two operators Xˆ and Yˆ , with different eigenstates, are characterized by their commutator operator defined as h
i Xˆ , Yˆ = Xˆ Yˆ − Yˆ Xˆ .
(9)
Thus, for a time-independent envelope, single frequency Fourier mode, the non-Hermitian photon annihilation operator aˆ and its adjoint the photon creation operator aˆ + , satisfy the commutation relation (Loudon, 1983) a, ˆ aˆ + = 1.
(10)
For this single frequency Fourier mode, the photon annihilation operator aˆ and its adjoint aˆ + can be expressed as the sum of an in-phase component aˆ I and a quadrature Hermitian component aˆ Q in the form aˆ = aˆ I + jaˆ Q
aˆ + = aˆ I − jaˆ Q
(11)
where we have obviously aˆ I = (aˆ + aˆ Ď )/2 aˆ Q = (aˆ − aˆ Ď )/2j.
(12)
Since the operators aˆ I and aˆ Q are Hermitian, as it will be shown in Section 2.1.3, their commutator is imaginary. It is easily deduced from Equation (10) j aˆ I , aˆ Q = . 2
(13)
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Physical variables for which the commutator operators are equal to j, as is the case within a real multiplicative constant for the two optical field quadratures, are referred to as canonically conjugate observables. For a time-dependent envelope, which is the superposition of multiple frequency Fourier components over the time duration T , the Fourier mode coefficients aˆ k , are without unit. It is the photon annihilation operator for the mode of angular frequency ωk = ω0 + 2kπ/T , fulfilling the commutation relation h i Ď aˆ l , aˆ m = δlm .
(14)
By using the standard Fourier expansion of the periodic Dirac delta function +∞ X n=−∞
δ(t − nT ) =
+∞ 1 X 2π exp j kt T k=−∞ T
(15)
its is easy to derive, from Equation (14), the more general corresponding commutation relation in the form h i aˆ (t) , aˆ Ď t 0 = T δ t − t 0 .
(16)
This relation is in fact a local (i.e. equal-space) commutation relation (Haus and Lai, 1990; Kozlov, 2003), which has been verified for linear optical ¨ Scheel and Welsch, 2001) and media (Huttner and Bernett, 1992; Knoll, can be considered as a good approximation for fiber optics engineering (Kozlov, 2003), while the nonlinearity and the dispersion remain relatively weak. For the in-phase and quadrature Hermitian components of this time dependant envelope, the superposition of multiple frequency Fourier components, the corresponding commutation relation is given by j aˆ I (t) , aˆ Q t 0 = T δ t − t 0 . 2
(17)
Similar relations hold for each time-independent Fourier coefficient also expressed as the sum of an in-phase and quadrature components aˆ k = aˆ k,I + j aˆ k,Q , for which we have the commutation relations as j aˆ l,I , aˆ m,Q = δlm . 2
(18)
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Information communication and processing engineering generally deal with time varying modulated signals. However, assuming a linear channel, it is usual to represent a modulated signal as a statistical mixing of single Fourier modes for which the commutation relations under the form of Equations (10) and (13) may be used.
Separating aˆ into a classical component A = aˆ , including signal modulation and eventually classical noise, and a quantum fluctuation component 1a, ˆ we have aˆ = A + 1a. ˆ
(19)
Since the classical parts commute, only the quantum fluctuations fulfill the commutation relationships 1a, ˆ 1aˆ + = 1
and
j 1aˆ I , 1aˆ Q = . 2
(20)
It is easy to check that we have 1a1 ˆ aˆ + = (1aˆ I )2 + (1aˆ Q )2 +
1 2
1 2 (21)
and 1aˆ + 1aˆ = (1aˆ I )2 + (1aˆ Q )2 −
and therefore 1 (1a1 ˆ aˆ + + 1aˆ + 1a) ˆ = (1aˆ I )2 + (1aˆ Q )2 2
(22)
meaning that the sum of the in-phase and out-of-phase square fluctuation operators is obtained by the symmetrization of the Hermitian operator 1aˆ + 1a, ˆ given by the right-hand side of Equation (22). This symmetrization turns the normal quantum operator order 1aˆ + 1aˆ into a symmetrical situation that is sometime referred to as Weyl’s order. Since 1aˆ + 1aˆ is already Hermitian, this symmetrization is not made here to obtain a self-adjoint extension of the operator, but stands for a quantity which is non-directly observable, but whose effects are, as will be discussed in Section 2.4. 2.1.3 From Commutation to Uncertainty Relationship Let us consider two Hermitian operators Xˆ and Yˆ . Let us consider the linear combination of the corresponding fluctuation operators 1aˆ = 1 Xˆ + jλ1Yˆ
(23)
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where λ is a real number. We have h i 1aˆ + 1aˆ = 1 Xˆ 2 + λ2 1Yˆ + jλ 1 Xˆ , 1Yˆ h i h i with 1 Xˆ , 1Yˆ = Xˆ , Yˆ = Xˆ Yˆ − Yˆ Xˆ .
(24)
Since 1aˆ Ď 1aˆ is Hermitian and semi-positive definite, its average value over any quantum state |ψi is nonnegative h i hψ| 1a + 1a |ψi = (δ X )2 + λ2 (δY )2 + jλ hψ| Xˆ , Yˆ |ψi ≥ 0
(25)
1/2 where δ X = 1X 2 . The last term in Equation (25) is mandatory real, implying an imaginary commutator and the result is nonnegative, implying a negative determinant i.e. 2 h i hψ| Xˆ , Yˆ |ψi − 4(δ X )2 (δY )2 ≤ 0.
(26)
Since the result depends only on the commutator eigenvalues, it does not depend explicitly on the quantum state and we have (δ X )2 (δY )2 ≥
1 Dh ˆ ˆ iE 2 X, Y . 4
(27)
So the uncertainty products appear directly related to the commutator. As a direct consequence of Equation (27), when it is applied to the photon annihilation operator aˆ and its adjoint aˆ + or to the in-phase and inquadrature Hermitian component operators aˆ I and aˆ Q , we have D ED E 1 (1a)2 (1a + )2 = (δa)2 (δa + )2 ≥ 4 D ED E 1 and 1a 2I 1a 2Q = (δa I )2 (δa Q )2 ≥ 16
(28)
2.2 Quantum States of Optical Fields We will mainly restrict our discussion to a single mode of the field. As already mentioned, for the considered mono-mode field, we will always omit the subscript and the time argument of the photon number operator Nˆ and photon annihilation operator aˆ to avoid any possible ambiguity. We will briefly present pure photon number states and coherent states before we introduce mixed states.
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2.2.1 Photon Number States The photon number states |ni introduced by Dirac are the eigenstates of the photon number operator Nˆ = aˆ Ď aˆ Nˆ |ni = n |ni .
(29)
As a direct consequence of its definition the photon number operator Nˆ = aˆ Ď aˆ is Hermitian and nonnegative definite. As will be discussed in Section 2.3 for any quantum state |ψi, the probability that the eigenstate |ni is observed with the eigenvalue n for the photon number is p(n) = |hn|ψi|2 = hψ|ni hn|ψi = hψ| ρˆn |ψi where ρˆn = |ni hn| is the projection operator over the state |ni i.e the corresponding density operator. The eigenstates |ni of the photon number constitute an obviously complete orthonormal vector set, i.e., hm|ni = δmn
and
∞ X
|ni hn| = 1
(30)
n=0
and its eigenvalues n are discrete, real nonnegative integers. It can be shown that, if |ni is an eigenstate of Nˆ = aˆ Ď a, ˆ aˆ |ni is also an eigenstate associated to the eigenvalue n − 1, and aˆ + |ni is also an eigenstate associated to the eigenvalue n + 1. Therefore, it is easy to show the following two relations hold aˆ |ni =
√ n − 1 |n − 1i
and aˆ Ď |ni =
√
n + 1 |n + 1i .
(31)
The solutions of these equations are in the general form aˆ +n |ni = √ |0i . n!
(32)
2.2.2 Coherent States ¨ First introduced by Erwin Schrodinger, coherent states play a key role in the quantum theory of light (Glauber, 1963) and its applications. They are easily produced at low cost by reliable optical coherent sources such as semiconductor lasers (SLC) and are the closest quantum states to a classical pure sinusoidal wave. More precisely, as will be discussed in Sections 2.4.3 and 5.6.3, thanks to phase diffusion, laser sources
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only produce coherent states over a time duration corresponding to the coherence time. Furthermore, as discussed in Section 4.4.2, noisy coherent optical pulses naturally turn into coherent states through attenuation, because their excess of noise naturally disappears. A coherent state, usually denoted by the ket |αi, may be defined as the eigenstate of the photon annihilation photon operator a. ˆ aˆ |αi = α |αi .
(33)
Since the photon annihilation operator aˆ is non-Hermitian, its eigenvalues α are complex and they are also continuous. Since a coherent state is turned to a new one by the annihilation (i.e. the detection) of a photon, the probability of detection of a second photon is unchanged. The statistics of photon detection is therefore Poissonian, as will be derived in Section 2.4.2. A coherent state is the most classical and more wave-like state of the light, in opposition to a single photon Fock’s state which is the more particle one. A coherent state |αi can be expanded on the set |ni as (Glauber, 1963) ∞ 1 2 X αn |αi = exp − |α| √ |ni . 2 n! n=0
(34)
The square of the coefficient of this expansion is obviously the Poisson probability distribution, as will be discussed more extensively in Section 2.3.2. Two coherent eigenstates are not orthogonal, and their overlap can be found as ∞ X ∞ 1 2 2 X αn αm |hα1 |α2 i| = exp − α1 + α2 √ √ hn|mi 2 n=0 m=0 n! m! = exp − |α1 − α2 |2 .
2
(35)
We see, therefore, that the farther apart α1 and α2 are, the more nearly orthogonal are the coherent states |α1 i and |α2 i (Helstrom, 1976). Since the coherent states are not orthogonal the integration of the projection operator ρˆα = |αi hα| over all the possible α values is not equal to 1 and follows the closure relation 1 π
Z
+∞ Z +∞
−∞
−∞
|αi hα| dα 2 = 1.
(36)
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The double integration sign stands for integration over the complex plane. A pure state can be decomposed on the set |αi as |ψi =
1 π
Z
+∞ Z +∞
−∞
|αi hα|ψi dα 2 .
(37)
−∞
2.2.3 Mixed States Signals widely encountered in engineering applications are usually modulated and the corresponding states are not pure states but often in the form of mixed states, which are statistical mixtures of pure states and therefore cannot be represented by the state vectors. In this case, the density operator is necessary for describing the system. The density operator is generally of the form ρˆ =
s X
pn |ψn i hψn | =
n=1
s X
pn ρˆψn
(38)
n=1
where |ψ1 i , |ψ2 i , . . . , |ψs i is a set of normalized state vectors which are not necessarily, and not usually, orthogonal and pn is the proportion of the system which is in the corresponding state |ψn i. These proportions are nonnegative and add up to 1, i.e.,
pn ≥ 0
and
s X
pn = 1.
(39)
n=1
While the density operator of any pure state |ψi, defined as ρˆ = |ψihψ| is characterized by a trace trρˆ = 1, and by trρˆ 2 = 1, we have for a mixed state trρˆ = 1 but trρˆ 2 < 1. Moreover, the density operator is non-negative definite. As an example, for a mono-mode field in harmonic oscillation with the Hamiltonian Hˆ = h¯ ω Nˆ + 1/2 , its thermal state density operator is given by ρˆth = exp( Hˆ /kT )/tr[exp( Hˆ /kT )]
(40)
where k and T are, respectively, the Boltzmann constant and the absolute temperature. Since the Hamiltonian Hˆ and the photon number operator Nˆ have the same eigenstates, it can be easily shown that, in the photon
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number representation, ρˆth can be written as (Helstrom, 1976) ρˆth =
∞ X
pn |ni hn| , with pn =
n=0
n 1 N th (ω) 1 + N th (ω) 1 + N th (ω)
(41)
where Nth (ω) =
1 exp(h¯ ω/kT ) − 1
(42)
is the average thermal photon number and is known as the Planck distribution. According to Equation (41), it is easy to show that the proportions pn verify Equation (38). Another useful form of ρˆth is its P-representation (Glauber, 1963) in terms of the coherent states, given by ρˆth =
Z
+∞ Z +∞
Pth (α) |αi hα| d2 α, " # |α|2 1 exp − . with Pth (α) = π Nth (ω) Nth (ω) −∞
−∞
(43)
Since Pth (α) is a normalized Gaussian function, it verifies the integral form of Equation (37). It is to be noted, however, that the function P(α) of the P-representation, under the form of the first equation of Equation (43), cannot generally be considered as a proportion-function that must verify Equation (39), because P(α) may be negative in some regions on the complex plane.
2.3 Quantum Probability Distributions 2.3.1 Average Value as a Function of the Density Operator Let us consider first the standard expansion of the state P P ket |ψi, over an orthonormal base in the form |ψi = n Cn |u n i with n Cn∗ Cn = 1. Using the same base, the average value of an observable h Xˆ i can be expressed in the form D E D E XX D E Xˆ = ψ Xˆ ψ = ρ pn X np with X np = u p Xˆ u n n
p
and ρ pn = C ∗p Cn = u p |ψ hψ|u n i = u p ρˆ u n
(44)
∗ since the density operator ρˆ = |ψihψ| of the considered where ρ pn = ρnp P P state is Hermitian. Furthermore we have tr (ρ) ˆ = p ρˆ pp = n Cn∗ Cn = 1
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Finally the average value Xˆ can be expressed as a function of the density operator D E X Xˆ = (ρˆ Xˆ ) pp = tr(ρˆ Xˆ ).
(45)
p
This relation is easily generalized to a state mixture. 2.3.2 Positive Operator-valued Measure (POVM) According to quantum-mechanical principles, when we observe with a certain instrument a number of independent quantum systems of the same kind and prepared in the same state, or, according to ergodicity, when we observe repeatedly a single system that is re-prepared in the same state after each observation, the observation results present generally random variations that cannot be entirely attributed to the imprecision of the instrument or the preparation of the system. In effect, there is an irreducible purely quantum-mechanical uncertainty factor, because of which we must reject the exact prediction of the observations that were believed to be achievable in classical physics. For this reason, what we are able to do most precisely is to determine the probability distribution of our observation results at the quantum limit. The measurement can only be performed on the observables that are described by the Hermitian operators having real eigenvalues. In an observation by measurement with a given instrument, each numerical outcome of the instrument is a real scalar or vector random variable in a space R of suitable dimensionality. To calculate the probability that the outcome lies in a region 1 of R, designed by Pr (1), which is a kind of mapping associating a nonnegative number with the region 1 (Papoulis and Pillai, 2002), we can use the probability-operator measure (POM), also ˆ (1), through known as the positive operator-valued measure (POVM), 5 (Helstrom, 1976) ˆ (1)]. Pr (1) = tr[ρˆ 5
(46)
Any POVM must be Hermitian and nonnegative definite, following ˆ (R) = 1, 5 ˆ (∅) = 0 and, 5 ˆ (11 + 12 ) = 5 ˆ (11 ) + 5 ˆ (12 ) where ∅ is 5 the empty element of R, and 11 and 12 are disjoint regions (Helstrom, 1976). It is well known that the eigenvectors of a Hermitian operator can constitute a complete orthonormal set, which means, for a Hermitian
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operator Bˆ with a discrete spectrum {bn }, hbm |bn i = δmn
and
X
|bn i hbn | = 1
(47)
n
and for a Hermitian operator Xˆ with a continuum spectrum {x},
x|x 0 = δ x − x |
Z
+∞
and
|xi hx| dx = 1.
(48)
−∞
Consequently, the operators ˆ (bn ) = |bn i hbn | 5
ˆ (1) = and 5
Z 1
|xi hx| dx
(49)
are both POVM and known as orthogonal, for their evident orthogonality, or projection-valued POVM (Helstrom, 1976). 2.3.3 Coherent State Photon Number Probabilty From Equations (44), (48) and (34), the probability of measuring n photons from a coherent source, characterized by the projection operator ρˆ = |αihα| ¯
Pr (n) = hn|αi hα|ni = e− N
N¯ n n!
(50)
where N¯ = |α|2 is the average photon number. This is just Poisson distribution that can be considered as resulting from the random and independent arrivals of the photons in the corpuscular picture of the light. 2.3.4 Thermal Light Photon Number Probabilty Another example is the probability of measuring n photons from a thermal source. Using Equation (41) we can straightforwardly find n 1 N th (ω) Pr (n) = 1 + N th (ω) 1 + N th (ω)
(51)
which is just the Bose–Einstein distribution and has been experimentally verified for a source of amplified spontaneous emission (Wong, Haus, Jiang, Hansen and Margalit, 1998).
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2.3.5 Coherent State Quadrature Probabilty In this section we will derive probability distributions for the two quadratures a I and a Q of a coherent state and will point out in this that the uncertainty product attains its minimum value. As a direct consequence of the Cauchy–Schwartz theorem, Gaussian functions minimize their quadratic width, meaning that the last property to be derived already includes the first, meaning that the probability distributions for a I and a Q are both Gaussian p(a I /Q ) =
a I /Q − A I /Q 1 √ exp − 2σ I2/Q σ I /Q 2π
2 (52)
where the quantum average values are A = A I + jA Q ,
with A I = aˆ I
A Q = aˆ Q
and
(53)
where the departures from the average value are 1a I /Q = a I /Q − A I /Q with the variance σ I2/Q = δa I /Q
2
=
D
1a I /Q
2 E
.
(54)
According to Equation (27) we have for a minimum uncertainty product δa I .δa Q =
1 1 aˆ I , aˆ Q = . 2 4
(55)
Assuming no fluctuation squeezing and therefore equipartition of uncertainties between the two quadratures of the field, we have a complex circular Gaussian noise characterized by δa I = δa Q =
1 . 2
(56)
Let us now rigorously derive the probability distribution. According to Equation (45), since |xihx| dx is an infinitesimal element of the POVM, the probability density function (PDF) of the measurement outcome on Xˆ is given by p (x) = tr ρˆ |xi hx| = hx| ρˆ |xi
(57)
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R +∞ from which, by using −∞ |xihx|dx = 1 and noting that the orders of the trace and the integral can be exchanged, we can find the moment generating function (MGF) (Papoulis and Pillai, 2002) of the observable X as 8 (s) =
Z
+∞
Z
+∞
p (x) exp (sx) dx = tr ρˆ |xi hx| exp (sx) dx −∞ Z +∞ |xi hx| exp s Xˆ dx = tr ρˆ −∞ h i = tr ρˆ exp s Xˆ .
−∞
(58)
Exponentials of the operator are defined by analogy with the standard exponential function expansion. Therefore, using Glauber’s relation, ˆ = exp( A) ˆ exp( B) ˆ exp − 1 [ A, ˆ B] ˆ exp( Aˆ + B) 2
(59)
ˆ B] ˆ is assumed to commutate with Aˆ and B, ˆ which is obviously where [ A, ˆ B] ˆ is scalar, we can find the MGF of the observable obtained when [ A, of the quadrature operator aˆ Q = aˆ − aˆ Ď /2 j for the coherent state ρˆ = |αihα| as ! aˆ − aˆ Ď |αi 8 Q (s|α) = hα| exp s aˆ Q |αi = hα| exp s 2i ! ! s aˆ Ď s aˆ s2 |αi exp = hα| exp − exp 2j 2j 8 ! s2 = exp + s AQ 8
(60)
where a Q is the quadrature component of α = a = a I + ja Q , with quantum average values A = A I + jA Q , with A I = haˆ I i and A Q = haˆ Q i and where the operators in the normal quantum theory operator order, meaning that the creation operator is placed to the left of the annihilation operator, have been set to their average values. Since the probability density function (PDF) is the inverse Fourier transform of the characteristic function 8 (−js), the PDF of the quadrature component can
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be found as Z +∞ 1 exp jsa Q 8 Q (−js|α) ds 2π −∞ r h 2 i 2 = exp −2 a Q − A Q . π
p(a Q ) =
(61)
Equally, in agreement, with Equations (52) and (55), we can find the PDF of the observable of the in-phase component formally identical to that of the quadrature component, i.e., r p (a I ) =
h i 2 exp −2 (a I − A I )2 . π
(62)
Therefore, for coherent states, the PDFs of the outcomes of the independent measurements on the in-phase and quadrature components, 2 respectively, are both a Gaussian function with variance σa,I /Q = 1/4. Figure 1 presents the standard picture of a coherent state as the sum of a deterministic field vector and an uncertainty disk. In this sense, an optical signal in a coherent state can be considered as the sum of a deterministic part and a stochastic circular Gaussian noise. The elusive noise added to the deterministic (and eventually equal to 0) part A = A I + jA Q of a coherent state is the so-called zero point fluctuation, or vacuum fluctuation. As a direct consequence of Equation (27), coherent states are the minimum uncertainty states, we have D ED E 1 (1a)2 (1a + )2 = (δa)2 (δa + )2 = 4 D ED E 1 and 1a 2I 1a 2Q = (δa I )2 (δa Q )2 = . 16
(63)
2.4 Engineering Description of Quantum Noise 2.4.1 Spectral Density of Quantum Noise In digital communication theory the signal x(t) is usually normalized in such a manner that the power P is the squared signal modulus. Such representation may be used in optical engineering when we assume that both the signal and the optical noise refer to the same polarization and can be represented by scalar notation. The corresponding signal operator is related to the photon annihilation quantum operator aˆ by r x(t) ˆ =
hν a(t). ˆ T
(64)
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FIGURE 1 A coherent state is the addition of a circular Gaussian noise to its deterministic part
FIGURE 2 Bidimensional noise addition to a deterministic signal
Digital communication theory also deals with a narrow-band signal with thermal noise, usually depicted as an additive Gaussian white noise (AGWN) N (t), with a flat single-sided spectral power density S N (usually denoted N0 = kT ) over the considered pass-band bandwidth B O . In Rice’s representation (Rice, 1948) the complex envelope of a random signal, including the modulated signal X and an additive random noise 1X , is written as the sum x(t) = x I (t) + jx Q (t) of an in-phase and an inquadrature component (Figure 2). The noise component 1x(t) = 1x I (t) + j1x Q (t) is Gaussian as well as its independent in-phase 1x I (t) and inquadrature 1x Q (t) components. These two components are uncorrelated base-band noise processes with bandwidth Bo /2 and have the same single-sided spectral power density S N as the total noise 1x(t). The probability distributions for the independent in-phase and quadrature
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noise components are p(x I /Q ) =
1
√
σx,I /Q 2π
exp −
2 x I /Q − x I /Q
2 2 with σx,I = σx,Q = SN
2σ I2/Q BO . 2
(65)
For a coherent state, according to Equations (21) and (22) we have
1a1 ˆ aˆ + = 1,
+ 1aˆ 1aˆ = 0
and
1 1 (1a1 ˆ aˆ + + 1aˆ + 1a) ˆ = (1aˆ I )2 + (1aˆ Q )2 = . 2 2
(66)
The symmetrization of the Hermitian operator, cancelling out the commutator contributions, allows us to calculate the sum of the inphase and out-of-phase square noise power which is the equivalent of the classical total noise power, but is not directly observable. However it produces amplitude noise (or intensity noise, i.e. shot noise) as will be discussed in Section 2.3.3 or phase noise (and therefore line width though phase diffusion), as will be discussed in Section 5.6.3. According to Equations (64) and (66), and by denoting by B O = 1/T the bandpass bandwidth centred on the carrier frequency associated with a perfect integrator over the time duration T , we have PˆN = (1x1 ˆ xˆ + + 1xˆ + 1x) ˆ = (1xˆ I )2 + (1xˆ Q )2 .
(67)
The total average noise power is obtained as D E D E D E hν BO . PN = PˆN = (1xˆ I )2 + (1xˆ Q )2 = 2
(68)
This additive noise, which accompanies any optical field, is usually referred in quantum electrodynamics, to the zero-point field fluctuations or the vacuum fluctuations. The addition of the zero-point field fluctuations to a classical deterministic field is an intrinsic property making the coherent state of the light a stochastic process whether the light is modulated or not. Note that by using normal operator order, meaning that the creation operator is placed to the left of the annihilation operator in the expression, the directly observable optical noise power is obtained as
+ 1xˆ 1xˆ = 0
(69)
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173
meaning non-classical behavior of the quantum noise whose power may appear to be equal to zero. It is an elusive noise not directly observable and therefore cannot be detected alone. Equation (68) allows us to consider quantum fluctuations as produced by an AGWN with singlesided spectral density, in the considered signal polarization mode (70)
S N 0 = hν/2.
This noise is only observable through its cross term product with another signal, but not directly. By using B O = 1/T , the energy hν/2 can be interpreted as the minimum detectable value of energy for an observation time T . This value is also the minimum value E 0 = hν/2 of the quantified energy E n = (n + 1/2)hν of a harmonic oscillator, which is always present but not available for exchange. For the so-called communication wavelength of 1550 nm, i.e. ν = 193 Thz, corresponding to the minimum minimorum of silica fiber and to the centre of amplification range of EDFA, the zero-point field spectral density S N 0 is 0.65 10−19 W/Hz. Vacuum fluctuations are additive for the field and are already present at any (evenly unused) signal input optical amplifier, an optical coupler etc. They include phase and amplitude (or intensity) noise as well. According to Section 2.3, for a coherent state, the probability density p(a I /Q ) = ha I /Q |αihα|a I /Q i is a Gaussian distribution with variance 2 σa,I /Q = 1/4. In agreement with Equations (55) and (66) we have p(a I /Q )da I /Q = p(x I /Q )dx I /Q and therefore 2 2 σx,I /Q = hν B0 σa,I /Q = S N 0
BO . 2
(71)
2.4.2 Optical Power Noise Since the vacuum fluctuations are additive for the optical field itself, they are not additive for the usual optical intensity measurement. However, in a standard square-law detection they produce cross terms with the signal, usually called the shot noise. The shot noise is not a consequence of using the corpuscular description of light, but a counterpart of fundamental optical field fluctuations themselves. Any un-modulated normalized optical field may be considered as the sum of a deterministic √ ˆ = 1xˆ I (t) + j1xˆ Q (t). part P exp jϕ and a Gaussian quantum noise 1x(t) The power fluctuations are governed by addition of the in-phase 1xˆ I and out-of-phase 1xˆ I noise components (Figure 2). Since the quantum fluctuations are invariant through phase rotation, without loss of generality, we can assume√that the deterministic part of the field refers to a single quadrature X I = P. Therefore the instantaneous power is the
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squared field sum ˆ Pˆ = X 2I + 2X I 1xˆ I + 1xˆ + 1x.
(72)
Since 1xˆ + 1xˆ = 0 for a coherent state, the last term of Equation (74) does not contribute to the quantum average power in the mode D E P = Pˆ = X 2I .
(73)
It does contribute, however, to the power fluctuation 1 Pˆ = 2X I 1xˆ I + 1xˆ + 1xˆ but makes no contribution to the mean-square value
1 Pˆ
2
D 2 E = 4P 1xˆ I = 4P PN 0I = 2P PN 0
(74)
where PN 0I is the noise power of the in-phase quadrature, i.e. half of the total noise power PN 0 . The in-phase quadrature appears to be only responsible for the mean-square power fluctuation. This result of a quantum approach is only obtainable in the classical way by using the approximation of relatively weak noise (Nilsson, 1994; Gallion, 1999, 2002). The in-phase quadrature averaged noise power is PN 0I =
D
1xˆ I
2 E
= hν B O
D
1aˆ I
2 E
= SN 0
BO . 2
(75)
If we assume that the optical fluctuations are completely described by their second order momentum, we approximate the chi-square distribution by a Gaussian one. The quantum noise is usually taken into account in terms of the photocurrent fluctuation by using the electrical form of the well-known Schottky relation. Assuming a photodetector responsivity R equal to 1, which implies that the photocurrent fluctuation exactly duplicates the optical power fluctuation, the optical and electrical Schottky relations, easily derived from Equation (68), are respectively written as D E D E (76) (1P)2 = 2hν P B E and (1I )2 = 2eI B E where B E is the base-band observation bandwidth. Using the proportionality relation between the number of photons and the optical
received power N = P T / hν, the Poisson fluctuation relation (1N )2 = N is recovered.
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In optical communication engineering where the square-law direct detection is, up to now, widely used, it is common to describe the signal-to-additional-noise cross-term as a beat noise. This heuristic interpretation, leading to the same results, considers that the optical noise spectral components within the total spectral range B O , corresponding to a frequency departure, on each side of the optical carrier frequency, are equal to the observation (base-band) bandwidth B E . Only frequency components within these two limits are able to produce beating within the electrical bandwidth and the total optical noise bandwidth contribution is determined by B O = 2B E . Since we are only concerned here by inphase fluctuations, only the half of the total noise power is concerned. The shot noise appears clearly as the consequence of the beating of the deterministic part of the signal of the in-phase components and optical noise with a spectral density S N 0 . 2.4.3 Optical Phase Noise Derivation of a Hermitian phase noise quantum operator is a tedious and incompletely solved problem that will be evoked in Section 6. Starting with a heuristic approach and considering the addition x = X + 1x of a relatively√weak noise 1x(t) = 1x I (t) + j1x Q (t) to a single quadrature X = X I = P of the deterministic part of the field, we have xQ j x 1ϕ = − ln ∗ ≈ 2 x XI
(77)
where x ∗ stands for the complex conjugate and we have a standard ln expansion up to the first order, assuming that the noise is weak when compared to the signal. As a consequence of this approximation the phase noise is considered here to be produced only by the quadrature noise component, as already suggested by Figure 2. This relation can be easily generalized to a quantum one for the corresponding Hermitian operators. The mean-squared phase fluctuation is thus expressed as D D
1ϕˆ
2 E
≈
1xˆ Q P
2 E =
PN 0Q PN 0 = P 2P
(78)
where PN 0Q is the noise power in the in-phase quadrature, i.e. half of the total noise power PN 0 . For a coherent state with spectral density S N 0 , the product of the power noise, given by Equation (74), and the phase noise fluctuations, given by Equation (78), are independent of the classical part
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of the signal and fulfill the uncertainty product relationship D ED E D 2 E D 2 E 1xˆ Q (1P)2 (1ϕ)2 = 1xˆ I = 4PN 0I PN 0Q = (PN 0 )2 = (S N 0 B0 )2 .
(79)
Again using the proportionality relation between the number of photons during time T and the received optical power N = P T / hν, the uncertainty product relationship given by Equation (79) may finally be written as D
ED E 1 (1N )2 (1ϕ)2 = . 4
(80)
2.4.4 Channel Capacity The channel capacity may be first derived in the particular case of a multilevel power detection receiver. The output power is assumed to be the sum of the signal power PS and noise power PN and to be first observed during a unit of time. The number of distinguishable output energy levels is limited by noise to (PS + PN ) /PN and the corresponding number of information bits is therefore limited to log2 [(PS + PN ) /PN ]. Assuming a limited signal bandwidth equal to B O , the observation time may be reduced down to 1/B O , allowing to sample the channel B O time per second, the well-known Shannon’s channel capacity is obtained. PS . C = B O log2 1 + PN
(81)
A more general derivation of Equation (81) may be obtained from information theory (Shannon, 1948) Bandwidth is not the only key issue in increasing channel capacity, since the noise power is in general a function of bandwidth. Assuming white noise with a constant spectral density S N = 2σ N2 , the noise power PN = S N B O linearly increases with the bandwidth Bo , the channel capacity only increases with the bandwidth up to the limit C = log2 e
Ps Ps = 1.44 . SN SN
(82)
The theoretical value of the channel capacity, obtained by continuous signal assumption, must be considered as an ultimate limit. It should be added that information theory provides no information on any practical means of achieving this limit. Figure 3 shows the theoretical value
Non-commutating Quadrature Measurements
177
FIGURE 3 Theoretical value of an optical channel capacity submitted to fundamental quantum noise impairments only, as a function of signal bandwidth
of an optical channel in the presence of fundamental quantum noise impairments only, indicating a spectral density of optical noise S N 0 , as a function of bandwidth for different signal power levels. The saturation of the channel capacity when the signal bandwidth increases appears for low values of the signal power. Optical bandwidth increase requires the control of fiber dispersion and nonlinearity, the development of large bandwidth devices and multiplexing techniques. As the power signal level is limited by the nonlinearity control, the improvement of the channel capacity also also requires control of the accumulation of noise generated by lossy fibers, amplifiers and other devices. It is convenient to define the spectral efficiency of a communication channel as the ratio C/B O of the channel capacity to the signal bandwidth. Figure 4 shows the theoretical value of the spectral efficiency of an optical channel submitted to the fundamental quantum noise impairments only, as a function of the signal power, for different values of the bandwidth.
3. NON-COMMUTATING QUADRATURE MEASUREMENTS AND QUANTUM STATE DISTINGUISHABILITY For transmission in an optical communications channel, every information symbol is encoded into the complex amplitude of an optical field which is governed by quantum principles, thus producing a set of states that must be discriminated (i.e. distinguished) at the receiver in the
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FIGURE 4 Theoretical value of the spectral efficiency of an optical channel submitted to the fundamental quantum noise impairments only, as a function of the signal power
presence of channel impairments; this is a central issue in optical communications theory. Diverse criteria have been proposed for distinguishing among quantum states: Fuchs and van de Graaf (1998) present a survey of the distinguishability measures from the quantum cryptography point of view, while Chefles (2004) and Bergou, Herzog and Hillery (2004) study the discrimination of quantum states from two points of view: unambiguous detection at the expense of allowing inconclusive outcomes (useful to quantum cryptography), and conclusive results only at the expense of finite bit error rate (useful for quantum communications). Waiting for reliable single photon sources the quantum states used in optical communications and cryptography are, in general, nonorthogonal faint pulse coherent states and deriving the optimum quantum measurement is a central task for receiver design and implementation.
3.1 Non Commutating Quadrature Measurements For complete information about the optical fields, we must measure both their in-phase and quadrature components, whereas the standard deviations of the outcomes of the measurement on these two noncommutating observables, aˆ I , aˆ Q = j/2, are subject to the Heisenberg uncertainty relation derived in Section 2.1.3 D ED E 1 2 1 1aˆ 2I 1aˆ 2Q ≥ aˆ I , aˆ Q = . 4 16
(83)
Non-commutating Quadrature Measurements
179
FIGURE 5 Symmetries S1 and S2 of four port optical combiner
From Equation (83), we see that the separated measurements on the two components in coherent states correspond to the minimum
2 D 2 E case 1aˆ I 1aˆ Q = 1/16. The separated measurements are, however, frequently not satisfactory, not only because we generally need information on both components, but also because measurement on one of the two components can significantly disturb the measured field by adding noise or changing the state of the field. For this last reason, the separated measurements here should be generally understood as measurements on separated systems, or specimens, of optical fields in the same state or in nearly the same state. 3.1.1 4 Port Coupler, Combiner and Beam Splitter Let us consider an ideal reversible beam splitter/combiner. We will use a general black box description to make our analysis independent of the different technological implementations (beam splitter in bulk optics, optical fiber couplers, fiber Bragg gratings, etc.). As presented in Figure 5, two input beams and two output beams are involved in a four-port description. The general input-to-output transfer matrix is written as b2 a a b a1 =A 1 = . b2 a2 c d a2
(84)
According to purely geometrical symmetry with respect to S2 we have b = c, meaning A = At , and a = d. According to the Hermitian symmetry with respect to S1 we have also A+ = A−1 2
2
with a ∈ R and b ∈ I cos θ j sin θ A= . j sin θ cos θ
a −b =1
(85) (86) (87)
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FIGURE 6 Quadrature measurement arrangement
Assuming a lossless 50/50 coupler we have |a| = |b|
and θ = ±
π mod π. 4
(88)
Selecting one of the four solutions we have 1 1 −j A= √ . 2 −j 1
(89)
A more general input–output relation is discussed in the appendix, including phase sensitive attenuation and amplification and degenerate parametric amplification. 3.1.2 Physical Implementation Let us use the ideal reversible beam splitter/combiner discussed in Section 3.1.1 to split the input signal aˆ 1 into two parts on which we expect to make at the output separated measurements of the two quadratures. There is no user signal at the second input aˆ 2 , which is in the vacuum state As presented in Figure 6, the quantum input-output relation for this lossless 50/50 coupler is given by 1 bˆ1 = √ (aˆ 1 − jaˆ 2 ) and 2
1 bˆ2 = √ (−jaˆ 1 + aˆ 2 ). 2
(90)
At the input we have the commutation relations [aˆ 1 , aˆ 1+ ] = [aˆ 2 , aˆ 2+ ] = 1
and [aˆ 1 , aˆ 2 ] = [aˆ 1 , aˆ 2+ ] = 0.
(91)
Non-commutating Quadrature Measurements
181
This device divides the input signal into two parts but it is worth noting that, although not directly used, the second input aˆ 2 must be taken into account in order to preserve the commutation relations. It is easy to verify that this coupler preserves the commutation relation, i.e., that we have at the output [bˆ1 , bˆ1+ ] = [bˆ2 , bˆ2+ ] = 1
and [bˆ1 , bˆ2 ] = [bˆ1 , bˆ2+ ] = 0.
(92)
By separating aˆ 1 and aˆ 2 into in-phase and quadrature components, we have 1 bˆ1 = √ aˆ 1I + aˆ 2Q + j aˆ 1Q − aˆ 2I and 2 1 bˆ2 = √ aˆ 1Q + aˆ 2I − j aˆ 1I − aˆ 2Q . 2
(93)
Thus, the measurements of the in-phase components of bˆ1 and bˆ2 allow access to aˆ 1I and aˆ 1Q . We can verify that the operators 0 aˆ 1I =
√ 2bˆ1I = aˆ 1I + aˆ 2Q
0 and aˆ 1Q =
√ 2bˆ1Q = aˆ 1Q + aˆ 2Q
(94)
are commutating, by 0 0 [aˆ 1I , aˆ 1Q ] = [aˆ 1I , aˆ 1Q ] + [aˆ 2Q , aˆ 2I ] = j/2 − j/2 = 0
(95)
which means that these two Hermitian operators can be measured simultaneously without additional noise. Since the second input port aˆ 2 is in the vacuum state, we have the average values
0 i = aˆ 1I + aˆ 2Q = aˆ 1I haˆ 1I
and
0 i = aˆ 1Q + aˆ 2I = aˆ 1Q (96) haˆ 1Q
which are the same average values as for the separated measurements on the two components of the input signal, aˆ 1I and aˆ 1Q that we were expecting to measure. 3.1.3 Additional Quantum Noise Let us consider now the corresponding fluctuations we obtain 0 1aˆ 1I = 1aˆ 1I + 1aˆ 2Q
0 and 1aˆ 1Q = 1aˆ 2Q + 1aˆ 2I .
(97)
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By adding the square average on the two uncorrelated noise contributions we obtain D E D 2 E D 2 E 0 2 1aˆ 1I = 1aˆ 1I + 1aˆ 2Q 2 D E D E = 1 + 1 = 1 . (98) 2 2 0 4 4 2 1aˆ 1Q = 1aˆ 2Q + 1aˆ 2I Comparing Equations (98) and (62), we find that the outcome variances of the non-commutating quadrature measurements, on coherent states, are twice those of the separated measurements on the two components. This uncertainty augmentation is in fact due to the simultaneous measurement of the two non-commutating observables and the same result will be obtained for the same reasons when amplifying the two field quadratures in a phase insensitive linear amplifier, as will discussed in Section 4, or for simultaneous two quadrature detection in a heterodyne arrangement, as will be discussed in Section 6. It has been shown that the joint probability density function (PDF) of the outcomes, a I and a Q , of the simultaneous quantum-mechanical limited quadrature measurement in the field is given by (Helstrom, 1976) 1 1 f a I , a Q = hα| ρˆ |αi = tr ρˆ |αi hα| , π π with α = a = a I + ja Q .
(99)
Thus, following the same procedure as in Section 2.3.5, we can show that their MGF is given by: 8 (u, v) =
Z
+∞ Z +∞
f a Q , a I exp ua I + va Q da I da Q −∞ −∞ h i = tr ρˆ exp w ∗ aˆ exp waˆ Ď
(100)
where w = (u + jv) /2. We launch the signal into the input aˆ 1 , and there is no input at aˆ 2 , which is in the vacuum state. The total density operator is, therefore, ρˆ = ρˆ1 ⊗ ρˆ2 ,
with ρˆ2 = |0i22 h0| .
(101)
Now, we can show that this configuration of the lossless 50/50 coupler is a physical implementation of the quantum-mechanical limited non0 commutating quadrature measurement. Indeed, since the operators aˆ 1I 0 and aˆ 1Q are commutating, they possess common eigenstates, denoted by
Non-commutating Quadrature Measurements
183
0 |x, yi = y|x, yi. Because they are 0 |x, yi = x|x, yi and a ˆ 1Q |x, yi, with aˆ 1I both Hermitian, their common eigenstates |x, yi constitute a complete orthogonal set,
x 0 , y 0 |x, y = δ x − x 0 δ y − y 0 Z +∞ Z +∞ |x, yi hx, y| dxdy = 1
−∞
and (102)
−∞
which means that the joint PDF of the outcomes x and y is given by: p (x, y) = hx, y|ρ|x, ˆ yi, as shown in Section 2.3. Again following the same procedure as in Section 2.3.4, we can show that their MGF is 8 (u, v) = 0 + va 0 )], and then using Equation (59) and Equation (94), we tr[ρˆ exp(u aˆ 1I ˆ 1Q can find: h i Ď Ď 8 (u, v) = tr ρˆ exp w ∗ aˆ 1 + waˆ 1 exp jw ∗ aˆ 2 − jwaˆ 2 h i i h Ď Ď = tr ρˆ1 exp w ∗ aˆ 1 + waˆ 1 tr |0i22 h0| exp jw∗ aˆ 2 − jwaˆ 2 h i Ď = tr ρˆ1 exp w ∗ aˆ 1 + waˆ 1 exp(|w|2 /2) h i Ď = tr ρˆ1 exp w∗ aˆ 1 exp waˆ 1 (103) with w = (u + jv) /2. But this is just the MGF defined by Equation (100). This shows clearly that the lossless 50/50 coupler, dividing the input signal aˆ 1 into two parts, is a physical implementation of the quantummechanical limited non-commutating quadrature measurement, where the second input aˆ 2 is called as an ancilla (Helstrom, 1976). Using Equation (35), for a coherent state ρˆ = |αi hα|, we can find: 1 p a I , a Q = exp − |a − A|2 . π
(104)
Now, integrating the in-phase or quadrature component in Equation (103), we obtain the PDF of the other component, given by: r
h 2 i 1 0 exp − a1I − AI and π r 2 1 0 0 p(a1Q )= exp − a1Q − AQ π
0 p(a1I )
=
(105)
which is a Gaussian function with variance σ 2 = 1/2. We find that the outcome variances of the non-commutating quadrature measurement of
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coherent states are twice those of the separated measurements of the two components. This uncertainty augmentation is in fact due to the simultaneous measurement of the two non-commutating observables. This can be understood more concretely by the fact that, although not used directly, the ancilla, or second input aˆ 2 , making aˆ 1I and aˆ 1Q commutating, introduces additional noise of vacuum fluctuation to the outputs. So although simultaneous measurement of the non-commutating in-phase and quadrature components of the optical field is possible, additional noise with variance 1/4 is inevitably added to the measurement results, without violation of the Heisenberg principle. A heterodyne detection which permits simultaneous measurement of in-phase and quadrature components has such limitation. Heterodyne detection is a band-pass process requiring twice the bandwidth of the base band. The signal and the image band fluctuations are the fundamental noise limitations which lead to doubling of the noise level of homodyne detection.
3.2 Quantum Detection The pioneering works on quantum detection and estimation theory by Helstrom (1976), Yuen, Kennedy and Lax (1970), Hirota and Tsushima (1989) and Belavkin (1975), for the digital channel, were based on testing the quantum hypothesis. In this theory the detection process consists of a generalized quantum measurement, which is mathematically described by a probability operator value measurement (POVM) in the following way: given an M-ary received signal, whose states have a-priori probabilities ξm and density operators ρˆm , m = 1, . . . , M consisting of a unit trace, non-negative Hermitian operators, that is: ρˆm ≥ 0, ∀m and trρˆm = 1, ∀m, where tr stands for the trace of the operator matrix, the ˆ l , l = 1, . . . , M possessing the properties: POVM is a detection operator 5 ˆ l ≥ 0∀l −Positiveness: 5 M X ˆ l = Iˆ −Completeness: 5
(106) (107)
l=1
where Iˆ is the identity operator. The conditional probability of inferring that a measurement output signal in the m state corresponds to a signal in the l state is: ˆ l ρˆm ). Pr(l/m) = tr(5
(108)
This is the probability of choosing the hypothesis Hm when Hl is true. Therefore the probability of error, also called the Bit Error Ratio (BER), is
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obtained in terms of the POVM:
BER = 1 −
M X
ˆ m. ξm trρˆm 5
(109)
m=1
Helstrom (1976) introduced the quantum statistical detection theory for optimal decision among several hypotheses: for the case of binary signals consisting of pure states ψ0 and ψ1 , there are two density operators labelled ρ0 = |ψ0 ihψ0 | and ρ1 = |ψ1 ihψ1 |, with prior probabilities ξ0 and ξ1 , with ξ0 + ξ1 = 1; the two hypotheses are labeled H0 and H1 . n o ˆ Iˆ − 5 ˆ is applied, obtaining the In this approach a POVM operator 5, following probabilities: Pr(H1 /H1 ) = tr (ρ1 5) : detection probability
(110)
Pr(H1 /H0 ) = tr (ρ0 5) : false alarm probability
(111)
The average probability of error is obtained from: BER = ξ0 Pr(H1 /H0 ) + ξ1 [1 − Pr(H1 /H1 )] .
(112)
Based on the statistical Neyman Pearson criterion for the maximization of detection probability, Helstrom (1976) finds the minimum attainable probability of error, the so-called Helstrom bound q 1 2 BER = 1 − 1 − 4ξ0 ξ1 |hψ1 |ψ0 i| . 2
(113)
Thus depending on the inner product hψ1 |ψ0 i, this probability is lower the farther apart, i.e. the more nearly orthogonal, are the quantum states.
3.3 Symmetrical Coherent States In the case of symmetrical coherent state fields used, for instance, for binary phase-shift keying (BPSK) signal, the two signal coherent states are |αi and | − αi with average signal photon number Ns = |α|2 , the quantum state overlap is, according to Equation (36) |hψ1 |ψ0 i|2 = |hα| − αi|2 = exp(− |2α|2 ) = exp −4N S .
(114)
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Assuming equally probable prior states, ξ0 = ξ1 = 0.5, the probability of error is the binary coherent Helstrom bound BER =
o p 1n 1 − 1 − exp(−4Ns ) . 2
(115)
Ban, Kurokawa, Momose and Hirota (1997), based on a Bayes strategy, have also studied the problem of discrimination among symmetrical quantum states, arriving at the same bound for coherent states. They considered the case of quantum estimation, which is important when the state includes unknown parameters and found that the corresponding optimum estimation POVMs possessed a similar structure to those for data detection. In addition, asymptotic solutions for the probability of error and for mutual information for higher order PSK and QAM constellations, have been derived by Kato, Osaki, Sasaki and Hirota (1999). The POVM is a generalization of a Von Neuman projection value measurement (PVM) of a signal observable by projections on to orthogonal states, as treated by Huttner, Muller, Gautier, Zbinden and Gisin (1996), which leads to conclusive results only, but with finite error probability. The projection operators correspond to the standard (classical) receivers, for example: photon counters, heterodyne, homodyne, etc. for different signal observables such as photon number, complex amplitude, signal quadrature, and they even include an optical Costas loop based on homodyne (Momose, Osaki, Ban, Sasaki and Hirota, 1996). Therefore their ultimate probability of error corresponds to the standard quantum limit (SQL). For binary symmetric states we have BER =
p 1 erfc (2Ns ) 2
(116)
√ R∞ where erfc [x] = 2/ π x exp(−t 2 ) dt is the complementary error function, (here we are using the notation of Sasaki, Usuda and Hirota (1995)). The POVM measurement can provide superior performance to that of classical receivers, however, not only its physical implementation faces considerable challenges, but also its physical interpretation is a subject of research. Osaki, Ban and Hirota (1996) have derived and interpreted the optimum detection operators for binary, ternary and quaternary optical phase-shift-keying modulated fields, based on the criterion of minimum probability of error, they interpret the beating of the SQL as a “quantum interference” phenomenon. While the POVM gives the probabilities of measurement of a quantum state, no indication about the structure of the physical device is suggested.
Non-commutating Quadrature Measurements
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FIGURE 7 Heuristic quantum receiver structure for the binary optical channel. Only detector 1 (photon counting) is used in Kennedy receiver (open loop) and Dolinar receiver (closed loop). Vilnrotter receiver uses both detector 1 and detector 2 (coherent)
Myers and Brandt (1997), Banaszek (1999) and Brandt (2003) have investigated how to mechanize a photonic implementation of a POVM with applications to quantum information processing and quantum cryptography. As the optimal structures are difficult to mechanize, a practical detector must trade off optimal performance and physical implementability.
3.4 Open Loop Quantum Receivers In another approach, several heuristic suboptimum structures have been proposed as early as the 1970s in the search for implementable receivers that are close to the Helstrom bound, originally for the binary channel with coherent states using photon counting techniques such as the Kennedy receiver (Kennedy, 1973). As shown in Figure 7 (in the open loop configuration), prior to detection, a locally generated field of precisely controlled amplitude is superimposed on the received field, via an (ideal) coupler with negligible coupling coefficient, in order to preserve the received amplitude at the detector. Therefore a powerful local field must be used to compensate for the low coupling. As the complex amplitude of the local laser is the negative of one of the two signals, the Kennedy receiver performs a displacement of the signal constellation in order to null one of the two hypotheses while doubling the other. In this “nulling receiver” with perfect photon counter (zero dark count, no background) a “Z” channel is obtained, as in Bendjaballah (1995). For a transmitted “0” the receiver selects hypothesis Ho with unit probability, since no photoelectrons are emitted by the vacuum state, however, for a transmitted “1”, there is a finite probability of emitted no photoelectron since all coherent states overlap with the vacuum, this is why the Kennedy receiver does not reach the Helstrom bound. However
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this near optimum receiver has a better probability of error than the SQL for large signal photon numbers, as Kennedy derived using a conditional Poisson process at the photon counter BER =
1 exp(−4Ns ). 2
(117)
The photon number required for the probability of error for the Kennedy receiver is twice the photon number that appears in the asymptotic limit BER = (1/2) exp(−2Ns ) of the SQL. This can be understood by considering that, for the SQL, as will be shown in Section 5, the power of the incoming signal is spread on two detectors and that the two detector outputs, proportional to the signal amplitude, are subtracted, recovering a factor of 2, while for the Kennedy receiver, the spread signal amplitude is doubled on one detector and the power is therefore squared. In this last case, the signal contribution on the other detector is canceled. Although it is a heuristic structure, it is interesting to investigate which operator is measured; Shapiro (1980) and later Osaki, Ban and Hirota (1996) have inferred that the detection operator that is measured by the Kennedy receiver is Iˆ − ρ0 (only miss errors) while the optimum receiver measures u(ρ1 − ρ0 ) (false alarm as well as miss errors), where u(x) is the Heaviside step function. Ban (1997) formalizes the quantum mechanical displacement operators in the Kennedy receiver, and arrives at the same measured operator, he also considers the effects of thermal noise and non-ideal couplers and detectors. Similarly, Vilnrotter and Lau (2001) have also formulated a quantum detection theory for the free space channel, obtaining the measurement operators for the binary channel with applications to PSK and OOK modulations in the presence of background noise, as well as M-ary modulations with no background noise. Looking for implementable structures, Van Enk (2002) proposes a generalized Kennedy measurement by using a realistic coupler and a finite reference field, such as a local oscillator in telecom applications or a time multiplexed reference for quantum cryptography (Hirano, Yamanaka, Ashikaga, Konishi and Namiki, 2003), therefore accessing the two output ports is possible. Analytically this structure performs better than Kennedy’s for a certain range of parameters. Furthermore, he also derives the eigenstates for optimum measurement, however their physical implementation remains an open question. Later, Vilnrotter and Lau (2005) proposed, for the free space channel, a practical extension of the Kennedy receiver, implementable with realistic directional couplers, using the coupled signal component to improve signal detection with an additional coherent detector on the other port. This “vector” receiver uses signal processing of the two ports for data
Non-commutating Quadrature Measurements
189
detection and closed loop precise control of the local laser. Numerical evaluations demonstrated improved performance over Kennedy when realistic elements are taken into account. They also investigated the effects of background noise and the impact of low density parity check (LDPC) coding on receiver performance. Takeoka and Sasaki (2007), observing that the amount of displacement in the Kennedy receiver is not optimal, propose an open loop receiving structure that beats the SQL for any value of photon number, using optimal displacements with photon counting detectors.
3.5 Feedback Quantum Receivers Another heuristic receiving structure, that in principle can reach the Helstrom bound, was also proposed as early as the 1970s by Dolinar (1973), it is for the binary channel with coherent states using photon counting techniques (in the absence of thermal noise). The Dolinar receiver is based on the Kennedy receiver, also with ideal couplers. It incorporates an adaptive strategy to implement a feedback approximation to the optimum POVM and operates with a local oscillator LO whose amplitude can be controlled and phase switched over the bit duration, in order to dynamically null one hypothesis depending on the observed count process in a real time feedback scheme, as shown in Figure 7 (in closed loop configuration). This constitutes a “conditionally nulling” receiver that progressively annuls the more probable signal according to the measured counts. As time goes on the phase is switched less frequently until a final count, after which the input field is almost canceled. In a Dolinar receiver, the decision is carried out by the number of counts at the end of the symbol — hypothesis 1 for odd count and hypothesis 0 for even count. Alternative derivations of the conditionally nulling receiver, for optimal discrimination, have been reported by Holevo (2003), supposing infinitely fast feedback and by Takeoka, Sasaki and Van Loock (2005), proposing an implementation using linear optics and continuous measurements. Geremia (2004) interprets the Dolinar receiver as an optimum control problem that implements a channel state estimation, taking advantage of the spatio-temporal extension of a quantum signal. Geremia reports simulations that predict the possibility of operation with imperfect parameters for a realistic channel, i.e. non-unit quantum efficiency detection and finite feedback loop time. Bondurant (1993) has extended the Dolinar receiver principle, predicting near quantum optimum operation for the quaternary phase-quadrature channel. Furthermore, taking advantage of the development of the quantum feedback theory and techniques as well as postdetection digital signal
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processing (Stockton, Armen and Mabuchi, 2002), the binary quantum receiver concept has been experimentally demonstrated with the implementation of a Dolinar receiver for the free space channel (Lau, Vilnrotter, Dolinar, Geremia and Mabuchi, 2006), with a secondary light source for precise phase synchronization. Cook, Martin and Geremia (2007) performed experimental adaptive quantum measurements on an optical fiber Dolinar receiver structure, reaching a BER behavior close to the Helstrom bound by using precisely controlled (weak) coherent states (i.e. a strongly attenuated laser source) and an elaborated feedback control policy from the photon counting process.
4. OPTICAL AMPLIFICATION Nowadays, the measurements of optical fields are generally performed optoelectronically, which means using photo-detection devices, i.e. photodiodes or photo-counters associated with passive and/or active optical devices. The detected signals are then processed electrically. In this technique, both the photon–electron conversion process and the electric signal processing circuit add noise. When a weak optical signal is to be detected and processed, the optical amplification is generally useful for elevating the signal to a level higher than the noises induced by the optoelectronic processing. The optical amplification is, however, also limited by the Heisenberg principle. As we will discuss in this section, it cannot improve, but only deteriorate, the quantum-mechanically limited signal-to-noise ratio (SNR). There are two main types of optical amplifier. One is based on the population inversion that creates a surplus of stimulated emission, turning the initially absorbing media into an amplifying one by changing the sign of the imaginary part of the linear optical susceptibility. The other is based on the coherent multi-wave mixing process in a nonlinear optical media, which splits the pump photon energy into signal photon energy, and, possibly, other bosons, e.g., idler photons and phonon energy. For example, the optical semiconductor amplifiers (SOA) and the Erbium doped fiber amplifiers (EDFA) belong to the former, and the fiber Raman amplifiers (FRA) and the non-degenerate and degenerate optical parametric amplifiers (NDOPA and DOPA) belong to the latter. Quantum noise in optical amplifiers has been already widely discussed from a theoretical point of view (Haus and Mullen, 1962; Yamamoto and Inoue, 2003). In the small-signal regime, the SOA, EDFA, FRA and NDOPA are all linear and phase-insensitive amplifiers, and the DOPA is a linear and phase-sensitive amplifier. In this section, we discuss essentially the quantum noise of the former, and the latter will be briefly discussed at the end of the section.
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191
4.1 Minimum Output Additive Noise 4.1.1 Canonical Conjugation Preservation Let us consider first an ideal linear and phase-insensitive optical amplifier. Its input–output operator relation for one mode can be written in its most general form, as √ aˆ OUT =
G aˆ IN + bˆ
(118)
where G is the optical power gain of the amplifier and is assumed to be a real positive number. Otherwise the difference can only be a constant phase that has no impact on our analysis. The operator bˆ is the intrinsic additive noise operator, belonging to the amplifier internal degrees of freedom and, hence, commutating with the input aˆ in . One can easily verify that the presence of the additive noise operator bˆ is mandatory to preserve the commutation relations of the field operators at the input and output, Ď Ď [aˆ IN , aˆ IN ] = [aˆ OUT , aˆ OUT ] = 1, and therefore its commutation relation is given by ˆ bˆ Ď ] = [aˆ IN , aˆ Ď ] − G[aˆ OUT , aˆ Ď ] = 1 − G. [b, IN OUT
(119)
Now, in terms of the in-phase and quadrature operators, Equation (118) can be rewritten as √ aˆ OU T,I /Q =
G aˆ I N ,I /Q + bˆ I /Q .
(120)
The noise variances on the two components are therefore given by (Yamamoto and Inoue, 2003) h1aˆ 2OU T,I /Q i = Gh1aˆ 2I N ,I /Q i + h1bˆ 2I /Q i.
(121)
Since we have from Equation (119), " [bˆ I , bˆ Q ] =
bˆ + bˆ Ď bˆ − bˆ Ď , 2 2j
# =
j (1 − G) , 2
(122)
and using the Heisenberg’s minimum uncertainty principle, we find 2 D ED E 1 1 1bˆ 2I 1bˆ 2Q ≥ [bˆ I , bˆ Q ] = (1 − G)2 . 4 16
(123)
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Thus, since the linear phase-insensitive amplifier should have equal noise partition on the two components (Yamamoto and Inoue, 2003), their minimum variances are given by D E D E 1 1bˆ 2I = 1bˆ 2Q = |G − 1| . 4
(124)
4.1.2 Classical Equivalent Added Noise As in Section 2.4, the symmetrization of an operator, cancelling out the commutator contributions, allows us to calculate the power operators for the sum of the in-phase and out-of-phase added square noise powers, which is the equivalent of the classical total added noise h i ˆ bˆ + + 1bˆ + 1b)hν ˆ Pˆ A = (1b1 B0 = (1bˆ I )2 + (1bˆ Q )2 hν B0 .
(125)
According to Equation (124), the total average classical equivalent added noise power is obtained as D E hν PA = Pˆ A = |G − 1| B0 . 2
(126)
The total output noise power is obtained by adding the amplified (or attenuated when G is smaller than the unit) input vacuum fluctuations PN = PA + G PN 0 (2G − 1) hν B = (G − 1) hν B + hν B when G > 1 0 0 0 2 2 (127) = hν B0 when G ≤ 1. 2 Notice here that the vacuum fluctuation are considered as an input noise and their contribution is already included in the output noise. 4.1.3 Output Noise Added to the Vacuum Fluctuations Let us consider now the power of added noise involved in Equation (118). The average output photon number is given by Ď
Ď
ˆ haˆ IN aˆ IN i = Ghaˆ OUT aˆ OUT i + hbˆ Ď bi
(128)
ˆ the directly observable average By using the normal order operator hbˆ Ď bi, ˆ = 0, output additive noise photon number is obtained. Assuming hbi
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193
ˆ we can find, from Equations (122) and (124), the which implies bˆ = 1b, Ď ˆ ˆ minimum value of hb bias D E ˆ = 1bˆ 2I + 1bˆ 2Q + j[bˆ I , bˆ Q ] hbˆ Ď bi =
1 G − 1 when G > 1 [|G − 1| + (G − 1)] = 0 when G ≤ 1. 2
(129)
Thus, for an optical bandwidth B O , the total corresponding average output additive noise power PA/N 0 , in excess of the vacuum fluctuation level, is PA/N 0 = PN − PN 0 =
(G − 1)hν B0 when G > 1 0 when G ≤ 1.
(130)
The vacuum fluctuations are not considered here as an input noise and their contribution is to be added to the output noise. For G ≤ 1, we see that the minimum average added noise power PA is zero, which in fact corresponds to the case where there is no amplification but attenuation or a beam splitting. In this case, according to the fluctuation dissipation theorem, the added noise just compensates for the attenuation of vacuum fluctuations to maintain their input noise level. A beam partition noise has been introduced by Ross (1966) to account for attenuation noise in a corpuscular description of the light. For G > 1, PA is proportional to (G − 1). This result is in agreement with the standard amplified spontaneous emission (ASE) average power derivation (Simon, 1983; Desurvire, 1994), where the average ASE power is given by PAS E = Nsp (G − 1)hν B0 , with Nsp ≥ 1 being the spontaneous emission factor. This noise contribution is independent of the input and is to be added to the amplified signal power, as shown by Equation (130). From Equation (121), we see that the two noise-contributions at the output are the amplified input noise and the additive noise. In Section 2.3, 2 i = h1a 2 i = 1/4. we have shown that, for coherent states, h1aˆ in,I ˆ in,Q According to Equations (126) and (127), when the input is incoherent state and for high values of the gain G, these two contributions are closed. Therefore, in the high gain limit, G 1, the noise figure (NF) of a linear and phase-insensitive optical amplifier is approaching the fundamental limit of 2, i.e., 3 dB, as first pointed out by Heffner (1962) and as will be discussed in more detail in Section 4.2. Indeed, this result is valid for any type of linear phase-insensitive amplifier. In the SOAs, for example, the output additive noise is the ASE, and the minimum value of the NF is only obtained when a perfectly population-inverted medium is assumed, i.e., Nsp = 1. In the NDOPAs, where the amplification is provided by splitting
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pump photon energy into signal photon energy and idler photon energy and these last ones are in a different mode from that of the signal, the output additive noise is the amplified zero-point fluctuations entering by the idler channel, and thus, their noise figure is always 3 dB in the large gain limit when other classical noises are neglected.
4.2 Attenuation and Amplification Noises As expressed by Equation (123), the noise generation is unavoidable so long as there is attenuation and/or amplification, i.e. G 6= 1, as is well understood, in the radio frequency range, thanks to the Nyquist theorem. In fact, these noise generation processes, related to the attenuation and amplification at any operating frequency, have been unified by the fluctuation–dissipation theorem. The difference is that, while the Nyquist noise is the radio frequency thermal noise and is well described as classical, it become negligible in the optical frequency range, where the quantum noise cannot, in general, be neglected. Let us now consider the light amplification through propagation in an elementary slice of width 1z of a amplifying medium, with the gain per unit of length β. The minimum noise generation can be deduced from Equation (126) by using G = 1 + β1z. According to the discussion in Section 4.1, the contribution to the single-sided spectral density of noise of the elementary slice, associated with amplification, is therefore, given by 1S =
hν β1z. 2
(131)
Similarly, by using G = 1 − α1z, for the light attenuation through an elementary slice of width 1z in a medium with attenuation per unit of length α, the single-sided spectral density of the elementary noise contribution is expressed as 1S =
hν α1z. 2
(132)
Therefore, for an amplifier medium, including both gain per unit of length β and attenuation per unit of length α, the single-sided spectral density S N of optical noise follows the propagation equation dS N = dz
(β − α)S N | {z }
Noise Amplification
hν + (β + α) . | {z 2} Noise Generation
(133)
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195
In the general case the gain coefficient β decays, due to pump absorption and pump depletion and the general solution of Equation (133) is expressed as incomplete gamma functions. Under the assumption of an amplification L shorter than the effective length corresponding to pump absorption and depletion, the gain as well as the loss does not depend on the coordinate z, the solution of Equation (133) at the output is S N (L) = K (G − 1)
hν + G S N (0) 2
(134)
where G is the net gain, defined as G = exp(β − α)L, and K is the multiplicative noise excess factor compared to the minimum added amplification noise (G − 1)hν/2, expressed as K = (β + α)/(β − α).
(135)
Using, in the optical domain, the standard definition of the noise factor F of a linear amplifier as the excess of amplification of the noise compared to that of the signal, and making reference to a coherent state input, i.e. to vacuum fluctuations input noise with a spectral density S N 0 (0) = hν/2, the total output of fundamental quantum noise (FQN) is S N (L) = F G
hν . 2
(136)
The noise factor F, denoted as noise figure (NF) when expressed in dB, is F=K
G−1 +1≈ K +1 G
for G 1.
(137)
Introducing the built-in internal loss attenuation A = exp(−αL), the noise factor given by Equation (137) can also be expressed as F=
ln G − 2 ln A G − 1 + 1. ln G G
(138)
The noise figure, for various values of the achieved net gain and as a function of fiber loss, is shown on Figure 8. The noise figure is obviously found to be less than the 3 dB high gain limit for the low values of the built-in loss and of the achieved net gain. For a purely attenuating medium with β = 0, implying G = A, the noise figure is F = 1/A. For an exact and local attenuation and gain compensation β = α, implying an overall gain G = 1, the signal propagation is kept at a constant level
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FIGURE 8 Noise figure as a function of fiber loss for various values of the net gain
along propagation, and the noise figure is F = 2αL + 1. The noise figure tends to the 3 dB high gain limit when the overall gain increases, making noise enhancement by the built-in loss negligible. Note that the optical noise figure concept is only relevant when the input noise is specified and when it is defined in the optical domain and for an unsaturated amplifier for which the gain is independent of the signal and for which no noise regression occurs. According to Equation (133), it is easy to see that a signal with an excess of noise when compared to a coherent state turns to a coherent state, since the excess of noise, when compared to vacuum fluctuation, naturally disappears through attenuation with the same decay as the signal, while the attenuated vacuum fluctuation is exactly replenished by attenuation noise up to its nominal level.
4.3 Quantum Langevin Approach Assuming that the input–output operator relation, discussed in Section 2.3, is applicable to all elementary slices of width 1z, implying that the local commutation relation [aˆ (z) , aˆ Ď (z)] = 1 is preserved everywhere in the amplifier, we can easily find that the evolution of the field operator, aˆ (z), is described by the following quantum Langevin equation d 1 aˆ (z) = g (z) aˆ (z) + fˆ (z) dz 2
(139)
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where g (z) is the gain coefficient and fˆ (z) is, therefore, the so-called quantum Langevin force, with [ fˆ (z) , fˆĎ z 0 ] = −g (z) δ z − z 0 .
(140)
Note that dispersion and nonlinearity effects are neglected. The solution of Equation (139) is given by " aˆ (z) =
p
G (z) aˆ (0) +
Z 0
z
# fˆ (x) dx , √ G (x)
with G (z) = exp
Z
z
g (x) dx
0
(141) being the accumulated local gain. It is easy to verify using Equations (139) and (140) that, if [aˆ (0) , aˆ Ď (0)] = 1, the commutation relation [aˆ (z) , aˆ Ď (z)] = 1 is indeed preserved everywhere in the amplifier. Besides, according to Equation (140), we can introduce the rate of additive noise contribution, n¯ f (z), by formally writing the following relations: E fˆĎ z 0 fˆ (z) = n¯ f (z) g (z) δ z − z 0 and E D fˆ z 0 fˆĎ (z) = n¯ f (z) − 1 g (z) δ z − z 0 . D
(142)
Using Equation (140), we can evaluate the average and the variance of the photon number Nˆ (z) = aˆ Ď (z) aˆ (z). For the first, we can find Z E ˆ N (z) = N (z) = G (z) N (0) + D
0
z
n¯ f (x) g (x) dx . G (x)
(143)
Derivation of Equation (143), with respect to z, gives the evolution equation of the average photon number: dN (z) = g (z) N (z) + n¯ f (z) g (z) . dz
(144)
We can immediately identify this equation as the standard amplified spontaneous emission formula, and n¯ sp (z) = n¯ f (z) as the spontaneous emission factor. This last depends generally on z, and its explicit expression depends obviously on the amplification mechanism. We can now rewrite Equation (143) as N (z) = G (z) N (0) + N AS E (z)
(145)
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with the average ASE photon number defined as N AS E (z) = G (z)
Z 0
z
n¯ sp (x) g (x) dx. G (x)
(146)
If the spontaneous emission factor is a constant independent of z, n¯ sp (z) ≡ Nsp , we find the usual expression N AS E = Nsp [G (z) − 1]. On the other hand, for the photon number variance, we can also find the following well-known expression E2 E D D σ N2 (z) = Nˆ 2 (z) − Nˆ (z) h i = G 2 (z) σ N2 (0) − N (0) 2 + N (z) + 2G (z) N (0) N AS E (z) + N AS E (z)
(147)
where the first term vanishes when the input signal is exactly in a coherent state, the second is due to the shot noise, and the third and fourth are, respectively, the beating terms of signal-ASE and ASE-ASE. Using Equation (140), we can also evaluate the single-side spectral density of optical noise and we have 1 S N (z) = G (z) [N (0) − |A (0)|2 ]hν + N AS E (z) + hν 2
(148)
where A (0) = aˆ (0) = aˆ I (0) + j aˆ Q (0) . When the input is in a coherent state, the first term of Equation (148) vanishes. The derivative of Equation (148) with respect to z gives the evolution equation of S N (z) as d 1 S N (z) = g (z) S N (z) + g (z) n sp (z) − hv dz 2
(149)
which deserves more development. In Equation (149), the last term, n¯ sp (z) g (z) is non-negative and can be interpreted as the rate of spontaneous emission. So, according to the well-known Einstein theory of spontaneous and stimulated emissions, n¯ sp (z) g (z) is also the stimulated emission coefficient. Since the stimulated emission or, in general, the stimulated photon creation, is the physical mechanism of optical amplification, we can consider β (z) = n¯ sp (z) g (z)
(150)
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as the total amplification coefficient. The net gain coefficient g (z) should be always smaller than β (z), because there are still optical absorption, scattering losses, etc. We can thus define α (z) = [n¯ sp (z) − 1]g (z)
(151)
as the total attenuation coefficient. With these two coefficients, the net gain coefficient is g (z) = β (z) − α (z) .
(152)
Thus, using Equations (150) and (151), we can rewrite Equation (149) as d S N (z) = [β (z) − α (z)] S N (z) + [β (z) + α (z)] hν/2 dz
(153)
which is just Equation (133)(Bristiel, Jiang, Gallion and Pincemin, 2006).
4.4 Equivalent Lumped Amplifier Noise Factor The performance of a distributed amplifier is usually expressed in terms of the noise factor FLUMP , for a hypothetical lumped amplifier, localized after the corresponding attenuating propagation section, and producing the same amount of ASE power. Observing that the noise figure FFIBER of a pure attenuation fiber is related to its attenuation coefficient by FFIBER = 1/A and using the standard Friis cascading noise factor formula, this noise factor is expressed as FLUMP = AF. This value is strongly dependent on the attenuation of the fiber and may be obviously less than the 3 dB (N F = 2) high gain limit of an ideal amplifier for which K = n S P = 1. Of course this equivalent noise factor may also be negative when expressed in dB, this is a quite misleading, but widely used, concept whose usage is now impossible to uproot. It only expresses the benefit of the gain distribution of the DRA amplifier, for instance, as compared to a more localized amplifier such as the EDFA.
4.5 Pre-amplified Optical System Applications The PIN photodiodes, widely used in optical receivers, are only sensitive to optical power. The simultaneous reception of the signal and the noise in optical fields generates squared and cross-term products with frequencies within the observation bandwidth B E , making the noise no longer additive. When the signal is amplified the noise power PN is usually far larger than PN 0 which corresponds to the vacuum fluctuations.
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4.5.1 Signal Against Optical Noise Beating The cross-term products are usually referred as beat noise or signal against optical noise beating. Thanks to amplification noise we usually have PN PN 0 in pre-amplified systems. Using the same small-noise approximation as in Section 2, the power fluctuations are mainly governed by the inphase noise component x I (t), leading to the instantaneous power which is the squared sum of the deterministic field and the in-phase component of the noise. The resulting optical power fluctuates around its average value hPi = A2 with mean-squared fluctuation written as in Section 3 D E D E (1P)2 ≈ 4P (x I (t))2 = 4P PN I = 2P PN
(154)
where PN = S N B0 is the total averaged noise power of the singlesided spectral density in the signal polarization mode S N , of which only one half is concerned, since PI = PN /2. Observing that only the optical noise spectral components within the spectral range B E on each side of the optical carrier frequency produce beating within the electrical bandwidth, the optical noise bandwidth contribution for the beating, usually considered in optical communication engineering, is determined by B O = 2B E , or observing that, as usually considered in digital communication engineering, x I is a base-band process with spectral density S N , the noise power is PI = S N B E . Denoting the response of the detector by R, the variance of the corresponding photocurrent fluctuations is 2 σSIG−NOISE ≈ 4R 2 P S N B E .
(155)
Signal against noise beating obviously only occurs when a signal field is present on the photodetector and is not to be considered, for instance, during the transmitted “0” of on-off keyed modulated signals with a perfect extinction ratio. 4.5.2 Optical Noise Against Optical Noise Beating Whatever the signal level, the power fluctuations resulting from squaring the noise term alone, which is not the vacuum fluctuation anymore, are to be considered also, including both the in-phase and quadrature components. This contribution is usually referred to as noise- againstnoise beating and has been widely discussed by numerous authors including Yamamoto (1980), Simon (1983), Olsson (1989), Steele, Walker and Walker (1991), Desurvire (1994). The optical noise bandwidth B O to be considered is, in this case, the total noise bandwidth of the optical amplifier itself or the bandwidth of the optical filter used. As for any
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201
Gaussian process, the mean-square of the power fluctuations is equal to the square of the mean power D E (1P)2 = PN2 = PI2 + PQ2 = (S N B O )2 .
(156)
As proposed by Kingston (1978), the power spectral density of noise can be found directly by considering that the spectral spread of the power is twice that of the noise, since squaring multiplies the frequency by 2, that a linear frequency roll-off is mandatory and that integration of the spectrum has to produce the mean-square of the power fluctuations. The power spectral density of optical power fluctuations is therefore expressed as S1P ( f ) =
S N2 (B O − | f |) for | f | < B O 0 for | f | > B O .
(157)
More rigorously, assuming a flat optical noise spectrum within a bandwidth B O , the auto-correlation function of the power fluctuation is expressed as R P N (τ ) =
R 2N
(0) + 2R 2N
(τ ) =
PN2
sin π B O τ + BO π BO τ
2
S N2
(158)
in which R N (τ ) is the correlation function of the noise process itself. By using the Wiener–Kintchine theorem, the power spectrum of the power fluctuations can then easily be calculated by its Fourier transform. On the right hand side of Equation (158), the first term corresponds to the squared mean noise power, and the second one provides the power spectrum of power fluctuations given by Equation (157). The photo-receiver directly converts these fluctuations into current fluctuations with a base-band filtering over the electrical bandwidth B E , leading to the photo-current variance 2 = R2 σNOISE−NOISE
Z
+B E −B E
S1P ( f ) d f = 2m R 2 S N2
BO −
BE 2
B E (159)
in which R is the responsivity and m, equal to 1 or 2, is the number of orthogonal polarization modes producing the noise contribution. In the usual case of optical fiber communication engineering, optical noise is non-polarized and the two polarization components produce the same noise-against-noise beating while only one of them is to be considered for the beating against a polarized signal contribution. In this case m must be set to 2 since PN refers to a single polarization.
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4.6 Technical Noise and Impairment in Optical Amplification In addition to the ASE, there are, in practice, some technical factors, that have impacts on the noise performances of optical amplifiers. First of all, the injection-loss, mainly due to the coupling losses between two optical waveguides, can not only lower the gain of optical amplifiers but also degrade the noise factor. The discrete and distributed reflections in and out of the optical amplifiers, e.g., the reflections at the ends of an EDFA and Rayleigh backscattering in the optical transmission-fibers, can also induce intra-band crosstalk by means of multi-path interference. The nonlinear effects, such as four-wave mixing, cross-phase modulation, and self-phase modulation, etc., can also induce cross-talks and signal pulse distortion. Finally, attention should also be paid to the transient gain effects. Apart from the cross- and self-gain modulation, it can also induce relative intensity noise transfer in, for example, Raman fiber amplifiers and saturated EDFAs, where the optical pump intensity fluctuation can be transferred to the signal (Fludger, Handerek and Mears, 2001; Mermelstein, Headley and Bouteiller, 2002). It has also been shown that the polarization fluctuation acts as an additional noise transfer mechanism (Lin and Agrawal, 2003; Jiang and Gallion, 2007).
4.7 Linear and Phase-Sensitive Amplification Now we will discuss briefly the minimum noise of linear and phasesensitive amplification (Haus, 1987). According to Section 2.3, the most general input–output operator relation, for linear and phase-sensitive amplifiers, can be written as Ď
aˆ OUT = η1 aˆ IN + η2 aˆ IN + bˆ
(160)
or equivalently, in terms of the in-phase and quadrature operators, as aˆ OUT,I /Q =
p
G I /Q aˆ IN,I /Q + bˆ I /Q
(161)
where G I and G Q must be real numbers, because the operators are Hermitian. Similar to the linear and phase-insensible amplifiers, the operator bˆ is the intrinsic additive noise operator. Preservation of the commutation relations at the input and output of the field operator, [aˆ IN,I , aˆ IN,Q ] = [aˆ OUT,I , aˆ OUT,Q ] = j/2, gives its commutation relation as [bˆ I , bˆ Q ] = [aˆ OUT,I , aˆ OUT,Q ] − p j = (1 − G I G Q ). 2
p
G I G Q [aˆ IN,I , aˆ IN,Q ] (162)
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In the same way, the variances of the two components are given by 2 2 ˆ2 h1aˆ OUT,I ˆ IN,I /Q i + h1b I /Q i /Q i = G I /Q h1a
(163)
and Heisenberg’s minimum uncertainty principle gives 2 D ED E 1 p 1 1bˆ 2I 1bˆ 2Q ≥ [bˆ I , bˆ Q ] = (1 − G I G Q )2 . 4 16
(164)
We are interested in the particular case of G I = 1/G Q , where the commutation relation, corresponding to Equation (122) collapses, and, hence, there is no intrinsic internal noise that should be added to the 2 signal, which means h1bˆ 2I i = h1bˆ 2Q i = 0, hence, h1aˆ OUT,I /Q i = 2 G I /Q h1aˆ IN,I /Q i. Then, if the signal is carried on the one of the two, i.e. the in-phase or quadrature, components the noise factor is given by
FI /Q =
2 h1aˆ OUT,I /Q i 2 G I /Q h1aˆ IN,I /Q i
= 1,
(165)
which shows clearly that a theoretical quantum limit of 0 dB is possible for linear and phase-sensitive amplification, but at the expense of using only one of the two components to carry the information.
4.8 Optical Amplification Sensitivity Assuming a large value of gain, the amplification process overcomes the thermal noise limitation. Since some noise contributions may depend on the received optical power, the photocurrent fluctuations are also functions of the transmitted symbol. The output photocurrent fluctuates from one bit to another, around an average value hi 1 i with the variance σ12 when the symbol 1 is transmitted and hi 0 i with the variance σ02 when the symbol 0 is transmitted. For an OOK signal for instance, assuming a perfect extinction ratio, the average photon number per bit N S is half of the average photon number during symbol 1. Assuming a photodetector sensitivity R = 1A/W, the BER is expressed as (Agrawal, 1997; Desurvire, 1994; Gallion, 2002) BER =
1 Q erfc √ 2 2
with Q =
hi 1 i − hi 0 i . σ1 + σ0
(166)
The variance σ12 is obtained by adding the signal-against-noise beating contribution to the noise-against-noise beating contribution given,
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Signal and Quantum Noise in Optical Communications and Cryptography
respectively, by Equations (155) and (159), the variance σ02 is the noiseagainst-noise beating contribution. The average photon number per bit N S , required to obtain a given Q-factor, is F NS = Q Q + 2
s
m 2
2Bo 1 − RB 2
!
.
(167)
Assuming a noise factor F = 2, a number of polarization mode noise contributions m = 2, and a BER equal to 10−9 which leads to Q = 6, N S is found to be 43.3 photon/bit for an optical bandwidth B0 = R B , and 38 photons/bit for B0 = R B /2, as first shown by Henry (1989) and confirmed by Humblet and Azizoglu (1991) and later by Jacobsen (1994). So the quantum limit for an optically preamplified receiver is 38 photons/bit for OOK signals under matched filtering conditions, and 20 photons/bit for differential phase-shift keyed (DPSK) signals using balanced detection as shown by Chinn, Boroson and Livas (1996). On the experimental side for ASK systems at 1550 nm wavelength, Caplan and Atia (2001) have reported a sensitivity of 43 photons/bit for a 5 Gb/s system in a matched communication link, using a Return-toZero OOK (RZ-OOK) modulation format. Atia and Bondurant (1999) have reported a sensitivity of 60 photons/bit for a 10 Gb/s system using a RZOOK format, neglecting the optical-preamplifier input-isolator insertion loss. This result is less than 2 dB from the theoretical quantum limit of 38 photons/bit. Winzer, Gnauck, Raybon, Chandrasekhar, Su and Leuthold (2003) have reported a sensitivity of 78 photons/bit for a 40 Gb/s system using an OOK format in an RZ Alternate Mark Inversion version.
5. SINGLE QUADRATURE HOMODYNE DETECTION The mechanization of the POVMs resulting from quantum detection theory is a difficult task, even in the binary case, requiring a sophisticated preparation of the quantum state to exhibit, for instance, the quantum interference proposed by Osaki, Kozuka and Hirota (1996). On the other hand, the heuristic structures, mentioned in Section 3, require ideal components such as sources, couplers and photon counters, and elaborated feedback policies. Now, despite a sustained progress observed in photon counting technologies, in general they use cooled avalanche photodiodes that exhibit low quantum efficiency, slow frequency response, high dark count and other disturbances such as after-pulse events, especially at the socalled telecommunication wavelength of 1550 nm. Therefore, an alternative technique that has been used for decades in the classical optical telecommunications field is heterodyne or homodyne
Single Quadrature Homodyne Detection
205
detection, based on known receiver configurations. This “structured approach” is not the result of the optimization of a suitable probabilistic function, therefore it not able to reach the Helstrom bound. This near optimal structure is widely used when phase-dependent measurements of the optical field are needed, and is not limited to communications but is used in other applications such as fast tomographic measurements, as mentioned above (Raymer, Cooper and Carmichael, 1995). It has the advantage of using standard room temperature PIN photodiodes, made of InGaAs for telecommunications wavelength operation, that exhibit a very high quantum efficiency and fast frequency response. Furthermore, homodyne receiver structures are flexible and are designed for phase diversity, polarization diversity, and polarization division multiplexing, either with separated local oscillators (LO) or in self-homodyne configurations. On the other hand, in binary detection applications, the photon counter possesses an almost built-in decision characteristic, i.e. its output consists of the photoelectron count process and dark counts for which a user threshold is difficult to implement. For the homodyne detection, having the benefit of a noise-free mixing gain, the output process consists of the signal plus random noise, easily processed, i.e. amplified, filtered or synchronized, and on which user thresholds are easy to implement. Fast digital signal processing techniques for performing these operations have been reported experimentally, enabling the correction of several impairments in the optical channel. Phase synchronization in a feedforward configuration has been the most successful application, but the possibility of compensation for fiber linear and nonlinear phase dispersion has been already reported by Kazovsky, Kalogerakis and Shaw (2006), Noe (2005) and Kikuchi (2006). Thus for the problem of discrimination among quantum states, homodyne detection consists of a quantum mechanical Gaussian operation, that is, mapping Gaussian input states into Gaussian states ρ = ξ1 |αi hα| + ξ2 |−αi h−α| for the binary channel. Takeoka and Sasaki (2007) and Nha and Carmichael (2005) have proved that homodyne measurement is the best strategy to discriminate among binary coherent states with Gaussian operations, and that any classical operation, such as conditional dynamics, does not increase the distinguishability. For the case of more complex constellations with coherent states, i.e. M-ary modulations, further analysis is necessary, but simulations, as reported by Bargatin (2005) showed that adaptive homodyne measurements may increase the mutual information, with a trade-off in the computational complexity of the adaptive algorithm, since distinguishing among an ensemble of states can be conceptualized as a problem of optimization
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over all possible POVMs, the one that realizes minimum overlap among states is optimal. The balanced homodyne detection (BHD) scheme detects the field superposition at the two output ports of the 50/50 coupler and the electronic subtraction cancels the sum of the number of photons at the input ports from the detected fields. Semi classical analysis of the BHD has been made by Abbas, Chan, and Yee (1983), demonstrating the property of cancelling the excess noise of the local oscillator, but still interpreting the quantum limit as the result of the LO shot-noise. Yuen and Chan (1983), Shumaker (1984), and also Collett, Loudon and Gardiner (1987), introduced a quantum mechanical treatment for fields of general quantum state, interpreting the BHD as cancelling both the LO excess and quantum noise and demonstrating that the quantum limit is the signal quantum fluctuation. Extensions to imperfect time/frequency overlap between signal and local oscillators have been analyzed by Grosshans and Grangier (2001). In their operational formulation of homodyne detection, Grabowski (1992) for single-ended HD, and Tyc and Sanders (2004), show that BHD performs the POVM given by the projection |xi hx| to the |xi eigenstate of the quadrature phase operator. Under the conditions of ideal coupler and detectors and LO in a strong coherent state, with a phase θ relative to the received signal, the quantum observable is the field quadrature xˆθ =
exp(jθ )aˆ + + exp(−jθ )aˆ . √ 2
(168)
Thus the ideal BHD is the mechanization of a quantum measurement of the field quadrature defined as the intrinsic homodyne quantum observable |xθ i, consisting of the signal quadrature distribution, rescaled by the amplitude of the LO as the statistical outcome of an ideal experiment xˆθ |xθ i = xθ |xθ i . Therefore it refers to the intrinsic properties of the detection system, independent of the devices used for measurement. For imperfect BHD, lossy couplers and non-unit efficiency detectors add vacuum noise that would result in Gaussian spread of the quadrature measurement. Extensions of the theory have been made by Braunstein (1990), for different signal states and for finite LO power. Banasek and Wodkiewicz (1996) also proposes a family of homodyne operators, that depend on the experimental situation, including random phase conditions. Coherent detection is a well-known method using combining and nonlinear mixing, usually with a square-law detector, of the signal field to be detected with a reference field, the so-called local field or LO.
Single Quadrature Homodyne Detection
207
FIGURE 9 Homodyne detection arrangement
When the frequencies of the two mixed fields are different, it is referred to as a heterodyne detection and as a homodyne detection when the two frequencies are identical. Heterodyne detection allows a band-pass recovery of the signal information centred at the frequency difference between the two mixed signals, the so-called beating frequency or intermediate frequency (IF), allowing easier post-detection information processing free of low-frequency noise and fluctuations. In optical interferometry, homodyne corresponds to the generation of the local oscillator field by a previously derived part or by beam splitting from the same source from which the beam producing signal, after scattering for instance, originated. Such an arrangement is insensitive to fluctuations in the frequency of the common field source except for those occurring on a time scale smaller than the usually short delay due to unbalanced path beams recombination. Decoherence resulting from time delay will be discussed in Section 5. In coherent detection the signal field is usually weak, when compared to the local one, and a strong and noise free mixing gain overcoming thermal noise is obtained. Homodyne detection and heterodyne detection represent, respectively, ideal measurements for a single quadrature and simultaneous two-quadrature components of the signal (Oliver, Haus and Townes, 1962; Yuen and Shapiro, 1980). For this reason heterodyne detection will be discussed in Section 6
5.1 Quantum Theory of Homodyne Detection Let us consider the homodyne detection of a signal with local oscillator ˆ fields described by the quantum photon annihilation operators sˆ and l. The two fields are first assumed to be combined by using a lossless and perfectly balanced coupler as depicted in Figure 9. According to the coupler transfer matrix given by Equation (89) the resulting fields on detector D1 and D2 are 1 ˆ aˆ 2 = √1 (−jˆs + l) ˆ aˆ 1 = √ (ˆs − jl) 2 2
(169)
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Signal and Quantum Noise in Optical Communications and Cryptography
and the associated photon number operators are respectively 1 + sˆ sˆ + lˆ+lˆ − j sˆ +lˆ − lˆ+ sˆ and 2 1 + sˆ sˆ + lˆ+lˆ + j sˆ +lˆ − lˆ+ sˆ . aˆ 2+ aˆ 2 = 2
aˆ 1+ aˆ 1 =
(170)
The coherent subtraction of the two photocurrent outputs allows us to take advantage of all the signal interactions spread on the two detectors by the combiner. Assuming perfect quantum efficiency for the two detectors, the photoelectron number operator is equal to the photon number operator for each detector and the electron number operator for the subtraction output current is given by Nˆ = aˆ 1+ aˆ 1 − aˆ 2+ aˆ 2 = −j sˆ +lˆ − lˆ+ sˆ .
(171)
Using signal and local field operator expansions in terms of in-phase and quadrature Hermitian components sˆ = sˆ I + jˆs Q
and lˆ = lˆI + jlˆQ
(172)
we have Nˆ = 2 sˆ I lˆQ − sˆ Qˆ lˆI .
(173)
Note that, in this description, sˆ and lˆ refer to the field at the input of the optical combiner and that, despite their different index, sˆ I and lˆQ ( or sˆ Q and lˆI ) refer to the same quadrature at the detector input, according to the phase shift property of the optical combiner. Despite the natural character of this description when optical fiber couplers are used, we can significantly simplify the notation, by referring the phase of the local oscillator field to the detector 1, i.e by a simple phase reference ˆ into the above equations. We obtain with this change turning −jlˆ in l, notation (Yuen and Chan, 1983) Nˆ = sˆ +lˆ + lˆ+ sˆ = 2 sˆ I lˆI + sˆ Q lˆQ .
(174)
Including the coupler phase rotation, this phase reference convention is in agreement with those used in the field of quantum optics. Nˆ is twice the projection of the signal operator on the local field operator. We will
Single Quadrature Homodyne Detection
209
restrict our analysis to the case where both the local and signal fields are single coherent states. Braunstein (1990) has addressed the case of a signal under the form of a coherent state superposition. Assuming that the signal and the local fields are coherent states, we can separate the classical and quantum contributions for the two quadratures of the signal and the local oscillator field in the form
sˆ I = S I + 1ˆs I with S I = sˆ I D E lˆI = L I + 1lˆI with L I = lˆI
and sˆ Q = S Q + 1ˆs Q with S Q = sˆ Q D E (175) and lˆQ = L Q + 1lˆQ with L Q = lˆQ .
To detect S I (or S Q ) we have to set L Q (or L I ) to zero. Assuming that S I is to be detected, we set L Q to zero. Therefore the local oscillator acts as the phase reference for the in-phase and quadrature definitions and we obtain Nˆ = 2
L I + 1lˆI
S I + 1ˆs I + 1lˆQ S Q + 1ˆs Q .
(176)
2 , the dominating For strong local oscillator level N L = L 2I N S = S I2 + S Q term is
Nˆ = 2ˆs I lˆI = 2 S I + 1ˆs I L I .
(177)
We have neglected quantum fluctuations of the local oscillator since they are added to its deterministic part and, therefore, have no cross product with it. The output signal of a balanced homodyne detection arrangement is proportional to the quadrature S I and its additional quantum noise 1ˆs I . Its input signal is found to be amplified by the deterministic part of the in-phase local oscillator quadrature on the detectors which act as a noise-free mixing gain. This in-phase local oscillator quadrature of the detectors corresponds to the orthogonal quadrature of the local oscillator when referred to the input of the optical combiner, according the phase shift property of the optical combiner already discussed. In homodyne detection only one quadrature is measured and no noise addition to the zero-point fluctuation of the signal field is introduced. As reported by Yuen and Chan (1983) the input signal quantum noise is therefore the only noise limitation. The local oscillator noise has a negligible influence and the output noise is only governed by the vacuum fluctuation entering the signal port. As a consequence, homodyne detection is only limited by the quantum fluctuation of the signal itself. The limitation of the output noise of a homodyne detector at quantum level has been experimentally confirmed by Machida and Yamamoto (1986). They point out the difficulty
Signal and Quantum Noise in Optical Communications and Cryptography
210
of verifying that the quantum noise of a local oscillator wave can be canceled as well as its excess of noise, when the signal and the local waves possess the same amount of quantum noise. A squeezed state input signal is required in order to clarify this point and completely refute the semiclassical description based on the local oscillator shot noise.
5.2 Second Order Momentum and Signal to Noise Ratio Assuming a perfectly phase matched local oscillator, free of relative phase fluctuations with respect to the signal, the overall photon number operator is D E D E Nˆ = Nˆ + 1 Nˆ with Nˆ = 2L I S I
and
1 Nˆ = 2L I 1ˆs I .
(178)
The average photon number is equal to the average electron number, assuming unit quantum efficiency. The square electron number is D
E D 2 E Nˆ 2 = 4L 2I S I2 + 1ˆs I .
(179)
Assuming that signal and local are coherent states, denoted by |α S i and |α L i, respectively, that a constant envelope modulation is used for the signal and introducing the average signal and local photon number N S and N L defined as
N S = sˆ + sˆ = |α S |2
and
D E N L = lˆ+lˆ = |αL |2 ,
(180)
we obtain D
E D E2 2 1 2 2 2 ˆ ˆ ˆ N = N + 1N = 4L I S I + = 4N L N S + N L . 4
(181)
The first term on the left hand side of Equation (181) is the averaged square of the signal photon number, while the second term is the averaged square of the photon number fluctuations D E2 Nˆ = 4N L N S
and
1 Nˆ L
2
= NL .
(182)
The last part of Equation (182) is the well-known Poisson fluctuation relationship, in agreement with the classical theory of homodyne detection, which will be discussed below and for which the fundamental
Single Quadrature Homodyne Detection
211
noise limitation is interpreted as the local oscillator shot noise. The signalto-noise ratio (SNR) is given by SNR =
4N L N S = 4N S NL
(183)
In digital communication systems it is common to express the SNR in terms of the energy per bit divided by the noise power spectral density, i.e. the ratio E B /N0 , where E B is the energy per bit and N0 the singlesided spectral density of the noise. It is a normalized signal-to-noise ratio measure, also known as the SNR per bit. Denoting by T the bit duration, i.e. the observation time, assuming that a matched electrical filter with equivalent bandwidth B E = 1/2T is used and keeping in mind that, since the homodyne beating signal is base-band, the optical bandwidth B O is identical to the electrical one, the signal-to-noise ratio given by Equation (183), may also be expressed as a function of the averaged optical signal power PS , in the form (Proakis, 1995). SNR =
PS hν 2
BO
=2
EB . N0
(184)
The single-sided spectral density of noise N0 is equal to the zero point fluctuation of the optical field S N 0 = hν/2, as shown in Section 2. The signal-to-noise ratio per bit is especially useful when comparing the BER performance of different digital modulation schemes without taking the bandwidth consideration into account. It is equal to the SNR divided by the link spectral efficiency in bit/s/Hz, which is R/B0 = 2 in our case, where the bit rate R = 1/T refers to data, independently of error correction overhead or modulation symbols.
5.3 Comparison with the Classical Approach Result In the classical approach, the homodyne detection of a quantum noise free classical signal field with two quadratures s I and s Q is considered by using a classical local oscillator with the two quadratures l I and l Q . Using the same phase convention as in Section 5.2 and assuming that s I (or s Q ) is to be detected and that as a consequence, l Q (or l I ) is set to 0, the electron number is N = 2s I l I .
(185)
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Assuming than l I is larger than s I the shot noise associated the DC photon rate on each detector is D E l2 (1Ni )2 = hNi i = I 2
(186)
and adding the two detector noise contributions, the fluctuations at the output of the adder are D E (1N )2 = hN i = l 2I .
(187)
Assuming a constant envelope modulation of the signal and introducing the average signal and local N S and N L photon numbers defined as N S = s I2 N L = l 2I ,
(188)
the SNR, given by Equation (183), is thereby obtained. This result is the same as that obtained from the quantum theory in Section 5.2. However the local oscillator shot noise should be produced by the vacuum fluctuation entering through the local port and it has been shown that this has no influence on the output noise. The coherent subtraction of the two photocurrent outputs also allows rejection of the classical and quantum fluctuations of the local oscillator. However, the classical approach may be considered as a na¨ıve description, conceptually wrong but leading to valid numerical results in most experimental situations.
5.4 Coherent Optical System Applications Okoshi, Emura, Kikuchi and Kersten (1981); Okoshi (1984, 1987); Okoshi and Kikuchi (1988) have already widely discussed the application of coherent detection to optical fiber communications. As mentioned in Section 3, the different coherent states transmitted by conventional light sources are not orthogonal. Because of the non-commutation of the non-orthogonal state projective measurements, a simple Von Neuman projective measurement cannot conclusively distinguish the different states. For the sake of conciseness, here we will only consider the case of binary phase-shift keying (BPSK) in which the two modulated binary symbols (“0”, “1”) correspond to two antipodal states for the phase (0, π ). This corresponds to the simplest constant envelope modulation, using antipodal coherent signals and obviously maximizing the signal distance, minimizing the two state overlap and therefore minimizing also the BER
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to be expected. The average received power is the same when the symbol “1”, or the symbol “0”, are transmitted. Note that the results will remain valid for phase encoded quantum cryptography using the Bennett and Brassard quantum key distribution (QKD) protocol, the so called BB84 protocol, in which the two binary symbols are represented by two antipodal signals on two different orthogonal bases forming a quadrature phase-shift keying (QPSK) constellation. In this case the base coincidences turn into a BPSK detection. Base anti-coincidences are not to be considered since their results are discarded and so do not contribute to the BER. For sake of simplification and for maximization of the transmitted message entropy, we assume that the probabilities p(1) and p(0) of transmitting “1” and “0”, respectively, are equal to 1/2. The transmitted signal is the equal probability statistical mixing of the two to quantum antipodal coherent states with the average density operator ρˆ ρˆ =
1 (|α S i hα S | + |−α S i h−α S |) . 2
(189)
Since, according to Equation (177), the operator density for the electron number Nˆ and the operator for the projection sˆ I of the received coherent state on the selected local oscillator quadrature differ by a factor 2L I , the probability density to find a given value N of the electron number at the output of the balanced detection arrangement fulfills p(N ) dN = p(s I ) ds I
and
p(s I ) = hs I |α S i hα S |s I i .
(190)
As, according to Equation (62), for a coherent state, we have r p(s I ) =
2 exp −2 (s I − S I )2 , π
(191)
the probability density p(±N ) to find a given value N of the electron number in one of the current signs at the output of the balanced detection arrangement is p(±N ) = √
2 √ N ∓ 2 NS NL 1 exp − . 2N L 2π N L
(192)
The observed output photon N fluctuates, from one bit to another, D Enumber √ √ around an average value Nˆ = 2 N L N S with the r.m.s. value σ = N L 1
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D E D E √ = −2 N L N S = − Nˆ when the symbol “1” is transmitted and Nˆ 0
1
with the same r.m.s. value when the symbol 0 is transmitted. At the decision time t D , determined by a clock recovery circuit, the decision circuitD compares E D the E observed current value N with a threshold value N D . ˆ ˆ Since N = − N the threshold value is set to 0. When N is found above 0
1
0 the firm decision that a “1” was transmitted can be made. When N is found below 0 the firm decision can be made that a “0” was transmitted. The performance of a digital communication system is expressed in terms of bit error probability, also called the bit error ratio (BER), this is defined as the ratio of the number of the wrong decisions to the number of transmitted bits. In optical communication, the bit error ratio is also frequently referred to as the bit error rate. However, this name is somewhat misleading since it is not the number of errors per unit of time that is being considered, but the error probability. An error occurs due to the signal noise, when, with the probability P(0/1), N is found to be above 0 whereas the symbol “0” has been transmitted, or in the same way, when with the probability P(1/0), N is found below 0 whereas the symbol “1” has been transmitted. The BER is therefore expressed as BER = p(1)P(0/1) + p(0)P(1/0) =
1 (P(0/1) + P(1/0)) . 2
(193)
Since the communication channel is symmetrical here, the error probabilities of the two transmitted symbols are identical and the BER is written as 1 BER = √ 2π N L
Z 0
+∞
2 # √ N + 2 NS NL exp − dN . 2N L "
(194)
which can be written either by using the complementary error function or the Marcum function (Ziemer and Peterson, 2000) √ √ 1 BER = Q(2 N S ) = erfc 2N S 2
(195)
in which erfc(x) = 1−erf(x) is the complementary error function and Q(x) is defined as 2 erfc (x) = √ π
∞
Z x
h i exp −u 2 du
1 x Q(x) = erfc √ . 2 2
(196)
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This standard result, referred as the SQL, is in good agreement with the classical analysis given by Proakis (1995) and Saltz (1986). For large photon numbers by using the standard asymptotic approximation x 2 1 Q(x) ≈ √ exp − . 2 x 2π
(197)
it is shown that we have BER <
1 exp(−2N S ). 2
(198)
Since a complete differentiation of the transmitted states is not possible an inherently finite error rate is obtained. Such a value is far lower than the practical error rate achievable with photon counters. Phase-shift-keying (PSK) systems using homodyne detection are a promising technique by allowing at the same time a good approach to the quantum noise limited sensitivity and excellent spectral efficiency. The price to pay is obviously the availability of a strong phase reference at the receiver. In the late 1980s early experiments employed optical phaselock loops (OPLL) to lock the phase of a local oscillator to that of the incoming signal (Kazovsky and Atlas, 1990). The phase noise sensitivity of coherent receiver has been widely discussed in the 1980s and 1990s and later reviewed by Kazovsky, Kalogerakis and Shaw (2006). It is pointed out that OPLL receivers are very sensitive to the phase noise, requiring laser line width 10−6 times lower than the bit rate. Due to the difficulty in implementing an optical phase lock loop (OPLL) to generate the local oscillator, homodyne detection has not yet been used in trial optical communication systems. The main issue in optical communication is not to find the modulation and associated demodulation techniques having the better quantum limited sensitivity, but to find those which are able to operate closest to this limit under practical system impairments. Let us consider, for instance, an ideal amplitude shift keying (ASK) with a perfect extinction ratio, also referred as OOK. Since the probability P(1/0) is equal to zero, the optimal BER is obtained by setting the decision threshold close to 0 and by using Poisson statistics for P(0/1) directly. The detection is degenerated in this case since no error occurs for the transmitted “0”. The BER is BER =
1 exp(−2N S ). 2
(199)
The average photon number per bit N S is the half of the number of photons during the optical pulse. According to Equation (199), the
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theoretical required value N S to obtain a BER equal to 10−9 is 10 photons/bit, which is in practice unachievable due to thermal noise limitation level which is 650 times larger (Gallion, 2002). The quantum noise limitation can unfortunately be obtained only for a high signal level with an associated shot noise overcoming the thermal noise limitation, or under low temperature operation, which is of little interest for communications systems. One of the more promising techniques in optical communications includes in-line or optical preamplifiers in association with the differential phase shift keying (DPSK) and delay line demodulation. This scheme is a self-delay homodyne (or interferometric) arrangement using a weak local oscillator which is similar to the super homodyne arrangement, i.e. the Kennedy receiver, discussed in Section 3. In the DPSK format the transmitted signal during a bit time is a replica of the signal sent during the previous bit time, to which a phase shift π is added or not according to the “1” or “0” value of the bit symbol. The basic receiver consists of an optical filter of bandwidth B O , a Mach-Zehnder optical interferometer, including in one of its branches a differential delay T precisely adjusted so the νT is an integer. At each of the output branches of the Mach-Zehnder filter, a photodetector and integrator are used. At a given time t, half the sum of the optical signal inputs at times t and t − T appears at one output of the Mach-Zehnder, while half the difference appears at the other output. So, in a Mach-Zehnder delay interferometer and balance detection, the information signal appears only at one of the two Mach-Zehnder outputs, depending on whether the signal phase was shifted or not with respect to the previous bit. DPSK is well known to avoid local phase reference generation or recovery and to relax laser line width requirements, since only the optical phase diffusion over one bit duration impairs the mixing. The finite line width impairment in unbalanced path systems will be discussed in Section 5.6.3. Because the thermal noise is neglected the sensitivity advantage of a DPSK system, as compared to an OOK one, is a 3 dB reduction of the required OSNR to achieve a given BER. Because a LO is no longer necessary, DPSK transmission in conjunction with differential detection and differential encoding has attracted considerable interest in the past few years due to its good robustness to laser line width impairments, high tolerance to fiber propagation impairment and thanks to its narrow modulation bandwidth and constant envelope signal. At the 1550-nm wavelength, using a return-to-zero DPSK (RZ-DPSK) signal at 10 Gb/s and a balanced-photodiode detection scheme, the experimental sensitivity of 30 photons/bit, has been reported by Atia and Bondurant (1999). As expected, this result is 3 dB better than the OOK one at the same bit rate. Gnauck, Chandrasekhar, Leuthold and Stulz (2003) have reported sensitivity of 45 photons/bit at 42.7 Gb/s. At the same
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bit rate, a sensitivity of about 39 photons/bit has been reported using RZ-DPSK by Sinsky, Adamiecki, Gnauck, Burrus, Leuthold, Wohlgemuth and Umbach (2003) and also by Idler, Klekamp, Dischler, Lazaro and Konczykowska (2003). This is approximately 3 dB better than the best OOK results of 78 photons/bit at the same bit rate.
5.5 Application to Cryptography Homodyne detection is an alternative method to photon counting for detecting weak light pulses. The photon counting method suffers from a technical limitation resulting from thermal effects such as the dark-count, which can only be reduced with a trade-off in the reduction of quantum efficiency At present, there exists no efficient photon counter operating at the optical communication wavelength 1550 nm. Thanks to its noise free mixing gain and its flexible post-detection decision(s) implementation of homodyne detection appears today to be a promising technique for achieving quantum limit applications. 5.5.1 Phase-Encoded Cryptographic Systems Quantum Cryptography, as the protocol proposed by Bennett and Brassard in 1984 for QKD, is based on the quantum properties of single photons and promises unconditional security (Bennett and Brassard, 1984). Waiting for the availability of single photon sources the use of faint coherent state optical pulses is an attractive alternative since they are easily produced by well-stabilized conventional, reliable and inexpensive light sources. Utilization of the telecommunication wavelength allows us to take advantage of the low fiber attenuation of the telecommunications infrastructure, and of the stable low-cost integrated telecommunications devices and circuitry that are mandatory for trial deployment on a large scale. The key issue for the system implementation is the detection of very weak optical pulses. For optical fiber applications, phase modulation appears more robust to transmission impairments than the initially proposed polarization encoding. The simplest implementation for the Qbit modulation in the BB84 protocol is obtained when Alice encodes her Q-bits in two orthogonal bases with two antipodal symbols on each base, leading to a QPSK modulation format. The transmitted signal density operator ρˆ is in this case expressed as ρˆ =
1 (|α S i hα S | + |−α S i h−α S | + |jα S i hjα S | + |−jα S i h−jα S |) . 4
(200)
When the base choice is announced, during the reconciliation process, the results of the measurements of signals orthogonal to the selected reception base are discarded and the problem reverts to a PSK detection for the remaining signal, reducing the density operator to Equation (189).
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5.5.2 Interferometric and Photon Counting Detection Interferometric arrangements are sometimes used to perform the implementation of phase detection. The key issue is to obtain a phase reference at the receiver. The separate transmission of phase reference by a separate fiber leads to the difficult stabilization of an interferometer over the span of the transmission link. Using a single route and a single path configuration, to avoid the penalty of a round trip (Muller, Herzog, Huttner, Tittel, Zbinden and Gisin, 1997), is mandatory. Using DPSK is another way to provide phase reference by relaxing the phase stabilization on a time scale of the same order as the bit duration. Demodulation by the delay line method has been extensively discussed during the early days of optical communications (Okoshi and Kikuchi, 1988; Kazovsky, 1986a,b; Chikama, Watanabe, Naito, Onaka, Kiyonaga, Onoda, Miyata, Suyama, Seino and Kuwahara, 1990) and more recently (Xu, Liu and Wei, 2004; Gnauck and Winzer, 2005). This technique implements a super homodyne nulling receiver but suffers, especially for signals at the quantum level, from the penalty due to the time splitting of the received bit energy. An attractive dual implementation in the frequency domain has been proposed by addition/subtraction of in-phase or anti-phase coherent modulation side bands (M´erolla, Mazurenko, Goedgebuer, Porte and Rhodes, 1999a; M´erolla, Mazurenko, Goedgebuer, Porte, and Rhodes, 1999b). However, for all these techniques, the lack of an optical mixing gain makes the use of photon counters for detection of quantum level pulses mandatory. 5.5.3 Double Threshold Homodyne Detection By using a standard PIN photodiode and taking advantage of mixing with a strong local optical field, the homodyne detection technique is potentially able to overcome undesirable effects such as after-pulses and dark counts, characteristics of single photon detection measurement (SPDM). Furthermore balanced homodyne detection can be implemented by using high efficiency, high bandwidth and low cost PIN photodiodes operating at room temperature at the fiber telecom wavelength. Balanced homodyne detection (BHD) of phase encoding signals using a high power local oscillator for operation near the quantum limit, is commonly used in classical optical fiber communications and can be implemented for the detection of suitable quadrature for quantum level signals (Hirano, Yamanaka, Ashikaga, Konishi and Namiki, 2003) such as weak coherent pulses. Homodyne detection also allows phase signal encoding, which is more suitable for optical fiber communications than polarization encoding in one-way systems. In addition to its high quantum efficiency, the BHD system is free from undesired counts and allows implementation of a flexible decision threshold which is
Single Quadrature Homodyne Detection
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completely independent of the optical detection process. The coherent homodyne detection is also sensitive to phase and polarization matching, thus leading to a channel selection scheme that is very useful for background radiation rejection in free space application and for the compatibility with the current WDM networks. Intrinsic performances of the balanced homodyne detection are inferior to those of the super homodyne Kennedy receiver and may compromise the quantum security in standard decision implementation. However the problems of optical communication and quantum cryptography differ significantly. In classical digital communications using standard forward error coding (FEC), with transmission overhead usually less than 10%, each bit restoration is usually attended for the maximization of the BER. The overhead is a priori added to the transmitted message and is only used to recover from channel erasures. Decision abandon in QKD is turning into efficiency reduction since the corresponding information can be suppressed during the reconciliation. Decision abandons, allowing inconclusive results for weak values of post-detection signals, may be easily implemented by using the flexibility threshold control at the strong post-detection electrical output signal level. The inconclusive results reduce the efficiency of the key production but it may remain far above the photon counter efficiency at the telecommunication wavelength. In this case, the reconciliation has to simultaneously include the base coincidence and the diffusion of the rank of the inconclusive bits on the public channel. Some security improvement (Koashi, 2004) may be expected since Eve cannot abandon any decision result before reconciliation, and because when the attacks lead more to a Bob’s signal degradation than a substitution, the probability of inconclusive Bob decisions is enhanced, making the inclusive bit erasures more efficient than the suppression of base non-coincidence of the basis, which is independent of Eve’s intervention. Assuming the simple symmetrical decision rule 1 if N > X N S bit value = 0 if N < −X N S inconclusive otherwise,
(201)
where X is the double threshold separation parameter, the post-detection efficiency ρ is the probability of obtaining a conclusive judgment: h i h i ρ = 1/2erfc (2N S )1/2 (X + 1) + 1/2erfc (2N S )1/2 (X − 1) ≤ 1.
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The post selection BER is finally expressed as h i BER = (1/2ρ) erfc (2N S )1/2 (X + 1) .
(202)
The BER, as a function of the threshold parameter X and for different values of the average number of photons per pulse, is presented in Figure 10. The BER can be reduced to an arbitrarily low value by sacrificing the key exchange efficiency, as shown in Figure 11. Figure 12 shows the BER, as a function of the average number of photons per pulse, for different values of the threshold parameter X . The more pertinent parameter to describe the influence of quantum noise is the information throughput h h ii I AB = BER + erfc (2N S )1/2 (X − 1) !# " BER × 1+ H 1+ BER + erfc (2N S )1/2 (X − 1)
(203)
where H is the binary entropy defined as H ( p) = − p log2 p − (1 − p) log2 (1 − p).
(204)
Obviously the differential of mutual information between Alice and Bob and, for instance Alice and Eve, as a function of the attack strategy is more oriented to the security application (Hirano, Yamanaka, Ashikaga, Konishi and Namiki, 2003).
5.6 Technical Noise and Impairment in Homodyne Detection 5.6.1 Local Oscillator Static Phase Mismatch In homodyne detection, the local oscillator acts as the phase reference for in-phase and quadrature definitions. With a phase mismatch θ of the local oscillator, sˆθ = sˆ exp (−jθ ) is observed instead of sˆ and the photon number operator is Nˆ = 2 |L| sˆθ I = 2 |L| sˆ I cos θ + sˆ Q sin θ .
(205)
While the quantum noise associated with the signal is invariant through phase rotation, the output noise power also remains unchanged, leading to the signal-to-noise ratio 2 S = 4 S I cos θ + S Q sin θ . N
(206)
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FIGURE 10 BER as a function of the threshold parameter X for different values of the average number of photons per pulse
FIGURE 11 Key exchange efficiency: a function of the threshold parameter X and different values of the average number of photons per pulse
Assuming a single quadrature S I informative signal, we have S = 4N S cos2 θ. N
(207)
The static phase mismatch appears as a simple attenuation of the incoming signal leading to a higher signal photon number to achieve a given performance level. 5.6.2 Classical Phase Fluctuation Let us consider now the effect of classical phase fluctuations from the technical origin around the quiescent value θ = 0. Limiting our discussion
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FIGURE 12 BER as a function the average number of photons per pulse for different values of the threshold parameter X
to the signal change induced by the first order 1θ of the phase fluctuation, Equation (205) leads to 1 Nˆ = 1N = 2L S Q 1θ.
(208)
Since S is the maximum value of S Q , taking the root-mean-square value, we obtain (δ N )Phase = 2L Sδθ
(209)
The phase noise is multiplied by the signal and also by the local amplitudes, and the local field acts as a lever for the classical phase noise effect. Since the obtained output signal is also proportional to the received signal, the noise is no longer additive and the signal-to-noise ratio concept must be used very carefully. The stabilization requirement may be evaluated by considering that this additional fluctuation term has to be kept far below the quantum phase noise contribution (δ N )quantum noise = L, leading to δθ
1 . 2S
(210)
The classical fluctuation appears therefore as a limitation for the signal level, which is an important limitation for strong signal level applications, for instance for QKD using continuous variables.
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5.6.3 Phase Diffusion and Line Width Shawlow and Townes (1958) and Lax (1967a,b) first pointed out the role of fluctuations in the phase of the optical field on the laser line width. When considering a semi-conductor laser (SLC), Henry (1982) began with a corpuscular point of view, in which the instantaneous changes of the phase of the optical field are caused by discrete spontaneous emission events, which discontinuously alter the phase and intensity of the lasing field. Henry’s 1982 approach pointed out the importance for SCL of the phase amplitude coupling, resulting from deviation of the imaginary part of the refractive index from its steady-state value caused by the change of its real part associated with the gain change, and including an additional phase shift of the laser field. This effect has been previously pointed out by Lax (1967a,b), but was of negligible effect for the laser considered. As intensity fluctuations are smoothed out by gain saturation, the averaged optical phase diffuses in a Brownian motion due to the lack of a restoring force, under the direct and the phase amplitude coupling induced phase changes. Quantum phase noise has been already discussed in Section 2.4.3 for coherent states. However field line width is not a property of quantum noise alone, but is also the result of phase diffusion. Electromagnetic wave packets are the classical counter part of the electromagnetic field quantum states. The coherence time is the inverse of the laser line width which is determined by the laser source and corresponding population inversion, built-in losses, cavity Q factor and phase coupling factor. The laser line width has also been discussed by Nilsson (1994) in terms of quantum noise filtering by the lasing cavity. A CW laser, far above threshold, generates a sequence of consecutive nearly coherent states with individual finite time occupancy corresponding to the coherence time. Through the coherent states succession, the phase goes though a random walk, i.e. a Brownian motion, except when a restoring force is applied by using, for instance, injection-locking techniques (Gallion, Nakajima, Debarge and Chabran, 1985). Anyway, phase can never be controlled within the photon numberphase, Heisenberg uncertainty will be discussed in Section 6. Following Nilsson’s approach (1984) the laser fluctuation is filtered by the laser cavity bandwidth. Let us assume a perfect integrator over the cold-cavity photon lifetime τ P , with the associated band-pass equivalent optical noise bandwidth 1/τ P and a noise power spectral density S N = hν inside the cavity equal to twice the zero point spectral density, to account for optical amplification and equivalent internal and external noise as well. According to Equation (78), we have D E (1ϕ)2 =
hν 1 = 2 hPi τ P 2 hI i
(211)
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where hI i is the average photon number inside the cavity. For an observation time τ much larger than τ P , by adding the τ/τ P average square values concerned, we obtain D E D E 1ϕ 2 (τ ) = [ϕ (t + τ ) − ϕ (t)]2 =
1 τ . 2 hI i τ P
(212)
Assuming an amplitude stabilized field in the form E(t) = E 0 exp j[ω0 t − ϕ(t)] and for which the spectrum broadening is only produced by phase noise, the standard procedure is to derive the spectrum by using the Wiener Khintchine theorem, starting with the derivation of the field autocorrelation function (Yariv, 1989; Agrawal and Dutta, 1986)
R E (τ ) = E(t)∗ E(t + τ ) = E 0 hexp j1ϕ(t, τ )i exp jω0 .τ.
(213)
By using a standard signal analysis relationship (Rowe, 1965) we have E 1D R E (τ ) = E 02 exp − 1ϕ 2 (τ ) exp jω0 τ 2
(214)
and the standard Lorentzian spectral profile is finally obtained by Fourier transform of Equation (214), using Equation (165) S E (ω) =
1ω L /2π R E (τ ) exp(−jωτ )dτ = E 02 2 2 (ω − ω 0 ) + (1ω L /2) −∞
2 1ϕ (τ ) with 1ω L = . τ Z
+∞
(215)
By including the incomplete inversion factor n S P , to account for the ratio between the total laser gain and the net gain, and including the phase amplitude coupling effect (Lax, 1967a,b; Henry, 1982), the full laser line width at half maximum (FWHM) is found as n S P 1 + α2 1ω L = 1ν = 2π 2 hI i τ P
(216)
where α is the phase amplitude coupling parameter i.e. the ratio between the real and the imaginary parts of the index change. This result was obtained by Henry (1982) using a corpuscular representation in the form of random phase photon addition to the lasing mode, but is usually obtained by using the standard Lorentzian spectral profile for the cavity,
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with a FWHM 1 f FWHM = (2π τ P )−1 , where τ P is the cold cavity photon lifetime at 1/e, rather than the equivalent band pass bandwidth. Phase diffusion is a natural limitation for interferometric arrangements, in the case of unbalanced recombination of the path acting as local oscillator and the path acting as signal. As first shown by Armstrong (1966), and discussed in detail by Gallion and Debarge (1984) the homodyne photocurrent spectrum consists, in this case, of two terms, whose behavior relates closely to the normalized time delay, and phase mismatch values. A frequency Dirac function, corresponding to the DC component, stands for the incoherent addition of the two optical powers to the amount of remaining phase correlation between the two mixed beams. When the time-delay is large, as compared the coherence time, we have completely uncorrelated mixed fields and the dependence on the phase matching vanishes out. When the time-delay is small as compared to the coherence time, it becomes very sensitive to phase mismatch values and it no longer depends on the spectral spread, turning into a pure Dirac function, whatever the spectral width is. The second term takes the form of an approximately Lorentzian line shape. It vanishes for a close-to-zero delay value. For large delay values, this term is no longer dependent on the phase matching and stands for the optical mixing of two independent fields and it becomes rigorously Lorentzian with a FWHM of twice the original laser line width, because the detector acts as an optical product detector, whose output is the autocorrelation product of the laser field spectrum. 5.6.4 Unbalanced Homodyne Detection and Imperfect Quantum Efficiency Kennedy’s (Kennedy, 1973) binary coherent-state signals receiver uses a homodyne-like configuration with a weak local oscillator whose amplitude is matched to the signal to produce an unconditional nulling of one of its antipodal values. Dolinar (1973) extended Kennedy’s results by allowing the local oscillator to depend on the observed output of the photodetector. This structure is an explicit realization of the optimum quantum receiver for PSK signals. For such a “one output port” single detector homodyne arrangement, also discussed by √ Yuen and Shapiro √ (1980), the amplitude splitting coefficients ε and 1 − ε are selected to be equal to 1 and 0, respectively,respectively and a theoretically infinite power of a local oscillator is required to surmount thermal noise. For the two port arrangement, proposed by Yuen and Chan (1983) and also discussed by Shapiro, Yuen, and Machado Mata (1979) and Shapiro (1985), the exact intensity cancellation condition of a local oscillator noise is ε = 1/2. Any attenuation in a balanced homodyne arrangement can be expressed, without loss of generality, by an intensity transmission
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coefficient T < 1. Attenuation destroys the balance in the two arms and produces an amplitude attenuation of the measured quadrature by √ a factor T , introducing an additional Gaussian attenuation noise with a √ spectral density, 1 − T hν/2, keeping the noise level constant. A balanced mixer achieves the quantum-limited operation only when the two photodetector quantum efficiencies η are identical and equal to unity. For non-ideal photodetectors with quantum efficiencies η1 and η2 corresponding signal penalties are introduced. However Machida and Yamamoto (1986) point out that an exact local oscillator intensity noise cancellation condition can be preserved by using η1 (1 − ε) = η2 ε.
6. IN-PHASE AND QUADRATURE MEASUREMENTS In-phase and quadrature measurements are required in coherent optical reception. In optical communications, this kind of detection has been studied for several decades, due to its unique features concerning the use of complex amplitude modulations, that allow lower photon numbers for a given post-detection BER. As shown in Section 5, when a strong LO is used, SQL reception is easily attainable in the homodyne configuration. Furthermore, the use of constant envelope modulation formats, in opposition to traditional on-off keying (OOK), is more tolerant to nonlinear effects in the optical fiber channel (Mitra and Stark, 2001; Kahn and Ho, 2001). Additionally, for multi-channel systems, the highly selective spectral transposition into a base-band signal is of great interest for de-multiplexing the carrier. Other applications of coherent detection have been intensively researched such as interferometric sensors, radio-over-fiber (RoF), coherent reflectometry, coherent spectral analysis and multi-port optical networks. Free space applications in communications and lidar systems use the additional characteristics of highly selective spectral and spatial filtering for background radiation rejection. In communications applications, since coherent detection is sensitive to the instantaneous field complex amplitude, reception relies on accurate synchronization of the optical carrier phase with respect to the reference wave, usually an LO. This constitutes a difficult task, particularly for constellations with suppressed carriers, as required for efficient transmission. Thus the fiber optic channel imposes considerable challenges on coherent systems, due to unavoidable fluctuations in the instantaneous optical phase caused by fundamental noises in the signal and LO, as well as fluctuations in the thermomechanical state of the fiber, polarization fluctuation and other in-line components, thus constituting an optical channel with unknown stochastic parameters.
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In this section, we present the concepts of homodyne phase estimation techniques, as well as the detection of the in-phase I and the quadrature Q components of a coherent state signal prepared for digital transmission using homodyne reception, for optical phase estimation and synchronization, with application both for communications and cryptographic channels, which is particularly important in suppressed carrier and optical phase coded systems. Finally we mention several structures for homodyne detection with I–Q measurements.
6.1 Phase Estimation with Classical Signals Efficient modulation formats produce a suppressed-carrier optical signal, requiring elaborate phase synchronization techniques since conventional carrier tracking structures such as phase locked loops (PLLs) are not applicable since there is nothing to lock on to. For the classical telecommunication channel, diverse heuristic receiver structures for phase synchronization have been proposed and experimentally implemented, such as the use of imperfect modulation and residual carrier tracking loops (Glatt, Schreiblehner, Haider and Lebb, 1996), local oscillator phase dithering (Herzog, Kudielka, Erni and Bachtold, 2006), or the insertion of extra synchronization bits with 90◦ phase (Wandernoth, 1992). All of these inherently induce a power penalty and/or a reduction of the useful data rate (Kudielka and Klaus, 1999). More formally, to perform synchronization from the information bearing signal itself (with suppressed carrier for power economy), diverse approaches, inherited from the statistical detection and estimation theory for the RF channel, have been proposed, as a result of the mechanization of a suitable, in general nonlinear, likelihood function (Meyr, Moeneclaey and Fechtel, 1998). Their fundamental bounds on the optical detection and estimation parameters have been studied for the classical optical channel in the presence of additive noise and phase noise (Georghiades and Snyder, 1985). For fully suppressed carriers, decision directed loops, Costas loops or higher order loops, have been derived and experimentally demonstrated for QPSK and higher order constellations. For the reception of BPSK signals, I–Q detection is also used, with trade offs related to the power splitting fractions used for data detection and phase lock, respectively (Kazovsky, 1986a,b). Even in the reception of differentially modulated fields, such as DPSK or DQPSK, where an improved tolerance to phase error is obtained at the expense of less efficient coding (Ho, 2005; Kahn, 2004), as demodulation is performed using delayed self homodyne configurations, no strict phase coherence exists between the interacting fields, and synchronization is necessary in order to track the relative phase fluctuations due to source
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FIGURE 13 Received binary signal and local oscillator phasors. Indicated are the contours of the coherent states and schematically the diffused states due to phase noise in laser sources
phase noise, interferometric drifts, etc. (Winzer and Essiambre, 2006; Xu, Liu and Wei, 2004). Finally, as homodyning produces a spectral transposition directly into the base-band, post-detection tasks can be simplified and some I–Q loops allow the possibility of AC coupled front ends. As data rates increase dramatically, delays in the reception loops significantly degrade the carrier synchronization performance, therefore fast data acquisition and digital signal processing electronic (DSP) systems have been proposed for (feed-forward) optical carrier estimation and demodulation (Kazovsky, Kalogerakis and Shaw, 2006). Additionally DSP possesses attractive potentials for electronic compensation of linear and nonlinear fiber dispersions and other channel impairments (Noe, 2005; Kikuchi, 2006). In general, I–Q reception structures are based on the splitting of the incoming optical signal into its in-phase I and quadrature Q components and their separate beating with a strong local oscillator, as shown in Figure 13. The optical phase is usually modeled as a stochastic Wiener process resulting from a flat frequency noise spectrum, which is characteristic of an amplitude stabilized laser operating well above threshold dφ = (1ν)1/2 w (t) dt
(217)
where 1ν is the field FWHM and w(t) is a normalized white frequency noise.
In-phase and Quadrature Measurements
229
FIGURE 14 A generic 90◦ optical hybrid. Received optical signal and local oscillator (LO) are split into two parts; when mixed in the lower branch balanced homodyne detector, the signal beats with the in-phase component of the LO; in the upper branch it beats with the quadrature component of the LO
A structure to perform this task is known as an optical 90◦ hybrid (Gnauck and Winzer, 2005; Seimetz and Weinert, 2006). As shown in Figure 14 it employs two identical balanced homodyne detectors (BHD) and a 90◦ phase shifter either in the coupled local oscillator port (as shown), or in the coupled signal port. In the classical analysis, a received signal of power PS , modulated phase φmod (t) and random phase, φ S (t) is I–Q homodyned with a strong LO of power PL and phase φ L (t) in an optical hybrid, whose detectors have responsivity R. The electrical current processes at the in-phase and quadrature outputs are, for perfect 50%–50% couplers, i X (t) = i 0 sin[φmod (t) + φ S (t) − φ L (t)] + i n X (t) i Y (t) = i 0 cos[φmod (t) + φ S (t) − φ L (t)] + i nY (t),
(218) (219)
where the current signal amplitudes are p i 0 = R PL PS /2.
(220)
The quantum treatment of the 90◦ coupler will be presented in Section 6.4.
6.2 Phase Estimation with Quantum Signals The previous researches have being made for the reception of classical light fields, i.e. those carrying many photons per symbol, which are typically encountered in the above mentioned applications, thus allowing a diversity of configurations for suppressed carrier phase synchronization and demodulation such as an LO laser in the loop or free running in
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FIGURE 15
Homodyne phase estimation structure
feed-forward demodulation. Even injection locking schemes have been proposed (Bordonalli, Walton and Seeds, 1999). Now, numerous new applications of coherent reception techniques are being developed in this telecommunications wavelength, that work with photon numbers substantially lower than those used in classical transmission, such as quantum cryptography (Gisin, Ribordy, Tittel and Zbinden, 2002), long distance free space communications (Chan, 2006), highly sensitive sensors (Pulford, 2005), homodyne tomography (Zavatta, Bellini, Ramazza, Marin and Arecchi, 2002), and other instrumentation and scientific applications (McKenzie, Mikhailov, Goda, Lam, Grosse, Gray, Mavalvala and McClelland, 2005), that operate with very few average photons per observation time. The problem of measurement of the instantaneous phase of an optical field has been the subject of numerous researches at the theoretical level for a phase operator (Barnett and Pegg, 1989; Hall, 1991), (Vogel and Welsch, 1994, Chapter 4) as well as formulation of the POVM and optimum phase estimation (D’Ariano, Macchiavello and Sacchi, 1998; Shapiro and Sheppard, 1991). Since optical phase cannot be measured directly, its detection is based on the measurement of some other related variable, which produces additional uncertainty. Using the homodyne approach, an indirect measurement of phase can be done by projection on to the local oscillator phasor, and number-phase uncertainty relations for optical fields have been verified by (Opatrny, Dakna and Welsch, 1998) using BHD. Experimental measurements for coherent states have been reported by Smithey, Beck, Cooper and Raymer (1993), including the number-phase commutator. Their measurements are close to the uncertainty limit 1N S 1θ ≥ 1/2, for large photon numbers. In the search for optimum phase estimation Wiseman (2002) has researched the adaptive measurement of optical phase, both from CW and pulsed quantum signals, proposing feedback structures for estimation with diverse feedback algorithms, as shown schematically in Figure 15. These estimation techniques take advantage of the spatio-temporal extent of the quantum signal under analysis. The quantum feedback
In-phase and Quadrature Measurements
231
takes place during the pulse duration (Berry and Wiseman, 2002), and theoretically can yield phase estimates closer to the uncertainty limits than open loop techniques, when an optimum feedback algorithm is used. The BHD output can be expressed as i(t) = i 0 sin[φ S (t) − φ L (t)] + i n (t).
(221)
A usual feedback algorithm of the form dφ L (t) i(t) =G dt i0
(222)
with gain G, when applied to classical signals, its optimum gain is infinite (in the SQL), however Berry and Wiseman (2002) found a finite optimum value in the quantum regime, when a Wiener fluctuation model for the phase was used. This feedback algorithm yields a phase error variance √ 2 that scales as σadapt = 1/2 N S . √ This adaptive estimation provides a factor of 1/ 2 improvement √ over 2 the non-adaptive heterodyne technique that scales as σhetero = 1/ 2N S . Armen, Au, Stockton, Doherty and Mabuchi (2002) have developed experiments on adaptive phase estimation techniques for weak coherent states, mainly for long duration signals, obtaining phase variances closer to the uncertainty limit than standard open loop techniques. In this approach, as the leading edge of each pulse is measured, the photodetector output is used to make a quick estimate of the signal phase. The local oscillator can then be adjusted so as to be optimal for the remainder of the pulse. However these adaptive techniques face an important challenge for fast telecommunications signals, requiring very short feedback loop times. In fact in classical communications, phase synchronization is usually performed by integrating a series of measurements on the data string.
6.3 Heterodyne Detection Historically heterodyne detection was the first used since it allows the simple measurement of both quadratures. It is easy to perform since it does not need phase lock in frequencies much lower than intermediate frequency. However, heterodyne estimates the complex amplitude of the signal field subject to image band noise. The signal image band quadrature fluctuation gives the fundamental noise limit in heterodyne detection (Yuen and Chan, 1983; Hall, 1991). That is, both the in-phase and the quadrature relative to the LO contribute to the output observations, half of the quantum noise comes through the signal field operator and the
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other half comes through the image field operator (Shapiro and Sheppard, 1991) . Heterodyne is wasteful because it obtains information about amplitude as well as phase, because these are complementary quantities, obtaining information on amplitude halves the information on phase that could be obtained (Wiseman, 2002). In digital reception the LO phase is rapidly swept over 2π, e.g. leaving the LO of Figure 15 in free running mode sampling the interference process many times over the bit duration. Sampling uniformly the two quadratures is therefore subject to additional uncertainty (Mabuchi and Khaneja, 2005). However, Collett, Loudon and Gardiner (1987) propose that this 3dB disadvantage can be removed if the total signal flux is divided into two equal contributions whose frequencies match the signal and image band frequencies of the heterodyne detector. In heterodyne detection the signal and the local oscillator have optical frequencies, differing only by the intermediate angular frequency ω Heterodyne detection appears as a linear change in time of the local oscillator phase mismatch θ = ωt leading to the electron number operator Nˆ = 2 |L| sˆθ I = 2 |L| sˆ I cos ωt + sˆ Q sin ωt
(223)
where ω is the beating frequency. Writing the two signal quadratures as the sum of a quiescent and a fluctuating part, the averaged square output is expressed as D
E D E E 2 D 2 sin2 ωt . Nˆ 2 = 4 |L|2 S I cos ωt + S Q sin ωt + 1ˆs I2 cos2 ωt + 1ˆs Q
(224) Assuming that the signal is in a coherent state: D
E 1 2 Nˆ 2 = 4 |L|2 S I2 cos2 ωt + S Q sin2 ωt + . 4
(225)
Both signal quadratures contribute to the electron output. Taking the time average over the beat frequency ω we obtain D
E 1 2 = 2N S N L + N L . Nˆ 2 = 2 |L|2 S I2 + S Q + 2
(226)
The signal-to-noise ratio per bit is S 2N L N S = = 2N S . N NL
(227)
In-phase and Quadrature Measurements
233
This signal-to-noise ratio is two times lower than in the homodyne case (Kazovsky, 1985). This can be interpreted since the noise is the same, but the time averaged signal power is also two times lower since we have moved from a base-band output signal to a band-pass signal at the beating frequency. Another interpretation is to consider that the noise bandwidth includes the image frequency band.
6.4 I–Q Measurements Suitable structures for the phase estimation from I–Q measurements from quantum signals, such as multiport interferometers, have been treated in Mandel and Wolf (1995, Chapter 10), Perina (1998, Chapter 4), introducing an operational phase concept that defines the operators of the trigonometric functions (sine and cosine) of the phase difference between two interacting fields, corresponding to intensity measurements with photon counters (Noh, Foug`eres and Mandel, 1992), or with homodyne detectors (Raymer, Cooper and Beck, 1993). These allow the simultaneous measurement of the two quadratures of the input field at the price of introducing noise due to the vacuum fields that enter via the unused input ports, as shown in Figure 14. The quantum formulation of the I–Q receiver √ in Figure 14 is performed following the analysis of Section 5. The 1/ 2 splitting of the signal and local oscillator fields gives the following field operators at the four outputs 1 ˆ aˆ 2 = √1 (−jˆs + l) ˆ aˆ 1 = √ (ˆs − jl) 2 2 1 ˆ aˆ 4 = √1 (−ˆs + l) ˆ aˆ 3 = √ (− j sˆ − j l) 2 2
(228)
corresponding to the 270◦ , 90◦ , 0◦ and 180◦ relative phases, respectively. Performing photodetection and subtraction as indicated in Figure 14, we obtain the following number operators for the standard quantum limited X (in phase) and Y (quadrature) outputs 1 Nˆ X = (−ˆs +lˆ + lˆ+ sˆ ) 2
and
1 Nˆ Y = (−ˆs +lˆ − lˆ+ sˆ ). 2j
(229)
If θ is the relative phase between signal and local oscillator fields, following the analysis of the BHD of Section 5, we obtain the numbers 1 Nˆ X = |L| [ˆs + exp(jθ ) + sˆ exp(−jθ )] = |L| (ˆs I cos θ + sˆ Q sin θ ) 2
(230)
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Signal and Quantum Noise in Optical Communications and Cryptography
and 1 Nˆ Y = |L| [ˆs + exp(jθ ) − j sˆ exp(−jθ )] = |L| (ˆs I sin θ − sˆ Q cos θ ). (231) 2 These numbers correspond to the measurement of sˆθ = sˆ exp(−jθ ) and sˆθ+π/2 = sˆ exp[−j(θ + π/2)], respectively, these are, the in-phase and quadrature components relative to the local oscillator. The signal field splitting produces the following SNR at the X and Y outputs SNR X = (S I cos θ + S Q sin θ )2
and SNRY = (S I sin θ − S Q cos θ )2 . (232)
Walker (1987) demonstrated that in this 4 × 4 port coupler, the operators that correspond to the measurement of N X and NY commute, therefore simultaneous measurement of (non commutating) I and Q signal quadratures are possible, allowing the determination of the optical phase. However the price is the unavoidable introduction of vacuum noises entering by the two unused ports, in agreement with the uncertainty principle. Note that in the four-port balanced homodyne receiver of Section 5, no vacuum leaks in, however only one quadrature is measured.
6.5 Suppressed Carrier Phase Estimation In classical theory, an optimum receiver structure can be designed from the results of the application of the concepts maximum likelihood estimation (MLE) of a signal in the presence of noise, resulting in known receiving structures such as phase locked loops (PLL) for pilot carrier modulations, or Costas loops, squarer loops, decision directed loops, etc., for suppressed carrier modulations (Djordjevic, 2002). Another approach consists of the mechanization of the analytical expressions MLE with no a priori suppositions on their structure. Several optimum receivers have been proposed by Georghiades and Snyder (1985) or Arvizu and Mendieta (1998), for instance, that do not attempt to remove modulation, but rather perform maximum likelihood phase estimates. In the structured approach, homodyne provides the signal quadrature corresponding to the LO phasor Now, in a diversity of telecommunications and cryptography applications, we need to measure both quadratures, such as in coherent reception of QPSK or higher order and even in BPSK where one quadrature serves for data detection and the other one for phase-lock (Kazovsky, 1986a,b). For the classical communications channel, heuristic structures for detection based on in-phase and quadrature homodyne measurements
In-phase and Quadrature Measurements
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FIGURE 16 An I–Q detection receiver: the nonlinear multiplying operation eliminates the modulation from the suppressed-carrier received signal and generates the phase error signal for optical phase synchronization.
have been extensively studied and experimentally realized since the pioneer works by Oliver (1965), they employ an (unbalanced) I–Q receiver that uses a small portion of signal for phase-lock. Since then numerous contributions have been reported, which constitute an elegant and efficient technique for demodulation of suppressed carrier signals. In classical optical communications, measurement of the two quadratures of the received optical field is a standard operation, such as in the QPSK demodulation or in the synchronization of suppressed carrier BPSK, as mentioned above (Norimatsu, 1995). Figure 16, shows a generic I–Q detection receiver, with a similar structure as in RF (Stiffler, 1971, Chapter 8), using an optical hybrid such as that in Figure 14. As shown in Figure 16, the purpose of the nonlinear multiplying block is to suppress the modulation (Djordjevic, 2001), leaving only a function of the phase error φe (t) = φ S (t) − φ L (t) with additive noise. For small phase errors, a linearized analysis of the Costas loop receiver allows the obtention of the phase error statistics. When a first order feedback loop with natural frequency f n is used, the phase error variance can be obtained as (Kazovsky, Benedetto and Willner, 1996) σe2 = 0.7
1ν Tb f n + 3.3 . fn NS
(233)
Tb is the bit period (NRZ signaling is supposed). This represents the tradeoff between the phase noise and shot noise effects. The larger the loop bandwidth, the faster the system tracks the signal phase, but the noisier the feedback and an optimum bandwidth is easily computed from this equation. However in the low photon number regime the introduction of splitting elements is not as straightforward as in the classical channel and quantum behavior must be taken into account in the modeling of these devices.
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FIGURE 17 Telecommunications channel model with diffused phase states and baseband observation
6.6 In-phase and Quadrature Measurements Applications 6.6.1 Telecommunications Applications When a signal is homodyned with a separate local oscillator, the receiver/channel model must include the relatively fast phase fluctuations process in the two independent processes, plus the relatively slow phase fluctuations of the in-line elements in the fiber channel due to the fluctuations in the thermomechanical states, and in the free space channel, because of turbulence (Vilnrotter and Lau, 2001, 2005). This severely degrades the absolute PSK or QPSK modulations, however, differential modulations such as DPSK, DQPSK, are more tolerant to the phase noise since only the differential phase accumulated between a symbol and its precedent participate in the fluctuation, at the expense of a less efficient coding. For quantum coherent states, the Gaussian model must be completed by the addition of the phase fluctuations which are unavoidable in any laser oscillator, as in Figure 13, producing a non minimum uncertainty diffused phase state (Yamamoto and Haus, 1986) that can be modeled as a Wiener diffusion process (Gallion, Mendieta and Leconte, 1982; Gallion and Debarge, 1984; Hall, 1995). Figure 17 shows a schematic diagram of the telecommunications system model with optical fields undergoing phase diffusion (Arvizu and Mendieta, 1998). The optical phase is modeled as a stochastic process resulting from a flat frequency noise spectrum, which is characteristic of an amplitude stabilized laser operating well above threshold given by Equation (217). Starting from the base-band observable, Kazovsky (1986a,b) has studied the homodyne reception of phase modulated signals arriving at the optimum loop dynamics, in terms of trading off the phase noise tracking capability and the noise in the loop bandwidth.
In-phase and Quadrature Measurements
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FIGURE 18 A sequential I–Q detection receiver, requiring only one balanced homodyne detector. Optical signal data are replicated to beat sequentially with the local oscillator whose phase is sequentially switched between 0◦ and 90◦ at the symbol rate
Now, even in the classical channel stabilization of the 90◦ optical hybrids is a difficult task (Seimetz and Weinert, 2006), therefore alternative schemes that do not require a hybrid but only one BHD have been proposed, such as the use of deterministic time switching phase diversity at the receiver (Fabrega and Prat, 2007) and also Habbab, Kahn and Greenstein (1988), switching at the transmitter end. For the quantum channel (D’Ariano, Paris and Seno, 1996) proposes a sequential measurement based on couples of independent homodyne measurements of commuting quadratures. Every measurement is prepared in the same input state before every detection step, in the source’s coherence time. All these schemes cause a “bit rate penalty” of 1/2 since modulation data are replicated to beat with the local oscillator in I and Q sequentially, as shown in Figure 18. 6.6.2 Cryptographic Applications Soon after the first reported experiments on QKD with faint pulses in the visible or near infrared spectrum, based mainly on photon polarization, numerous experiments have been reported in the fiber channel at telecom wavelengths, introduction coding other than polarization, such as phaseshift-keying, QPSK, BB84, or working generally in differential mode so that interferometric detection at the receiver acts as a demodulator. This QKD channel requires the carrier phase synchronization, and different techniques have been proposed for stabilization of the interferometers, although some experiments have been reported with open loop interferometers, such as photonic light circuits (PLC) (Diamanti, Takesue, Langcrock, Fejer and Yamamoto, 2006), the multiplexing of bright impulses at a different wavelength (Hughes, Morgan and Peterson, 2000) or sending training frames at intervals (Elliott, Pikalo, Schlafer and Troxel, 2003; Makarov, Brylevski and Hjelme, 2004).
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FIGURE 19 A “synthetic” I–Q detection receiver in a self-homodyne configuration using an interferometric configuration. The error signal is fed back into a phase shifter to maintain the interferometer in-quadrature
Balanced homodyne detection has been used for detection in the cryptographic channel (Hirano, Yamanaka, Ashikaga, Konishi and Namiki, 2003). The LO can be generated locally or can be produced by Alice, so the key signal and LO travel together in a time multiplexed scheme based on a delay interferometer at Alice’s end, while the coherent superposition at Bob’s end is produced by another similar delay interferometer (Xu, Costa e Silva, Danger, Guilley, Bellot, Gallion and Mendieta, 2007). Interferometers must be precisely stabilized in the presence of perturbations, constituting a carrier synchronization problem (Xu, Costa e Silva, Gallion and Mendieta, 2007), since effects of differential drifts and even dispersions must be considered (Grosshans and Grangier, 2002). Xu, Costa e Silva, Gallion and Mendieta (2008) propose a detection structure based on sequential I–Q detection with weak coherent states, using a switched phase LO, as shown in Figure 19, constituting a synthetic Costas loop. They study its theoretical performance in terms of the trade off between phase precision measurement and available photon number. They perform experiments in a fiber optic cryptographic set up at 1550 nm, with BPSK modulation with time multiplexed strong reference and delayed interferometric detection (Xu, Costa e Silva, Arvizu, Gallion and Mendieta, 2008). They perform measurements on BER and phase uncertainty and discuss the departures from the uncertainty product 1N S 1θ ≥ 1/2.
7. CONCLUSION Starting from quantum optics and communications theory principles, this chapter presents the basic concepts related to optical signals and quantum noise and their applications and repercussion on the transmission
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of information in optical channels, both for telecommunications and cryptography. Quantum noise becomes the ultimate limitation in the performance of these systems, when thermal noise is surmounted by employing suitable detection and/or amplification systems, such as cooled photon counters or room temperature heterodyne and homodyne detectors with strong reference fields. In the detection and estimation of optical signals, the manifestation of quantum phenomena sets the fundamental bounds in error or in fidelity, which are of great interest in present and new applications that operate with few photons per observation time. We presented in Section 2 the basics of the quantum description of light. By using a symmetrized noise energy operator, a pragmatic additive Gaussian white noise (AGWN) description may be introduced, strengthening the links with digital communication theory, and capitalizing its well-established notions. The symmetrization of the Hermitian operator (Weyl’s order), canceling out the commutator contributions, allowed us to calculate the sum of the in-phase and outof-phase square noise power, which is the equivalent to the classical total noise power, but is not directly observable. Throughout this work we focused on the quantum coherent state models of the radiation field. As introduced in Section 2, they represent the quantum states that most closely resemble the classical description of a classical field, and many concepts are understandable in a straightforward way by elaborating from this representation, such as power noise and phase noise manifestations as a result of commutation and uncertainty relations, as well as other important concepts such as channel capacity, that were presented in Section 2, from an engineering point of view. The attractive feature of coherent states is that they are the most similar to real laser sources. However, they are non-orthogonal therefore they cannot be discriminated perfectly, thus a finite error rate is unavoidable if only conclusive measurements are allowed, as in telecommunications applications, whereas in cryptographic applications, inconclusive measurements are allowed at the price of bit decision abandons. Consequently, in Section 3, we presented the attempts to find the fundamental detection bounds for quantum state discrimination and to interpret the quantum mechanical operators that are concerned, as well as the receiver structures inspired by them. These are extremely difficult to find in the general case. However we discussed the bounds in error probability for the detection of coherent state symmetric binary signals, that are intensively used in classical and quantum information transmission.
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TABLE 1 Comparison of different quantum receiver implementations and performances Quantum receiver
BER
Helstrom limit and Dolinar receiver
1 2
Super homodyne Kennedy receiver
1 exp (−4N ) S 2
Homodyne detection
1 erfc 2
Heterodyne detection
1 erfc 2
Direct detection of OOK signals
1 exp(−2N ) S 2
Asymptotic approximation
1−
p 1 − exp (−4N S ) 41 exp (−4N S ) Theoretical bound
√
2N S
NS
Comments
Idealist combiner Single Photon Counter
Conditional nulling receiver Phase and amplitude feed-back control Unconditional nulling receiver Thermal effect limitation
1 exp (−4N ) S 2 Super Quantum Limit
Idealist combiner Single Photon Counter
1 2 exp(−2N S )
Realistic balanced combiner 2 PIN photodiodes
Strong local oscillator mixing gain overcoming thermal noise
Realistic balanced combiner 2 PIN photodiodes Single PIN photodiode
Strong local oscillator mixing gain overcoming thermal noise Usually strongly limited by thermal noise
Standard Quantum Limit √
Devices
1 2 exp(−N S )
1 exp(−2N ) S 2 Standard Quantum Limit
Besides, in parallel, we presented several heuristic receiver structures that approach the fundamental bounds when ideal optical components are used, constituting nulling receivers in a Z channel, i.e. no photoelectrons in the vacuum state, operating both in open loop configurations as well as in closed loop and adaptive schemes that take advantage of the spatio-temporal extent of the transmitted signals to perform a quantum feedback. A comparison of different quantum receiver implementations and performances for the detection of binary PSK signals is shown in Table 1. In Section 3 we also presented the challenges to implementing these kinds of heuristic detectors in the high bit rate transmissions, due to the fact that feedback needs to take place during the bit period, however much progress is expected in the post-detection signal processing field, such
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as that experienced presently for the classical multi-Gigabit per second communications. Figure 20 is a plot of the bit error ratio as a function of the average photon number for the different quantum receivers of Table 1. In Section 4, devoted to optical amplification, the phase insensitive linear optical amplifier minimum added noise has been derived from the commutations relationship, independently from the amplifier structure, leading to the well-known 3dB minimum value of the noise figure, in the high gain limit. Performances of lumped and distributed amplifiers have been compared in terms of noise factor and equivalent noise figure. The pre-amplified direct detection optical systems performances and applications have been addressed. Finally, technical noises and their relative importance were discussed, as well as noise generation in-phase sensitive and nonlinear amplification. Pre-amplification direct OSNR measurements are not possible, since the vacuum fluctuations cannot be directly observed, and the noise figure discussions are done in terms of ESNR. The corresponding OSNR measurement, referring to the equivalent spectral density of vacuum fluctuation, allows a discussion independent of the detection arrangement and filtering considerations. Coming back to signal detection, as physical realizability of the receiver structures inspired by the quantum detection theory of Section 3 is a difficult task, we presented in Section 5, on single quadrature measurement, the performance bounds of the standard receiver structures such as photon counting, heterodyne and homodyne detection systems. Their corresponding operators are, in general, projection valued operators that are bounded by the standard quantum limit. Thus in Section 5, we elaborated on homodyne detection, a continuousvariable measurement extensively developed for classical coherent reception systems, due to its interesting characteristics concerning noisefree conversion gain to approach the standard quantum limit, using room temperature standard PIN photodetectors, such as those extensively used in classical transmission, that exhibit a high quantum efficiency, superior speed, dynamic range, resolution and input mode selectivity. Besides communications and cryptography, these characteristics are attractive in a diversity of applications such as radio-on-fiber, free space coherent communications, lidar, optical sensors, coherent optical instrumentation, precise interferometry and diverse engineering instrumentation and scientific applications. The balanced homodyne detector is the paradigm of phase sensitive reception systems in quantum detection, it detects the field superposition and the electronic subtraction cancels the photon number. Under a strong local oscillator it measures the single quadrature phase amplitude of a signal field, i.e. it performs the projective measurements corresponding to the quadrature phase operator. It cancels both the local
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FIGURE 20 Bit error ratio as a function of the average photon number for different quantum receivers
oscillator excess noise and field fluctuations. In the theory of balanced homodyne detection, approaches based on coarse models involving mixed corpuscular and wave nature of light gave satisfactory results, employing the local oscillator shot noise interpretation. However in recent applications processing few photons per observation time in communications and cryptography and the other applications mentioned above, a more detailed description is necessary, we therefore presented, also in Section 5, a more accurate representation in a full quantum treatment explaining the fact that the noise in balanced homodyne detection (as well as in heterodyne) is due to fundamental quantum fluctuations at the signal port. Since a single quadrature is measured, this detection is limited by the uncertainty principle with no additional noise required. As a comparison, while in binary detection, the photon counting detector possesses a built-in decision characteristic, the balanced homodyne detector output process consists of a signal-plus-noise process, that must be properly processed. Gaussian statistics allow an easy transit from the deep physical concept of commutations relationships to the
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TABLE 2 BER and the required photon number to achieve a B E R = 10−9 for various receivers and modulation format Modulation format
OOK or ASK
Receiver
Theoretical BER
Theoretical photon number N s for BER = 10−9
Direct detection
1 exp(−2N ) S 2
10
Homodyne detection
1 erfc√ N S 2
18
Heterodyne detection Optical preamplification PSK
Homodyne detection Heterodyne detection
DPSK
Optical preamplification and optical delay line demodulation
√
1 erfc N /2 S 2 No simple expression 1 erfc√2N S 2 1 erfc√ N S 2
36 38
1 exp(−N ) S 2
20
9 18
measurable quantities in the communications channel such as signal-tonoise ratios, bit error ratios, and channel capacity. We thus elaborate on this detection technique for diverse applications such as phase coded transmissions, differential schemes with delay interferometric detection and phase coded cryptographic systems with threshold detection, where the use of homodyne systems allows easy threshold setting to optimize performance with a trade off between bit error rate and inconclusive decisions. For comparison with the photon counting structures, the receiver implementations and performances for these homodyne and heterodyne receivers with BPSK modulation are also included in Table 1, and the bit error ratio as a function of the average photon number for the homodyne receiver is also included in Figure 20. Table 2 presents the BER and the required photon number to achieve a given BER for various receivers and different modulation formats. Balanced homodyne detection measures only a single quadrature. Now, as we mentioned in Section 6, in a diversity of applications the measurement of in-phase I and quadrature Q components is necessary, such as in coherent systems for phase synchronization, and it constitutes a problem, presently under research, in the high speed PSK and higher order modulations with suppressed carrier for power economy, and also in other applications that rely on the phase lock to a local oscillator or in interferometric receivers for quadrature control. Consequently we considered diffused quantum states, that represent more realistically the emission of lasers, i.e. amplitude stabilized free running lasers undergoing random walk phase drift. These consist of non-minimum
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uncertainty states that must be in general synchronized for detection and demodulation. After briefly mentioning adaptive phase estimation measurements for pulsed quantum level signals, using fast feedback, we then reviewed heterodyne techniques that sample equally both quadratures, and are therefore subject to additional uncertainty. Noise is two times higher than in homodyning because of the signal and image band contributions. Then, also in Section 6, we reviewed the 90◦ optical hybrid as the paradigm of the multiport detection of (non-commutating) I and Q quadratures simultaneously, that has been extensively researched for classical suppressed carrier phase synchronization as well as for higher order constellations. We extended our previous analysis of balanced homodyne detection systems to this I–Q measurement system using the quantum formalism, obtaining the photoelectron numbers at the output electric ports, which are found to commute, therefore the simultaneous measurement of I and Q signal quadratures is possible, allowing determination of the optical phase. However the price is the unavoidable introduction of vacuum noises entering by the unused ports, in agreement with the uncertainty principle. The simultaneous measurement of these two conjugated observables inevitably introduces additional noise to solve the non-commutability. The uncertainty product for the measurement is 3 dB larger than the one dictated by the Heisenberg limit. Finally in Section 6, we presented alternative techniques for I–Q measurements, particularly sequential detection that employs only one balanced homodyne detector, but requires replicating the information signal to beat with the local oscillator whose phase is periodically switched, thus resulting in a bit rate penalty of 1/2.
ACKNOWLEDGMENTS The authors would like to thank Govind Agrawal, Arturo Arvizu, Jean Claude Belfiore, Marcia Costa e Silva, Yves Jaou¨en, Jorge Rodriguez Guisantes, C´edric Ware, Qing Xu, for comments, suggestions, and discussions on improving the manuscript content. It would have been impossible to write this section without the many years of fruitful interaction with engineering degree students and PhD students of TELECOM ParisTech, Ecole Nationale Sup´erieure de T´el´ecommunications. While the authors alone remain responsible for the bugs in this chapter they are grateful to Govind Agrawal, Benjamin Lefaudeux and Hamed Mousavi who carefully read the manuscript. This work was partially funded by the French Agence Nationale de la Recherche (ANR), in the fame of the HQNET project.
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FIGURE A.1 Schematic illustration of a four-port optical device
APPENDIX A.1 General Quantum Field Input–Output Relations Let us consider a four-port optical device, as illustrated in Figure A.1, at each port of which we have an incoming or outgoing quasimonochromatic optical field that can be expressed in the form n o of Equation (4). The two input–output groups Eˆ L ,in (t) , Eˆ R,out (t) and o n Eˆ R,in (t) , Eˆ L ,out (t) have their own carrier frequencies, which are not necessarily the same. We can decompose each quasi-monochromatic field into its respective modal components verifying Equation (14). Thus, without loss of generality, we can assume that there are two groups of operators, aˆ in,k and aˆ out,k , respectively, for the input and the output, that verify their own commutation relations h i h i Ď Ď aˆ in,l , aˆ in,m = δlm and aˆ out,l , aˆ out,m = δlm . (A.1) We will present these two groups of annihilation operators using the T column vectors of the operators, aˆ in = . . . , aˆ in, j , aˆ in,k , aˆ in,l , . . . and T Ď Ď aˆ out = . . . , aˆ out, j , aˆ out,k , aˆ out,l , . . . . Therefore, aˆ in and aˆ out are row vectors, whose elements are Hermitian conjugated to those of aˆ in and aˆ out . The output is generally a function of the input, i.e., we have aˆ out = Ď Ď Ď ˆ F(ˆain , aˆ in ) and aˆ out = Fˆ Ď (ˆain , aˆ in ). However, these relations or functions are not completely arbitrary because they must preserve the commutation relations Equation (A.1). The most general relations must therefore be derivable from a unitary transformation, which means we can write aˆ out = Uˆ Ď aˆ in Uˆ
Ď Ď and aˆ out = Uˆ Ď aˆ in Uˆ
(A.2)
Ď where the unitary operator Uˆ = Uˆ (ˆain , aˆ in ), Uˆ Uˆ Ď = Uˆ Ď Uˆ = 1, is a function of the input and possibly other degrees of freedom belonging to
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the device. Because we are in the Heisenberg picture, the density operator of the total system ρˆ is invariant from the input to the output. It is an operator defined in the Hilbert space H = Hin ⊗ H D , where Hin is the Hilbert space of the input field and H D is the Hilbert space in which are defined the operators and the states of other degrees of freedom belonging to the device. We assume the operation of the device is independent of the input, so that its working state is uncorrelated with the input. Therefore, the density operator of the total system ρˆ can be written as ρˆ = ρˆin ⊗ ρˆ D
(A.3)
where ρˆin and ρˆ D describe, respectively, the states of the input and the device. Therefore any device is completely described by its unitary transformation operator Uˆ , and the density operator ρˆ D describes the working state of the device. Using the well-known Baker–Hausdorff formula iii ii i h h h h h h ˆ ˆ − Aˆ ˆ Bˆ ˆ Bˆ + 1 A, ˆ A, ˆ Bˆ + 1 A, ˆ A, ˆ A, + ··· = Bˆ + A, e A Be 2 3! (A.4) We have enumerated in Table 3 the unitary transformations for some linear devices, and their corresponding input–output relations. From the first three types of device, we see that, for linear and phase-insensitive devices, the following relations are verified aˆ out = S · aˆ in + V · bˆ + W · cˆ Ď ,
with VVĎ − WWĎ = I − SSĎ
(A.5)
where bˆ and cˆ , commutating, are vectors of the operators that are independent of the input and verify both their own boson-like commutation relations of Equation (A.1). Now, if we carry out a measurement at the output on an observable Xˆ , Ď which is purely a function of the output, Xˆ = X (ˆaout , aˆ out ), the expectation of the measurement on Xˆ is given by: h Xˆ i = tr (ρˆ Xˆ ). According to Equation (A.2). Ď Xˆ = Uˆ Ď X (ˆain , aˆ in )Uˆ
(A.6)
Ď h Xˆ i = tr[Uˆ ρˆ Uˆ Ď X (ˆain , aˆ in )]
(A.7)
we have therefore
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247
TABLE 3 Different input–output transformation Unitary transformation Uˆ Ď exp j aˆ in · M · aˆ in Ď Ď exp bˆ · M · aˆ in − aˆ in · M · bˆ
Input–output relation
Device type
aˆ out = exp ( jM) · aˆ in
Lossless beam splitter/coupler
aˆ out = cos (M) · aˆ in − sin (M) · bˆ
Phaseinsensitive attenuator
T Ď Ď Ď exp bˆ · M · aˆ in − (ˆain )T · M · bˆ aˆ out = cosh (M) · aˆ in − sinh (M) · (bˆ )T Ď Ď exp κ aˆ in aˆ in e jθ − aˆ in aˆ in e− jθ
Ď
aˆ out =
Ď cosh (2κ) aˆ in − e− jθ sinh (2κ) aˆ in
Phaseinsensitive amplifier Degenerate parametric amplifier
The matrix M is Hermitian. Parameters κ and θ are real.
Ď
Noting that X (ˆain , aˆ in ) is an operator on Hin , we can rewrite Equation (A.7) as Ď h Xˆ i = tr[ρˆeff X aˆ in , aˆ in ], with ρˆeff = tr (D) (Uˆ ρˆ Uˆ Ď ) (A.8) where tr(D) stands for the trace on H D , and, thus, ρˆeff is an operator on Hin . So, we see that, for the total system state given by Equation (A.3), Ď the measurement on Xˆ = X (ˆaout , aˆ out ) at the output is equivalent to Ď the measurement on X (ˆain , aˆ in ) at the input with the optical field in the ¨ effective state ρˆe f f . In the Schrodinger picture, the operators are invariant aˆ in = aˆ out = aˆ , and the states evolve from the input to the output. Thus, we can also say that the input state is given by ρˆin , and the output state is given by ρˆout = ρˆe f f = tr (D) (Uˆ ρˆin ⊗ ρˆ D Uˆ Ď ). So, at the output, the expectation of an observable X (ˆa, aˆ ) is given by h Xˆ iout = tr[ρˆout X (ˆa, aˆ Ď )].
A.2 Abbreviation Index AGWN Additive Gaussian white noise ASE Amplified spontaneous emission ASK amplitude shift keying BER Bit error ratio BHD Balanced homodyne detection BPSK Binary phase shift keying BS Beam splitter DOPA Degenerate optical parametric amplifiers DPSK Differential phase shift keying
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DQPSK Differential quadrature phase shift keying DSP Digital signal processing EDFA Erbium doped fiber amplifiers ESNR Electrical signal-to-noise ratio FEC Forward error coding FQN Fundamental quantum noise FRA Fiber Raman amplifier FWHM Full width at half maximun FWM four-wave mixing IF Intermediate frequency IQ in-phase and in-quadrature LO local oscillator MGF Moment generation function MLE Maximum likelihood estimation NDOPA Non-degenerate optical parametric amplifiers NF Noise figure OOK On-off keying OPLL Optical phase locked loop OSNR Optical signal to noise ratio PBS Polarization beam splitter PDF Probability density function PIN P doped intrinsic N doped PLL Phase locked loop PLC Photonic light-wave circuit POM Probability-operator measure POVM Positive operator value measurement PVM Projection value measurement PSK Phase shift keying PVM Projection value measurement QBER Quantum bit error ratio QPSK Quadrature phase shift keying RF Radio frequency RoF Radio over fiber SNR Signal-to-noise ratio SOA Semiconductor optical amplifier SCL Semiconductor laser SPDM Single photon detection measurement SQL Standard quantum limit WDM Wavelength division multiplexing
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A.3 Notation Index a (t) Complex slowly varying envelope a I (t) and a Q (t) In-phase and quadrature components of a (t) ak , ak,I and ak, Q Time-independent Fourier components of a (t) aˆ (t) and aˆ Ď (t) Time-dependent annihilation and creation operators Ď
aˆ k and aˆ k Annihilation and creation operators of mode k aˆ k,I and aˆ k, Q In-phase and quadrature operators of mode k
A and 1 aˆ Classical component A = aˆ and quantum fluctuation component of aˆ 1 aˆ I and 1 aˆ Q In-phase and quadrature operators of 1aˆ bˆ Additive noise operator B E photodetector electronic bandwidth B O Signal optical bandwidth or optical pass-band C Channel capacity erfc (x) Complementary error function E (t) Scalar optical field En Quantified energy of a harmonic oscillator, E n = (n + 1/2)hν fˆ Quantum Langevin force 1 fFWHM Full width at half maximum frequency fn Natural frequency F Noise Figure g Net gain coefficient per unit of length G Optical power gain of the amplifier h Planck’s constant, h = 6.63 · 10−34 J s h¯ Reduced Planck’s constant (h divided by 2 π ) Hˆ Hamiltonian operator H( p) Binary entropy of p i(t) Electrical current hi 0 i and hi 1 i Average currents for the symbol “0” and “1” Iand1I Photocurrent and its fluctuation I AB Information throughput k Boltzmann’s constant, k = 1.38 · 10−23 J/K K Multiplicative noise excess factor ˆ lˆI and lˆQ Local oscillator operator and its in-phase and quadrature l, operators 1lˆI and 1lˆQ Local oscillator in-phase and quadrature fluctuation operator L I , L Q Local oscillator in-phase and quadrature classical components L Amplification length
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n¯ sp (z) and Nsp Local and global spontaneous emission factor ˆ N and 1 Nˆ Photon number operator, its average and fluctuation N, operator N L and N S Local oscillator and signal photon numbers N X and NY X (in-phase) and Y (quadrature) outputs photon number (BHD) NASE ASE photon number Nth (ω) Thermal photon number at optical frequency ω p (x) Probability density function ˆ P and 1 Pˆ Optical power operator, its average and fluctuation P, operator Pe Average probability of error Pˆ A and PA Total added noise power operator and its average PAS E ASE noise power Pˆ N and PN Total noise power operator and its average PS Signal power Pr Probability Q Quality factor R Detector responsivity [A/W] R B Bit rate R E (τ ) Optical field autocorrelation function S/N Signal to noise ratio sˆ , sˆ I and sˆ Q Signal operator and its in-phase and quadrature operators 1ˆs I and 1ˆs Q Signal in-phase and quadrature fluctuation operators SI and S Q Signal in-phase and in-quadrature classical components SN Single sided noise spectral power density SE (ω) Spectral density of the optical field tr (·) Trace T Time interval/Absolute temperature Tb Bit duration w(t) Normalized white frequency noise ˆ x(t) Quantum signal operator x(t) and 1x(t) Signal complex amplitude and its noise x I (t) and x Q (t) Signal in-phase and an quadrature components 1x I (t) and 1x Q (t) Noise in-phase and an quadrature components √ xˆ θ Field quadrature quantum observable, xˆθ = (aˆ Ď ejθ + ae ˆ −jθ )/ 2 8 (s) and 8 (−js) Moment generation function and characteristic function ˆ (1) POM / POVM 5 α and β Attenuation and gain coefficients
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δl m Kronecker’s delta δ t − t 0 Dirac’s delta function √ √ ε and 1 − ε Amplitude splitting coefficients φ L (t) Local oscillator phase φmod (t) and φ S (t) Signal modulated phase and random phase ϕ and 1ϕ Optical signal phase and its fluctuation ν and ω0 Central carrier frequency, ω0 = 2π ν 1ν Field full-width-half-maximum spectral spread θ Relative phase between signal and local oscillator ρˆ Operator density matrix σ 2 Variance τ P Cold-cavity photon lifetime ξ0 and ξ1 Prior probabilities
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CHAPTER
4 Invisibility Cloaking by Coordinate Transformation Min Yan, Wei Yan and Min Qiu 1 Department of Microelectronics and Applied Physics, Royal Institute of Technology (KTH), Electrum 229, 164 40 Kista, Sweden
Contents
Introduction Coordinate Transformation in Electromagnetism Principle and Construction of Invisibility Cloaks Arbitrarily-shaped Invisibility Cloaks 4.1 Outer Boundary of Cloak 4.2 Inner Boundary of Cloak 5. Cylindrical Invisibility Cloak 5.1 Approximate Ideal Cylindrical Cloak 5.2 Material Simplification 6. Spherical Invisibility Cloak 7. Other Related Works and Some Practical Issues 8. Conclusion Acknowledgement References
1. 2. 3. 4.
261 262 266 268 269 270 274 277 282 295 299 300 301 301
1. INTRODUCTION Recently the advent of artificial electromagnetic (EM) materials, referred to as metamaterials (Smith, Pendry and Wiltshire, 2004; Shalaev, 2006), has opened up many new ways for us to interact with or control EM waves. EM phenomena that do not exist in nature have been demonstrated experimentally. Negative refraction (Shelby, Smith and Schultz, 2001) 1 Tel.: +46 8 7904068; fax: +46 8 7904090.
E-mail address:
[email protected] (M. Qiu). c 2009 Elsevier B.V. Progress in Optics, Volume 52 ISSN 0079-6638, DOI 10.1016/S0079-6638(08)00006-1 All rights reserved.
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and negative-index slab superlenses (Pendry, 2000) are representative applications of this class of engineered materials. With metamaterials, not only are we able to tailor their permittivity and permeability values at will, but we also have precise control over the anisotropy of the material and how all parameters vary with spatial location. While it is not a problem to engineer each individual metamaterial unit, placing the units in the appropriate order to achieve a desired macroscopic optical phenomenon remains a challenge. In simple language, with trees of various sizes and color at our disposal, how does one create an enchanting forest? The recently proposed coordinate transformation method provides an ideal recipe for solving such a design problem. This paper focuses on the design of a special type of EM device, the invisibility cloak, deduced from the coordinate transformation technique. The structure of our presentation is as follows. First, the theory of coordinate transformation for Maxwell’s equations will be summarized in Section 2. The principle and formation of invisibility cloaks based the technique of coordinate transformation will be described in Section 3. In Section 4, the perfect invisibility performance of arbitrarily shaped cloaks will then be confirmed by using full-wave analytical analysis. The material parameters of arbitrarily shaped cloaks will be examined closely. After covering the general arbitrarily shaped cloaks, we will study cylindrical and spherical cloaks (Sections 5 and 6) in detail. Special attention will be paid to the cylindrical invisibility cloak since it is arguably the simplest structure in terms of realization. Some practical issues concerning invisibility cloaks, as well as other related studies, will be discussed in Section 7. Finally, a summary will be given in Section 8.
2. COORDINATE TRANSFORMATION IN ELECTROMAGNETISM The theory of transformation optics has its roots in the covariance property of the Maxwell equations. The best mathematical description of such a covariance property is by using differential geometry (Post, 1962), a machinery that is commonly adopted for formulating the theory of general relativity (Leonhardt and Philbin, 2006). Strict derivation of the theory of transformation optics in 4D Minkowski space can be found in Leonhardt and Philbin (2008). In this paper we directly lay out the most important conclusions in a mathematically more accessible form. Since most practical applications of transformation optics are static or moving slowly compared to the speed of light, we can always choose our working frame properly, and therefore can consider only spatial coordinate transformation. Time will not be involved in coordinate transformations discussed in this paper.
Coordinate Transformation in Electromagnetism
263
In a flat 3D Euclidean space, the macroscopic Maxwell’s equation can be written as ∇ ×E=−
∂B , ∂t
∇ ×H=
∂D + j, ∂t
∇ · D = ρ,
∇ · B = 0.
(1)
E and H are the electric and magnetic fields, respectively; D and B are the electric and magnetic flux densities, respectively; j is the electric current density; and ρ is the charge density. The Maxwell equations are completed by two constitutive relations D = 0 ε · E,
B = µ0 µ · H,
(2)
where ε and µ are 3×3 permittivity and permeability tensors, respectively. Consider a coordinate transformation from the Cartesian space (x, y, z) to an arbitrary curved space described by coordinates (q1 , q2 , q3 ) with x = f 1 (q1 , q2, q3 ),
y = f 2 (q1 , q2, q3 ),
z = f 3 (q1 , q2, q3 ).
(3)
The Jacobian transformation matrix 3 is written as ∂x ∂x ∂x ∂q1 ∂q2 ∂q3 ∂y ∂y ∂y . 3= ∂q1 ∂q2 ∂q3 ∂z ∂z ∂z ∂q1 ∂q2 ∂q3
(4)
The length of a line element in the transformed space is given by dl 2 = [dq1 , dq2 , dq3 ]g[dq1 , dq2 , dq3 ]T , where g = 3T 3 is the space metric tensor, and the superscript “T ” denotes matrix transposition. The volume of a space element is expressed as dv = det(3)dq1 dq2 dq3 , where det(3) represents the determinant of 3. Then Maxwell’s equations in the curved space can take their invariant form as (Ward and Pendry, 1996) ∇q × b E=−
∂b B , ∂t
b= ∇q × H
∂b D b + j, ∂t
∇q · b D=ρ b,
∇q · b B=0
(5)
with new constitutive equations as b D = 0b ε ·b E,
b b B = µ0b µ · H,
(6)
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where all variables in the new coordinate system have been denoted by a hat. In order to keep such invariant forms of Maxwell’s equations, the new permittivity and permeability tensors have to fulfill b ε = det(3)(3)−1 ε3−T ,
b µ = det(3)(3)−1 µ3−T ,
(7)
where the superscript “−1” denotes matrix inversion, and “−T ” denotes matrix transposition and inversion. The fields and the sources in the new coordinate system can be directly obtained from their respective distributions in the original coordinate system as b b = 3T H, E = 3T E, H b j = det(3)(3)−1 j, ρ b = det(3)ρ.
(8) (9)
It is seen from the above equations that a change of coordinate system does not prevent us from solving exactly the same set of equations, provided that the permittivity and permeability tensors have been defined differently. When the permittivity and permeability in the original Cartesian coordinate system are isotropic, the newly obtained permittivity and permeability in Equation (7) can be further written as b ε = det(3)g −1 ε,
b µ = det(3)g −1 µ.
(10)
It should be mentioned that more than one coordinate transformation can take place successively. The relationship between the EM system in the initial coordinates and that in the final coordinates can be characterized by the global Jacobian coordinate transformation matrix, which is computed by multiplying Jacobian matrices for individual coordinate transformations (Zolla, Guenneau, Nicolet and Pendry, 2007). The global Jacobian matrix can then be applied for obtaining material and field distributions in the final coordinate system. One useful application of this strategy occurs when the coordinate transformation can be described much more easily in some coordinate system other than the Cartesian system. As an example, under a cylindrical coordinate system, a coordinate transformation may involve only a mapping of the radial coordinate component, i.e. from (r, θ, z) to (r 0 , θ 0 , z 0 ) with θ = θ 0 and z = z 0 . To interpret the transformation in a Cartesian coordinate system, the global Jacobian transformation matrix is 3 = 3xr 3rr 0 3r 0 x 0 ,
(11)
Coordinate Transformation in Electromagnetism
265
where 3xr characterizes the change from Cartesian coordinates to cylindrical coordinates, 3rr 0 characterizes the transformation from the original cylindrical coordinates to new cylindrical coordinates, and 3r 0 x 0 denotes change of the new cylindrical coordinates back to Cartesian coordinates. There are notably two important applications of the covariance property of electrodynamics described above. First, the geometries of some complex structures can be simplified greatly if they are described in their “natural” coordinates. Such an alternative geometrical description of complex structures can facilitate more efficient theoretical treatment of the EM problem, usually at the expense of more complex material parameter distributions. One example is to numerically deal with a helically twisted optical waveguide (Nicolet, Zolla and Guenneau, 2004; Shyroki, 2008), where a 3D problem can be simplified into a 2D one. The second application of the covariance property is for designing optical devices that can achieve novel phenomena. From the equations presented above, the new EM field, if interpreted from the original Cartesian coordinate system, will appear distorted. The field distortion is characterized by the Jacobian transformation matrix, and that in turn is characterized by how we choose the curved coordinate system. It should be noticed that the structure, including its material parameters obtained through coordinate transformation in the curved coordinate system, has been interpreted back on to the original Cartesian system. This is the socalled “material interpretation” of the covariance property of Maxwell’s equations (Schurig, Pendry and Smith, 2006b). When the newly obtained material parameters are interpreted in the original Cartesian coordinate system, one obtains an effective medium, which is referred to as the transformation medium. The transformation medium mimics the effect of curved spatial coordinates, and facilitate bending of light. In many cases, distortion of the EM field is what most optical devices are trying to achieve. Typical examples include waveguide bends, where change of beam direction is desired to facilitate dense optical integration, and lenses, where a spatially divergent field from a point source is refocused to a spot, etc. The fact that we are able to set out from certain desired EM wave behaviour and obtain some specific complex medium using the method of coordinate transformation suggests that designs of many novel devices can be within our reach. The metamaterial technology would complete the final fabrication step according to derived material parameter distributions. In the published literature, there are several proposed devices based on the theory of coordinate transformation, namely the invisibility cloak (Leonhardt, 2006; Pendry, Schurig and Smith, 2006), perfect lenses (Leonhardt and Philbin, 2006; Schurig, Pendry and Smith, 2007; Tsang
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and Psaltis, 2008; Kildishev and Shalaev, 2008; Kildishev and Narimanov, 2007; Yan, Yan and Qiu, 2008a), EM field rotator (Chen and Chan, 2007), EM concentrators (Rahm, Schurig, Roberts, Cummer, Smith and Pendry, 2008b), EM beam splitter (Rahm, Cummer, Schurig, Pendry and Smith, 2008a), antenna (Kong, Wu, Kong, Huangfu, Xi and Chen, 2007), and EM wormholes (Greenleaf, Kurylev, Lassas and Uhlmann, 2007a) etc. When time is also involved in the transformation, as in Minkowski space, the theory of coordinate transformation also helps to explain the EM analogues of the event horizon as well as the so-called optical AharonovBohm effect (Leonhardt and Philbin, 2006). In this paper, we will focus particularly on the EM cloaking devices. Not only because the fact that the proposal of the EM cloaking phenomenon has revived the covariance principle of electrodynamics in the design of novel devices, but also because the invisibility cloaking technology itself is of great interest to many for military and civilian applications. Invisibility cloaking has remained as a very popular concept in the world of science fiction or fantasy novels. The possibility of an EM invisibility device has recently inspired theoretical investigations on cloaking of sound (Milton, Briane and Willis, 2006; Cummer and Schurig, 2007; Cummer, Popa, Schurig, Smith, Pendry, Rahm and Starr, 2008; Norris, 2008; Chen, Yang, Luo and Ma, 2008), matter waves (Zhang, Genov, Sun and Zhang, 2008d; Greenleaf, Kurylev, Lassas and Uhlmann, 2008), and even surface wave in a liquid (Farhat, Enoch, Guenneau and Movchan, 2008). It should be mentioned that some studies on invisibility cloaking existed before the paper by Pendry and his co-workers (Pendry, Schurig and Smith, 2006). Notably, a technique similar to the coordinate transformation was deployed for achieving invisibility in the static limit (Greenleaf, Lassas and Uhlmann, 2003). Furthermore, there have been proposals on invisibility cloaking devices based on techniques other than the coordinate transformation approach. Among such works, surface plasmon resonance has been utilized to hide small particles (see e.g. Alu` and Engheta (2005); Milton and Nicorovici (2006); Alu` and Engheta (2008)), and also in Miller (2006) a network of sensors and active sources are used to achieve invisibility.
3. PRINCIPLE AND CONSTRUCTION OF INVISIBILITY CLOAKS The possibility of invisibility cloaking arises when a newly chosen curved coordinate system contains a void region. A simple way to visualize this is to lay out the curved coordinate lines in the original Cartesian coordinate system according to the relation as defined by Equation (3) (in fact, it will be much clearer to see from the inverse of the functions, i.e. q1 = q1 (x, y, z) etc). A spatial region in the Cartesian space that
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is not passed through by any curved coordinate lines is a void region. This situation is illustrated in Figure 1, where Figure 1(a) and (b) depict coordinates before and after transformation, respectively. Notice that the coordinates change only for a finite spatial region bounded by S1 in both panels. As a consequence, the material outside S1 remains the same in Figure 1(a) and (b) due to the identity transformation in that region of space. In Figure 1(b), a void region has appeared, which is bounded by the boundary S2 . The void region corresponds to a point of zero size (or a line, if the structure is invariant in the direction normal to the page) in the original coordinate system shown in Figure 1(a). The annular shape, as bounded by S1 and S2 in Figure 1(b), is electromagnetically equivalent to the region bounded by S1 in Figure 1(a). However their material parameters are different owing to the change of coordinates. A medium obtained through coordinate transformation, such as the annular shape in Figure 1(b), is referred to as a transformation medium. Due to the presence of a void region in Figure 1(b), hiding of objects is possible since light will be unable to penetrate the void region. In the case where the original space in Figure 1(a) is empty, the resulting transformation medium is an object with anisotropic material properties, which is invisible and at the same time provides shielding to other objects enclosed in S2 . As a side note, at exactly the S1 boundary in Figure 1, the transformed coordinate components are kept the same as the original Cartesian coordinate components, i.e. x| S1 = q1 | S1 etc. In Yan, Yan and Qiu (2008b), such coordinate continuation at a finite transformation medium’s boundary is found to be a necessary and sufficient condition in order to achieve an impedance matching condition for an arbitrarily shaped structure. The method of coordinate transformation can not only design complex transformation media, it also directly defines the field profile in the transformation media through Equation (8). Here, for ease of illustration, we consider the systems shown in Figure 1 which are two-dimensional, i.e. they are invariant in the direction normal to the page (say z). Under this condition, E z , the electric field in original space and E z0 0 , the electric field in transformed space should be equal in amplitude, except that the distribution of E z0 0 should be distorted compared with E z according to the particular coordinate transformation deployed. Assume also that the original space is free space. In the case of a right-propagating plane wave with E z polarization, the electric field distributions before and after transformation are shown in Figure 2(a) and (b), respectively. In short, to construct a cloak, one usually starts by compressing a finite space whilst keeping the outer boundary S1 unchanged, and the inner boundary S2 is obtained by blowing up a line (straight or curved) or a point. Thus the cloaks are divided into two classes: line-transformed cloaks
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FIGURE 1 Illustration of coordinate transformation for obtaining a cloak of virtually arbitrary shape in 2D. (a) Original Cartesian coordinates; (b) Transformed curved coordinates. Coordinate lines before and after transformation are in red. The region bounded by S1 and S2 is the transformation medium that constitutes a cloak
FIGURE 2
E z field before transformation. (b) E z field after transformation
and point-transformed cloaks. The permittivity and permeability tensors of the cloak in a Cartesian coordinate system are given in Equation (7).
4. ARBITRARILY-SHAPED INVISIBILITY CLOAKS The material parameters of an arbitrarily-shaped cloak are very complex. The medium is not only anisotropic, but also spatially gradient. In this section, we will examine the performance and properties of such invisibility cloaks using an analytical technique. Both the line-transformed cloak and the point-transformed cloak are studied. Analytical study of such arbitrarily-shaped cloaks is possible since the eigenfunctions of the
Arbitrarily-shaped Invisibility Cloaks
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wave equations in the cloak medium are in fact quite simple, and related to the eigenfunctions in the original space through Equation (8). In order to achieve perfect invisibility, the cloak should be able to exclude light from a protected object without perturbing the outer field. Thus for a perfect invisibility cloak, two conditions must be fulfilled: (1) at the outer boundary S1 , external incident light should excite no reflection; (2) at the inner boundary S2 , no reflection is excited, and no light can penetrate the cloaked region. In the following subsections, we will prove the invisibility of the cloak by investigating the behaviour of waves at the cloak’s outer and inner boundaries. For simplicity, the invisibility cloak is considered to be placed in free space. That is, the relative permittivity and permeability of the cloak in Equation (10) will be simplified to b ε=b µ = det(3)g−1 .
(12)
4.1 Outer Boundary of Cloak In this subsection, it will be proved that no reflection is excited at the outer boundary for either line-transformed cloaks or point-transformed cloaks. i bi (before interacting The transmitted electric field b E and magnetic field H with the inner boundary S2 ) are expressed as i b E = 3T Ei ,
bi = 3T Hi , H
(13)
where Ei and Hi represent the electric and magnetic fields of the external incident waves. According to Equation (8), it is easily seen that the fields expressed in Equation (13) satisfy Maxwell’s equations in the cloak medium. Thus in order to prove that no reflection is excited at S1 , one i only needs to confirm that the tangential components of Ei (Hi ) and b E i b ) remain continuous across S1 . (H i i bi = [ H bi into b bti , E bti ] and H bni , H bti , H bti ], bni , E Decomposing b E and H E = [E 1 2 1 2 where the subscript n represents the surface normal in the direction pointing outward from S1 ; t1 and t2 represent two surface tangent directions, which are perpendicular to each other, Equation (13) can be equivalently expressed as bni E bi n,b t1 , b t2 ]−1 3T Ei , E t = [b
1
bti E 2
bni H bi n,b t1 , b t2 ]−1 3T Hi , Ht = [b 1
bti H 2
(14)
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where b n, b t1 , b t2 represent the unit vectors in n, t1 , and t2 directions, respectively. At the outer boundary S1 , q1 = x, q2 = y, and q3 = z. So f i (q1 , q2 , q3 ) − qi = 0 (i = 1, 2, 3) characterize the outer boundary S1 . Therefore, it is bi (i = 1, 2, 3) lie in the same line as the obvious that the vectors ∇q f i − C b1 = b b2 = b b3 = b normal direction n of S1 , where C x, C y, and C z. For the bi = 0, ∇q f i − C bi can be expressed as 0b special case when ∇q f i − C n , i.e., the vector with zero magnitude in the n direction. Therefore, 3T on S1 can be expressed as b3b 3T = [F1b n +b x , F2b n +b y, F n +b z],
(15)
with s |Fi | =
2 ∂ fi ∂ fi 2 ∂ fi 2 −1 + + , ∂qi ∂q j ∂qk
(16)
where i, j, k = 1, 2, 3 and i 6= j 6= k; Fi = |Fi | when the direction of bi is the same as the n direction, and Fi = −|Fi | if the direction ∇q f i − C bi is opposite to the n direction. Substituting Equation (15) into of ∇q f i − C Equation (14) and noticing that b n, b t1 , and b t2 are orthogonal to each other, it is easily shown that at S1 bti = Ei · b E t1 , 1 i i b b E t2 = E · t2 ,
bti = Hi · b H t1 , 1 i i b b Ht2 = H · t2 ,
(17) (18)
i bi which indicates that the tangential components of Ei (Hi ) and b E (H ) are continuous across S1 . Thus, it is proved that no reflection is excited at the outer boundary for both line-transformed cloaks and point-transformed cloaks.
4.2 Inner Boundary of Cloak In this subsection, it will be proved that at the inner boundary S2 , no reflection is excited and no field can penetrate into the cloaked region. As discussed in the above subsection, the inner boundary is constructed by blowing up a line or a point, as seen in Figure 3(a) and (b). So in the following, two cases: (1) line-transformed cloaks, (2) point-transformed cloaks, will be discussed separately. 4.2.1 Line-transformed Cloaks Assume that x = b1 (s), y = b2 (s) and z = b3 (s) characterize the line, which is mapped to the inner boundary S2 . Thus, we have f 1 (q1 , q2 , q3 ) = b1 (s)
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FIGURE 3 Illustration of inner boundary of cloak for (a) a line-transformed cloak; (b) a point-transformed cloak
and f 2 (q1 , q2 , q3 ) = b2 (s), and f 3 (q1 , q2 , q3 ) = b3 (s) at S2 . Each point (b1 , b2 , b3 ) on the line maps to a closed curve on S2 . The parameter s can be expressed as a function of q1 , q2 , q3 with s = u(q1 , q2 , q3 ). ∇q s = ∂u/∂q1b x+ ∂u/∂q2b y + ∂u/∂q3b z is the gradient of s, which points in the direction of the greatest rate of increase of s. For ∇q bi , we have ∇q bi = ∂bi /∂s∇q s, where i = 1, 2, 3. Thus, the vectors ∇q bi and ∇q s have the same (or antiparallel) direction. For ease of discussion, we again decompose the transmitted fields i bni , E bti , E bti ], expressed in Equation (13) at the inner boundary as b E = [E 1 2 i b = [H bni , H bti , H bti ], where the subscript n denotes the surface normal H 1 2 direction pointing outward from S2 ; t1 and t2 denote two surface tangent directions of S2 , with t1 perpendicular to the plane determined by two vectors in n and ∇q s directions, and t2 perpendicular to t1 . Since s varies on the surface S2 , the direction of ∇q s (denoted by sq in Figure 3(b)) should not be parallel to the normal direction of S2 . Thus, the plane determined by the vectors in n and ∇q s directions always exists. The n, t1 , and t2 directions i bi at S2 can also be expressed are unique, as illustrated in Fig. 3(a). b E and H in Equation (14), however, with different definitions of n, t1 and t2 . Since f i (q1 , q2 , z) − bi (s) = 0 characterizes the inner boundary S2 , ∇q f i − ∇q bi characterizes the normal direction of S2 . Then 3T at S2 can be written as 3T = [F1b n + ∇q b1 , F2b n + ∇q b2 , F3b n + ∇q b3 ],
(19)
v u 3 uX |Fi | = t (∂ f i /∂q j − ∂bi /∂q j )2 ,
(20)
with
j=1
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where Fi = |Fi | when the direction of ∇q f i − ∇q bi is the same as the direction of n, and Fi = −|Fi | when the direction of ∇q f i −∇q bi is opposite to the direction of n. Notice that t1 is orthogonal to both n and sq , when b s denotes the unit vector in the direction of ∇q s. Substituting Equation (19) into Equation (14), it is easily derived that at the inner boundary bi = H bi = 0. E t1 t1
(21)
However, the other components of the fields are not zero. In particular, bi = (b E s ·b t2 )[B1 , B2 , B3 ]Ei , t2 bi = (b H s ·b t2 )[B1 , B2 , B3 ]Hi , t2 bni = [F1 + B1 (b E s ·b n ), F2 + B2 (b s ·b n ), F3 + B3 (b s ·b n )]Ei ,
(22)
= [F1 + B1 (b s ·b n ), F2 + B2 (b s ·b n ), F3 + B3 (b s ·b n )]H ,
(25)
bni H
i
(23) (24)
with v u 3 uX Bi = t (∂bi /∂q j )2 .
(26)
j=1
To further investigate how the waves interact with the inner boundary, the values of the permittivity and permeability at S2 are needed. Considering 3 as expressed in Equation (19), it is easily seen that 3b t1 = 0, t1 = 0. Thus one of the eigenvectors of g is indicating that gb t1 = 3T 3b b t1 with the eigenvalue λt1 = 0, implying det(3) = 0. Because g is a symmetrical matrix, the other two eigenvectors denoted by b a and b b should be orthogonal to each other and in the n − t2 plane. The corresponding eigenvalues are denoted by λa and λb , respectively, with b× b λa λb = |b n ×b s|2 | F B|2 ,
(27)
b = F1b where F x + F2b y + F3b z, b B = B1b x + B2b y + B3b z. Since λt1 λa λb = det(g) = det(3)2 , λt1 = det(3)2 /(λa λb ). By observing Equation (12), we obtain gb = gb µ = det(3). Therefore, b ε and b µ can be expressed as b ε=b µ = diag[λa λb / det(3), det(3)/λa , det(3)/λb ],
(28)
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273
where the diagonal elements correspond to the principle axes b t1 , b a , and b b, respectively. Since det(3) = 0, we have a = µa = b = µb = 0, indicating that the cloak medium at S2 is isotropic in the n − t2 plane. Therefore, b n and b t2 can also be considered as the principle axes with n = µn = t2 = µt2 = 0. It is seen that t1 and µt1 have infinitely large values. Thus, the inner boundary operates similarly to a combination of a PEC (perfect electric conductor) and a PMC (perfect magnetic conductor), which can support both electric and magnetic surface displacement currents in the t1 direction (Zhang, Chen, Wu, Luo, Ran and Kong, 2007; Greenleaf, Kurylev, Lassas and Uhlmann, 2007c,b). In order to have zero reflection at S2 , the boundary conditions at this PEC and PMC combined layer require that the incident electric (magnetic) fields in the t1 direction and normal electric (magnetic) displacement fields are all zero. From Equation (21), bti = H bti = 0. Since n = µn = 0, we obtain D bni = b we have E Bni = 0. 1 1 i bi Therefore, it is proved that no reflection is excited at S2 , and b E and H expressed in Equation (13) are just the total fields in the cloak medium. The PEC and PMC combined layer guarantees that no field can penetrate into the cloaked region. It is worth noting that the induced displacement bti and H bti at S2 go down to surface currents in the t1 direction make E 2 2 bti and H bti are not zero at the location approaching S2 in zero. However, E 2 2 the cloak medium and thus are discontinuous across the inner boundary S2 (Zhang, Chen, Wu, Luo, Ran and Kong, 2007; Greenleaf, Kurylev, Lassas and Uhlmann, 2007c,b). 4.2.2 Point-transformed Cloaks In this case, a point with the coordinate [c1 , c2 , c3 ] maps to a closed surface S2 as the cloak’s inner boundary. At S2 , we have f 1 (q1 , q2 , q3 ) = c1 , f 2 (q1 , q2 , q3 ) = c2 , and f 3 (q1 , q2 , q3 ) = c3 . The transmitted electric and i magnetic fields at the inner boundary can be decomposed into b E = bi = [ H bni , E bti , E bti ] and H bni , H bti , H bti ], where the definition of the subscript [E 1 2 1 2 00 n 00 denotes the normal direction of S , which is outward from the cloaked 2 region; “t1 ” and “t2 ” represent the two tangential directions of S2 , which are perpendicular to each other, as shown in Figure 3(b). Consider 3T at the inner boundary, which can be expressed as 3T = diag[F1b n , F2b n , F3b n ],
(29)
with |Fi | =
q
(∂ f i /∂q1 )2 + (∂ f i /∂q2 )2 + (∂ f i /∂q3 )2 ,
(30)
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where i = 1, 2, 3. Then considering Equation (29), it can be derived that at S2 bti = H bti = 0, E 1 1 bti = H bti = 0, E 2 2 i b E n = [F1 , F2 , F3 ]Ei ,
(31) (32) bni = [F1 , F2 , F3 ]Hi . H
(33)
Unlike the line-transformed cloak, in this case the tangential fields are all zero, implying that no field discontinuity exists at S2 . Analyzing g similarly, as in the line-transformed cloak case, we obtain that b n is an eigenvector of g with the eigenvalue λn = F12 + F22 + F32 . While the other two eigenvectors are b t1 and b t2 with corresponding eigenvalues λt1 = λt2 = 0. This indicates that det(3) = 0. Considering λn λt1 λt2 = q 2 2 det(3) , we have λ = λ = det(3)/ (F + F 2 + F 2 ). Therefore, b ε and b µ t1
t2
1
2
3
are given as b ε=b µ = diag[det(3)2 /(F12 + F 2 + F 3 ),
q
F12 + F 2 + F 3 ,
q
F12 + F 2 + F 3 ], (34)
where the diagonal elements are in the principle axes b n, b t1 , and b t2 , respectively. Since det(3) = 0, n = µn = 0. Considering n = µn = 0, and that the tangential components of the incident fields at S2 are zero, we conclude that no reflection is excited at S2 , and no field can penetrate into the cloaked region. The fields expressed in Equation (13) are the total fields in the cloak medium. In this section, it has been proved that no reflection is excited at either the outer or the inner boundaries of an arbitrarily shaped cloak, and no field can penetrate into the cloaked region. Therefore, invisibility cloaks of arbitrary shape constructed by general coordinate transformations are confirmed.
5. CYLINDRICAL INVISIBILITY CLOAK In this section, we will comprehensively study the performance of cylindrical invisibility cloaks, including both ideal ones and those with simplified material parameters. The cylindrical form is by far the most extensively studied invisibility cloaking structure. Some previous studies include Cummer, Popa, Schurig, Smith and Pendry (2006), Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith (2006a), Cai, Chettiar, Kildishev and Shalaev (2007a), Ruan, Yan, Neff and Qiu (2007), Yan, Ruan and Qiu
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(2007a), Greenleaf, Kurylev, Lassas and Uhlmann (2007c), Yan, Ruan and Qiu (2007b), Zhang, Chen, Wu, Luo, Ran and Kong (2007), Cai, Chettiar, Kildishev, Shalaev and Milton (2007b), Yan, Yan, Ruan and Qiu (2008d,c). A cylindrical cloak is in fact a special type of line-transformed cloak. It is constructed by compressing EM fields in a cylindrical region r 0 < b into a concentric cylindrical shell a < r < b. Its inner boundary is blown up by a straight line. Here we consider a generalized coordinate transformation in cylindrical coordinates in which r 0 = f (r ) with f (a) = 0 and f (b) = b, while θ and z are kept unchanged. The permittivity and permeability of the cloak expressed in the cylindrical coordinate system are as follows r = µr =
f (r ) , r f 0 (r )
θ = µθ =
r f 0 (r ) , f (r )
z = µz =
f (r ) f 0 (r ) , r
(35)
where the superscript 0 denotes differentiation. i
i
Consider the fields E and H incident upon the cloak. The fields in the cloaked medium can be obtained directly from Equation (13) br (r, θ, z) = f 0 (r )Eri ( f (r ), θ, z), H br (r, θ, z) = f 0 (r )Hri ( f (r ), θ, z), (36) E bθ (r, θ, z) = f (r ) E i ( f (r ), θ, z), H bθ (r, θ, z) = f (r ) H i ( f (r ), θ, z),(37) E θ θ r r bz (r, θ, z) = E zi ( f (r ), θ, z), H bz (r, θ, z) = Hzi ( f (r ), θ, z), E (38) where [Eri , E θi , E zi ] and [Hri , Hθi , Hzi ] are components of the incident fields expressed in cylindrical coordinate form. At the inner boundary, one can easily see that, no matter what f (r ) is, θ and µθ are infinitely large, and the other components are zero, due to the fact that it is a line-transformed cloak. Field components E θ and Hθ , Dr and Br are all zero at the inner boundary r = a, which guarantees that no reflection is excited at the inner boundary, as analyzed above. The surface displacement currents are induced to make E z and Hz tend to zero at the inner boundary (Zhang, Chen, Wu, Luo, Ran and Kong, 2007; Greenleaf, Kurylev, Lassas and Uhlmann, 2007c,b). By choosing different radial mapping functions, we can obtain different parameters for the cloak medium. One possible class of functions can take the form of r0 =
b (r − a)n , (b − a)n
(39)
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FIGURE 4 E z field distributions in r θ plane. (a) Virtual EM space; (b) Cloak with n = 1; (c) Cloak with n = 3. Colormap: E z . Green lines: Poynting vectors. Invariant coordinate lines are imposed. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
where n is the transformation order. The corresponding parameters of the cloaks are εr = µr =
r −a , nr
εθ = µθ =
nr , r −a
εz = µz =
nb2 (r − a)2n−1 . (40) (b − a)2n r
When n = 1, it is the well-known linear cylindrical cloak as proposed by Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith (2006a). The interactions between an in-plane propagating plane wave with E z polarization and cylindrical cloaks designed with different n are shown in Figure 4. Figure 4(a) shows a plane wave propagating in the original free EM space. Figure 4(b) and (c) show interactions of the plane wave with two cloaks, designed with n = 1 and n = 3, respectively. Comparing Figure 4(b) and (c), we see that the Poynting vectors (rays) are bent much more heavily in the cloak medium when n = 3 than when n = 1. As mentioned earlier, the inner boundary of a cylindrical cloak is inherently singular, i.e., the components of the permittivity and permeability in the θ direction are infinitely large, which are impossible to obtain in reality. To avoid this, there are two approaches: one is to remove a thin layer from the inner boundary, i.e., to construct an approximate ideal cloak; the other is to simplify the parameters of an ideal cloak, which not only circumvents the singular inner surface problem, but also makes implementation much easier. However, such material simplification is done at the expense of degradation of invisibility performance, owing to a change in the wave equation. In the following, we will first examine the properties of an approximate ideal cloak, then some material simplification methods will be summarized and compared.
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FIGURE 5 The schematic of a near-ideal cylindrical cloak: The distribution of the material parameter is the same as the ideal one shown in Equation (35), and the outer boundary is still fixed at r = b. However, the actual inner boundary is at r = a + δ, where δ is a very small positive number
5.1 Approximate Ideal Cylindrical Cloak The material parameters for a general radially transformed cylindrical cloak are derived in Equation (35). To avoid divergence at the inner boundary r = a, a thin layer is removed from the inner boundary, as illustrated in Figure 5. The goal here is to find the scattering performance of such an approximate cylindrical cloak. Without loss of generality, here we focus on the TM polarization (the magnetic field of the wave is polarized purely along the z axis). The relevant material parameters are therefore only µz , r and θ . Throughout the following discussion, the time dependence e−iωt is assumed. As shown in Equation (8), the eigenfunction of the wave equations in the cloak is related to the eigenfunction in free space by the matrix 3T . Thus, the general form of the Hz component outside the cloak and in the cloak medium can always be expressed as
Hz = Hz =
+∞ X n=−∞ +∞ X
anin Jn (k0r )einθ + ansc Hn (k0r )einθ
(b < r ),
an1 Jn (k0 f (r ))einθ + an2 Hn (k0 f (r ))einθ
(41)
(a + δ < r < b).
n=−∞
(42) In Equations (41) and (42), Jn and Hn represent Bessel functions of the first kind and Hankel functions of the first kind, respectively, both of the order n; anin and ansc are the expansion coefficients for the incident and scattered fields outside the cloak, respectively; ani (i = 1, 2) are the expansion coefficients for the field in the cloak shell; and k0 = ω/c is the wavenumber in free space. Hz and E θ in the region r < a + δ can be
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expressed as Hz =
X
an3 Sn (r )einθ ,
(43)
k0 an3 Tn (r )einθ /(iω0 ),
(44)
n
Eθ =
X n
respectively, where 0 is the permittivity of the free space, and Sn (r ) and Tn (r ) are determined by the media in the cloaked region. For isotropic homogeneous media with relative permittivity u and permeability µu , √ √ Sn (r ) = Jn (ku r ) and Tn (r ) = Jn0 (ku r ) µu /u , where ku = ω µu u /c. However, when the cloaked region is composed of inhomogeneous or anisotropic media, close forms of Sn (r ) and Tn (r ) are difficult to obtain. The boundary conditions require that tangential fields should be continuous across the interfaces at r = b and r = a + δ. The boundary conditions lead to the set of equations: anin Jn (k0 b) + ansc Hn (k0 b) = an1 Jn (k0 b) + an2 Hn (k0 b), f 0 (b)k0 1 0 k0 anin Jn0 (k0 b) + k0 ansc Hn0 (k0 b) = a J (k0 b) θ (b) n n f 0 (b)k0 2 0 a H (k0 b), + θ (b) n n
(45)
an1 Jn (k0 f (a + δ)) + an2 Hn (k0 f (a + δ)) = an3 Sn (a + δ), f 0 (a + δ)k0 2 0 f 0 (a + δ)k0 1 0 an Jn (k0 f (a + δ)) + a H (k0 f (a + δ)) θ (a + δ) θ (a + δ) n n
(47)
= k0 an3 Tn (a + δ),
(46)
(48)
From Equations (45)–(48), one can easily obtain the expression of the scattering coefficient for each order, which is defined as Cnsc = ansc /anin . It can be proved that Cnsc → 0 when δ → 0 (see Ruan, Yan, Neff and Qiu (2007) in the case of a linearly tranformed cloak). That is, the ideal cylindrical cloak is a perfect invisibility cloak for generalized coordinate transformations. Here we pay particular attention to the effect of a tiny perturbation δ. We notice that, when δ a and k0 δ 1, Cnsc can be approximated as Cnsc ≈
a Jn (k0 δ f )Tn (a + δ) − δ f Jn0 (k0 δ f )Sn (a + δ) , δ f Hn0 (k0 δ f )Sn (a + δ) − a Hn (k0 δ f )Tn (a + δ)
(49)
where δ f = f p (a)δ p , f p (a) = d p f (r )/dr p |r =a , with p ≥ 1 and f q (a) = 0 for q < p. For the special case when a PEC layer or a PMC layer is put at
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r = a+δ, Equation (49) is still valid by setting Sn (a+δ) = 1 and T (a+δ) = 0 for the PEC case, or setting Sn (a + δ) = 0 and T (a + δ) = 1 for the PMC case. In the following, based on the scattering equation given in Equation (49), we will show that the zeroth order scattering term is very sensitive to the perturbation, while the high order scattering terms are not so sensitive. A method is then proposed to overcome the high sensitivity problem for the zeroth order wave. 5.1.1 zeroth Order Scattering Coefficient For the zeroth order term, since J00 = −J1 and H00 = −H1 , we obtain C0sc ≈
a J0 (k0 δ f )T0 (a + δ) + δ f J1 (k0 δ f )S0 (a + δ) . −δ f H1 (k0 δ f )S0 (a + δ) − a H0 (k0 δ f )T0 (a + δ)
(50)
Considering the asymptotic forms of the Bessel functions, we have J0 (k0 δ f ) ≈ 1 and δ f J1 (k0 δ f ) ≈ k0 δ 2f /2. Therefore we know that the numerator of Equation (50) will be dominated by a J0 (k0 δ f )T0 (a +δ) except for the case where a PEC layer is put at the cloak’s inner surface and T0 (a+δ) is always equal to zero. In view of this, we will present, separately, the following two different cases: (1) no PEC layer is at r = a + δ and (2) there is a PEC layer at r = a + δ. Case (1): no PEC layer at r = a + δ In this case, the numerator of Equation (50) will be dominated by a J0 (k0 δ f )T0 (a + δ), so C0sc can be further simplified to C0sc ≈
a J0 (k0 δ f )T0 (a + δ) , i2S0 (a + δ)/(π k0 ) − a H0 (k0 δ f )T0 (a + δ)
(51)
where δ f H1 (k0 δ f ) ≈ −i2/(πk0 ) is considered. To verify the validity of Equation (51), we take an example with f (r ) = b(r − a)/(b − a), a = 1.2π/k0 , b = 2π/k0 , and the region inside the cloak shell being air. Using Equation (51), |C0sc | versus δ for small values of δ is plotted in Figure 6. It is seen that the result calculated from Equation (51) fits very well with the exact result obtained directly from Equations (45)–(48). Observing Equation (51) and noting that H0 (k0 δ f ) diverges with the decrease of δ, we find that the value of C0sc can be approximated by J0 (k0 δ f )/H0 (k0 δ f ) ∝ −1/ ln(δ), which is independent of the media inside the cloak, under the circumstance that no PEC layer is at the inner boundary. This results in a very slow convergence rate for C0sc , approaching zero as δ decreases (also see Ruan, Yan, Neff and Qiu (2007)). Thus, C0sc is sensitive to perturbation in this case. In other words, a
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FIGURE 6 |C0sc | versus δ for a cylindrical cloak with f (r ) = b(r − a)/(b − a), a = 1.2π/k0 , b = 2π/k0 , and the region inside the cloak shell is assumed to be air
noticeable scattering will be induced even by a tiny perturbation, for example, from Figure 6, |C0sc | is 0.1017 even when δ/a = 10−8 . However, by choosing an appropriate f (r ), the noticeable scattering induced by a tiny perturbation can be reduced to some extent. Observing Equation (51), we find that |C0sc | decreases when δ f decreases. Since δ f = f p (a)δ p , we know that by choosing an f (r ) with a small f p (a) and a large p we can reduce the scattering coefficient. To verify this idea, the zeroth order scattering coefficients are derived for three types of cylindrical cloaks. The parameters a, b, and the medium inside the cloak shell are kept the same as in Figure 6. k0 = 0.064/π. The function f (r ) is chosen consecutively for three case studies as f (r ) = b(r − a)/(b − a) with p = 1 and f 0 (a) = 2.5, f (r ) = b(r 10 − a 10 )/(b10 − a 10 ) with p = 1 and f 0 (a) = 0.1, and f (r ) = b/2 − b/2 cos(π/(b − a)(x − a)) with p = 2 and f 00 (a) = 0.1. |C0sc | versus δ for three cloaks are plotted in Figure 7. It is seen that |C0sc | decreases with the decrease of f 0 (a) and the increase of p. However the reduction in the scattering coefficient is not significant because the overall convergence speed determined by −1/ ln(δ) can not be improved. Case (2): with a PEC layer at r = a + δ When a PEC layer is present, Equation (51) can be simplified to C0sc = −
J1 (k0 δ f ) . H1 (k0 δ f )
(52)
In Equation (52), the value of C0sc is determined by J1 (k0 δ f )/H1 (k0 δ f ) ∝ δ 2 p . δ 2 p converges to zero much faster than −1/ ln(δ), therefore, in this case C0sc also converges to zero much faster, i.e., C0sc is not sensitive to
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FIGURE 7 |C0sc | versus δ for p = 1 and f 0 (a) = 2.5, p = 1 and f 0 (a) = 0.1, p = 2 and f 00 (a) = 0.1. Other parameters are the same as those in Figure 6
FIGURE 8 |C0sc | versus δ for the case (1) without a PEC layer at r = a + δ and the case (2) with a PEC layer at r = a + δ. Other parameters are the same as those in Figure 6
the perturbation. To show the improvement of the convergence speed of C0sc , we calculate |C0sc | versus δ with f (r ) = b(r − a)/(b − a), a = 1.2π/k0 , b = 2π/k0 , for both case (1) and case (2). The region inside the cloak shell is air for case (1). The result is shown in Figure 8. It clearly indicates that the convergence speed of C0sc is improved drastically by introducing a PEC layer at r = a + δ (Yan, Yan, Ruan and Qiu, 2008d; Greenleaf, Kurylev, Lassas and Uhlmann, 2007c). 5.1.2 High Order Scattering Coefficients For the high order terms, considering the asymptotic forms of the Bessel function and Hankel function for small arguments when δ f Jn0 (k0 δ f ) ≈
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|n|Jn (k0 δ f )/k0 , and δ f Hn0 (k0 δ f ) ≈ −|n|Hn (k0 δ f )/k0 , we arrive at Cnsc ≈ −An Bn ,
(53)
where An =
k0 aTn (a + δ) − |n|Sn (a + δ) , k0 aTn (a + δ) + |n|Sn (a + δ)
Bn =
Jn (k0 δ f ) . Hn (k0 δ f )
(54)
From Equation (54), it can be seen that An is nearly a constant, which characterizes the properties of the cloaked region. However, Bn is affected significantly by the perturbation. Therefore, the value of Cnsc is mainly determined by Bn , i.e., Jn (k0 δ f )/Hn (k0 δ f ) ∝ δ 2|n| p , which is, unlike the case for the zeroth order, independent of the situation whether a PEC layer is put at the inner boundary or not. To verify the validity of Equation (53), we take again the example with f (r ) = b(r − a)/(b − a), a = 1.2π/k0 , a = 2π/k0 , and the region inside the cloak shell being air. |Cnsc | for n = 1 and n = 2 when 10−8 a < δ < 10−2 a are plotted in Figure 9(a) and (b), respectively. We see that the results calculated from Equation (53) agree with the exact results almost perfectly. In addition, by comparing Figures 6 and 9, it is obvious that the high order scattering coefficients converge much faster than the zeroth order coefficient. Interestingly, the convergence speed of C1sc determined by δ 2 p is the same as that of C0sc if a PEC layer is put at r = a + δ.
5.2 Material Simplification To implement an ideal cylindrical invisibility cloak, two restrictions exist. One is that some material parameters require an infinitely large value at the inner boundary, and the other is that each anisotropic material parameter varies with its spatial location. The first restriction can be solved by removing a thin layer from the inner boundary, as mentioned in the above subsection. However, the existence of the second restriction still makes practical implementation difficult. Therefore, it is useful to simplify the material parameters of an ideal cylindrical invisibility cloak. The first simplified model was proposed in Cummer, Popa, Schurig, Smith and Pendry (2006) and later demonstrated in Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith (2006a). In these preliminary works, the radial transformation function is considered to be a linear function r 0 = f (r ) = b(r − a)/(b − a). The corresponding ideal material parameters and simplified material parameters are listed in Table 1. It is obvious that the implementation of such a cloak will become much easier after simplification: no material parameter requires an infinite value; and only εr and µr are inhomogeneous, while the other components
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FIGURE 9 (a) |C1sc | versus δ; (b) |C2sc | versus δ. Other parameters are the same as those in Figure 5
TABLE 1 Material parameters for linear cloaks Ideal
εr = µr = r −a r r εθ = µθ = r −a 2 b r −a εz = µz = b−a r
Simplified (Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith, 2006a) 2 εr = µr = r −a r εθ = µθ = 1 2 b εz = µz = b−a
are all constant. In the case where normal incidence (the k vector is perpendicular to the cloak cylinder axis) and single TE polarization (the electric field only exists in the z direction) are considered, we need only account for three material components, µr , µθ and z . Among these three
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FIGURE 10 2D microwave cloaking structure (Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith, 2006a)
parameters, we need only pay special attention to µr when it comes to engineering the necessary metamaterial. In Ref. (Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith, 2006a), such a simplified cloak for TE polarization at microwave frequency is experimentally demonstrated. As shown in Figure 10, the native cylindrical coordinate has been used as the construction coordinate. The key component µr is engineered by a split ring resonator (SRR) structure, and its variation along the radial direction is achieved by tuning the geometrical parameters of the SRRs. The experimental results are shown in Figure 11. Simplified cylindrical invisibility cloaks make the implementation much easier. However, this is at the expense of loss of the perfect invisibility performance, since the material parameters deviate from the ideal values. The arguments in Cummer, Popa, Schurig, Smith and Pendry (2006) and Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith (2006a) are presented mostly through examining numerically simulated or experimentally recorded field distributions. The exact scattering induced by their simplified cloak model is not quantified. In the following, we will show a more precise quantitative analysis of simplified cloaks and show some optimal simplified models. 5.2.1 Simplified Linear Cloak Consider a simplified cylindrical cloak with the material parameters listed in Table 1. Both domains inside and outside the cloak are assumed to be air. The structure is in general a three-layered cylindrical scatterer. We refer to the layers from inside to outside as layer 1, 2 and 3, respectively.
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FIGURE 11 Snapshots of time-dependent, steady-state electric field patterns, with stream lines indicating the direction of power flow (i.e., the Poynting vector). The fields shown are (a) simulation of the cloak with the exact material properties, (b) simulation of the cloak with the reduced material properties, (c) experimental measurement of the bare conducting cylinder, and (d) experimental measurement of the cloaked conducting cylinder (Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith, 2006a)
The analysis is on normal incidence with TE polarization. The TM polarization case can be easily derived by the duality principle. By default, we choose the cloak’s cylindrical coordinate as the global coordinate. In a homogeneous material region (i.e. the cloak interior and exterior) the general solution is expressible in Bessel functions. Within the cloak medium, the general wave equation that governs the E z field can be written as 1 ∂ r ∂ Ez 1 ∂ 1 ∂ Ez + 2 + k02 εz E z = 0. r ∂r µθ ∂r r ∂θ µr ∂θ
(55)
In the case of the simplified cloak, the invariant µθ can be taken out of the differential operator in Equation (55). This is however not true for the ideal cloak medium. Therefore the wave behaviour within the cloak shell is altered when compared to that in an ideal cloak. Since the material parameters of such simplified cloaks are azimuthally invariant (which is also true for the ideal parameter set), we can use the variable separation E z = 9(r )2(θ ). Equation (55) can then be decomposed into d2 2 + m 2 2 = 0, dθ 2
(56)
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286
d dr
r d9 µθ dr
+ k02r εz 9 − m 2
1 9 = 0, r µr
(57)
where m is an integer. The solution to Equation (56) is exp(imθ ). Equation (57) is a second-order homogeneous differential equation so two independent solutions are expected. At this moment, we assume the solution to Equation (57), for a fixed m, can be written in general as Am Q m +Bm Rm , where Am and Bm are constants. Q m and Rm are functions of r . Now valid field solutions in the three layers (denoted by superscripts) can be described by E z1 =
X
Am1 Jm (k0r ) exp(imθ ),
(58)
m
E z2 =
X 2 {Am2 Q m + Bm Rm } exp(imθ ),
(59)
m
E z3 =
X 3 {Am3 Jm (k0r ) + Bm Hm (k0r )} exp(imθ ).
(60)
m
Hm is the Hankel function, which represents an outward-travelling cylindrical wave. The Jm and Hm terms in the 3rd layer physically correspond to the incident and scattered waves, respectively. Hence, it is most important to solve the scattering problem, with Am1 (transmitted 3 (scattered field) subject to a given incidence A 3 .2 field) and Bm m 2 are physically less interesting. The coefficients Comparatively Am2 and Bm are solved by matching the tangential fields, E z and Hθ , at the layer interfaces. Due to the orthogonality of the function exp(imθ ), the cylindrical waves in different orders decouple. Hence, we can examine the transmission and scattering of the cloak for each individual order number m. By substituting the simplified material parameters into Equation (57), we obtain 2d
(r − a)
29
dr 2
" # 2 b (r − a)2 d9 2 2 2 + (r − a) k0 − m 9 = 0. (61) + r dr b−a
Equation (61) has two non-essential singularities at r = 0 and r = a for m 6= 0. It is worthwhile mentioning that, with the ideal parameters,
2 For example, the right-travelling incidence plane wave exp(−jk x) can be expanded in Bessel 0 P functions, or a generalized Fourier series, as m (−i)m Jm (k0 r ) exp(imθ). Therefore Am3 can be determined beforehand. See (Felbacq, Tayeb and Maystre, 1994).
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Equation (61) can be written, instead, as (r
d2 9 − a)2 2 dr
# " 2 d9 b 2 2 2 + (r − a) + (r − a) k0 − m 9 = 0. (62) dr b−a
When m = 0, Equation (61) can be further simplified to r2
d2 9 d9 +r + r 2 k02 dr dr 2
b b−a
2
9 = 0.
(63)
This is the zeroth-order Bessel differential equation. Its non-essential singularity remains at r = 0. Equation (63) suggests that an incoming zeroth-order cylindrical wave would effectively see the simplified cloak as a homogeneous isotropic medium whose effective refractive index is b n eff = b−a . Its transmission through the cloak shell is therefore determined by the etalon effect of the finite medium. When m 6= 0, the wave solution is governed by Equation (61). In the following, the scattering coefficient sm , subject to individual cylindrical wave incidences, will be derived. The scattering coefficient in each 3 /A 3 . Analytic derivation of s cylindrical order is defined as sm = Bm m m can be done if two solutions to Equation (61) (i.e. Q m and Rm ) are known in closed form. However, despite its analogy to Equation (62), Equation (61) fails the analytic Frobenius method (Arfken, 1970). Here we tackle the problem by the finite-element method (FEM). The scattering problem is computed numerically. The field outside the cloak is then used for 3 through a fitting procedure. s is known deriving coefficients Am3 and Bm m in turn. Solutions with different azimuthal orders are obtained by varying the azimuthal dependence of a circular current source outside the cloak. The scattering problem is numerically manageable since the functionals θ εz µθ and µ µr in Equation (57), unlike in the ideal cloak case, are both finite and do not possess any removable singularity for the simplified medium. The commercial software COMSOL was deployed to carry out the calculations. In our case study, we fixed a = 0.1 m and the operating frequency f = 2 GHz. The performance of the cloak was examined as b was increased from 0.2 m. Similar parameters are also found in (Cummer, Popa, Schurig, Smith and Pendry, 2006). The scattering coefficients in different azimuthal orders as a function of b are shown in Figure 12. As expected, the zeroth-order scattering coefficient is quite distinct from the others due to the different governing wave equation. In fact, an analytic solution exists when m = 0. The excellent agreement between the numerical and analytic results for the zeroth-order scattering coefficient confirms the validity and accuracy of
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FIGURE 12 Variation of the scattering coefficients, examined in each cylindrical wave order, as a function of b. For m = 1, 2, the curves are fitted using the Savitzky-Golay smoothing filter according to numerically derived data points (dots)
our approach. It has been argued that the invisibility performance of a simplified cloak improves with increase of b. Here, using the analytic technique, the zeroth-order scattering coefficient is found to converge to 0.867 with respect to b. Despite the fact that the effective index of the cloak approaches 1 as b increases, the phase variation of the zeroth-order b (b − a) = k0 b. wave within the cloak medium is increasing, as k0 b−a This explains why the scattering coefficient converges to a value other than 0. The existence of the zeroth-order scattering coefficient effectively disqualifies the cloak from being completely invisible. Compared to the zeroth-order scattering coefficient, the high-order scattering coefficients (only those for m = 1, 2 are shown in Figure 12) are seen to behave in a similar oscillatory fashion, but in general are much smaller. The scattering coefficient tends to converge to a value closer to zero when the order number m increases. We should attribute the relatively small high-order scattering coefficients to the cloak’s partial relation to the ideal cloak based on coordinate transformation. Having said that, however, our numerical result shows that the high-order scattering coefficients do not converge to zero even when the cloak wall is very thick. For comparison purposes, it should be mentioned that the scattering coefficient in any order of an annular cylinder varies between 0 and 1 as a function of either its refractive index (as the geometry is fixed) or its outer radius (as the material and inner radius are fixed). Besides the requirement of zero scattering (invisibility), a device also needs to possess a spatial region which is in complete EM isolation from the outer world in order to be an invisibility cloak. Therefore it is meaningful to know how much the field penetrates into the simplified cloak, subject to a foreign EM illumination. Again, the problem is studied
Cylindrical Invisibility Cloak
FIGURE 13
289
The zeroth-order transmission coefficient
by examining the individual cylindrical wave components separately. The transmission coefficient, defined as tm = Am1 /Am3 , is used to characterize the field transmission. When m = 0, the amount of field transmitted into the cloak interior can be analytically derived, this is shown in Figure 13 as a function of b. The transmission is seen to be oscillatory, and converging to 1 as b increases. Numerical calculation is also superimposed for validation. When m 6= 0, the FEM calculations show that the field inside the cloak is almost zero. The corresponding transmission coefficients are exclusively smaller than 0.005 and hence are not plotted in Figure 13. This indicates that the contour r = a provides an insulation between its enclosed domain and the outer domain, but only for all high-order cylindrical waves. Therefore, any objects placed inside the cloak are exposed to the zeroth-order cylindrical wave component. Conversely, the zeroth-order wave component of an EM source placed within the cloak (or a wave scattered by objects inside the cloak) will transmit out. As a result, objects enclosed by a simplified cloak may be sensed by a foreign detection unit. 5.2.2 Optimal Models of Simplified Cloaks For the simplified cloak presented above, the material parameters, as listed in Table 1, do not satisfy the perfectly-matched layer (PML) condition at the outer interface. The reflective outer interface becomes a factor causing scattering. One can see that, to keep the functionals µθ εz and µr εz invariant with respect to the ideal values, more than one solution exists. Here, we show another set of material parameters of a simplified linear transformed cloak, r = µr = (
r −a 2 b ) , r b−a
θ = µθ =
b , b−a
z = µz =
b . (64) b−a
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FIGURE 14 (a) Material parameters for an ideal linear cloak (black curves), a simplified linear cloak (green curves), and an improved linear cloak (red curves). (b) Material parameters for an ideal quadratic cloak (black curves), and an simplified quadratic cloak (blue curves). Solid line: µr ; dashed line: µθ ; dotted line: εz . All cloaks have a = 1 and b = 3 (a.u.)
For the new parameter set, at r = b, we have εz = µθ = 1/µr = b/(b − a), i.e. the PML condition. Therefore, with the impedance-matched outer surface, it is expected that a cloak with the new set of parameters would induce smaller scattering. The improved parameters are plotted in Figure 14(a) in red curves. Notice the unchanged material values at r = b as compared to those for the ideal cloak (black curves). The quadratic coordinate transformation was also proposed for the cloak design (Cai, Chettiar, Kildishev, Shalaev and Milton, 2007b). It is found that the simplified version of the quadratic cloak also matches the outer free space in impedance. The quadratic coordinate transformation takes the form r=
b−a + p(r 0 − b) r 0 + a, b
(65)
where p can be arbitrary as long as | p| < (b − a)/b2 is valid for keeping the spatial mapping monotonic. The ideal material parameters, according to such a coordinate transformation, are shown in the lefthand column of Table 2. Since p is free to change here, the material parameters can dr be varied. Interestingly, at p = a/b2 , we have dr 0 |r =b = 1. Under this condition, the material parameters are all valued at 1 at r = b. Electromagnetically, the resulting cloak does not even have an interface at r = b, i.e., the material parameters are the same as in free space. The anisotropic ratio of the cloak material then grows at radial positions
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TABLE 2 Material parameters for quadratic cloaks Ideal
0
εr = µr = rr
h
p(2r 0 − b) + b−a b
i
Simplified (Cai, Chettiar, Kildishev, Shalaev and Milton, 2007b) 0 2 εr = µr = rr
εθ = µθ = ε1r
εθ = µθ = h
0
εz = µz = 1
1 εz = µz = rr p(2r 0 −b)+ b−a b
1 p(2r 0 −b)+ b−a b
i2
closer to the cloak’s inner surface. Notice this particular choice of p value, together with the monotonicity condition, imposes a constraint on the ideal cloak’s wall thickness, which is b > 2a. The material parameters for a sample ideal cloak are plotted in Figure 14(b) (black curves). Again a divergent µθ is observed at r = a. The cloak can also be simplified by keeping the functionals µθ εz and µr εz invariant. The simplified cloak has the material parameters as given in the righthand column of Table 2. The simplified material parameters are plotted in Figure 14(b) (also with p = a/b2 ). Notice that the “interfaceless” property at r = b is inherited by the simplified cloak. In the following, a detailed analysis of the above two optimal models of simplified cloaks will be given. Similar to the previous analysis, normal incidence and TE polarization are considered. As shown above, a simplified cloak allows a certain percentage of a foreign field to penetrate into the cylindrical shell. A quick and reliable fix to this problem is to introduce a PEC lining on the inner surface of the cloak shell. It follows that the invisibility performance is only characterized by the scattering induced by the cloak. Three types of cylindrical cloaks, including the previously proposed simplified linear cloak (in Table 1), improved simplified linear cloak (Equation (64)), and the simplified quadratic cloak (Table 2) are presented here. All cloaks consist of a PEC lining. We examine their invisibility performance as a function of parameters a, b and λ. When the effect of b is analyzed, we fix a = 0.1 m and λ = 0.15 m (or f = 2 GHz). When the effect of a is analyzed, we fix b = 0.6 m and the frequency is unchanged. When the the effect of λ is analyzed, we fix a = 0.2 m and b = 0.6 m. The calculated scattering coefficients for the three types of simplified cloak are summarized in Figure 15. Panels in each row in Figure 15 are for a particular type of cloak, while panels in each column are for the effect of a particular parameter. Only the scattering coefficients for the first four
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FIGURE 15 The scattering coefficients in each cylindrical order as a function of either b, a or λ. (a1)–(a3) Previously proposed simplified linear cloak; (b1)-(b3) Improved simplified linear cloak; (c1)–(c3) Simplified quadratic cloak. In each row, the left panel shows the effect of b as a = 0.1 m and λ = 0.15 m; the middle panel shows the effect of a as b = 0.6 m and λ = 0.15 m; the right panel shows the effect of λ as a = 0.2 m and b = 0.6 m.
cylindrical orders are presented. In general, for all cloaks the scattering is smaller when the order number m is larger. From Figure 15, it is quite evident that the originally proposed simplified linear cloak [Figure 15(a1)–(a3)] induces relatively larger scattering in all cylindrical orders compared to the other two cloaks. The scattering in each order varies greatly as a, b or λ changes, suggesting the relatively strong cavity effect of the cloak. In a dramatic contrast, the scattering caused by the improved linear cloak [Figure 15(b1–b3)] has only a very slight dependence on either a, b or λ. The geometry-insensitive scattering characteristics should be attributed to the perfect impedance matching at the cloak’s outer surface. The EM system now does not have a closed region bounded by reflective interfaces, therefore a cavity can’t be formed. For the simplified quadratic cloak [Figure 15(c1–c3)], the scattering characteristics are quite similar to those of the improved linear
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cloak when its geometry fulfills the b > 2a condition. The relatively small scattering is again due to the outer surface being matched with the outer space. With a relatively thin cloak wall (b < 2a), incoming cylindrical waves, in general, experience heavy scattering. This result confirms the minimum thickness restriction, i.e. b > 2a, for the simplified quadratic cloak. One interesting observation from Figure 15 is that, for all three types of cloak, the scattering coefficient in each cylindrical order tends to converge to some specific value when b a. For example, the zerothorder scattering coefficient s0 is approaching ∼0.7, and s1 is approaching ∼0.16, etc. As the range of cloaks analyzed in this study is considered to be quite wide, the values suggest that the performances of simplified cloaks may have an upper limit. For the improved linear cloak and the simplified quadratic cloak, scattering is dominated by the zeroth-order component. For the originally proposed simplified linear cloak, relatively heavier scattering exists even when b a. In addition, its scattered field is likely to comprise a significant portion of high-order cylindrical waves. Knowing their inherent scattering properties, it is meaningful to verify the findings by some numerical experiments. Notice that scattering patterns obtained in such numerical experiments are source-dependent. Nevertheless they can qualitatively reflect the performances of the cloaks. Here a plane wave of unit amplitude is chosen as the incident wave in all calculations. The wave has λ = 0.15 m and travels from left to right. Three types of cloak are studied, including the previously proposed simplified linear cloak, improved linear cloak, and the simplified quadratic cloak, all with a PEC lining. For each type of cloak, two scales are examined. While a is kept constant at 0.1 m, b is at 0.15 m or 0.3 m. The calculated far field scattering patterns are summarized in Figure 16. For reference, the scattering pattern for a bare PEC cylinder with a 0.1 m radius is also imposed in the panels. All scattered far fields are normalized to the maximum scattered far field by the bare PEC cylinder. For the previously proposed simplified linear cloak, the scattering remains high and varies wildly in angular direction, regardless of the cloak thickness. The sharp variation of the scattered field value in the angular direction suggests that the high-order cylindrical waves experience heavy scattering. By comparison, the improved linear cloak causes a relatively low scattering at all cloak thicknesses. The reduction in scattering is around 10dB in both the backward and forward directions as compared to the bare PEC cylinder. The relatively small angular dependence of the scattering pattern indicates that the scattering by such a cloak is dominated by the zeroth-order cylindrical wave. The quadratic cloak experiences heavy scattering when b = 0.15 m, which is almost as large as that caused by the bare PEC cylinder. This is due to violation of
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FIGURE 16 The far field scattering patterns by the simplified cloaks with an identical plane wave incidence (amplitude at 1). Green curves are for the previously proposed simplified linear cloak. Red curves are for the improved simplified linear cloak. Blue curves are for the simplified quadratic cloak. Dashed black curves are for the bare PEC cylinder. (a) a = 0.1 m, b = 0.15 m; (b) a = 0.1 m, b = 0.3 m. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
the thickness constraint b > 2a. At b = 0.3 m, the scattering drops greatly, and becomes comparable to that caused by an improved linear cloak. The E z field patterns, together with the Poynting vectors for the three types of cloak, all with a = 0.1 m and b = 0.3m, are shown in Figure 17(a)–(c). In all cases a PEC lining is present. It is seen that all simplified cloaks cause perturbations to the plane wave. Nevertheless, all simplified cloaks are observed to imitate their respective ideal versions by bending the Poynting vectors around the inner shell and then returning the vectors back to their original trajectories. The perturbation caused by the previously proposed simplified linear cloak [Figure 17(a)] is larger than that caused by the improved linear cloak [Figure 17(b)]. By a careful comparison of fields in Figure 17(b) and (c), one can see that the quadratic cloak [Figure 17(c)] bends the EM field more severely at locations closer to the cloak’s inner surface, whereas for the linear cloak [Figure 17(b)] the bending happens evenly at all radial locations. In all cases, the maximum field amplitude has increased from the ideal case, which is −1 → 1. The scattered E z fields by the three cloaks are plotted in Figure 17(d)–(f). The scattered field induced by the previously proposed simplified linear cloak [Figure 17(d)] is not only very high in amplitude, but also rather inhomogeneous angularly. By comparison, the scattered fields by the improved linear cloak [Figure 17(e)] and the simplified quadratic cloak [Figure 17(f)] are dominated by the zeroth-order cylindrical wave with
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FIGURE 17 Snapshots of E z fields around (a) a previously proposed simplified linear cloak, (b) an improved linear cloak, and (c) a simplified quadratic cloak. a = 0.1m, b = 0.3m. Poynting vectors are shown in green lines. The scattered E z fields of the three cloaks in (a)–(c) are shown respectively in (d)–(f). Domain for (a)–(c): 0.75 × 0.75 m2 , and for (d)–(f): 1.2 × 1.2 m2
significantly lower amplitude. These scattered near fields comply well with the results we obtained in Figures 15 and 16. We conclude this subsection by stating that simplified cloaks with impedance-matched outer surfaces, in general, have better invisibility performance. However, it should be kept in mind that realization of such optimal models of simplified cloaks can be more difficult when compared to the one suggested in Schurig, Mock, Justice, Cummer, Pendry, Starr and Smith (2006a).
6. SPHERICAL INVISIBILITY CLOAK Spherical cloaks have previously been studied in (Chen, Wu, Zhang and Kong, 2007b; Zhang, Chen, Wu and Kong, 2008a; Luo, Chen, Zhang, Ran and Kong, 2008). A three-dimensional spherical cloak can be constructed by compressing the EM fields in a spherical region r 0 < b into a spherical shell a < r < b. Here a generalized coordinate transformation where r 0 = f (r ) with
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Invisibility Cloaking by Coordinate Transformation
f (a) = 0 and f (b) = b is considered. Therefore, a spherical cloak is indeed a special type of point-transformed cloak. The permittivity and permeability of the cloak, depending on r , are given as r = µr =
f (r )2 , r 2 f 0 (r )
θ = µθ = φ = µφ = f 0 (r ).
(66)
Compared to ideal cylindrical cloaks, ideal spherical cloaks are more practical, since their material parameters don’t have infinitely large values. First, we look into the EM property of an ideal spherical cloak. Consider i i that fields E = [Eri , E θi , E φi ] and H = [Hri , Hθi , Hφi ] are incident upon the cloak. From Equation (13), the fields in the cloaked medium are obtained directly
br (r, θ, φ) = f 0 (r )Eri ( f (r ), θ, φ), E bθ (r, θ, φ) = f (r ) E i ( f (r ), θ, φ), E θ r f (r ) bφ (r, θ, φ) = E φi ( f (r ), θ, φ), E r
br (r, θ, φ) = f 0 (r )Hri ( f (r ), θ, φ), (67) H bθ (r, θ, φ) = f (r ) H i ( f (r ), θ, φ), (68) H θ r f (r ) bφ (r, θ, φ) = Hφi ( f (r ), θ, φ). (69) H r
At the inner boundary, observing Equations (68) and (69), the tangential components of the fields are zero. Combining this with r = µr = 0, it can be seen that no field can penetrate the cloaked region. For an ideal cylindrical cloak, it is known that a noticeable scattering will be induced if a thin layer is removed from the inner boundary. So we wonder whether the same phenomenon will occur for a spherical cloak. In the following, we will investigate such a problem. Considering Equation (66), we know that the wave in the cloak shell can be decoupled into spherical TE modes (the electric field has no rˆ component) and spherical TM modes (the magnetic field has no rˆ component). We consider the case where a TM wave is incident upon the cloak with a tiny perturbation. The results for a TE wave can be obtained in a similar way. The perturbation is again introduced by dislocating the cloak’s inner interface from the ideal r = a to r = a + δ. The media inside the cloak shell is unknown, but with the assumption that the wave can be separated into TE and TM components. It is known that the wave in each region can be expressed by −1
Debye potentials πe , i.e., BT M = ∇ ×(r πe )b r , DT M = i{∇ ×[µ ·∇ ×(r πe )b r ]}. From Maxwell’s equations, the Debye potentials in the outer free space
Spherical Invisibility Cloak
297
and the cloaked shell are obtained as πe =
πe =
n +∞ X X
in a(n,m) jn (k0r )Pnm (cos θ )eimφ
n=1 m=−n sc + a(n,m) h n (k0r )Pnm (cos θ )eimφ (b < r ), +∞ X n X f (r ) f 0 (r ) 1 [a(n,m) jn (k0 f (r )) r n=1 m=−n 2 + a(n,m) h n (k0 f (r ))]Pnm (cos θ )eimφ (a + δ
(70)
< r < b).
(71)
In Equations (70) and (71), jn and h n represent the nth order spherical Bessel function of the first kind and nth order spherical Hankel function of the first kind, respectively; Pnm represent the associated Legendre in sc polynomials with order n and degree m; a(n,m) and a(n,m) are the expansion coefficients for the Debye potentials of the incident and the i scattered fields outside the cloak, respectively; and a(n,m) (i = 1, 2) are the expansion coefficients for the the Debye potentials of the field in the cloakPshell. Hθ and E θ in the region r < a + δ are denoted +∞ Pn 3 m imφ /(sin θ µ ) and E by Hθ = θ = 0 m=−n a(n,m) m Sn (r )Pn (cos θ )e n=1 P+∞ Pn 3 m 0 imφ a T (r )P (cos θ )e /(−iω ), respectively, where S n (r ) 0 n m=−n (n,m) n n=1 and Tn (r ) are determined by the media in the cloaked region. For isotropic homogenous media with relative permittivity r and permeability µr , Sn (r ) = jn (ku r )/µr and Tn (r ) = ( jn (ku r )/r + ku jn0 (ku r ))/(u µu ). If the cloaked media are inhomogeneous or anisotropic, close forms of Sn (r ) and Tn (r ) are difficult to obtain. By considering the boundary conditions at r = b and r = a + δ, we in sc = a sc obtain the scattering coefficient Cn,m (n,m) /a(n,m) in an approximate form as sc C(n,m) ≈
δ f jn (k0 δ f )Tn (a + δ) − (k0 δ f jn0 (k0 δ f ) + jn (k0 δ f ))Sn (a + δ) ,(72) (k0 δ f h 0n (k0 δ f ) + h n (k0 δ f ))Sn (a + δ) − δ f h n (k0 δ f )Tn (a + δ)
where δ f is defined as the same as for the cylindrical cloak in Section 5.1. For the special case when a PEC (PMC) layer is put at r = a + δ, Equation (72) is also valid by setting Sn (a + δ) = 1 and T (a) = 0 for sc is the PEC case (Sn (a + δ) = 0 and T (a) = 1 for the PMC case). Cn,m sc only dependent on n, so in the following discussions Cn,m is denoted by Cnsc . Following the same procedure as for the cylindrical cloak, we find that the value of Cnsc for the spherical cloak is mainly determined by jn (k0 δ f )/ h n (k0 δ f ) ∝ δ (2n+1) p , which is independent of a, b, or the media in the cloaked region. To illustrate the convergence speed of Cnsc for the spherical cloak, we plot |Cnsc | for n = 1, 2 versus δ in Figure 18(a) and (b)
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Invisibility Cloaking by Coordinate Transformation
FIGURE 18 (a) |C1sc | versus δ; (b) |C2sc | versus δ. The spherical cloak is set f (r ) = b(r − a)/(b − a), a = π/k0 , b = 2π/k0 , and the region inside the cloak shell is air
with f (r ) = b(r − a)/(b − a), a = π/k0 , b = 2π/k0 , the region inside the cloak shell again being air. It is seen from Figure 18 that the exact result agrees well with the result calculated by Equation (72). The convergence speed of Cnsc is found to be fast. Therefore the spherical cloak is not as sensitive to perturbation as the cylindrical cloak. Despite the fact that the material parameters of a spherical cloak are all finite, the cloak’s response to an active source located within the cloaked region is intriguing. This has been reported in Zhang, Chen, Wu and Kong (2008a), where the authors discussed the field solutions for a spherical invisibility cloak with an active device inside the cloaked region. It is found that no fields can escape from the cloaked region, i.e., invisibility cloaks can also cloak active objects. More interestingly, the tangential fields across the inner boundary of the cloak are discontinuous. It is obvious that this field discontinuity is not related to induced surface displacement currents (which result in the field discontinuity for the cylindrical cloak), since the components of permittivity and permeability
Other Related Works and Some Practical Issues
299
at the inner boundary of the spherical cloak are all finite, as seen from Equation (66). The true physical explanation is that the normal fields at the inner boundary of the cloak act as delta functions,which are caused by infinite polarizations of the material at the inner boundary of the spherical cloak. This explains the extraordinary electric and magnetic surface voltages at the inner boundary, as noticed by the authors in the aforementioned reference. From Faraday’s law and Ampere’s law, this extraordinary surface voltage will lead to the field discontinuity.
7. OTHER RELATED WORKS AND SOME PRACTICAL ISSUES In this section, we mention some other related works on invisibility cloaks. Some practical issues concerning invisibility cloaks, including dispersion effects, material loss, and metamaterial technology development etc, will be addressed. Apart from the previously mentioned cylindrical and spherical cloaks, invisibility cloaks with other shapes, such as elliptic or square, have also been theoretically demonstrated in published papers (Kwon and Werner, 2008; Ma, Qu, Xu, Zhang, Chen and Wang, 2008; Rahm, Schurig, Roberts, Cummer, Smith and Pendry, 2008b). Usually, cloaks are considered to operate in a homogenous and isotropic background. If the background is not homogenous and isotropic, such as cloaking on a dielectric half-space and cloaking in a photonic crystal, invisibility cloaks can also be designed based on the coordinate transformation method (Zhang, Huangfu, Luo, Chen, Kong and Wu, 2008b; Zhang, Jin and He, 2008c). Our analyses presented in the above sections focus on the EM stablestate response of invisibility cloaks at a single operating frequency. The dispersion effects of cloaking materials are not considered. In practice, all material parameters in nature are dispersive. The effects induced by dispersion can’t be neglected. Dynamical response and operating bandwidth are two issues closely related to dispersion (Liang, Yao, Sun and Jiang, 2008; Chen, Liang, Yao, Jiang, Ma, and Chan, 2007a). In Liang, Yao, Sun and Jiang (2008), the dynamical response of invisibility cloaks is investigated by the finite-difference time-domain (FDTD) method (simulation of invisibility cloaks using FDTD method is also presented in detail in Zhao, Argyropoulos and Hao (2008)). It is found that there is a strong scattering process before a dispersive cloak achieves its stable state (cloaking state). This strong scattering phenomenon is due to the existence of material dispersion. The time length of this dynamical process, called ”relaxation time”, depends on the dispersion. A weaker dispersion leads to a shorter relaxation time. In a dynamical process, the EM field is usually not of a single frequency, but consists of a broad frequency spectrum. The ideal material parameters for achieving perfect invisibility can’t be
300
Invisibility Cloaking by Coordinate Transformation
satisfied at all frequency values, due to inevitable material dispersion and as a consequence, scattering is likely to occur. The presence of material dispersion is the key factor that technologically prevents broadband invisibility operation. To realize a practical invisibility cloak, the metamaterial technology can be applied to engineer material parameters. Metamaterials are usually constructed by periodic metal resonators with a periodicity much smaller than the wavelength. Each resonator operates effectively as an electric dipole or a magnetic dipole to induce electric or magnetic responses. So far, a variety of metamaterial structures have been proposed in the microwave regime (Falcone, Lopetegi, Laso, Baena, Bonache, Beruete, Marqu´es, Mart´ın and Sorolla, 2004; Pendry, Holden, Stewart and Youngs, 1996; Pendry, Holden, Robbins and Stewart, 1999). More recently, metamaterials operating at optical frequency have also been reported. Metamaterials with magnetic response at optical frequency have been reported in Dolling, Enkrich, Wegener, Zhou, Soukoulis and Linden (2005), Dolling, Enkrich, Wegener, Soukoulis and Linden (2006), Shalaev, Cai, Chettiar, Yuan, Sarychev, Drachev and Kildishev (2005) and Shalaev (2006). Thus, invisibility cloaks operating at optical frequency are possible. However, it should be pointed out that the resonance-based metamaterials are inherently lossy, especially at the optical frequency. The presence of imaginary parts in the permittivity and permeability values is another key restriction that limits the invisibility performance of cloaking devices. Engineering metamaterial with low loss is still a challenging task and is currently under extensive research.
8. CONCLUSION A detailed summary of invisibility cloaking technology, including theory, material simplification, and practical realization has been given in this article. Transformation optics provides a rather effective approach for designing such an invisibility cloak, and even for a broader class of functioning electromagnetic media, called transformation media. With the advances in metamaterial fabrication technology, it is now possible to realize the designed invisibility cloaks. The electromagnetic properties of a general ideal invisibility cloak have been investigated. By dividing cloaks into two types: line-transformed cloaks and point-transformed cloaks, we notice that: (1) No reflection is excited at both outer and inner boundaries of ideal invisibility cloaks, and no field can penetrate into the cloaked region. That is, ideal invisibility cloaks by coordinate transformation are perfectly invisible.
References
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(2) For a line-transformed ideal cloak, material parameters at the inner boundary always have infinitely large components. There exists a field discontinuity across the inner boundary. The surface displacement currents are induced to make this discontinuity self-consistent. (3) For a point-transformed ideal cloak, material parameters at the inner boundary do not have infinitely large components, the components in the normal direction of the inner boundary are always zero, and the fields are continuous across the inner boundary. Special attention has been paid to cylindrical invisibility cloaks, since they are arguably the simplest structure in terms of realization. The main properties of cylindrical cloaks are listed as follows: (1) An ideal cylindrical cloak is very sensitive to geometrical perturbations at the inner boundary. (2) This sensitive problem can be overcome by imposing a PEC (PMC) layer at the inner boundary of the cloak for TM (TE) polarization operation. (3) A simplified cloak is inherently visible. (4) If the outer boundary of a simplified cloak is matched with free space, scattering of high order cylindrical waves can be reduced dramatically. Spherical cloaks have also been discussed. It was concluded that ideal spherical cloaks are not sensitive to perturbations at the inner boundary. Some practical issues concerning invisibility cloaks, including dispersion, loss, and practical realization, have been addressed. The performance of invisibility cloaks is limited by both material dispersion and loss. Realization of high-performance invisibility cloaks relies critically on metamaterial design and fabrication technologies.
ACKNOWLEDGEMENT This work has been supported by the Swedish Foundation for Strategic Research (SSF) through the Future Research Leader program, the SSF Strategic Research Center in Photonics, and the Swedish Research Council (VR).
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AUTHOR INDEX FOR VOLUME 52
A Abbas, G.L., 206 Abdullaev, F., 129 Abdullaev, F.Kh., 84, 129 Abele, H., 30 Ablowitz, M.J., 74, 75, 90, 91, 98, 108 Abrahams, E., 6, 8, 9, 30, 34, 49, 50, 54, 55 Aceves, A.B., 87 Adamiecki, A., 217 Aegerter, C.M., 12, 13, 15, 19, 20, 34, 37–39, 42–44, 46, 47, 50, 51, 53–56 Agrawal, G., 64, 83 Agrawal, G.P., 202, 203, 224 Aitchison, J.S., 67, 69–75, 85, 87, 89–91, 125, 126 Akhmediev, N.N., 64, 83 Akkermans, E., 6, 7, 15, 16, 19–21, 43 Albiez, M., 89 Alexander, T.J., 89, 104, 105, 117 Alfimov, G.L., 88 Altshuler, B.L., 5 ` A., 266 Alu, Anderson, P.W., 4–6, 8, 9, 11, 30, 34, 37, 49, 50, 54, 55, 128 Anker, Th., 89 Ankiewicz, A., 64, 83 Arecchi, T., 230 Arfken, G., 287 Argyropoulos, C., 299 Arlt, J., 129 Armen, M., 190 Armen, M.A., 231 Armstrong, J.A., 225 Arsenovic, D., 102 Arvizu, A., 234, 236, 238 Ashikaga, M., 153, 188, 218, 220, 238 Ashkin, A., 67 Aspect, A., 29 Assanto, A., 94 Assanto, G., 65, 68, 71, 73–75, 83, 94, 95 Atia, W.A., 204, 216
Atlas, D.A., 215 Au, J.K., 231 Aubry, S., 83, 95, 103 Austin, M.W., 89, 90 Avidan, A., 128 Azizoglu, M., 204 B Bachtold, W., 227 Baena, J.D., 300 Baizakov, B.B., 91, 96, 98, 101, 103, 110, 117, 119 Ban, M., 186, 188 Banasek, K., 206 Banaszek, K., 187 Bandres, M.A., 78, 79, 108 Baney, D.M., 152 Bang, O., 87, 92 Barashenkov, I.V., 87, 89 Bargatin, I., 205 Barnett, S.M., 230 Bartal, G., 71, 100, 104–106, 126–128, 130 Bartlet, H., 74, 128 Bartolini, P., 35, 36 Baumgartner, R., 31 Bayer, G., 28, 29 Beck, M., 230, 233 Belavkin, B.P., 184 Belic, M., 102, 120 Belic, M.R., 105 Bellini, M., 230 Bellot, P., 238 Bendjaballah, C., 187 Benedetto, S., 235 Bennett, C.H., 153, 217 Berge, L., 94, 116 Berglind, E., 153 Bergmann, G., 5 Bergou, J.A., 178 Berkovits, R., 8, 10, 16, 31, 49, 50, 52, 53, 56 Bernard, J.-C., 25, 26 305
306
Author Index for Volume 52
Bernard, J.C., 26, 27 Bernett, S.M., 159 Berry, D.W., 231 Beruete, M., 300 Bezryadina, A., 77, 96, 98, 99, 101, 104, 124, 131 Bian, S., 68 Bidel, Y., 26, 27 Bishop, A.R., 83, 87, 88, 91, 95, 98, 100, 103, 126 Bjorkholm, J.E., 67 Bloch, I., 129 Bloembergen, N., 64 Blomer, D., 67 Bolger, J., 90 Bonache, J., 300 Bondurant, R.S., 189, 204, 216 Bordonalli, A.C., 230 Boroson, D.M., 204 Boucher, A., 30 Bouteiller, J.-C., 202 Bouyer, P., 29 Boyd, A.R., 67, 85 Boyd, R.W., 64 Brandt, H.E., 187 Brassard, G., 153, 217 Brauer, A., 67, 72, 73, 87, 126 Braunstein, S.L., 206, 209 Brazhnyi, V.A., 72, 75, 101, 126 Bret, B.P.J., 40, 41, 46–49, 52, 54 Briane, M., 266 Bristiel, B., 199 Bryant, H.C., 3 Brylevski, A., 237 Brzdakiewich, K.A., 73 Brzdakiewicz, K.A., 68, 94, 95 ¨ Buhrer, W., 20, 34, 37 Bui, L., 90 Buljan, H., 71, 126, 130 Burghoff, J., 67, 72 Burrus, C., 217 Buryak, A.V., 83 Busch, K., 17, 42, 44 C Cai, T., 88 Cai, W., 274, 275, 290, 291, 300 Campbell, D.K., 87 Campillo, M., 27, 28 Caplan, D.O., 204
Carmichael, H.J., 205 Carmon, T., 76, 84, 89, 130 Carretero-Gonzales, R., 75 Carretero-Gonzalez, R., 91, 95, 117 Castin, Y., 129 Cavalcanti, S.B., 74 Chabanov, A.A., 30–34, 50 Chabran, C., 223 Chaikin, P.M., 12 Champneys, A.R., 87 Chan, C.T., 266, 299 Chan, V.W.S., 152, 206, 208, 209, 225, 230, 231 Chandrasekhar, S., 204, 216 Chavez-Cerda, S., 78, 79, 108, 115 Chefles, A., 178 Chen, B., 299 Chen, F., 68, 89 Chen, H., 266, 273, 275, 295, 298, 299 Chen, H.-Y., 266 Chen, M., 102 Chen, W.H., 127 Chen, X., 100 Chen, Z., 68, 77, 96, 98–101, 103, 104, 110, 112, 120, 123–125, 127, 131 Chettiar, U.K., 274, 275, 290, 291, 300 Cheung, S.K., 33 Chikama, T., 218 Chinn, S.R., 204 Christiansen, P.L., 103 Christodoulides, D., 90 Christodoulides, D.N., 65, 68, 72–79, 83–85, 87–90, 92, 95–100, 102, 104, 109, 115, 119, 123, 124, 127, 128, 130, 131 Cianci, E., 75 Clausen, M., 12, 13, 15 Cl´ement, D., 29 Cohen, O., 71, 73, 90, 96, 100, 104–106, 126, 130 Collett, M.J., 206, 232 Conti, C., 83, 94 Cook, R.L., 190 Cooper, J., 205, 230, 233 Coskun, T.H., 130 Coskun, T.T., 130 Costa e Silva, M.B., 238 Crasovan, L.-C., 117, 118 Cronin-Golomb, M., 87 Cuevas, J., 87, 113 Cummer, S.A., 266, 274, 276, 282–285, 287, 295, 299
Author Index for Volume 52
D Dabrowska, B.J., 89, 91 Dakna, M., 230 Dalfovo, F., 84 Danger, J.L., 238 Dannberg, P., 73 D’Ariano, G.M., 230, 237 Darmanyan, S., 88, 90 Datta, S., 17, 42 Davenport, W.B., 152 De Angelis, C., 87 De Luca, A., 94 de Tomasi, F., 25, 26 de Valcarcel, G.J., 79 de Vries, P., 17 Debarge, G., 223, 225, 236 Deconinck, B., 88 Delande, D., 26, 27, 29, 43, 129 Denz, C., 68, 71, 78, 101, 102, 120, 121, 123, 124 Descartes, R., 3 Desurvire, E., 193, 200, 203 Desyatnikov, A.S., 73, 95, 102, 104, 105, 113, 121, 123, 124 Dholakia, K., 77, 79 Di Trapani, P., 83 Diamanti, E., 237 Dignam, M.M., 75 Dischler, R., 217 Djordjevic, I.B., 234, 235 Doherty, A.C., 231 Dolinar, S., 189, 190, 225 Dolling, G., 300 Donati, S., 151 Dong, L., 88 Dorda, G., 5 Douma, H., 27 Drachev, V.P., 300 Drake, J.M., 42, 46, 48 Dreischuh, A., 90, 123, 124 Dreisow, F., 67, 131 Drenkelforth, S., 129 Driben, R., 88 Droulias, S., 98 Dubbers, D., 30 Durnin, J., 78, 108 Dutta, N.K., 224 E Eberly, J.H., 78, 108
307
Edmundson, D., 92 Efremidis, N.K., 65, 68, 72, 76–78, 84, 88, 89, 96, 97, 99, 100, 109, 115, 119, 126, 130 Efros, A.L., 129 Eggleton, B.J., 90 Egorov, A.A., 98, 113, 114, 116, 131 Eiermann, B., 89 Eilbeck, J.C., 87 Eisenberg, H.S., 67, 69–73, 85, 87, 91, 125, 126 Eisenmann, C., 24 Elflein, W., 73 Elliott, B.B., 237 Emura, K., 212 Engheta, N., 266 Enkrich, C., 300 Enoch, S., 266 Erbacher, F.A., 23, 57 Erni, D., 227 Ertmer, W., 129 Essiambre, R-J., 228 Esslinger, T., 129 Etrich, C., 83 Eugenieva, E., 104 Eugenieva, E.D., 77, 96, 99, 101, 115, 123 F Fabrega, J.M., 237 Falcone, F., 300 Fallani, L., 29 Faraday, M., 21 Farhat, M., 266 Fechtel, S.A., 227 Fedele, F., 127 Fejer, M.M., 237 Felbacq, D., 286 Fernandez de Cordoba, P., 104 Ferrando, A., 104, 105, 114 Fiebig, S., 12, 13, 15, 20, 34, 37 Fischer, R., 90, 101 Fisher, R., 68, 71, 78, 113, 123, 124 Fishman, S., 128 Fitrakis, E.P., 85, 91 Flach, S., 83, 95 Fleischer, J.W., 65, 68, 73, 76–78, 84, 88–90, 96, 97, 99, 100, 104–106, 109, 119, 130 Fleischer, W., 71, 126, 127 Fleisher, J.W., 130 Fleury, P.A., 46 Fludger, C.R.S., 202
308
Author Index for Volume 52
Foglietti, V., 75 Fort, C., 29 Foug`eres, A., 233 Fraden, S., 37 Frantzeskakis, D.J., 75, 85, 91, 95, 99, 100, 103, 110, 117 Fraser, A.B., 3 Fratallocchi, A., 73 Fratalocchi, A., 68, 71, 73–75, 83, 94, 95 Freedman, B., 73, 100, 127 Frejlich, J., 68 Frigyik, B.A., 88 Frumker, E., 98 Fuchs, C. A., 178 G Gagliardi, R.M., 152 Gagnon, J., 100 Gaididei, Y.B., 103 Gallion, P., 152, 174, 199, 202, 203, 216, 223, 225, 236, 238 Gammal, A., 84 Gangardt, D.M., 29 Gao, Y.M., 123 Garanovich, I.L., 92, 93 Garcia, N., 31, 44 Garc´ıa-March, M.A., 105 Garcia-March, M.-A., 104, 114 Gardiner, C.W., 206, 232 Garmire, E., 71 Garnett, J.C.M., 17 Garnier, J., 129 Garvin, H.L., 71 Gautier, J.D., 186 Gavish, U., 129 Gehrels, T., 3 Geltenbort, P., 30 Genack, A.Z., 7, 30–34, 42, 44, 46, 48, 50 Genov, D.A., 266 Georghiades, C., 227, 234 Geremia, J.M., 189, 190 Gillner, L., 153 Giorgini, S., 84 Gisin, B.V., 88 Gisin, N., 186, 218, 230 Giuliani, G., 151 Glatt, W., 227 Glauber, R.J., 162, 163, 165 Gnauck, A., 217 Gnauck, A.H., 204, 216, 218, 229
Goda, K., 230 Goedgebuer, J.P., 218 Golubentsev, A.A., 21, 57 ´ Gomez Rivas, J., 35 Gordon, J.P., 153 Grabowski, M., 206 Grangier, P., 153, 206, 238 Gray, M.B., 230 Gredeskul, S.A., 128 Greenleaf, A., 266, 273, 275, 281 Greenstein, J.I., 237 Greiner, M., 129 Grˆet, A., 27 Grimmett, G.R., 129 Grischkowsky, D., 67 Gross, P., 12, 13, 15, 19, 46, 47, 50, 51, 56 Grosse, N., 230 Grosshans, F., 153, 206, 238 Gubeskys, A., 102, 119 Guenneau, S., 264–266 Guilley, S., 238 Guo, R., 123, 127 Gutierrez-Vega, J.C., 78, 79, 108, 115 H Ha, S., 90 Habbab, L.M.I., 237 Hache, A., 131 Hadzievski, A.L., 71 Hadzievski, L., 85–87 Haider, Ch., 227 Hall, M.J.W., 230, 231, 236 Handerek, V., 202 Hanna, B., 89, 91 H¨ansch, T., 129 Hansen, P.B., 167 Hao, Y., 299 Hapke, B.W., 4 H¨artl, W., 33 Hasegawa, A., 64 Haus, H.A., 151, 153–155, 159, 167, 190, 202, 207, 236 Havey, M.D., 25 He, S., 299 He, Y.J., 88, 113, 127 Headley, C., 202 Heffner, H., 193 Helstrom, C.W., 163, 165–167, 182–185 Hennig, D., 95 Henry, C.H., 152, 223, 224
Author Index for Volume 52
Henry, P.S., 204 Herbolzheimer, E., 12 Herrero, R., 79 Herring, G., 91 Herzog, F., 227 Herzog, T., 218 Herzog, U., 178 Hickmann, J.M., 74 Hikami, S., 21 Hillery, M., 178 Hirano, T., 153, 155, 188, 218, 220, 238 Hirota, H., 186 Hirota, O., 184, 186, 188, 204 Hizanidis, K., 92, 98, 125, 126 Hjelme, D.R., 237 Ho, K.P., 226, 227 Holden, A., 300 Holden, A.J., 300 Holevo, A.S., 189 Hopkins, V.A., 128 Hoq, Q.E., 110 Hu, B., 32, 44 Huangfu, J., 266, 299 Hudock, J., 72, 90, 96, 100, 102, 104, 131 Hugbart, M., 29 Hughes, R.J., 237 Humblet, P.A., 204 Hunsperger, R.G., 71 Huttner, B., 159, 186, 218 I Idler, W., 217 Igarashi, J.-I., 29 Ilan, B., 108 Imbrock, J., 123 Imhof, A., 40, 41, 46–49, 52, 54 Infeld, E., 64, 119 Inguscio, M., 29 Inoue, K., 190–192 Ioffe, A.F., 4, 13, 30 Ishimaru, A., 7, 12, 15 Iturbe-Castillo, M.D., 78, 108, 115 Iwanow, R., 72, 131 Iyer, R., 75 J Jablan, M., 130 Jacobsen, G., 204 Jagers, J., 121
309
Jander, P., 102 Jarmie, N., 3 Jiang, L.A., 167 Jiang, S., 199, 202 Jiang, X., 299 Jin, Y., 299 Johansson, M., 98, 103, 104 John, S., 5, 6, 8, 9, 11, 22, 23, 26, 34, 37, 55, 56 Johnson, P.M., 40, 41, 46–49, 52, 54 Jonckheere, T., 26 Joseph, R.I., 85 Jovanovic, R.D., 102 Jovic, D., 102 Jovic, D.M., 102, 105 Julien, K., 98 Justice, B.J., 274, 276, 282–285, 295 K Kahn, J.M., 226, 227, 237 Kaiser, F., 100, 105, 130 Kaiser, R., 25–27, 43 Kaliteevski, M.A., 128 Kalogerakis, G., 205, 215, 228 Kamchatnov, A.M., 84 Karp, S., 152 Karpierz, M.A., 68, 73, 94, 95 Kartashov, Y.V., 73, 75, 84, 87–92, 94, 98, 105, 107, 109, 111–118, 121, 122, 129, 131 Kato, K., 186 Kaveh, M., 8, 10, 16, 31, 49, 50, 52, 53, 56 Kazovsky, L.G., 205, 215, 218, 227, 228, 233–236 Keat, J., 128 Kennedy, R.S., 184, 187, 225 Kersten, R.Th., 212 Kevrekidis, I.G., 95 Kevrekidis, P.G., 75, 83, 85, 87, 88, 91, 95, 98–103, 110, 112, 113, 117, 128 Khaneja, N., 232 Kikuchi, K., 205, 212, 218, 228 Kildishev, A.V., 265, 274, 275, 290, 291, 300 Kingston, R.H., 151, 201 Kip, D., 68, 70, 71, 85, 87, 89, 91, 102, 131 Kirkpatrick, T.R., 28, 29 Kivshar, Y., 84, 86, 88 Kivshar, Y.S., 64, 67, 68, 70, 71, 73, 74, 77, 78, 83–85, 87–93, 95, 97, 100–102, 104, 105, 113, 117, 121, 123–126, 128, 130, 131 Kiyonaga, T., 218
310
Author Index for Volume 52
Klappauf, B., 26, 27 Klaus, W., 227 Klekamp, A., 217 Klinger, J., 120 Klitzing, K.v., 5 Klosner, J.M., 32, 44 ¨ L., 159 Knoll, Koashi, M., 219 Kobelke, J., 74, 128 Kobyakov, A., 88, 90 Kogan, E., 31 Koke, S., 102 Kolokolov, A.A., 84 Kominis, Y., 92, 125 Konczykowska, A., 217 Kong, F., 266 Kong, J.A., 266, 273, 275, 295, 298, 299 Konishi, T., 153, 188, 218, 220, 238 Konotop, V.V., 72, 75, 88, 101, 126 Kovalev, A.S., 128 Kozlov, V.V., 159 Kozuka, H., 204 Kramer, B., 55 Krolikowski, W., 68, 70, 71, 73, 77, 78, 84, 86–93, 97, 101, 102, 113, 123–125, 131 Kruse, J., 129 Kudielka, K., 227 Kudielka, K.H., 227 Kuga, Y., 12 Kuhn, R.C., 29, 129 Kulatunga, P., 25 Kupriyanov, D.V., 25 Kurokawa, K., 186 Kurylev, Y., 266, 273, 275, 281 Kutz, J.N., 88 Kuwahara, H., 218 Kuzmiak, V., 72, 101 Kwon, D.-H., 299 L Labeyrie, G., 25–27, 43 Lagendijk, A., 6, 7, 10–19, 21, 35, 36, 38–50, 52, 54 Lahini, Y., 98, 128 Lai, Y., 159 Lam, P.K., 230 Lambrianides, P., 55 Langcrock, C., 237 Laporta, P., 75 Larose, E., 27, 28
Laso, M.A.G., 300 Lassas, M., 266, 273, 275, 281 Lau, C.W., 188, 190, 236 Laughlin, R.B., 5 Law, K., 100 Lax, M., 184, 223, 224 Lazaro, J., 217 Lebb, W.R., 227 Leblond, H., 119 Leconte, R., 236 Lederer, F., 65, 67, 72–75, 83, 87, 88, 90, 91, 95, 117, 118, 126, 128 Lee, K.K., 68 Lee, P.A., 5 Lee, W.K., 68 Leeb, W.R., 151 Lehner, R., 23, 24 Lenke, R., 3, 4, 23, 24, 36, 46, 52, 57 Leonhardt, U., 153, 262, 265, 266 Leuthold, J., 204, 216, 217 Lewenstein, M., 129 Li, Y., 127 Li, Y.-P., 88 Liang, Z., 299 Licciardello, D., 6, 8, 9, 30, 34, 49, 50, 54, 55 Lifshitz, R., 127 Lin, Q., 202 Linden, S., 300 Liu, S., 127 Liu, S.M., 123 Liu, X., 218, 228 Liu, Z., 127 Liu, Z.H., 123 Livas, J.C., 204 Lobino, M., 75 Lodahl, P., 44, 45, 49 Longhi, S., 75, 91 Lopetegi, T., 300 Lopez-Aguayo, S., 113 Lopez-Mariscal, C., 78, 108 Lou, C., 100 Loudon, R., 158, 206, 232 Louis, P.J.Y., 74, 84, 85, 88 Lu, Y., 127 Luo, X.-D., 266 Luo, Y., 273, 275, 295, 299 Luther-Davies, B., 83 Lye, J.E., 29
Author Index for Volume 52
M Ma, H., 299 Ma, H.-R., 266 Mabuchi, H., 190, 231, 232 Macchiavello, C., 230 Machado Mata, J.A., 152, 225 Machida, S., 152, 209, 226 Mack, U., 3, 4 MacKinnon, A., 55 MacKintosh, F.C., 22, 23, 26 Makarov, V., 237 Makasyuk, I., 98, 99, 101, 104, 127 Maker, P.D., 68 Makris, K.G., 131 Malomed, B., 83 Malomed, B.A., 83, 86, 88, 89, 91, 96, 98–104, 110, 113, 116–119 Maluckov, A., 71, 85–87 Mandel, L., 233 Mandel, O., 129 Mandelik, D., 70, 74, 89, 91, 126 Manela, O., 68, 71, 79, 89, 100, 104–106, 126, 130, 131 Marangoni, M., 75 Maret, G., 3, 4, 6, 7, 11–13, 15, 16, 19, 20, 23, 24, 33, 34, 36–39, 42–44, 46, 47, 50–57 Margalit, M., 167 Margerin, L., 27, 28 Marin, F., 230 Marqu´es, R., 300 Mart´ın, F., 300 Martin, H., 77, 96, 99, 101, 104, 120, 123–125 Martin, P.J., 190 Marzlin, K.-P., 89 Masoller, C., 79 Matijevi´c, E., 33 Matsumoto, M., 64 Matuszewski, M., 89, 119 Mavalvala, N., 230 May-Arrioja, D.A., 72 Maynard, J.D., 128 Maynard, R., 6, 7, 15, 16, 19, 43 Maystre, D., 286 Mayteevarunyoo, T., 89, 98 Mazilu, D., 117, 118 Mazurenko, Y., 218 McCall, S.L., 46 McCarthy, K., 120 McClelland, D.E., 230
311
McGloin, D., 77, 79 McKenzie, K., 230 Mears, R.J., 202 Meegan, G.D., 128 Megens, M., 16, 39–42 Meier, J., 87, 90, 98 Melville, H., 77 Melvin, T.R.O., 87 Mendieta, F.J., 234, 236, 238 Merhasin, I.M., 88, 102, 119 Mermelstein, M.D., 202 M´erolla, J.M., 218 Merzlikin, A.M., 128 Meyr, H., 227 Mezentsev, V.K., 98 Miceli, J.J., 78, 108 Michaelis, D., 67, 73, 126 Michinel, H., 101, 105 Mie, G., 3, 30, 43 Mihalache, D., 83, 116–119, 131 Mikhailov, E., 230 Miller, D.A.B., 266 Miller, P.D., 87 Milton, G.W., 266, 275, 290, 291 Min, Y., 72, 131 Mingaleev, S.F., 126 Miniatura, C., 26, 27, 29, 43, 129 Miniatura, Ch., 25, 26 Mitchell, A., 89, 90, 131 Mitchell, M., 130 Mitra, P.P., 226 Miyata, H., 218 Mock, J.J., 274, 276, 282–285, 295 Modotto, D., 126 Modugno, M., 29 Moeneclaey, M., 227 Molina, M.I., 87, 131 Moloney, J.V., 64 Momose, R., 186 Monsoriu, J.A., 104 Montambaux, G., 7, 20, 21 Morales-Molina, L., 126 Morandotti, R., 67, 69–74, 85, 87, 89–91, 98, 125, 126, 128, 131 Morgan, J.L., 237 Morsch, O., 65, 84, 95, 116 Motzek, K., 90, 100, 130 Movchan, A.B., 266 Mullen, J.A., 190 Muller, A., 186, 218 ¨ Muller, C.A., 25, 26, 29, 43
312
Author Index for Volume 52
Muller, C.A., 129 Muller, R.E., 68 Muschall, R., 87 Musher, S.L., 98 Musslimani, Z.H., 74, 75, 90, 91, 96, 98, 103, 130 Myers, J.M., 187
Osrovskaya, E.A., 89, 91 Oster, M., 104 Ostrovskaya, E.A., 74, 84–86, 88–90, 100, 104, 105, 117, 123 Oxtoby, O.F., 87
N
Pacciani, P., 88 Papoulis, A., 166, 169 Paris, M.G.A, 237 Paul, A., 27 Peccianti, M., 68, 94 Pegg, D.T., 230 Peleg, O., 100 Pelinovsky, D.E., 83, 89, 90 Pendry, J., 266, 274, 282, 284, 287, 300 Pendry, J.B., 261–266, 274, 276, 282–285, 295, 299, 300 Pepper, M., 5 Perez-Garcia, V.M., 75, 126 Perina, J., 233 Personick, S.D., 151 Pertsch, T., 67, 72–75, 87, 91, 126, 128, 131 Peschel, T., 83, 87 Peschel, U., 67, 72–75, 83, 87, 91, 125, 126, 128 Peterson, C.G., 237 Peterson, R., 214 Petrovic, M., 102, 120 Petrovic, M.S., 102, 105 Petter, J., 120 Pezer, H., 130 Pezer, R., 130 Philbin, T.G., 262, 265, 266 Piestun, R., 108 Pikalo, O., 237 Pillai, U., 166, 169 Pincemin, E., 199 Pine, D.J., 12, 17 Pitaevskii, L., 84 Pitaevskii, L.P., 84 Popa, B.-I., 266, 274, 282, 284, 287 Porte, H., 218 Porter, M.A., 75 Post, E.J., 262 Pouget, J., 98 Pozzi, F., 128 Prat, J., 237 Proakis, J.G., 211, 215 ¨ Pronneke, L., 89
Naito, T., 218 Nakajima, H., 223 Namiki, R., 153, 155, 188, 218, 220, 238 Narimanov, E.E., 265 Neff, C.W., 274, 278, 279 Nelson, R.M., 4 Neshev, D., 84, 86, 88, 89, 92, 97, 104, 124, 125 Neshev, D.N., 68, 70, 71, 73, 77, 78, 89–93, 101, 102, 104, 113, 121, 123, 124, 131 New, G.H.C., 78, 108, 115 Newell, A.C., 64 Nha, H, 205 Nicolet, A., 264, 265 Nicorovici, N.-A.P., 266 Niederdr¨ank, T., 28, 29 Nieuwenhuizen, Th.M., 31, 44 Nikolaev, V.V., 128 Nikolov, N.I., 92 Nilsson, B.O., 153, 174, 223 Nilsson, T., 153 Noe, R., 205, 228 Noh, J.W., 233 Nolte, S., 67, 72, 74, 128, 131 Noordam, L.D., 40 Norimatsu, S., 235 Norris, A.N., 266 Nowinowski-Kruszelnick, E., 68 O Oberthaler, M., 65, 84, 95, 116 Oberthaler, M.K., 89 Oetking, P., 3 Okoshi, T., 212, 218 Oliver, B.M., 153, 207, 235 Olsson, N.A., 200 Onaka, H., 218 Onoda, Y., 218 Opatrny, T., 230 Osaki, M., 186, 188, 204
P
Author Index for Volume 52
Prvanovic, S., 102, 105 Prvanovich, S., 102 Psaltis, D., 265 Pulford, D., 230
313
Russell, P.St.J., 69, 87 ¨ Ruter, C.E., 68, 70, 71, 85, 89, 91, 102, 131 Ryzhenkova, I.V., 98 S
Q Qi, X., 127 Qiu, M., 265, 267, 274, 275, 278, 279, 281 Qu, S., 299 R Rahm, M., 266, 299 Ramakrishnan, T.V., 5, 6, 8, 9, 30, 34, 49, 50, 54, 55 Ramazza, P.L., 230 Ramirez, G.A., 78, 108, 115 Ramponi, R., 75 Ran, L., 273, 275, 295 Rasmussen, J.J., 92 Rasmussen, K.O., 83, 88, 95, 98, 103 Raybon, G., 204 Raymer, M.G., 205, 230, 233 Regel, A.R., 4, 13, 30 Reinke, D., 24 Reiter, G.F., 41 Rella, C.W., 40 Remoissenet, M., 98 Retter, J.A., 29 Rhodes, W.T., 218 Ribordy, G., 230 Rice, S.O., 152, 171 Richter, T., 100, 105 Rieth, T., 55 Righini, R., 35, 36 Rikken, G.L.J.A., 22 Ringhofer, K.H., 68 Ritzwoller, M.H., 27 Rivas, J.G., 40, 41, 46–49, 52, 54 Robbins, D., 300 Roberts, D.A., 266, 299 Rodas-Verde, M.I., 101, 105 Rodriguez-Dagnino, R.M., 78, 108, 115 Root, W.L., 152 Rosberg, C.R., 77, 89, 91–93, 131 Ross, M., 193 Rowe, H.E., 152, 224 Rowlands, R., 64 Ruan, Z., 274, 275, 278, 279, 281 Runde, D., 68, 89
Sacchi, M.F, 230 Sacha, K., 129 Sagemerten, N., 121, 123, 124 Sakaguchi, H., 89, 104, 113 Salamo, G., 87, 131 Salerno, M., 91, 96, 98, 101, 103, 110, 117, 119 Saltz, J., 215 Sanchez, A., 128 Sanchez-Palencia, L., 29 Sanders, B.C., 206 Sarychev, A.K., 300 Sasaki, M., 186, 189, 205 Savage, C.M., 84, 88 Scales, J., 27 Scharf, R., 87 Scheel, S., 159 Scheffold, F., 33, 34, 36 Schiek, R., 131 Schlafer, J., 237 Schmidt, E., 90 Schmidt, U., 30 Schmidt-Hattenberger, C., 87 Schonbrun, E., 108 Schreiber, M., 55 Schreiber, T., 67 Schreiblehner, M.A., 227 Schroder, J., 120 Schulte, T., 129 Schultz, S., 261 Schumaker, B.L., 152 Schurig, D., 265, 266, 274, 276, 282–285, 287, 295, 299 Schuster, H.G., 10, 55 Schuster, K., 74, 128 Schuurmans, F.J.P., 16, 39–42 Schwartz, T., 90, 128, 130 Sears, S., 68, 76, 77, 84, 88, 96, 109 Sebbah, P., 32, 44 Seeds, A.J., 230 Segev, M., 64, 65, 68, 71, 73, 76–79, 83, 84, 88–90, 96, 97, 99, 100, 104–106, 109, 119, 126–128, 130, 131 Seimetz, M., 229, 237 Seino, M., 218
314
Author Index for Volume 52
Seno, R., 237 Shalaev, V.M., 261, 265, 274, 275, 290, 291, 300 Shandarov, V., 68, 85, 89, 102, 131 Shannon, C.E., 176 Shapiro, J.H., 152, 153, 188, 207, 225, 230, 232 Shapiro, N.M., 27 Shaw, W.T., 205, 215, 228 Shawlow, A.L., 223 Shchesnovich, V.S., 74 She, W.L., 68 Shelby, R.A., 261 Sheppard, S.R., 230, 232 Shi, Z., 70 Shin, H.J., 123 Shklovskii, B.I., 129 Shlyapnikov, G.V., 29 Shore, H.B., 55 Shyroki, D.M., 265 Sibbett, W., 77 Sigwarth, O., 29, 129 Silberberg, A., 74 Silberberg, Y., 65, 67, 69–73, 83, 85, 87, 89–91, 95, 98, 125, 126, 128 Simon, J.C., 193, 200 Simoni, F., 68 Sinsky, J.H., 217 Siviloglou, G.A., 131 Skipetrov, S.E., 6, 10, 33, 34, 50 Skryabin, D.V., 83, 87, 97 Smerzi, A., 85, 126 Smirnov, E., 85, 89, 91, 102, 131 Smith, D., 266, 299 Smith, D.R., 261, 265, 266, 274, 276, 282–285, 287, 295 Smithey, D.T., 230 Smythe, W.D., 4 Snieder, R., 27 Snyder, D.L., 227, 234 Sohler, W., 72, 131 Sokolov, I.M., 25 Soljacic, M., 130 Somekh, S., 71 Song, D., 100 Song, T., 123 Sorel, M., 72, 74, 87, 126, 128, 131 Sorolla, M., 300 Soukoulis, C.M., 17, 42, 44, 300 Spalding, G.C., 77 Sparenberg, A., 22
Sprik, R., 40 Staliunas, K., 79 Stanley, C.R., 126 Stark, J.B., 226 Starr, A., 266 Starr, A.F., 274, 276, 282–285, 295 Steblina, V.V., 67 Steele, R.C., 200 Stegeman, G., 90 Stegeman, G.I., 64, 65, 72, 83, 87, 131 Stehly, L., 27 Stellmach, Ch., 30 Stepic, M., 68, 71, 85, 87, 89, 91, 102, 131 Sterke, C.M., 75 Stewart, W., 300 Stewart, W.J., 300 Stiffler, J.J., 235 Stockton, J., 190 Stockton, J.K., 231 ¨ Storzer, M., 12, 13, 15, 19, 20, 34, 37–39, 42–44, 46, 47, 50, 51, 53–56 Stoytchev, M., 30, 31, 34 Streppel, U., 67, 73, 126 Stringari, S., 84 Strinic, A., 120 Stulz, L., 216 Su, Y., 204 Sukenik, C.I., 25 Sukhorukov, A.A., 68, 70, 71, 73, 74, 77, 78, 89–93, 97, 101, 104, 113, 117, 123, 125, 126, 130 Sun, C., 266 Sun, X., 299 Suntsov, S., 131 Susanto, H., 85, 100, 101 Suyama, M., 218 Szameit, A., 67, 131 T Taglieber, M., 89 Takeoka, M., 189, 205 Takesue, H., 237 Tamga, J.M., 98 Tang, L., 100, 123 Tayeb, G., 286 Tepichin, E., 78, 108, 115 Terhalle, B., 123, 124 Tikhonenko, V., 67 Timothy, C.R., 153 Tip, A., 41–43, 46, 48
Author Index for Volume 52
Tittel, W., 218, 230 Tomio, L., 84 Torner, L., 73, 75, 83, 84, 87–92, 94, 95, 98, 102, 105, 107, 109, 111–118, 121, 122, 129, 131 Townes, C.H., 153, 207, 223 Tr¨ager, D., 68, 71, 78, 101, 102, 120, 121, 123, 124 Treutlein, P., 89 Trillo, S., 83, 87 Trippenbach, M., 89, 119 Trombettoni, A., 85, 126 Trompeter, H., 67, 73, 126 Troxel, G., 237 Trutschel, U., 87 Tsang, L., 7, 12, 15 Tsang, M., 265 Tsironis, G., 95 Tsopelas, I., 92 Tsushima, H., 184 Tucker, R.S., 152 ¨ Tunnermann, A., 67, 72, 74, 128, 131 Turitsyn, S.K., 98 Tweer, R., 36, 52 Tyc, T., 206 U Uhlmann, G., 266, 273, 275, 281 Umbach, A., 217 Umeton, C., 94 Usuda, T.S., 186 V Vakhitov, N.G., 84 van Albada, M.P., 6, 7, 11–19, 39, 41–43, 46, 48 van de Graaf, J., 178 van der Mark, M.B., 6, 7, 15, 16, 19, 39 Van Enk, S.J., 188 Van Loock, P., 189 van Rossum, M.C.W., 31, 44 van Tiggelen, B.A., 6, 10, 16, 17, 19, 21, 22, 27, 28, 33, 34, 38, 40–43, 46, 48, 50 Vanmaekelbergh, D., 16, 39–42 ´ A.F., 29 Varon, Vaujour, E., 26, 43 Vazquez, L., 128 Vellekoop, I.M., 44, 45, 49 Vicencio, R.A., 87, 91, 98, 126, 131
315
Vilnrotter, V.A., 188, 190, 236 Vinogradov, A.P., 128 Vladimirov, A.G., 97 Vlasov, Y.A., 128 Vogel, W, D.G., 230 Vollhardt, D. , 6, 10 Vreeker, R., 17 Vysloukh, V.A., 73, 75, 84, 87–92, 94, 105, 107, 109, 111–116, 121, 122, 129, 131 W Wabnitz, S., 87 Walker, G.R., 200 Walker, N.G., 200, 234 Walton, C., 230 Wan, J., 75 Wandernoth, B., 227 Wang, H.Z., 88, 113, 127 Wang, J., 88, 101, 127, 299 Wang, X., 100, 101, 110, 112, 131 Ward, A.J., 263 Watanabe, S., 218 Watson, G.H., 46 Wegener, M., 300 Wei, X., 218, 228 Weinert, C.M., 229, 237 Weinstein, M.I., 98 Weitz, D.A., 12, 17 Welsch, , 230 Welsch, D-G., 159 Welsch, D.G, 230 Werner, D.H., 299 Whitham, G.B., 64 Wiersma, D.S., 10, 13, 14, 16–19, 21, 29, 35, 36, 38, 40, 50 Wigner, E.P., 41 Wilkowski, D., 26, 27, 43 Will, M., 67, 72 Willis, C.R., 83, 95 Willis, J.R., 266 Willner, A., 235 Wilson, D.W., 68 Wiltshire, M.C.K., 261 Winzer, P.J., 151, 204, 218, 228, 229 Wise, F., 83, 116 Wiseman, H.M., 230–232 Wisniewski, J., 70, 71 Wodkiewicz, K., 206 Wohlgemuth, O., 217 Wolf, E., 233
316
Author Index for Volume 52
Wolf, P.E., 6, 7, 11, 12, 15, 16, 19, 43 ¨ Wolfle, P., 6, 10 Wong, W.S., 167 Wu, B.-I., 266, 273, 275, 295, 298, 299 Wyller, J., 92
Youngs, I., 300 Yuan, H.-K., 300 Yuen, H.P., 152, 153, 184, 206–209, 225, 231 Yulin, A.V., 87, 97 Z
X Xi, S., 266 Xu, C., 218, 228 Xu, J., 77, 96, 99–101 Xu, Q., 238 Xu, Z., 94, 115, 299 Y Yamamoto, Y., 151–154, 190–192, 200, 209, 226, 236, 237 Yamanaka, H., 153, 188, 218, 220, 238 Yan, M., 265, 267, 274, 275, 278, 279, 281 Yan, W., 265, 267, 275, 281 Yang, H., 87, 131 Yang, J., 70, 96, 98–101, 103, 104, 110, 127 Yang, T., 266 Yao, P., 299 Yariv, A., 71, 224 Ye, F., 88 Yee, T.K., 206 Young, J., 104
Zacares, M., 104, 114 Zakrzewski, J., 129 Zavatta, A., 230 Zbinden, H., 186, 218, 230 Zelenina, A.S., 87, 90 Zemlyanaya, E.V., 89 Zentgraf, T., 67, 72, 87 Zhang, B., 273, 275, 295, 298 Zhang, J., 295, 299 Zhang, P., 299 Zhang, S., 266 Zhang, T., 128 Zhang, X., 33, 266 Zhang, Z.Q., 31, 33, 50 Zhao, Y., 299 Zhou, J.F., 300 Zhou, L., 127 Zhu, J.X., 17 Zhu, N., 123 Ziemer, R.E., 214 Zolla, F., 264, 265 Zozulya, A.A., 67
SUBJECT INDEX FOR VOLUME 52
acoustic wave 28 additive noise 192 – – operator 202 Ampere’s law 299 amplified spontaneous emission 193 amplitude shift keying 215 Anderson localization 1–8, 17, 26, 29, 33, 37, 42, 55, 128 backscattering, coherent 1, 2, 4, 5, 10, 21 –, enhanced 3 BB84 protocol 217, 237 beam shaping 86 – splitter 179 – steering, all-optical 92 Bessel beam, nondiffracting 75, 78, 108, 110, 112, 113, 115 – lattice 108, 118 bit error ratio 184, 186, 214, 219 black body radiation 150 Bloch mode 71, 104 – momentum, angular 104 – oscillation 73 – wave 69, 71, 84, 96, 98, 101 – –, coherent 71 – –, nonlinear 75 Bohr’s principle of complementarity 151, 152 Bose–Einstein condensate 64, 65, 72, 81, 84, 116, 117 Bragg grating 89 – scattering 70, 104 Brillouin scattering 70 Cauchy–Schwartz theorem 168 channel capacity 176 cloaking 261 cloak, cylindrical 275–277, 282, 299, 301 –, line-transformed 268–270 –, point-transformed 268, 270 –, spherical 295 coherent state 162, 206, 227, 239 – – photon number probability 167 – – quadrature probability 168
– –, symmetrical 185 coordinate transformation in electromagnetism 262 cryptography 150, 217 Darboux transformation 123 Debye potential 296 defect mode 125 degenerate optical parametric amplifier 190 – parametric amplification 154 double-slit experiment 6 erbium-doped amplifier 152, 190 error function 214 estimation theory 184 Faraday effect 21 – law 299 – rotation 26 finite-difference time-domain method 299 Floquet–Bloch modes 70 ––– spectrum 88 four-wave mixing 71, 154 Glauber’s relation 169 glory 2, 4 Green–Kubo formalism 55 Hall effect for photons 22 – –, quantum 5 Helmholtz equation 78 Helstrom bound 185–187, 190, 205 heterodyne detection 152, 154, 155, 184, 204, 207, 231, 241 Hikami-box 21 hologram, dynamically controllable 77 homodyne detection 152, 154, 184, 204, 211, 217, 220, 241 – –, balanced 206, 209, 218, 238, 241, 244 – –, quantum theory of 207 – –, single quadrature 204 – –, unbalanced 225 input–output relation 180, 191, 202, 245 317
318
Subject Index for Volume 52
––– –, quantum 180 invisibility cloak 262, 265, 266, 295 Ioffe–Regel criterion 13, 25, 29, 37 Jacobian transformation 264 Josephson-junction array 129 Kennedy receiver 187 – –, super homodyne 219 Kerr medium 96, 116 – – nonlinearity 85, 90, 94, 98, 103, 109, 116 Kronig–Penney model 125 Langevin noise 196 laser cooling 24 liquid crystal 67, 73, 94 lithography 72 –, ultra-violet 73 localization of light waves 27 Mach-Zehnder interferometer 216 Marcum function 214 Mathieu beam 78, 108, 115, 116 – function 79 – lattice 115 matter wave 29 Maxwell’s equations 262, 263, 265, 269, 296 metamaterials 261, 300 Mie scattering 3 modulation instability 77 moment generating function 169 multiple scattering 4, 9, 25, 41, 57 – – of seismic wave 27 non-degenerate optical parametric amplifier 190 Ohm’s law 9 optical amplification 202 – communication 151–153, 177, 190 – lattice 66, 71, 72, 75–78, 88, 91 – –, diffraction control in 69 – – sensitivity 203 – –, two-dimensional 88 – phase noise 175 parabolic beam 108 – optical lattice 115 Peierls–Nabarro potential 94, 129 phase-insensitive amplifier 193, 241 —sensitive amplifier 202 – shift keying 215, 237 photon counting detection 218 – number state 161
photorefractive crystal 76, 81, 120 – material 68, 82 – medium 121, 123 – optical lattice 104 Planck distribution 165 positive operator-valued measure 166, 184, 186, 189, 204, 206 Poynting vector 294 P-representation 165 quadrature measurement, non-commutating 177, 178 quantum communication 178 – cryptography 153, 155, 178, 217, 219, 230 – detection 184 – efficiency 204, 208, 225 – fluctuation 160 – key distribution 213, 219, 237 – Langevin noise 196 – noise 151 rainbow 3 Raman amplifier 190 – distributed amplifier 152 Rayleigh–Debye–Gans theory 24 – scattering 2, 24 Schrodinger equation, nonlinear 64, 65, 80, 103, 109, 116, 123, 126 seismic wave 27 semiconductor laser 162 soliton 84, 109, 110, 116, 117 – array 119, 120 –, axially uniform 110 –, bright-bright 90 –, dark bright 90 –, – discrete 85 –, – matter 85 – dragging, topological 105 –, gap 88, 91, 100, 111, 123, 131 – in Bessel lattice 108 –, lattice 92, 94, 96, 100, 116, 131 –, multipole 88, 98, 113 –, necklace 99 –, one-dimensional 83, 97 – percolation 129 –, photovoltaic spatial 68 – steering 91 –, Townes 96 –, two-dimensional 66, 94, 96, 98 –, vector 90, 91, 101 –, vortex 66, 102–104, 108, 113, 115, 119 spectral density of quantum noise 170 square-law detector 206 squeezed state 210
Subject Index for Volume 52
standard quantum limit 153, 186 Talbot effect 72, 75 – –, discrete 72 thermal state 164, 167 tight-binding approximation 85 time-resolved measurements 45 uncertainty principle 150, 203 – relation 160, 192 vacuum fluctuations 192 Vakhitov-Kolokolov stability criterion 84
319
Verdet constant 22 Von Neuman projective measurement 212 waveguide array 71, 74, 87, 88, 125 Weyl’s order 160 Whittaker integral 79 Wiener fluctuation model 231 – Khintchine theorem 224 Zener tunneling 67 – –, photonic 73 zero point fluctuation 170, 172, 194
CONTENTS OF PREVIOUS VOLUMESI
VOLUME 1 (1961) 1
The modern development of Hamiltonian optics, R.J. Pegis
2
Wave optics and geometrical optics in optical design, K. Miyamoto
3
The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat
4
Light and information, D. Gabor
109–153
5
On basic analogies and principal differences between optical and electronic information, H. Wolter
155–210
6
Interference color, H. Kubota
211–251
7
Dynamic characteristics of visual processes, A. Fiorentini
253–288
8
Modern alignment devices, A.C.S. Van Heel
289–329
1– 29 31– 66 67–108
VOLUME 2 (1963) 1
Ruling, testing and use of optical gratings for high-resolution spectroscopy, G.W. Stroke
2
The metrological applications of diffraction gratings, J.M. Burch
3
Diffusion through non-uniform media, R.G. Giovanelli
109–129
4
Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi
131–180
5
Fluctuations of light beams, L. Mandel
181–248
6
Methods for determining optical parameters of thin films, F. Abel`es
249–288
1– 72 73–108
VOLUME 3 (1964) 1
The elements of radiative transfer, F. Kottler
1– 28
2
Apodisation, P. Jacquinot, B. Roizen-Dossier
29–186
3
Matrix treatment of partial coherence, H. Gamo
187–332
VOLUME 4 (1965) 1
Higher order aberration theory, J. Focke
2
Applications of shearing interferometry, O. Bryngdahl
3
Surface deterioration of optical glasses, K. Kinosita
4
Optical constants of thin films, P. Rouard, P. Bousquet
145–197
5
The Miyamoto–Wolf diffraction wave, A. Rubinowicz
199–240
1– 36 37– 83 85–143
I Volumes I–XL were previously distinguished by roman rather than by arabic numerals.
321
322
Contents of Previous Volumes
6
Aberration theory of gratings and grating mountings, W.T. Welford
241–280
7
Diffraction at a black screen, Part I: Kirchhoff’s theory, F. Kottler
281–314
VOLUME 5 (1966) 1
Optical pumping, C. Cohen-Tannoudji, A. Kastler
2
Non-linear optics, P.S. Pershan
3
Two-beam interferometry, W.H. Steel
145–197
4
Instruments for the measuring of optical transfer functions, K. Murata
199–245
5
Light reflection from films of continuously varying refractive index, R. Jacobsson
247–286
6
X-ray crystal-structure determination as a branch of physical optics, H. Lipson, C.A. Taylor
287–350
7
The wave of a moving classical electron, J. Picht
351–370
1– 81 83–144
VOLUME 6 (1967) 1
Recent advances in holography, E.N. Leith, J. Upatnieks
2
Scattering of light by rough surfaces, P. Beckmann
3
Measurement of the second order degree of coherence, M. Fran¸con, S. Mallick
4
Design of zoom lenses, K. Yamaji
105–170
5
Some applications of lasers to interferometry, D.R. Herriot
171–209
6
Experimental studies of intensity fluctuations in lasers, J.A. Armstrong, A.W. Smith
211–257
7
Fourier spectroscopy, G.A. Vanasse, H. Sakai
259–330
8
Diffraction at a black screen, Part II: electromagnetic theory, F. Kottler
331–377
1– 52 53– 69 71–104
VOLUME 7 (1969) 1
Multiple-beam interference and natural modes in open resonators, G. Koppelman
2
Methods of synthesis for dielectric multilayer filters, E. Delano, R.J. Pegis
3
Echoes at optical frequencies, I.D. Abella
139–168
4
Image formation with partially coherent light, B.J. Thompson
169–230
5
Quasi-classical theory of laser radiation, A.L. Mikaelian, M.L. Ter-Mikaelian
231–297
6
The photographic image, S. Ooue
299–358
7
Interaction of very intense light with free electrons, J.H. Eberly
359–415
1– 66 67–137
VOLUME 8 (1970) 1
Synthetic-aperture optics, J.W. Goodman
2
The optical performance of the human eye, G.A. Fry
3
Light beating spectroscopy, H.Z. Cummins, H.L. Swinney
133–200
4
Multilayer antireflection coatings, A. Musset, A. Thelen
201–237
5
Statistical properties of laser light, H. Risken
239–294
6
Coherence theory of source-size compensation in interference microscopy, T. Yamamoto
295–341
7
Vision in communication, L. Levi
343–372
8
Theory of photoelectron counting, C.L. Mehta
373–440
1– 50 51–131
Contents of Previous Volumes
323
VOLUME 9 (1971) 1
Gas lasers and their application to precise length measurements, A.L. Bloom
2
Picosecond laser pulses, A.J. Demaria
3
Optical propagation through the turbulent atmosphere, J.W. Strohbehn
4
Synthesis of optical birefringent networks, E.O. Ammann
123–177
5
Mode locking in gas lasers, L. Allen, D.G.C. Jones
179–234
6
Crystal optics with spatial dispersion, V.M. Agranovich, V.L. Ginzburg
235–280
7
Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, J. Petykiewicz
281–310
8
Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden
311–407
1– 30 31– 71 73–122
VOLUME 10 (1972) 1
Bandwidth compression of optical images, T.S. Huang
2
The use of image tubes as shutters, R.W. Smith
3
Tools of theoretical quantum optics, M.O. Scully, K.G. Whitney
4
Field correctors for astronomical telescopes, C.G. Wynne
137–164
5
Optical absorption strength of defects in insulators, D.Y. Smith, D.L. Dexter
165–228
6
Elastooptic light modulation and deflection, E.K. Sittig
229–288
7
Quantum detection theory, C.W. Helstrom
289–369
1– 44 45– 87 89–135
VOLUME 11 (1973) 1
Master equation methods in quantum optics, G.S. Agarwal
2
Recent developments in far infrared spectroscopic techniques, H. Yoshinaga
3
Interaction of light and acoustic surface waves, E.G. Lean
123–166
4
Evanescent waves in optical imaging, O. Bryngdahl
167–221
5
Production of electron probes using a field emission source, A.V. Crewe
223–246
6
Hamiltonian theory of beam mode propagation, J.A. Arnaud
247–304
7
Gradient index lenses, E.W. Marchand
305–337
1– 76 77–122
VOLUME 12 (1974) 1
Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto
2
Self-induced transparency, R.E. Slusher
3
Modulation techniques in spectrometry, M. Harwit, J.A. Decker Jr
101–162
4
Interaction of light with monomolecular dye layers, K.H. Drexhage
163–232
5
The phase transition concept and coherence in atomic emission, R. Graham
233–286
6
Beam-foil spectroscopy, S. Bashkin
287–344
1– 51 53–100
VOLUME 13 (1976) 1
On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment, H.P. Baltes
2
The case for and against semiclassical radiation theory, L. Mandel
27– 68
3
Objective and subjective spherical aberration measurements of the human eye, W.M. Rosenblum, J.L. Christensen
69– 91
1– 25
324
Contents of Previous Volumes
4
Interferometric testing of smooth surfaces, G. Schulz, J. Schwider
5
Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi
169–265
6
Aplanatism and isoplanatism, W.T. Welford
267–292
93–167
VOLUME 14 (1976) 1
The statistics of speckle patterns, J.C. Dainty
2
High-resolution techniques in optical astronomy, A. Labeyrie
3
Relaxation phenomena in rare-earth luminescence, L.A. Riseberg, M.J. Weber
4
The ultrafast optical Kerr shutter, M.A. Duguay
161–193
5
Holographic diffraction gratings, G. Schmahl, D. Rudolph
195–244
6
Photoemission, P.J. Vernier
245–325
7
Optical fibre waveguides – a review, P.J.B. Clarricoats
327–402
1– 46 47– 87 89–159
VOLUME 15 (1977) 1
Theory of optical parametric amplification and oscillation, W. Brunner, H. Paul
2
Optical properties of thin metal films, P. Rouard, A. Meessen
3
Projection-type holography, T. Okoshi
139–185
4
Quasi-optical techniques of radio astronomy, T.W. Cole
187–244
5
Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, J.E. Sipe
245–350
1– 75 77–137
VOLUME 16 (1978) 1
Laser selective photophysics and photochemistry, V.S. Letokhov
2
Recent advances in phase profiles generation, J.J. Clair, C.I. Abitbol
3
Computer-generated holograms: techniques and applications, W.-H. Lee
119–232
4
Speckle interferometry, A.E. Ennos
233–288
5
Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis
289–356
6
Light emission from high-current surface-spark discharges, R.E. Beverly III
357–411
7
Semiclassical radiation theory within a quantum-mechanical framework, I.R. Senitzky
413–448
1– 69 71–117
VOLUME 17 (1980) 1
Heterodyne holographic interferometry, R. D¨andliker
2
Doppler-free multiphoton spectroscopy, E. Giacobino, B. Cagnac
3
The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wilhelmi
163–238
4
Michelson stellar interferometry, W.J. Tango, R.Q. Twiss
239–277
5
Self-focusing media with variable index of refraction, A.L. Mikaelian
279–345
1– 84 85–161
Contents of Previous Volumes
325
VOLUME 18 (1980) 1
Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan
2
Photocount statistics of radiation propagating through random and nonlinear media, J. Pe˘rina
127–203
3
Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, V.U. Zavorotnyi
204–256
4
Catastrophe optics: morphologies of caustics and their diffraction patterns, M.V. Berry, C. Upstill
257–346
1–126
VOLUME 19 (1981) 1
Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow
2
Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy
3
Light scattering spectroscopy of surface electromagnetic waves in solids, S. Ushioda
4
Principles of optical data-processing, H.J. Butterweck
211–280
5
The effects of atmospheric turbulence in optical astronomy, F. Roddier
281–376
1
Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Court`es, P. Cruvellier, M. Detaille
2
Shaping and analysis of picosecond light pulses, C. Froehly, B. Colombeau, M. Vampouille
3
Multi-photon scattering molecular spectroscopy, S. Kielich
155–261
4
Colour holography, P. Hariharan
263–324
5
Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P. Stoicheff
325–380
1– 43 45–137 139–210
VOLUME 20 (1983) 1– 61 63–153
VOLUME 21 (1984) 1
Rigorous vector theories of diffraction gratings, D. Maystre
2
Theory of optical bistability, L.A. Lugiato
3
The Radon transform and its applications, H.H. Barrett
217–286
4
Zone plate coded imaging: theory and applications, N.M. Ceglio, D.W. Sweeney
287–354
5
Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J.C. Englund, R.R. Snapp, W.C. Schieve
355–428
1– 67 69–216
VOLUME 22 (1985) 1
Optical and electronic processing of medical images, D. Malacara
2
Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema
77–144
3
Spectral and temporal fluctuations of broad-band laser radiation, A.V. Masalov
145–196
4
Holographic methods of plasma diagnostics, G.V. Ostrovskaya, Yu.I. Ostrovsky
197–270
5
Fringe formations in deformation and vibration measurements using laser light, I. Yamaguchi
271–340
6
Wave propagation in random media: a systems approach, R.L. Fante
341–398
1– 76
326
Contents of Previous Volumes
VOLUME 23 (1986) 1
Analytical techniques for multiple scattering from rough surfaces, J.A. DeSanto, G.S. Brown
1– 62
2
Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka
3
Optical films produced by ion-based techniques, P.J. Martin, R.P. Netterfield
113–182
4
Electron holography, A. Tonomura
183–220
5
Principles of optical processing with partially coherent light, F.T.S. Yu
221–275
63– 111
VOLUME 24 (1987) 1
Micro Fresnel lenses, H. Nishihara, T. Suhara
2
Dephasing-induced coherent phenomena, L. Rothberg
3
Interferometry with lasers, P. Hariharan
103–164
4
Unstable resonator modes, K.E. Oughstun
165–387
5
Information processing with spatially incoherent light, I. Glaser
389–509
1– 37 39–101
VOLUME 25 (1988) 1
Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, L.M. Narducci
1–190
2
Coherence in semiconductor lasers, M. Ohtsu, T. Tako
191–278
3
Principles and design of optical arrays, Wang Shaomin, L. Ronchi
279–348
4
Aspheric surfaces, G. Schulz
349–415
VOLUME 26 (1988) 1
Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh
2
Nonlinear optics of liquid crystals, I.C. Khoo
105–161
3
Single-longitudinal-mode semiconductor lasers, G.P. Agrawal
163–225
4
Rays and caustics as physical objects, Yu.A. Kravtsov
227–348
5
Phase-measurement interferometry techniques, K. Creath
349–393
1–104
VOLUME 27 (1989) 1
The self-imaging phenomenon and its applications, K. Patorski
2
Axicons and meso-optical imaging devices, L.M. Soroko
109–160
3
Nonimaging optics for flux concentration, I.M. Bassett, W.T. Welford, R. Winston
161–226
4
Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C. Sibilia
227–313
5
Generalized holography with application to inverse scattering and inverse source problems, R.P. Porter
315–397
1–108
Contents of Previous Volumes
327
VOLUME 28 (1990) 1
Digital holography – computer-generated holograms, O. Bryngdahl, F. Wyrowski
2
Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa
87–179
3
The quantum coherence properties of stimulated Raman scattering, M.G. Raymer, I.A. Walmsley
181–270
4
Advanced evaluation techniques in interferometry, J. Schwider
271–359
5
Quantum jumps, R.J. Cook
361–416
1– 86
VOLUME 29 (1991) 1
Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall
2
Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev
3
Generation and propagation of ultrashort optical pulses, I.P. Christov
199–291
4
Triple-correlation imaging in optical astronomy, G. Weigelt
293–319
5
Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol
321–411
1– 63 65–197
VOLUME 30 (1992) 1
Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Fabre
2
Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P. Shchepinov
3
Localization of waves in media with one-dimensional disorder, V.D. Freilikher, S.A. Gredeskul
137–203
4
Theoretical foundation of optical-soliton concept in fibers, Y. Kodama, A. Hasegawa
205–259
5
Cavity quantum optics and the quantum measurement process, P. Meystre
261–355
1– 85 87–135
VOLUME 31 (1993) 1
Atoms in strong fields: photoionization and chaos, P.W. Milonni, B. Sundaram
2
Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov
139–187
3
Optical amplifiers, N.K. Dutta, J.R. Simpson
189–226
4
Adaptive multilayer optical networks, D. Psaltis, Y. Qiao
227–261
5
Optical atoms, R.J.C. Spreeuw, J.P. Woerdman
263–319
6
Theory of Compton free electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre
321–412
1–137
VOLUME 32 (1993) 1
Guided-wave optics on silicon: physics, technology and status, B.P. Pal
2
Optical neural networks: architecture, design and models, F.T.S. Yu
3
The theory of optimal methods for localization of objects in pictures, L.P. Yaroslavsky
1– 59 61–144 145–201
328
Contents of Previous Volumes
4
Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, V.U. Zavorotny
203–266
5
Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg
267–312
6
Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus
313–361
VOLUME 33 (1994) 1
The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin
2
Quantum statistics of dissipative nonlinear oscillators, V. Peˇrinov´a, A. Lukˇs
129–202
3
Gap solitons, C.M. De Sterke, J.E. Sipe
203–260
4
Direct spatial reconstruction of optical phase from phase-modulated images, V.I. Vlad, D. Malacara
261–317
5
Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt
319–388
6
Digital halftoning: synthesis of binary images, O. Bryngdahl, T. Scheermesser, F. Wyrowski
389–463
1–127
VOLUME 34 (1995) 1
Quantum interference, superposition states of light, and nonclassical effects, V. Buˇzek, P.L. Knight
2
Wave propagation in inhomogeneous media: phase-shift approach, L.P. Presnyakov
159–181
3
The statistics of dynamic speckles, T. Okamoto, T. Asakura
183–248
4
Scattering of light from multilayer systems with rough boundaries, I. Ohl´ıdal, K. Navr´atil, M. Ohl´ıdal
249–331
5
Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss
333–402
1–158
VOLUME 35 (1996) 1
Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov
2
Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis
3
Interferometric multispectral imaging, K. Itoh
145–196
4
Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo
197–255
5
Coherent population trapping in laser spectroscopy, E. Arimondo
257–354
6
Quantum phase properties of nonlinear optical phenomena, R. Tana´s, A. Miranowicz, Ts. Gantsog
355–446
1– 60 61–144
VOLUME 36 (1996) 1
Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti
2
Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders
3
Super-resolution by data inversion, M. Bertero, C. De Mol
129–178
4
Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, L.A. Apresyan
179–244
1– 47 49–128
Contents of Previous Volumes
5
Photon wave function, I. Bialynicki-Birula
329
245–294
VOLUME 37 (1997) 1
The Wigner distribution function in optics and optoelectronics, D. Dragoman
2
Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura
3
Spectra of molecular scattering of light, I.L. Fabelinskii
4
Soliton communication systems, R.-J. Essiambre, G.P. Agrawal
185–256
5
Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller
257–343
6
Tunneling times and superluminality, R.Y. Chiao, A.M. Steinberg
345–405
1
Nonlinear optics of stratified media, S. Dutta Gupta
2
Optical aspects of interferometric gravitational-wave detectors, P. Hello
3
Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W. Nakwaski, M. Osinski ´
165–262
4
Fractional transformations in optics, A.W. Lohmann, D. Mendlovic, Z. Zalevsky
263–342
5
Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L. Horner
343–418
6
Free-space optical digital computing and interconnection, J. Jahns
419–513
1
Theory and applications of complex rays, Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan
2
Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T. Opatrn´y
3
Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford
213–290
4
The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker
291–372
5
The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs
373–469
1
Polarimetric optical fibers and sensors, T.R. Wolinski ´
2
Digital optical computing, J. Tanida, Y. Ichioka
3
Continuous measurements in quantum optics, V. Peˇrinov´a, A. Lukˇs
115–269
4
Optical systems with improved resolving power, Z. Zalevsky, D. Mendlovic, A.W. Lohmann
271–341
5
Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, F. Wyrowski
343–388
6
Spectroscopy in polychromatic fields, Z. Ficek, H.S. Freedhoff
389–441
1– 56 57– 94 95–184
VOLUME 38 (1998) 1– 84 85–164
VOLUME 39 (1999) 1– 62 63–211
VOLUME 40 (2000) 1– 75 77–114
VOLUME 41 (2000) 1
Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang
2
Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur
3
Ellipsometry of thin film systems, I. Ohl´ıdal, D. Franta
1– 95 97–179 181–282
330
Contents of Previous Volumes
4
Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu
283–358
5
Quantum statistics of nonlinear optical couplers, J. Peˇrina Jr, J. Peˇrina
359–417
6
Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. S´anchez-Soto
419–479
7
Optical solitons in media with a quadratic nonlinearity, C. Etrich, F. Lederer, B.A. Malomed, T. Peschel, U. Peschel
483–567
VOLUME 42 (2001) 1
Quanta and information, S.Ya. Kilin
2
Optical solitons in periodic media with resonant and off-resonant nonlinearities, G. Kurizki, A.E. Kozhekin, T. Opatrn´y, B.A. Malomed
3
Quantum Zeno and inverse quantum Zeno effects, P. Facchi, S. Pascazio
147–217
4
Singular optics, M.S. Soskin, M.V. Vasnetsov
219–276
5
Multi-photon quantum interferometry, G. Jaeger, A.V. Sergienko
277–324
6
Transverse mode shaping and selection in laser resonators, R. Oron, N. Davidson, A.A. Friesem, E. Hasman
325–386
1– 91 93–146
VOLUME 43 (2002) 1
Active optics in modern large optical telescopes, L. Noethe
2
Variational methods in nonlinear fiber optics and related fields, B.A. Malomed
3
Optical works of L.V. Lorenz, O. Keller
195–294
4
Canonical quantum description of light propagation in dielectric media, A. Lukˇs, V. Peˇrinov´a
295–431
5
Phase space correspondence between classical optics and quantum mechanics, D. Dragoman
433–496
6
“Slow” and “fast” light, R.W. Boyd, D.J. Gauthier
497–530
7
The fractional Fourier transform and some of its applications to optics, A. Torre
531–596
1– 69 71–193
VOLUME 44 (2002) 1
Chaotic dynamics in semiconductor lasers with optical feedback, J. Ohtsubo
2
Femtosecond pulses in optical fibers, F.G. Omenetto
3
Instantaneous optics of ultrashort broadband pulses and rapidly varying media, A.B. Shvartsburg, G. Petite
143–214
4
Optical coherence tomography, A.F. Fercher, C.K. Hitzenberger
215–301
5
Modulational instability of electromagnetic waves in inhomogeneous and in discrete media, F.Kh. Abdullaev, S.A. Darmanyan, J. Garnier
303–366
1– 84 85–141
VOLUME 45 (2003) 1
Anamorphic beam shaping for laser and diffuse light, N. Davidson, N. Bokor
2
Ultra-fast all-optical switching in optical networks, I. Glesk, B.C. Wang, L. Xu, V. Baby, P.R. Prucnal
53–117
3
Generation of dark hollow beams and their applications, J. Yin, W. Gao, Y. Zhu
119–204
4
Two-photon lasers, D.J. Gauthier
205–272
5
Nonradiating sources and other “invisible” objects, G. Gbur
273–315
1– 51
Contents of Previous Volumes
6
Lasing in disordered media, H. Cao
331
317–370
VOLUME 46 (2004) 1
Ultrafast solid-state lasers, U. Keller
2
Multiple scattering of light from randomly rough surfaces, A.V. Shchegrov, A.A. Maradudin, E.R. M´endez
117–241
3
Laser-diode interferometry, Y. Ishii
243–309
4
Optical realizations of quantum teleportation, J. Gea-Banacloche
311–353
5
Intensity-field correlations of non-classical light, H.J. Carmichael, G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice
355–404
1–115
VOLUME 47 (2005) 1
Multistep parametric processes in nonlinear optics, S.M. Saltiel, A.A. Sukhorukov, Y.S. Kivshar
2
Modes of wave-chaotic dielectric resonators, H.E. Tureci, ¨ H.G.L. Schwefel, Ph. Jacquod, A.D. Stone
3
Nonlinear and quantum optics of atomic and molecular fields, C.P. Search, P. Meystre
139–214
4
Space-variant polarization manipulation, E. Hasman, G. Biener, A. Niv, V. Kleiner
215–289
5
Optical vortices and vortex solitons, A.S. Desyatnikov, Y.S. Kivshar, L.L. Torner
291–391
6
Phase imaging and refractive index tomography for X-rays and visible rays, K. Iwata
393–432
1– 73 75–137
VOLUME 48 (2005) 1
Laboratory post-engineering of microstructured optical fibers, B.J. Eggleton, P. Domachuk, C. Grillet, E.C. M¨agi, H.C. Nguyen, P. Steinvurzel, M.J. Steel
1– 34
2
Optical solitons in random media, F. Abdullaev, J. Garnier
3
Curved diffractive optical elements: Design and applications, N. Bokor, N. Davidson
107–148
4
The geometric phase, P. Hariharan
149–201
5
Synchronization and communication with chaotic laser systems, A. Uchida, F. Rogister, J. Garc´ıa-Ojalvo, R. Roy
203–341
35–106
VOLUME 49 (2006) 1
Gaussian apodization and beam propagation, V.N. Mahajan
2
Controlling nonlinear optical processes in multi-level atomic systems, A. Joshi, M. Xiao
3
Photonic crystals, H. Benisty, C. Weisbuch
4
Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield, C. Brosseau, A. Dogariu 315–380
5
Quantum cryptography, M. Duˇsek, N. Lutkenhaus, ¨ M. Hendrych
381–454
6
Optical quantum cloning, N.J. Cerf, J. Fiur´asˇek
455–545
1– 96 97–175 177–313
332
Contents of Previous Volumes
VOLUME 50 (2007) 1
From millisecond to attosecond laser pulses, N. Bloembergen
2
Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics, M.V. Berry, M.R. Jeffrey
13– 50
3
Historical papers on the particle concept of light, O. Keller
51– 95
4
Field quantization in optics, P.W. Milonni
5
The history of near-field optics, L. Novotny
137–184
6
Light tunneling, H.M. Nussenzveig
185–250
7
The influence of Young’s interference experiment on the development of statistical optics, E. Wolf
251–273
8
Planck, photon statistics, and Bose–Einstein condensation, D.M. Greenberger, N. Erez, M.O. Scully, A.A. Svidzinsky, M.S. Zubairy
275–330
1– 12
97–135
VOLUME 51 (2008) 1
Negative refractive index metamaterials in optics, N.M. Litchinitser, I.R. Gabitov, A.I. Maimistov, V.M. Shalaev
2
Polarization techniques for surface nonlinear optics, M. Kauranen, S. Cattaneo
3
Electromagnetic fields in linear bianisotropic mediums, T.G. Mackay, A. Lakhtakia
4
Ultrafast optical pulses, C.R. Pollock
211–249
5
Quantum imaging, A. Gatti, E. Brambilla, L. Lugiato
251–348
6
Assessment of optical systems by means of point-spread functions, J.J.M. Braat, S. van Haver, A.J.E.M. Janssen, P. Dirksen
349–468
7
The discrete Wigner function, G. Bj¨ork, A.B. Klimov, L.L. S´anchez-Soto
469–516
1– 67 69–120 121–209
CUMULATIVE INDEX – VOLUMES 1–52I
Abdullaev, F. and J. Garnier: Optical solitons in random media
48, 35
Abdullaev, F.Kh., S.A. Darmanyan and J. Garnier: Modulational instability of electromagnetic waves in inhomogeneous and in discrete media
44, 303
Abel`es, F.: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical frequencies
2, 249 7, 139
Abitbol, C.I., see Clair, J.J.
16, 71
Abraham, N.B., P. Mandel and L.M. Narducci: Dynamical instabilities and pulsations in lasers
25,
1
Aegerter, C.M. and G. Maret: Coherent backscattering and Anderson localization of light
52,
1
Agarwal, G.S.: Master equation methods in quantum optics
11,
1
Agranovich, V.M. and V.L. Ginzburg: Crystal optics with spatial dispersion
9, 235
Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers
26, 163
Agrawal, G.P., see Essiambre, R.-J.
37, 185
Allen, L. and D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett and M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefringent networks
9, 179 39, 291 9, 123
Anderson, R., see Carriere, J.
41, 97
Apresyan, L.A., see Kravtsov, Yu.A.
36, 179
Arimondo, E.: Coherent population trapping in laser spectroscopy
35, 257
Armstrong, J.A. and A.W. Smith: Experimental studies of intensity fluctuations in lasers
6, 211
Arnaud, J.A.: Hamiltonian theory of beam mode propagation
11, 247
Asakura, T., see Okamoto, T.
34, 183
Asakura, T., see Peiponen, K.-E.
37, 57
Asatryan, A.A., see Kravtsov, Yu.A.
39,
Babiker, M., see Allen, L.
39, 291
Baby, V., see Glesk, I.
45, 53
Baltes, H.P.: On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment
13,
Barabanenkov, Yu.N., Yu.A. Kravtsov, V.D. Ozrin and A.I. Saichev: Enhanced backscattering in optics
29, 65
1
1
I Volumes I–XL were previously distinguished by roman rather than by arabic numerals.
333
334
Cumulative Index – Volumes 1–52
Barakat, R.: The intensity distribution and total illumination of aberration-free diffraction images
1, 67
Barrett, H.H.: The Radon transform and its applications
21, 217
Bashkin, S.: Beam-foil spectroscopy
12, 287
Bassett, I.M., W.T. Welford and R. Winston: Nonimaging optics for flux concentration
27, 161
Beckmann, P.: Scattering of light by rough surfaces
6, 53
Benisty, H. and C. Weisbuch: Photonic crystals
49, 177
Beran, M.J. and J. Oz-Vogt: Imaging through turbulence in the atmosphere
33, 319
Bernard, J., see Orrit, M.
35, 61
Berry, M.V. and C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns
18, 257
Bertero, M. and C. De Mol: Super-resolution by data inversion
36, 129
Bertolotti, M., see Chumash, V.
36,
Bertolotti, M., see Mihalache, D.
27, 227
Beverly III, R.E.: Light emission from high-current surface-spark discharges
16, 357
Bialynicki-Birula, I.: Photon wave function
36, 245
Biener, G., see Hasman, E.
47, 215
¨ Bjork, G., A.B. Klimov and L.L. S´anchez-Soto: The discrete Wigner function
51, 469
Bloembergen, N.: From millisecond to attosecond laser pulses
50,
1
9,
1
Bloom, A.L.: Gas lasers and their application to precise length measurements
1
Bokor, N. and N. Davidson: Curved diffractive optical elements: Design and applications
48, 107
Bokor, N., see Davidson, N.
45,
Bouman, M.A., W.A. Van De Grind and P. Zuidema: Quantum fluctuations in vision
22, 77
Bousquet, P., see Rouard, P.
1
4, 145
Boyd, R.W. and D.J. Gauthier: “Slow” and “fast” light
43, 497
Braat, J.J.M., S. van Haver, A.J.E.M. Janssen and P. Dirksen: Assessment of optical systems by means of point-spread functions
51, 349
Brambilla, E., see Gatti, A.
51, 251
Brosseau, C. and A. Dogariu: Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield
49, 315
Brown, G.S., see DeSanto, J.A.
23,
Brown, R., see Orrit, M.
35, 61
Brunner, W. and H. Paul: Theory of optical parametric amplification and oscillation
15,
Bryngdahl, O.: Applications of shearing interferometry
1 1
4, 37
Bryngdahl, O.: Evanescent waves in optical imaging
11, 167
Bryngdahl, O., T. Scheermesser and F. Wyrowski: Digital halftoning: synthesis of binary images
33, 389
Bryngdahl, O. and F. Wyrowski: Digital holography – computer-generated holograms
28,
Burch, J.M.: The metrological applications of diffraction gratings
1
2, 73
Butterweck, H.J.: Principles of optical data-processing
19, 211
Buˇzek, V. and P.L. Knight: Quantum interference, superposition states of light, and nonclassical effects
34,
1
Cumulative Index – Volumes 1–52
335
Cagnac, B., see Giacobino, E.
17, 85
Cao, H.: Lasing in disordered media
45, 317
Carmichael, H.J., G.T. Foster, L.A. Orozco, J.E. Reiner and P.R. Rice: Intensity-field correlations of non-classical light
46, 355
Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi and M. Mansuripur: Principles of optical disk data storage
41, 97
Casasent, D. and D. Psaltis: Deformation invariant, space-variant optical pattern recognition
16, 289
Cattaneo, S., see Kauranen, M.
51, 69
Ceglio, N.M. and D.W. Sweeney: Zone plate coded imaging: theory and applications
21, 287
Cerf, N.J. and J. Fiur´asˇ ek: Optical quantum cloning
49, 455
Chang, R.K., see Fields, M.H.
41,
Charnotskii, M.I., J. Gozani, V.I. Tatarskii and V.U. Zavorotny: Wave propagation theories in random media based on the path-integral approach
32, 203
Chen, R.T. and Z. Fu: Optical true-time delay control systems for wideband phased array antennas
41, 283
Chiao, R.Y. and A.M. Steinberg: Tunneling times and superluminality
37, 345
Choi, J., see Carriere, J.
41, 97
Christensen, J.L., see Rosenblum, W.M.
13, 69
Christov, I.P.: Generation and propagation of ultrashort optical pulses
29, 199
Chumash, V., I. Cojocaru, E. Fazio, F. Michelotti and M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films
36,
Clair, J.J. and C.I. Abitbol: Recent advances in phase profiles generation
16, 71
Clarricoats, P.J.B.: Optical fibre waveguides – a review
14, 327
Cohen-Tannoudji, C. and A. Kastler: Optical pumping
1
1
5,
1
Cojocaru, I., see Chumash, V.
36,
1
Cole, T.W.: Quasi-optical techniques of radio astronomy
15, 187
Colombeau, B., see Froehly, C.
20, 63
Cook, R.J.: Quantum jumps
28, 361
Court`es, G., P. Cruvellier and M. Detaille: Some new optical designs for ultra-violet bidimensional detection of astronomical objects
20,
Creath, K.: Phase-measurement interferometry techniques
26, 349
1
Crewe, A.V.: Production of electron probes using a field emission source
11, 223
Cruvellier, P., see Court`es, G.
20,
Cummins, H.Z. and H.L. Swinney: Light beating spectroscopy
1
8, 133
Dainty, J.C.: The statistics of speckle patterns
14,
1
D¨andliker, R.: Heterodyne holographic interferometry
17,
1
Darmanyan, S.A., see Abdullaev, F.Kh.
44, 303
Dattoli, G., L. Giannessi, A. Renieri and A. Torre: Theory of Compton free electron lasers
31, 321
Davidson, N. and N. Bokor: Anamorphic beam shaping for laser and diffuse light
45,
Davidson, N., see Bokor, N.
48, 107
Davidson, N., see Oron, R.
42, 325
1
336
Cumulative Index – Volumes 1–52
De Mol, C., see Bertero, M.
36, 129
De Sterke, C.M. and J.E. Sipe: Gap solitons
33, 203
Decker Jr, J.A., see Harwit, M.
12, 101
Delano, E. and R.J. Pegis: Methods of synthesis for dielectric multilayer filters
7, 67
Demaria, A.J.: Picosecond laser pulses
9, 31
DeSanto, J.A. and G.S. Brown: Analytical techniques for multiple scattering from rough surfaces
23,
Desyatnikov, A.S., Y.S. Kivshar and L.L. Torner: Optical vortices and vortex solitons
47, 291
Detaille, M., see Court`es, G.
20,
Dexter, D.L., see Smith, D.Y.
10, 165
Dirksen, P., see Braat, J.J.M.
51, 349
Dogariu, A., see Brosseau, C.
49, 315
Domachuk, P., see Eggleton, B.J.
48,
1
Dragoman, D.: The Wigner distribution function in optics and optoelectronics
37,
1
Dragoman, D.: Phase space correspondence between classical optics and quantum mechanics
43, 433
Drexhage, K.H.: Interaction of light with monomolecular dye layers
12, 163
Duguay, M.A.: The ultrafast optical Kerr shutter
14, 161
¨ Duˇsek, M., N. Lutkenhaus and M. Hendrych: Quantum cryptography
49, 381
Dutta, N.K. and J.R. Simpson: Optical amplifiers
31, 189
Dutta Gupta, S.: Nonlinear optics of stratified media
38,
Eberly, J.H.: Interaction of very intense light with free electrons
1 1
1
7, 359
Eggleton, B.J., P. Domachuk, C. Grillet, E.C. M¨agi, H.C. Nguyen, P. Steinvurzel and M.J. Steel: Laboratory post-engineering of microstructured optical fibers
48,
Englund, J.C., R.R. Snapp and W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity
21, 355
Ennos, A.E.: Speckle interferometry
16, 233
Erez, N., see Greenberger, D.M.
50, 275
Essiambre, R.-J. and G.P. Agrawal: Soliton communication systems
37, 185
Etrich, C., F. Lederer, B.A. Malomed, T. Peschel and U. Peschel: Optical solitons in media with a quadratic nonlinearity
41, 483
Fabelinskii, I.L.: Spectra of molecular scattering of light
37, 95
Fabre, C., see Reynaud, S.
30,
Facchi, P. and S. Pascazio: Quantum Zeno and inverse quantum Zeno effects
42, 147
Fante, R.L.: Wave propagation in random media: a systems approach
22, 341
Fazio, E., see Chumash, V.
36,
Fercher, A.F. and C.K. Hitzenberger: Optical coherence tomography
44, 215
Ficek, Z. and H.S. Freedhoff: Spectroscopy in polychromatic fields
40, 389
Fields, M.H., J. Popp and R.K. Chang: Nonlinear optics in microspheres
41,
Fiorentini, A.: Dynamic characteristics of visual processes
1
1
1
1
1, 253
Fiur´asˇ ek, J., see Cerf, N.J.
49, 455
Flytzanis, C., F. Hache, M.C. Klein, D. Ricard and Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics
29, 321
Cumulative Index – Volumes 1–52
Focke, J.: Higher order aberration theory
337
4,
1
Forbes, G.W., see Kravtsov, Yu.A.
39,
1
Foster, G.T., see Carmichael, H.J.
46, 355
Franc¸on, M. and S. Mallick: Measurement of the second order degree of coherence
6, 71
Franta, D., see Ohl´ıdal, I.
41, 181
Freedhoff, H.S., see Ficek, Z.
40, 389
Freilikher, V.D. and S.A. Gredeskul: Localization of waves in media with one-dimensional disorder
30, 137
Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions
9, 311
Friesem, A.A., see Oron, R.
42, 325
Froehly, C., B. Colombeau and M. Vampouille: Shaping and analysis of picosecond light pulses
20, 63
Fry, G.A.: The optical performance of the human eye
8, 51
Fu, Z., see Chen, R.T.
41, 283
Gabitov, I.R., see Litchinitser, N.M.
51,
Gabor, D.: Light and information Gallion, P., F. Mendieta and S. Jiang: Signal and quantum noise in optical communications and cryptography Gamo, H.: Matrix treatment of partial coherence
1
1, 109 52, 149 3, 187
Gandjbakhche, A.H. and G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media
34, 333
Gantsog, Ts., see Tana´s, R.
35, 355
Gao, W., see Yin, J.
45, 119
Garc´ıa-Ojalvo, J., see Uchida, A.
48, 203
Garnier, J., see Abdullaev, F.
48, 35
Garnier, J., see Abdullaev, F.Kh.
44, 303
Gatti, A., E. Brambilla and L. Lugiato: Quantum imaging
51, 251
Gauthier, D.J.: Two-photon lasers
45, 205
Gauthier, D.J., see Boyd, R.W.
43, 497
Gbur, G.: Nonradiating sources and other “invisible” objects
45, 273
Gea-Banacloche, J.: Optical realizations of quantum teleportation
46, 311
Ghatak, A. and K. Thyagarajan: Graded index optical waveguides: a review
18,
Ghatak, A.K., see Sodha, M.S.
13, 169
Giacobino, E. and B. Cagnac: Doppler-free multiphoton spectroscopy
17, 85
Giacobino, E., see Reynaud, S.
30,
Giannessi, L., see Dattoli, G.
31, 321
Ginzburg, V.L.: Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena
32, 267
Ginzburg, V.L., see Agranovich, V.M. Giovanelli, R.G.: Diffusion through non-uniform media
1
1
9, 235 2, 109
Glaser, I.: Information processing with spatially incoherent light
24, 389
Glesk, I., B.C. Wang, L. Xu, V. Baby and P.R. Prucnal: Ultra-fast all-optical switching in optical networks
45, 53
338
Cumulative Index – Volumes 1–52
Gniadek, K. and J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics
9, 281 8,
1
Gozani, J., see Charnotskii, M.I.
32, 203
Graham, R.: The phase transition concept and coherence in atomic emission
12, 233
Gredeskul, S.A., see Freilikher, V.D.
30, 137
Greenberger, D.M., N. Erez, M.O. Scully, A.A. Svidzinsky and M.S. Zubairy: Planck, photon statistics, and Bose–Einstein condensation
50, 275
Grillet, C., see Eggleton, B.J.
48,
Hache, F., see Flytzanis, C.
29, 321
Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes
29,
Hariharan, P.: Colour holography
20, 263
Hariharan, P.: Interferometry with lasers
24, 103
Hariharan, P.: The geometric phase
48, 149
Hariharan, P. and B.C. Sanders: Quantum phenomena in optical interferometry
36, 49
Harwit, M. and J.A. Decker Jr: Modulation techniques in spectrometry
12, 101
Hasegawa, A., see Kodama, Y.
30, 205
Hasman, E., G. Biener, A. Niv and V. Kleiner: Space-variant polarization manipulation
47, 215
Hasman, E., see Oron, R.
42, 325
Heidmann, A., see Reynaud, S.
30,
Hello, P.: Optical aspects of interferometric gravitational-wave detectors
38, 85
Helstrom, C.W.: Quantum detection theory
10, 289
Hendrych, M., see Duˇsek, M.
49, 381
Herriot, D.R.: Some applications of lasers to interferometry
1
1
1
6, 171
Hitzenberger, C.K., see Fercher, A.F.
44, 215
Horner, J.L., see Javidi, B.
38, 343
Huang, T.S.: Bandwidth compression of optical images
10,
Ichioka, Y., see Tanida, J.
40, 77
Imoto, N., see Yamamoto, Y.
28, 87
Ishii, Y.: Laser-diode interferometry
46, 243
Itoh, K.: Interferometric multispectral imaging
35, 145
Iwata, K.: Phase imaging and refractive index tomography for X-rays and visible rays
47, 393
Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, P. and B. Roizen-Dossier: Apodisation
1
5, 247 3, 29
¨ Jacquod, Ph., see Tureci, H.E.
47, 75
Jaeger, G. and A.V. Sergienko: Multi-photon quantum interferometry
42, 277
Jahns, J.: Free-space optical digital computing and interconnection
38, 419
Jamroz, W. and B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation
20, 325
Cumulative Index – Volumes 1–52
339
Janssen, A.J.E.M., see Braat, J.J.M.
51, 349
Javidi, B. and J.L. Horner: Pattern recognition with nonlinear techniques in the Fourier domain
38, 343
Jiang, S., see Gallion, P.
52, 149
Jones, D.G.C., see Allen, L.
9, 179
Joshi, A. and M. Xiao: Controlling nonlinear optical processes in multi-level atomic systems
49, 97
Kartashov, Y.V., V.A. Vysloukh and L. Torner: Soliton shape and mobility control in optical lattices
52, 63
Kastler, A., see Cohen-Tannoudji, C.
5,
1
Kauranen, M. and S. Cattaneo: Polarization techniques for surface nonlinear optics
51, 69
Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems
37, 257
Keller, O.: Optical works of L.V. Lorenz
43, 195
Keller, O.: Historical papers on the particle concept of light
50, 51
Keller, U.: Ultrafast solid-state lasers
46,
Khoo, I.C.: Nonlinear optics of liquid crystals
26, 105
Khulbe, P., see Carriere, J.
41, 97
Kielich, S.: Multi-photon scattering molecular spectroscopy
20, 155
Kilin, S.Ya.: Quanta and information
42,
Kinosita, K.: Surface deterioration of optical glasses
1
1
4, 85
Kitagawa, M., see Yamamoto, Y.
28, 87
Kivshar, Y.S., see Desyatnikov, A.S.
47, 291
Kivshar, Y.S., see Saltiel, S.M.
47,
Klein, M.C., see Flytzanis, C.
29, 321
Kleiner, V., see Hasman, E.
47, 215
¨ Klimov, A.B., see Bjork, G.
51, 469
Klyatskin, V.I.: The imbedding method in statistical boundary-value wave problems
33,
1
Knight, P.L., see Buˇzek, V.
34,
1
Kodama, Y. and A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers
30, 205
1
Koppelman, G.: Multiple-beam interference and natural modes in open resonators
7,
1
Kottler, F.: The elements of radiative transfer
3,
1
Kottler, F.: Diffraction at a black screen, Part I: Kirchhoff’s theory
4, 281
Kottler, F.: Diffraction at a black screen, Part II: electromagnetic theory
6, 331
Kozhekin, A.E., see Kurizki, G.
42, 93
Kravtsov, Yu.A.: Rays and caustics as physical objects
26, 227
Kravtsov, Yu.A. and L.A. Apresyan: Radiative transfer: new aspects of the old theory
36, 179
Kravtsov, Yu.A., G.W. Forbes and A.A. Asatryan: Theory and applications of complex rays
39,
Kravtsov, Yu.A., see Barabanenkov, Yu.N.
29, 65
1
Kubota, H.: Interference color
1, 211
Kuittinen, M., see Turunen, J.
40, 343
340
Cumulative Index – Volumes 1–52
Kurizki, G., A.E. Kozhekin, T. Opatrny´ and B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities
42, 93
Labeyrie, A.: High-resolution techniques in optical astronomy
14, 47
Lakhtakia, A., see Mackay, T.G.
51, 121
Lean, E.G.: Interaction of light and acoustic surface waves
11, 123
Lederer, F., see Etrich, C.
41, 483
Lee, W.-H.: Computer-generated holograms: techniques and applications
16, 119
Leith, E.N. and J. Upatnieks: Recent advances in holography
6,
1
Letokhov, V.S.: Laser selective photophysics and photochemistry
16,
1
Leuchs, G., see Sizmann, A.
39, 373
Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H. and C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics
8, 343 41, 97 5, 287
Litchinitser, N.M., I.R. Gabitov, A.I. Maimistov and V.M. Shalaev: Negative refractive index metamaterials in optics
51,
Lohmann, A.W., D. Mendlovic and Z. Zalevsky: Fractional transformations in optics
38, 263
Lohmann, A.W., see Zalevsky, Z.
40, 271
Lounis, B., see Orrit, M.
35, 61
Lugiato, L., see Gatti, A.
51, 251
Lugiato, L.A.: Theory of optical bistability
21, 69
Luis, A. and L.L. S´anchez-Soto: Quantum phase difference, phase measurements and Stokes operators
41, 419
Lukˇs, A. and V. Peˇrinov´a: Canonical quantum description of light propagation in dielectric media
43, 295
Lukˇs, A., see Peˇrinov´a, V.
33, 129
Lukˇs, A., see Peˇrinov´a, V.
40, 115
¨ Lutkenhaus, N., see Duˇsek, M.
49, 381
Machida, S., see Yamamoto, Y.
28, 87
Mackay, T.G. and A. Lakhtakia: Electromagnetic fields in linear bianisotropic mediums
51, 121
M¨agi, E.C., see Eggleton, B.J.
48,
1
Mahajan, V.N.: Gaussian apodization and beam propagation
49,
1
Maimistov, A.I., see Litchinitser, N.M.
51,
1
Mainfray, G. and C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas
32, 313
Malacara, D.: Optical and electronic processing of medical images
22,
Malacara, D., see Vlad, V.I.
33, 261
Mallick, S., see Franc¸on, M.
1
1
6, 71
Malomed, B.A.: Variational methods in nonlinear fiber optics and related fields
43, 71
Malomed, B.A., see Etrich, C.
41, 483
Malomed, B.A., see Kurizki, G.
42, 93
Mandel, L.: Fluctuations of light beams
2, 181
Cumulative Index – Volumes 1–52
341
Mandel, L.: The case for and against semiclassical radiation theory
13, 27
Mandel, P., see Abraham, N.B.
25,
Mansuripur, M., see Carriere, J.
41, 97
Manus, C., see Mainfray, G.
32, 313
Maradudin, A.A., see Shchegrov, A.V.
46, 117
1
Marchand, E.W.: Gradient index lenses
11, 305
Maret, G., see Aegerter, C.M.
52,
Martin, P.J. and R.P. Netterfield: Optical films produced by ion-based techniques
23, 113
Masalov, A.V.: Spectral and temporal fluctuations of broad-band laser radiation
22, 145
Maystre, D.: Rigorous vector theories of diffraction gratings
21,
Meessen, A., see Rouard, P.
15, 77
Mehta, C.L.: Theory of photoelectron counting
1
1
8, 373
M´endez, E.R., see Shchegrov, A.V.
46, 117
Mendieta, F., see Gallion, P.
52, 149
Mendlovic, D., see Lohmann, A.W.
38, 263
Mendlovic, D., see Zalevsky, Z.
40, 271
Meystre, P.: Cavity quantum optics and the quantum measurement process
30, 261
Meystre, P., see Search, C.P.
47, 139
Michelotti, F., see Chumash, V.
36,
Mihalache, D., M. Bertolotti and C. Sibilia: Nonlinear wave propagation in planar structures
27, 227
Mikaelian, A.L.: Self-focusing media with variable index of refraction
17, 279
Mikaelian, A.L. and M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation
1
7, 231
Mills, D.L. and K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids
19, 45
Milonni, P.W.: Field quantization in optics
50, 97
Milonni, P.W. and B. Sundaram: Atoms in strong fields: photoionization and chaos
31,
Miranowicz, A., see Tana´s, R.
35, 355
Miyamoto, K.: Wave optics and geometrical optics in optical design Mollow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence
1
1, 31 19,
1
Murata, K.: Instruments for the measuring of optical transfer functions
5, 199
Musset, A. and A. Thelen: Multilayer antireflection coatings
8, 201
´ Nakwaski, W. and M. Osinski: Thermal properties of vertical-cavity surface-emitting semiconductor lasers
38, 165
Narayan, R., see Carriere, J.
41, 97
Narducci, L.M., see Abraham, N.B.
25,
Navr´atil, K., see Ohl´ıdal, I.
34, 249
Netterfield, R.P., see Martin, P.J.
23, 113
Nguyen, H.C., see Eggleton, B.J.
48,
1
Nishihara, H. and T. Suhara: Micro Fresnel lenses
24,
1
Niv, A., see Hasman, E.
47, 215
Noethe, L.: Active optics in modern large optical telescopes
43,
1
1
342
Cumulative Index – Volumes 1–52
Novotny, L.: The history of near-field optics
50, 137
Nussenzveig, H.M.: Light tunneling
50, 185
Ohl´ıdal, I. and D. Franta: Ellipsometry of thin film systems
41, 181
Ohl´ıdal, I., K. Navr´atil and M. Ohl´ıdal: Scattering of light from multilayer systems with rough boundaries
34, 249
Ohl´ıdal, M., see Ohl´ıdal, I.
34, 249
Ohtsu, M. and T. Tako: Coherence in semiconductor lasers
25, 191
Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback
44,
Okamoto, T. and T. Asakura: The statistics of dynamic speckles
34, 183
Okoshi, T.: Projection-type holography
15, 139
Omenetto, F.G.: Femtosecond pulses in optical fibers
44, 85
Ooue, S.: The photographic image
1
7, 299
Opatrny, ´ T., see Kurizki, G.
42, 93
Opatrny, ´ T., see Welsch, D.-G.
39, 63
Oron, R., N. Davidson, A.A. Friesem and E. Hasman: Transverse mode shaping and selection in laser resonators
42, 325
Orozco, L.A., see Carmichael, H.J.
46, 355
Orrit, M., J. Bernard, R. Brown and B. Lounis: Optical spectroscopy of single molecules in solids
35, 61
´ Osinski, M., see Nakwaski, W.
38, 165
Ostrovskaya, G.V. and Yu.I. Ostrovsky: Holographic methods of plasma diagnostics
22, 197
Ostrovsky, Yu.I. and V.P. Shchepinov: Correlation holographic and speckle interferometry
30, 87
Ostrovsky, Yu.I., see Ostrovskaya, G.V.
22, 197
Oughstun, K.E.: Unstable resonator modes
24, 165
Oz-Vogt, J., see Beran, M.J.
33, 319
Ozrin, V.D., see Barabanenkov, Yu.N.
29, 65
Padgett, M.J., see Allen, L.
39, 291
Pal, B.P.: Guided-wave optics on silicon: physics, technology and status
32,
Paoletti, D. and G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics
35, 197
Pascazio, S., see Facchi, P.
42, 147
Patorski, K.: The self-imaging phenomenon and its applications
27,
1
Paul, H., see Brunner, W.
15,
1
Pegis, R.J.: The modern development of Hamiltonian optics
1,
1
Pegis, R.J., see Delano, E.
7, 67
1
Peiponen, K.-E., E.M. Vartiainen and T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy
37, 57
Peng, C., see Carriere, J.
41, 97
Peˇrina Jr, J. and J. Peˇrina: Quantum statistics of nonlinear optical couplers
41, 359
Pe˘rina, J.: Photocount statistics of radiation propagating through random and nonlinear media
18, 127
Peˇrina, J., see Peˇrina Jr, J.
41, 359
Cumulative Index – Volumes 1–52
343
Peˇrinov´a, V. and A. Lukˇs: Quantum statistics of dissipative nonlinear oscillators
33, 129
Peˇrinov´a, V. and A. Lukˇs: Continuous measurements in quantum optics
40, 115
Peˇrinov´a, V., see Lukˇs, A.
43, 295
Pershan, P.S.: Non-linear optics
5, 83
Peschel, T., see Etrich, C.
41, 483
Peschel, U., see Etrich, C.
41, 483
Petite, G., see Shvartsburg, A.B.
44, 143
Petykiewicz, J., see Gniadek, K.
9, 281
Picht, J.: The wave of a moving classical electron
5, 351
Pollock, C.R.: Ultrafast optical pulses
51, 211
Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view
31, 139
Popp, J., see Fields, M.H.
41,
Porter, R.P.: Generalized holography with application to inverse scattering and inverse source problems
27, 315
Presnyakov, L.P.: Wave propagation in inhomogeneous media: phase-shift approach
34, 159
Prucnal, P.R., see Glesk, I.
45, 53
Psaltis, D. and Y. Qiao: Adaptive multilayer optical networks
31, 227
Psaltis, D., see Casasent, D.
16, 289
Qiao, Y., see Psaltis, D.
31, 227
Qiu, M., see Yan, M.
52, 261
Raymer, M.G. and I.A. Walmsley: The quantum coherence properties of stimulated Raman scattering
28, 181
Reiner, J.E., see Carmichael, H.J.
46, 355
Renieri, A., see Dattoli, G.
31, 321
Reynaud, S., A. Heidmann, E. Giacobino and C. Fabre: Quantum fluctuations in optical systems
30,
Ricard, D., see Flytzanis, C.
29, 321
Rice, P.R., see Carmichael, H.J.
46, 355
Riseberg, L.A. and M.J. Weber: Relaxation phenomena in rare-earth luminescence
14, 89
Risken, H.: Statistical properties of laser light
1
1
8, 239
Roddier, F.: The effects of atmospheric turbulence in optical astronomy
19, 281
Rogister, F., see Uchida, A.
48, 203
Roizen-Dossier, B., see Jacquinot, P.
3, 29
Ronchi, L., see Wang Shaomin,
25, 279
Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems
35,
Rosenblum, W.M. and J.L. Christensen: Objective and subjective spherical aberration measurements of the human eye
13, 69
Rothberg, L.: Dephasing-induced coherent phenomena
24, 39
Rouard, P. and P. Bousquet: Optical constants of thin films
1
4, 145
Rouard, P. and A. Meessen: Optical properties of thin metal films
15, 77
Roussignol, Ph., see Flytzanis, C.
29, 321
344
Cumulative Index – Volumes 1–52
Roy, R., see Uchida, A. Rubinowicz, A.: The Miyamoto–Wolf diffraction wave
48, 203 4, 199
Rudolph, D., see Schmahl, G.
14, 195
Saichev, A.I., see Barabanenkov, Yu.N.
29, 65
Saito, S., see Yamamoto, Y.
28, 87
Sakai, H., see Vanasse, G.A.
6, 259
Saleh, B.E.A., see Teich, M.C.
26,
1
Saltiel, S.M., A.A. Sukhorukov and Y.S. Kivshar: Multistep parametric processes in nonlinear optics
47,
1
¨ S´anchez-Soto, L.L., see Bjork, G.
51, 469
S´anchez-Soto, L.L., see Luis, A.
41, 419
Sanders, B.C., see Hariharan, P.
36, 49
Scheermesser, T., see Bryngdahl, O.
33, 389
Schieve, W.C., see Englund, J.C.
21, 355
Schirripa Spagnolo, G., see Paoletti, D.
35, 197
Schmahl, G. and D. Rudolph: Holographic diffraction gratings
14, 195
Schubert, M. and B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes
17, 163
Schulz, G.: Aspheric surfaces
25, 349
Schulz, G. and J. Schwider: Interferometric testing of smooth surfaces
13, 93
¨ Schwefel, H.G.L., see Tureci, H.E.
47, 75
Schwider, J.: Advanced evaluation techniques in interferometry
28, 271
Schwider, J., see Schulz, G.
13, 93
Scully, M.O. and K.G. Whitney: Tools of theoretical quantum optics
10, 89
Scully, M.O., see Greenberger, D.M.
50, 275
Search, C.P. and P. Meystre: Nonlinear and quantum optics of atomic and molecular fields
47, 139
Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework
16, 413
Sergienko, A.V., see Jaeger, G.
42, 277
Shalaev, V.M., see Litchinitser, N.M.
51,
Sharma, S.K. and D.J. Somerford: Scattering of light in the eikonal approximation
39, 213
Shchegrov, A.V., A.A. Maradudin and E.R. M´endez: Multiple scattering of light from randomly rough surfaces
46, 117
Shchepinov, V.P., see Ostrovsky, Yu.I.
30, 87
Shvartsburg, A.B. and G. Petite: Instantaneous optics of ultrashort broadband pulses and rapidly varying media
44, 143
Sibilia, C., see Mihalache, D.
27, 227
Simpson, J.R., see Dutta, N.K.
31, 189
Sipe, J.E., see De Sterke, C.M.
33, 203
Sipe, J.E., see Van Kranendonk, J.
15, 245
Sittig, E.K.: Elastooptic light modulation and deflection
10, 229
Sizmann, A. and G. Leuchs: The optical Kerr effect and quantum optics in fibers
39, 373
Slusher, R.E.: Self-induced transparency
12, 53
1
Cumulative Index – Volumes 1–52
345
Smith, D.Y. and D.L. Dexter: Optical absorption strength of defects in insulators
10, 165
Smith, R.W.: The use of image tubes as shutters
10, 45
Snapp, R.R., see Englund, J.C.
21, 355
Sodha, M.S., A.K. Ghatak and V.K. Tripathi: Self-focusing of laser beams in plasmas and semiconductors
13, 169
Somerford, D.J., see Sharma, S.K.
39, 213
Soroko, L.M.: Axicons and meso-optical imaging devices
27, 109
Soskin, M.S. and M.V. Vasnetsov: Singular optics
42, 219
Spreeuw, R.J.C. and J.P. Woerdman: Optical atoms
31, 263
Steel, M.J., see Eggleton, B.J.
48,
Smith, A.W., see Armstrong, J.A.
Steel, W.H.: Two-beam interferometry
6, 211
1
5, 145
Steinberg, A.M., see Chiao, R.Y.
37, 345
Steinvurzel, P., see Eggleton, B.J.
48,
Stoicheff, B.P., see Jamroz, W.
20, 325
¨ Stone, A.D., see Tureci, H.E.
47, 75
Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy
1
9, 73 2,
1
Subbaswamy, K.R., see Mills, D.L.
19, 45
Suhara, T., see Nishihara, H.
24,
1
Sukhorukov, A.A., see Saltiel, S.M.
47,
1
Sundaram, B., see Milonni, P.W.
31,
1
Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams
12,
1
Svidzinsky, A.A., see Greenberger, D.M.
50, 275
Sweeney, D.W., see Ceglio, N.M.
21, 287
Swinney, H.L., see Cummins, H.Z.
8, 133
Tako, T., see Ohtsu, M.
25, 191
Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets
23, 63
Tana´s, R., A. Miranowicz and Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena
35, 355
Tango, W.J. and R.Q. Twiss: Michelson stellar interferometry
17, 239
Tanida, J. and Y. Ichioka: Digital optical computing
40, 77
Tatarskii, V.I. and V.U. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium
18, 204
Tatarskii, V.I., see Charnotskii, M.I.
32, 203
Taylor, C.A., see Lipson, H. Teich, M.C. and B.E.A. Saleh: Photon bunching and antibunching
5, 287 26,
1
Ter-Mikaelian, M.L., see Mikaelian, A.L.
7, 231
Thelen, A., see Musset, A.
8, 201
Thompson, B.J.: Image formation with partially coherent light
7, 169
Thyagarajan, K., see Ghatak, A.
18,
Tonomura, A.: Electron holography
23, 183
1
346
Cumulative Index – Volumes 1–52
Torner, L., see Kartashov, Y.V.
52, 63
Torner, L.L., see Desyatnikov, A.S.
47, 291
Torre, A.: The fractional Fourier transform and some of its applications to optics
43, 531
Torre, A., see Dattoli, G.
31, 321
Tripathi, V.K., see Sodha, M.S.
13, 169
Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering
2, 131
¨ Tureci, H.E., H.G.L. Schwefel, Ph. Jacquod and A.D. Stone: Modes of wave-chaotic dielectric resonators
47, 75
Turunen, J., M. Kuittinen and F. Wyrowski: Diffractive optics: electromagnetic approach
40, 343
Twiss, R.Q., see Tango, W.J.
17, 239
Uchida, A., F. Rogister, J. Garc´ıa-Ojalvo and R. Roy: Synchronization and communication with chaotic laser systems
48, 203
Upatnieks, J., see Leith, E.N.
6,
1
Upstill, C., see Berry, M.V.
18, 257
Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids
19, 139
Vampouille, M., see Froehly, C.
20, 63
Van De Grind, W.A., see Bouman, M.A.
22, 77
van Haver, S., see Braat, J.J.M.
51, 349
Van Heel, A.C.S.: Modern alignment devices Van Kranendonk, J. and J.E. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vanasse, G.A. and H. Sakai: Fourier spectroscopy
1, 289 15, 245 6, 259
Vartiainen, E.M., see Peiponen, K.-E.
37, 57
Vasnetsov, M.V., see Soskin, M.S.
42, 219
Vernier, P.J.: Photoemission
14, 245
Vlad, V.I. and D. Malacara: Direct spatial reconstruction of optical phase from phase-modulated images
33, 261
Vogel, W., see Welsch, D.-G.
39, 63
Vysloukh, V.A., see Kartashov, Y.V.
52, 63
Walmsley, I.A., see Raymer, M.G.
28, 181
Wang Shaomin, and L. Ronchi: Principles and design of optical arrays
25, 279
Wang, B.C., see Glesk, I.
45, 53
Weber, M.J., see Riseberg, L.A.
14, 89
Weigelt, G.: Triple-correlation imaging in optical astronomy
29, 293
Weisbuch, C., see Benisty, H.
49, 177
Weiss, G.H., see Gandjbakhche, A.H.
34, 333
Welford, W.T.: Aberration theory of gratings and grating mountings
4, 241
Welford, W.T.: Aplanatism and isoplanatism
13, 267
Welford, W.T., see Bassett, I.M.
27, 161
Cumulative Index – Volumes 1–52
Welsch, D.-G., W. Vogel and T. Opatrny: ´ Homodyne detection and quantum-state reconstruction Whitney, K.G., see Scully, M.O. Wilhelmi, B., see Schubert, M. Winston, R., see Bassett, I.M. Woerdman, J.P., see Spreeuw, R.J.C. Wolf, E.: The influence of Young’s interference experiment on the development of statistical optics ´ Wolinski, T.R.: Polarimetric optical fibers and sensors Wolter, H.: On basic analogies and principal differences between optical and electronic information Wynne, C.G.: Field correctors for astronomical telescopes Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Turunen, J.
347
39, 63 10, 89 17, 163 27, 161 31, 263 50, 251 40, 1
1, 155 10, 137 28, 1 33, 389 40, 343
Xiao, M., see Joshi, A. Xu, L., see Glesk, I.
49, 97 45, 53
Yan, M., W. Yan and M. Qiu: Invisibility cloaking by coordinate transformation Yan, W., see Yan, M. Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y., S. Machida, S. Saito, N. Imoto, T. Yanagawa and M. Kitagawa: Quantum mechanical limit in optical precision measurement and communication 28, 87 Yanagawa, T., see Yamamoto, Y. Yaroslavsky, L.P.: The theory of optimal methods for localization of objects in pictures Yeh, W.-H., see Carriere, J. Yin, J., W. Gao and Y. Zhu: Generation of dark hollow beams and their applications Yoshinaga, H.: Recent developments in far infrared spectroscopic techniques Yu, F.T.S.: Principles of optical processing with partially coherent light Yu, F.T.S.: Optical neural networks: architecture, design and models
52, 261 52, 261
Zalevsky, Z., D. Mendlovic and A.W. Lohmann: Optical systems with improved resolving power Zalevsky, Z., see Lohmann, A.W. Zavorotny, V.U., see Charnotskii, M.I. Zavorotnyi, V.U., see Tatarskii, V.I. Zhu, Y., see Yin, J. Zubairy, M.S., see Greenberger, D.M. Zuidema, P., see Bouman, M.A.
22, 271 6, 105 8, 295
28, 87 32, 145 41, 97 45, 119 11, 77 23, 221 32, 61
40, 271 38, 263 32, 203 18, 204 45, 119 50, 275 22, 77
