WAVELENGTH TUNABLE DEVICES BASED ON HOLOGRAPHIC POLYMER DISPERSED LIQUID CRYSTALS
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WAVELENGTH TUNABLE DEVICES BASED ON HOLOGRAPHIC POLYMER DISPERSED LIQUID CRYSTALS
A dissertation Submitted to Kent State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY
By Hailiang Zhang May, 2008
Dissertation written by Hailiang Zhang B.S., Xiangtan University, 1987 M.S., Kent State University, 1999 Ph.D., Kent State University, 2008
Approved by , Jack R. Kelly, Professor, Chair, Doctoral Dissertation Committee , Gregory P. Crawford, Professor, Members, Doctoral Dissertation Committee , Deng-Ke Yang, Professor, Members, Doctoral Dissertation Committee , Eugene C. Gartland, Jr, Professor, Members, Doctoral Dissertation Committee , Qi-Huo Wei, Assistant Professor, Members, Doctoral Dissertation Committee , Donald L. White, Professor, Members, Doctoral Dissertation Committee Accepted by Oleg D. Lavrentovich
, Chair, Liquid Crystal Institute
Timothy S. Moerland
, Dean, College of Arts and Sciences
ii
ZHANG, HAILIANG, Ph.D, May 2008 WAVELENGTH
TUNABLE
CHEMICAL PHYSICS
DEVICES
BASED
ON
HOLOGRAPHIC
POLYMER
DISPERSED LIQUID CRYSTALS, (222) Director of Dissertation: Jack Kelly
Wavelength tunable devices have generated great interest in basic science, applied physics, and technology and have found applications in Lidar detection, spectral imaging and optical telecommunication. This thesis focuses on the physics, technology and application of several wavelength tunable devices based on liquid crystal technology, especially on Holographic Polymer Dispersed Liquid Crystals (HPDLC). HPDLCs
are
formed
through
the
photo-induced
polymerization
process
of
photopolymerizable monomers, and self-diffusion process and phase separation process of the mixture of liquid crystals and monomers, when the mixtures of liquid crystals and monomers are exposed to the interfering monochromatic light beams. The infomation from the interfering pattern is recorded into the holographic liquid crystal/polymer composites, which are switchable or tunable upon external electric fields. Based on the electrically controllable beam steering capability of transmission HPDLCs, novel switchable circular to point converter (SCPC) devices are demonstrated for selecting and routing the wavelength channels discriminated by a Fabry-Perot interferometer, with application in Lidar detection, spectral imaging and optical telecommunication. SCPC devices working in both visible and near infrared (NIR) wavelength ranges are demonstrated. A random optical switch can be created by integrating a Fabry-Perot interferometer with a stack of SCPC units. iii
Liquid crystal Fabry-Perot (LCFP) Products have been analyzed, fabricated and characterized for application in both spectral imaging and optical telecommunication. Both single-etalon system and twin-etalon system are fabricated. Finesse of more than 10 in visible wavelength range and finesse in more than 30 in NIR are achieved for the tunable LCFP product. The materials, fabrication and characterization of lasing emission of dye doped HPDLCs are discussed. Lasing from different modes of HPDLCs is studied and both the switching and tunability of the lasing function is demonstrated. Lasing from two-dimensional HPDLC based Photonic Band Gap (PBG) materials will also be demonstrated. Finally, lasing from polarization modulated grating is discussed.
iv
TABLE OF CONTENTS
LIST OF FIGURES ........................................................................................................................ ix LIST OF TABLES......................................................................................................................xviii ACKNOWLEDGEMENT ............................................................................................................ xix CHAPTER I. INTRODUCTION TO LIQUID CRYSTALS ......................................................... 1 Physical Properties of Liquid crystals.............................................................................................. 1 Liquid crystal phases........................................................................................................................ 1 Anisotropic properties of liquid crystals.......................................................................................... 5 The Frank Free Energy and the continuum theory .......................................................................... 9 Surface Alignment of Liquid crystal.............................................................................................. 11 Modeling of Director Configuration of Liquid Crystals ................................................................ 13 Director configuration in case of infinite surface anchoring ......................................................... 13 Director configuration in case of finite surface anchoring............................................................. 21 CHAPTER 2. LIGHT PROPAGATION IN STRATIFIED MATERIALS .................................. 24 Introduction.................................................................................................................................... 24 Jones matrix method ...................................................................................................................... 26 Berreman's 4-by-4 matrix method ................................................................................................. 29 Light Propagation in Periodic Media ............................................................................................. 37 Introduction to grating ................................................................................................................... 37 Coupled Wave Theory .................................................................................................................. 40
v
Coupled wave theory for transmission Gratings............................................................................ 42 Coupled wave theory for reflection Gratings................................................................................. 48 CHAPTER 3. HOLOGRAPHIC POLYMER DISPERSED LIQUID CRYSTAL........................ 52 Introduction to Holography............................................................................................................ 52 Introduction to holographic polymer dispersed liquid crystals...................................................... 54 Transmission Mode HPDLCs ........................................................................................................ 57 Reflection Mode HPDLC .............................................................................................................. 60 Variable-Wavelength HPDLC ....................................................................................................... 62 HPDLC Materials .......................................................................................................................... 65 UV Mixtures .................................................................................................................................. 65 Visible Mixtures ............................................................................................................................ 66 Summary ........................................................................................................................................ 67 CHAPTER 4. LIQUID CRYSTAL FABRY-PEROT ................................................................... 68 Introduction.................................................................................................................................... 68 Introduction to Fabry-Perot interferometer.................................................................................... 68 Introduction to Liquid Crystal Fabry-Perot (LCFP) Tunable Filter............................................... 79 Fabrication and Testing of LCFP Tunable Filter ........................................................................... 84 Single LCFP system....................................................................................................................... 84 Twin LCFP system ........................................................................................................................ 91 Environment Test of LCFP............................................................................................................ 93 Summary and Conclusions ............................................................................................................ 98 CHAPTER 5 SWITCHABLE CIRCLE-TO-POINT CONVERTER............................................ 99 Introduction.................................................................................................................................... 99 Background: Introduction to HCPC............................................................................................... 99 vi
Principle of Operation of SCPC................................................................................................... 102 Optics Design of SCPC................................................................................................................ 104 First type (beam steering) SCPC.................................................................................................. 104 Second type (focusing) SCPC...................................................................................................... 107 Astigmatism in second type (focusing) SCPC............................................................................. 108 Fabrication and characterization of SCPC working in visible wavelengths ................................ 113 Single channel SCPC ................................................................................................................... 113 Fabrication and Characterization of SCPC working in NIR wavelengths ................................... 118 Material optimization for big-area SCPC working in NIR .......................................................... 118 Fabrication and Characterization of single channel SCPC working in NIR ................................ 121 Fabrication and Characterization of 32-channel SCPC working in NIR ..................................... 123 Summary and Conclusions .......................................................................................................... 133 CHAPTER 6. LASING OF DYE-DOPED HPDLC.................................................................... 134 Introduction.................................................................................................................................. 134 Introduction to Dye ...................................................................................................................... 134 Introduction to laser ..................................................................................................................... 139 Introduction to dye laser .............................................................................................................. 144 Introduction to Photonic Band Gap Materials ............................................................................. 149 Introduction to Lasing in Liquid Crystal Materials ..................................................................... 150 Introduction to Dye-Lasing in HPDLC........................................................................................ 152 Dye Lasing from HPDLC of Different Modes: Materials, Fabrications and Results .................. 154 Lasing of single reflective dye-doped HPDLC............................................................................ 154 Lasing of transmissive dye-doped HPDLC ................................................................................. 157 Multiple-Method for Lasing Tuning ............................................................................................ 167 vii
Lasing Tuning in Stack of HPDLCs ............................................................................................ 166 Lasing Tuning in chirped HPDLC ............................................................................................... 169 Two Dimensional Dye-Doped HPDLC Lasing ........................................................................... 172 Lasing of Polarization Grating..................................................................................................... 178 Summary and Conclusions .......................................................................................................... 188 CHAPTER 7 CONCLUSIONS AND CONSIDERATION ON FUTURE WORK ................... 189 BIBLIOGRAPHY........................................................................................................................ 191
viii
LIST OF FIGURES
Figure
Page
1.1
Schematic description of crystal, liquid crystal and liquid
2
1.2
Schematic description of smectic phase
3
1.3
Schematic description of director Orientation of Cholesteric state
3
1.4
Definition of θ used in equation (1-1)
4
1.5
Order parameter S changes with the temperature T
5
1.6
Molecular structure of three nematic liquid crystals
7
1.7
Director re-orientation in the external field
8
1.8
Three canonical elastic distortions: (a) bend (b) Twist and (c) Splay
10
1.9
Planar or homogeneous alignment of nematic liquid crystals
11
1.10 The direction n0 of a nematic liquid crystal on a solid surface, specified by the polar angle θ 0 and azimuthal angle φ0 . 1.11 Co-odinator system of director
13 14
1.12 Director configuration of 90° twist cell without applied voltage (a), and under applied voltage of 5V (b)
19
1.13 Director configuration of a chiral-doped homeotropicly aligned cell
20
1.14 Director configuration of a chiral-doped homeotropic-alignment cell with finite surface anchoring, with no applied voltage
22
1.15 Z-component of the director of a chiral-doped homeotropic-alignment cell at different surface anchoring strength
23 ix
2.1
Local coordinator system E x - E y - z in Jones matrix method
2.2
Spectral response of clock-wise circularly-polarized light
27
from two cholesteric cells
37
2.3
Diffraction of a monochromatic plane wave by an optically thick grating
38
2.4
Diffraction of a monochromatic plane wave by an optically thin grating
40
2.5
Diffraction by optically thick gratings. (a) Transmission grating; (b) Reflection grating
2.6
41
K-space of transmission gratings (a) Ideal phase match Δα = 0 ; (b) not ideal phase match Δα ≠ 0 .
2.7
43
Phase mismatch in transmission gratings due to (a) angular deviation; and (b) wavelength deviation
45
2.8
The diffraction efficiency decreases with the increase of phase mismatch.
46
2.9
Diffraction efficiency as a function of phase mismatch (Δβ/2k) for reflection gratings with different thickness kL.
48
2.10 Diffraction efficiency of a reflection grating as a function of κL for an ideally matched phase Δβ/2k.
50
2.11 Modeling of reflection grating at normal incidence. Berreman’s 4x4 method and coupled wave theory calculation are compared.
51
3.1
Optical setup for recording (a) and reconstructing (b) in-line hologram
53
3.2
The optical setup for recording (a) and reconstructing (b) off-line hologram
53
3.3
Holographic configurations for (a) reflective mode and
3.4
(b) transmissive mode HPDLCs
56
Transmission HPDLCs in the (a) on state; (b) off state
58
x
3.5
Polarization independent electro-optical device based on stacking of two polarization sensitive transmission HPDLCs (G1 and G2) and a polarization rotator (PR)
3.6
59
The SEM photograph of a transmission HPDLC operating at 1500 nm. top substrate is pealed before SEM photograph is taken
3.7
60
The SEM photograph of a reflective HPDLC operating at ~1500 nm. The image is of the cross section of a cell
61
3.8
Schematic illustration of a reflecting variable-wavelength HPDLC
63
3.9
(a) Reflectance and peak reflected wavelength as a function of applied voltage for variable wavelength HPDLC with a 5 μm cell gap. (b) Experimentally (points) and modeled (curves) reflectance spectra of variable wavelength HPDLC measured at 0, 120, and 220 V
4.1
Diagram of a plate with refraction index n immersed in the boundary media with refraction index of n'
4.2
64
69
(t ) (i ) Behavior of I / I as a function of the phase difference δ for various
values of finesse Ғ
72
4.3
Fabry-Perot interferometer
73
4.4
Image of the Fabry-Perot interference pattern with monochromatic incident light
74
4.5
Relation of the reflective finesse with the reflectivity
75
4.6
Spherical defects (a), surface irregularities (b), and parallelism defects (c)
75
4.7
Effective finesse changes with the defect finesse. FR represents the
4.8
reflective finesse
76
Modeling of twin etalon system with the gaps of 3 micron and 12 micron
78
xi
4.9
Structure of liquid crystal Fabry-Perot
4.10 The average refraction index changes with the applied voltages
80 81
4.11 Combination of polarization beam splitter and two LCFPs with alignment directions perpendicular to each other, to achieve the polarization-independent wavelength filtering
82
4.12 Two LC layers inside the Fabry-Perot Cavity to achieve the polarization independent wavelength filtering and tuning
83
4.13 Twist nematic Fabry-Perot
84
4.14 Spectral response of LCFP #1608, measured at 1.5 V
87
4.15 Spectral response of LCFP #1608, measured at 3.5 V
88
4.16 Spectral response of LCFP #1608, measured at 9.0 V
89
4.17 Electro-optical response of LCFP #1608, measured at 805 nm
90
4.18 LCFP #1608 in the housing with electrical connector
90
4.19 Photographs of the single etalon in the housing (right) and the twin etalon imaging filter (left)
91
4.20 Transmission as a function of wavelength for the 30 μm gap LCFP
92
4.21 Transmission as a function of wavelength for the 6 μm gap LCFP
93
4.22 Temperature versus time for the thermal vacuum testing of the LCFP
95
4.23 Transmission of the LCFP that underwent a Pegasus-level shake test for two different voltage settings (1 and 9 Volt)
96
4.24 Transmission of the LCFP that underwent thermal cycling, before and after the thermal cycling for two different voltage settings 5.1
97
The ray trace diagram of the holographic circular-to-point converter (HCPC) developed by McGill and co-workers xii
101
5.2
The cross-section drawing of a 4X2 switch employing two identical SCPC Elements
103
5.3
A random optical cross-switch can by stacking multiple SCPC units
103
5.4
The first type of SCPC: the diffracted beam is focused by a focal lens to a point.
5.5
104
Reading beam configuration (a) and recording beam configuration (b) of the beam steering HPDLLC for the first type of SCPC.
105
5.6
The holography setup for fabricating the second type SCPC
107
5.7
Recording beam profile across the HPDLC area using the setup in Figure 5.6
5.8
108
The diffraction beam profile of 1 inch HPDLCs fabricated using focal lenses with various focal length F
5.9
112
The left panel : the switch-off state of the SCPC (no voltage applied); the right panel : the switch-on state (voltage applied). In each panel, the holographic focal point is the point on the right side, and the “pass-through” light is on the left
113
5.10 Switching of a SCPC working in 532 nm
115
5.11 A schematic description of CAD design of a 10-pixel ITO pattern in SCPC
116
5.12 Switching of the center pixel of 10-pixel type-II SCPC
117
5.13 Switching of one non-center pixel of 10-pixel type-I SCPC
117
5.14 Switch on the center pixel of a beam-steering 10-channel SCPC
118
5.15 Holographic recording setup for fabricating the SCPCs working in 1550 nm range
121
5.16 Transmittance and diffraction efficiency as a function of voltage xiii
122
5.17 Switching of independent channels in the SCPC unit. The photos, show that the deactivation of the central pixel, the 5th pixel (count from the center), and the outmost pixel(32th), respectively.
124
5.18 The normalized transmittance and diffraction efficiency of the center channel of a SCPC unit as the function of voltage 5.19 Optical setup for measuring the wavelength dependence of the SCPC units
125 126
5.20 Transmittance and diffraction efficiency as a function of wavelength of the switch-off state of a SCPC sample: JL101404B
127
5.21 The fitting of the modeling result based on coupled wave theory and the refraction principle, with the measurement result, for the wavelength dependence of the diffraction efficiency
129
5.22 The transmission as a function of incident angle of the SCPC
130
5.23 The diffraction efficiency as a function of incident angle of the SCPC
131
5.24 Normalized transmittance is fitted to the formula for transmission grating derived using coupled wave theory
132
6.1
Absorption of positive dye (a) and negative dye (b)
135
6.2
Two basic kinds of dyes (a) azo dye (b) anthraquinone dye
136
6.3
Dye molecules inside liquid crystals
138
6.4
(a) Two-level energy system of laser medium. (b) three-level energy system
140
6.5
A four-level laser energy diagram.
143
6.6
Molecular structure of the lasing dye Pyrromethene 580(a) and DCM(b)
145
6.7
Emission spectrum of a dye molecule shifts from the absorption spectrum
147
6.8
“Littrow arrangement” tunes of the center peak of a laser xiv
by rotating the diffraction grating 6.9
148
Lasing emission from a reflection mode HPDLC (solid line) and transmission spectra of the same sample (dotted line)
6.10 Switching of the dye lasing emission from a reflection mode HPDLC
155 156
6.11 Two lens were used to generate the vertical line across the HPDLC grating in order to increase the area of the gain medium being pumped
157
6.12 Lasing emission of the sample with 0.5% Dye concentration as the pump beam polarization is changed from s-polarized to p-polarization.
159
6.13 Lasing emission of the sample with 1% Dye concentration as the pump beam polarization is changed from s-polarized to p-polarization.
160
6.14 Lasing emission of the sample with 2% Dye concentration as the pump beam polarization is changed from s-polarization to p-polarization
161
6.15 Dye molecules are distributed in the liquid crystal layers and are aligned with the liquid crystal in the surface.
162
6.16 Lasing emission at various pump energies in a sample with 0.5% dye.
163
6.17 Lasing emission at various pump energies in a sample with 1% dye.
164
6.18 Peak emission intensity at various pump energies. A threshold at ~18 µJ. Sample has dye concentration of 0.5%.
164
6.19 Effect of electric fields on lasing in a transmission HPDLC. Energy of pumping laser is 20 μJ.
165
6.20 Various modes of operation to tune the wavelength peak of the lasing.
167
6.21 Stacked grating for tunable lasing. The grating with the smaller pitch, lower reflection band in a zero voltage state, while the larger pitch grating has a field applied across it to switch off lasing. xv
167
6.22 Transmission of the two gratings used in the stack. A is doped with dye P580, and grating B is doped with dye DCM.
168
6.23 Tuning of a chirped HPDLC. Transmission at left (solid), middle (dashed) and right (dotted) points (top); and lasing emission at left (solid), middle (dashed) and right (dotted) points on the sample (bottom). 6.24 Switching of a reflection mode chirped HPDLC.
171 171
6.25 (a) Setup for creating 4-beam interference pattern and (b) the interference pattern; the bright (dark) regions represent areas of high (low) intensity.
174
6.26 SEM image of a HPDLC lattice generated by 4-beam interference. The designed period is 222nm.
174
6.27 (a) Setup for creating 6-beam interference pattern (b) the interference pattern; the bright (dark) regions represent areas of high (low) intensity.
175
6.28 (a) Isointensity plot for four-beam fabrication ; and (b) lasing from this structure doped with the laser dyes Pyrromethene 580 (solid line) and DCM (dotted line). Lasing emission is measured along x-direction. 6.29
176
(a) Isointensity plot for six-beam fabrication and subsequently lasing; and (b) lasing from this structure doped with the laser dyes Pyrromethene 580 (solid line) and DCM (dotted line).
6.30
a) Two linearly polarized beams with orthogonal polarization directions; (b)Two circularly polarized beams with opposite sense of clockwise
6.31
182
Microscope images of a cell of polarization grating between polarizers. (a) no voltage is applied; (b) 20 V voltage is applied
6.32
177
183
Writing beam and pump beam for fabrication and lasing emission testing of the polarization gratings
183 xvi
6.33
Lasing emission from a liquid crystal polarization holography grating.
184
6.34
Threshold of laser emission for the dye-doped polarization grating
184
6.35
lasing emission increases by 50% as the incident polarization is rotated
186
6.36
Effect of an applied electric field on a liquid crystal polarization grating pumped by p-polarized light.
xvii
187
LIST OF TABLES
4.1
Finesse and free spectral range of LCFP # 1608 at different voltages
86
4.2.
Testing result of tunable LCFP for tunable laser in NIR range
86
5.1.
Converging recording beam incident angle, Bragg reading and diffraction angles, diffraction angles with normal incident reading, and minimum distance between neighboring SCPC units.
110
5.2
Testing result of SCPC
115
5.3
Components of the HPDLC mixtures initially investigate
120
5.4
Material contents of Formula-SCPC
121
5.5
The transmittance of some channels of a SCPC unit
125
xviii
ACKNOWLEDGEMENT
I would like to thank my two advisors, Professor Gregory P. Crawford and Professor Jack Kelly, for all their education, instruction, and support. During the three years I studied and worked in the Liquid crystal Institute of Kent State University from 1996 to 1999, I had learned a lot from all the professors and teachers. I would like to express my gratitude to all of them. I would like to thank Scientific Solutions, Inc., the company I have worked for since late 2000, for providing a great research platform for me to continue my PhD research. I also thank the Display and Photonics lab of Brown University for the happy cooperation on all the research projects I have done during these years, especially, with special gratitute to Professor Gregory P. Crawford who has given me so many support, direction and inspiration. I got a lot of help from Dr. Haiqing Xianyu, Mr. Scott Woltman (PhD candidate), Mr. Jianhua Lian, Dr. Jun qi, and Dr. Matthew Sousa from Display and Photonics lab of Brown University. I would like to thank all of them for their assistance and helpful discussions. Special gratitude to my family, especially my parents, for their long-term support and encouragement. I would like to use my dissertation as a special gift to my lovely daughter, hopefully she will like it more than a toy. Finally, I would like to thank my Small Business Innovation Research (SBIR) project sponsors:
National
Science
Foundation,
xix
NASA
and
Department
of
Energy.
CHAPTER I
Introduction to Liquid Crystals
1.1 Physical Properties of Liquid Crystals The liquid crystal phase is an intermediate state between the solid crystalline phase and the isotropic liquid phase (Fig. 1.1). The distinguishing characteristic of the liquid crystalline state is the tendency of the molecules (mesogens) to point along a common axis, called the director. This is in contrast to molecules in the liquid phase, which have no intrinsic order. In the solid state, molecules are highly ordered and have little translational freedom. The characteristic orientational order of the liquid crystal state is between the traditional solid and liquid phases; this is the origin of the term mesogenic phase, used synonymously with the liquid crystal state. Liquid crystals exhibit some degree of fluidity, which may be comparable to that of an ordinary liquid; However they also exhibit anisotropies in their optical, electrical, magnetic and other physical properties like crystals.
1.1.1 Liquid Crystal Phases Liquid crystal phases are observed in certain organic compounds and usually are made up of elongated molecules. There are a number of distinct liquid crystal between the crystalline phase and the isotropic liquid. These intermediate transitions may be brought about by temperature variation; the compounds in which the liquid crystal phase is induced by a thermal process are known as thermotropic liquid crystals. The thermotropic liquid crystals are further classified into three types: nematic, smectic and cholesteric as proposed by Friedel [1]. This classification is
1
2
based on the molecular arrangement and the ordering of the molecules in the particular liquid crystal phases. Nematic liquid crystals have long-range orientational order but no long-range translational order. The average orientation of all of the molecules in the nematic liquid crystals is defined as the director, as shown as the arrow in Fig. 1.1(b). Smectic liquid crystals (Fig. 1.2) are different from nematics in that they have an additional degree of positional order. Smectics generally form layers within which there is a loss of positional order, while the orientational order is still preserved. There are several different categories to describe smectics. The two best known of these are Smectic A, in which the molecules tend to align perpendicular to the layer planes, and Smectic C, where the alignment of the molecules is at some arbitrary angle to the normal.
(a)
(b)
(c)
Fig. 1.1 Schematic description of crystal (a), liquid crystal (b) and liquid (c).
The cholesteric phase (or chiral nematic phase) is typically composed of nematic mesogenic molecules containing chiral center that produces intermolecular forces, which favor an alignment between molecules at a slight angle to one another. This leads to the formation of a structure that can be visualized as a stack of very thin 2-D nematic-like layers with the director in each layer twisted with respect to those above and below (Fig. 1.3). In this structure, the directors actually
3
form a continuous helical pattern about the layer normal. The black arrows in the Fig. 1.3 represent the director orientation in the succession of layers along the stack. An important characteristic, the pitch, p, is defined as the distance it takes for the director to rotate one full turn.
(a)
(b)
Fig.1.2. Schematic description of smectic phase. smectic A phase (a); smectic C phase (b).
Fig 1.3. Schematic description of director orientation of cholesteric state. Arrows represent the director directions in each layer.
4
nˆ θ
Fig.1.4. Definition of θ used in equation (1-1).
Nematic liquid crystals are usually uniaxial and are the most widely used liquid crystals in electro-optical applications such as for the twisted nematic effect, phase modulation, etc. The director determines the direction of the preferred orientation of the molecules but does not represent the degree of the orientational order. The order parameter S, proposed by Tsvetkov [2], provides us with a measure of the long range orientational order:
S=
3 cos 2 θ − 1 2
(1-1)
where θ is the angle between the axis of a molecule and the director of the liquid crystal (Fig.1.4); the angular brackets indicate an average over the complete system. For a perfect crystal S = 1 and for the isotropic phase S = 0. For nematics, S will have a value between 0 and 1, varying with the temperature. The critical temperature at the nematic to isotropic transition point is defined as T0 (Fig.1.5).
5
1.1.2 Anisotropic Properties of Liquid Crystals Liquid crystals show anisotropy in their magnetic, electrical, optical and other physical properties. A macroscopic anisotropy is found in liquid crystals because the molecular anisotropy does not average to zero, as is the case in an isotropic phase. For the interest of this thesis we only discuss their electrical and optical anisotropies. In uniaxial nematic liquid crystals the dielectric tensor ε can be diagonalized with eigenvalues
ε // and ε ⊥ , which refer to the dielectric constants parallel and perpendicular to the nematic director nˆ , respectively. The dielectric anisotropy is written as
Δε = ε // − ε ⊥
(1-2)
and the dielectric tensor is defined as:
ε αβ = ε ⊥δ αβ + Δεnα nβ
(1-3)
where nα and n β are the components of the director nˆ .
S 1.0
0.5
T0
T
Fig. 1.5. The order parameter S changes with the temperature T.
6
The theoretical consideration [3] suggests that
where ε =
ε // + 2ε ⊥
parameter. α //
3
ε // − 1 ρ α // = ε +2 3ε 0
(1-4)
ε ⊥ −1 ρ α⊥ = ε +2 3ε 0
(1-5)
is the average dielectric co-efficient that does not depend on the order
and α ⊥
are the average molecular polarizability when the applied field is
parallel or perpendicular to the director, respectively. For molecules without permanent dipoles,
2 3
(1-6)
1 3
(1-7)
α // = α + ΔαS α ⊥ = α − ΔαS where S is the order parameter and
α = (α // + 2α ⊥ ) / 3
(1-8)
Δα = α // − α ⊥
(1-9)
α // and α ⊥ are the molecular polarizability along the long molecular axis direction or perpendicular to the long molecular axis, respectively. For molecules with permanent dipoles, (1-6) and (1-7) should be modified to include the dipole term:
α // = α +
μ p // 2 + μ p⊥ 2 3KT
2 + (Δα + 3
1 2 KT
μ p // 2 − μ p⊥ 2
)S
(1-10)
7
μ p // + μ p⊥ 2
α⊥ = α +
2
1 − (Δα + 3
3KT
1 2 KT
μ p // 2 − μ p⊥ 2
)S
(1-11)
r
where μ p // or μ p⊥ are the components of permanent dipole μ along the long-molecular axis or perpendicular to the long-molecular axis, respectively.
C7H15
N C7H15
N
(a)
C7H15
CN
(b)
OC6H15 C2H5O C N
(c)
Fig.1.6. Molecular structure of three nematic liquid crystals: (a) a non-polar liquid crystal molecule; (b) a polar liquid crystal molecule with positive dielectric anisotropy; (c) a polar liquid crystal molecule with negative dielectric anisotropy.
8
From (1-10) and (1-11) we can see that when there is a large angle between the permanent
dipole and the long molecular axis direction, μ p // < μ p⊥ and Δα +
1 2 KT
μ p // 2 − μ p⊥ 2 may be
negative; therefore, α // < α ⊥ and from (1-4) and (1-5) we find ε // < ε ⊥ or Δε < 0 . Fig. 1.6 shows three kinds of nematics liquid crystals. (a) is a non-polar molecule while (b) is a polar molecule with a dipolar moment parallel to the long molecular axis, thus
Δε (b) > Δε (a) > 0 . In molecule (c) the CN group introduces a large permanent dipole moment at a large angle with the long molecular axis direction, so Δε (c) < 0 . The dielectric anisotropy introduces body torque on the molecules in the presence of an external field, which in turn gives rise to the director re-orientation. (Fig.1.7) This property can be used for liquid crystal materials with both positive and negative dielectric anisotropies. Under an external field, the director of a liquid crystal with a positive dielectric anisotropy tends to align parallel to the external field, while the director of a liquid crystal with a negative dielectric anisotropy tends to align perpendicular to the external field.
Electric Field
Fig.1.7 Director re-orientation in the external field
9
Liquid crystals are also found to have optical anisotropy, or birefringence, due to their anisotropic nature. They demonstrate double refraction, or light polarized parallel to the director has a different index of refraction (that is to say it travels at a different velocity) than light polarized perpendicular to the director. The optical anisotropy, or birefringence is given by:
Δn = ne − no
(1-12)
Where no is the ordinary index of refraction, and ne is the extraordinary index of refraction. The relation between the optical anisotropy and the dielectric anisotropy is given by:
ε ⊥ = no 2 ; and ε // = ne 2
(1-13)
1.2 The Frank Free Energy and the Continuum Theory In a liquid crystal system, the bulk free energy of an inhomogeneous sample has contributions from the elastic deformation of the system. The elastic properties of liquid crystals influence the behaviors of these materials in an electric or magnetic field. The simplest way to treat the deformation of a nematic liquid crystal is to consider it to be a continuous elastic medium, disregarding the details of the molecular structure. The state of the system is described by the director field n(r) , which determines the elastic free energy of the system. The stiffness of the system can be expressed by a fourth rank tensor [4]: Fel =
1 3 d xK ijkl ∇i n j ∇ k nl , where K ijkl is a 2∫
tensor that generally depends on the local director n(r) . Considering the symmetry of the nematic liquid crystal, the free energy should be invariant under the symmetry operation
n → −n . n is a unit vector; therefore, ni ∇ j ni is zero. These factors indicate that when the bulk energy is considered, K ijkl has three independent components, which can be designated as elastic
10
constants k11 , k 22 , k 33 , as depicted in Fig. 1.8. The first distortion, splay, is described by ∇ ⋅ n . The second kind of distortion, twist, is described by n ⋅ (∇ × n) . The third distortion, bend, is evaluated by n × (∇ × n) . In the continuum theory, first stated by Oseen [5] and Zocher [6], and completed by Frank [7], the Frank free energy density of a nematic liquid crystal medium with a curvature deformation in its director field is
1 f = {k11 (∇ • nˆ ) 2 + k 22 (nˆ • (∇ × nˆ )) 2 + k 33 (nˆ × (∇ × nˆ )) 2 } 2
(1-14)
Where k11 , k 22 and k 33 correspond to the elastic constants of splay, twist and bend, respectively. The surface elastic constants have been ignored in (1-14); they tend to play a larger role in highly confined liquid crystal systems [130]. This form of the Frank free energy density is minimized when the director is spatially uniform.
(a)
(b)
(c)
Fig.1.8. Three canonical elastic distortions: (a) bend (b) twist and (c) splay.
11
For cholesteric liquid crystals, there are spontaneous twists, which are originated by the chiral molecules. An additional term that takes into account the chirality of the molecules is introduced in the second term of (2-14) resulting in the expression:
1 f = {k11 (∇ • nˆ ) 2 + k 22 (nˆ • (∇ × nˆ ) + q 0 ) 2 + k 33 (nˆ × (∇ × nˆ )) 2 } 2
(1-15)
where q0 = 2π / p0 is the wave vector; p 0 is the pitch of cholesteric. Positive and negative values of q0 correspond to a left or right-handed helix, respectively.
1.3 Surface Alignment of Liquid Crystal In many liquid crystal devices, such as twisted nematic cells and waveplates, a uniform or well-defined orientation of the liquid crystal molecules is required. Without surface alignment and cell confinement, the liquid crystal cell will have multiple domains with different orientations, and boundary walls and defects between the domains. The multi-domain nature and the existence of numerous boundary walls and defects result in strong scattering. Specially treated surfaces are employed in order to ensure a single domain in the designated area.
(a)
( b)
Fig. 1.9. Planar or homogeneous alignment (a) and homeotropic alignment (b) of nematic liquid crystals.
12
Two types of surface alignment, as shown in Fig. 1.9, are widely used in liquid crystal devices, distinguished by the preferred orientation of the molecules on the surface. With planar or homogeneous alignment, the molecules are oriented in a direction parallel to the surface; whereas with homeotropic alignment, the molecules are oriented in a direction perpendicular to the surface. Planar alignment can be achieved by unidirectionally rubbing a coated polyimide layer [8], or by exposing photo-alignable polyimide to polarized UV light [9]. Homeotropic alignment is realized by depositing amphiphilic molecules such as lecithin [10], silane [11], or some polyimides, such as SE-7511 from Brewer Science, on the surface. A surface anchoring term is introduced into the free energy with the consideration of the alignment effect. In the vicinity of the treated surface, there is an energetically favorable direction given by a unit vector n 0 . In the model presented by Rapini and Papoular, the surface free energy density is given by [12]:
1 1 2 f surf = − w(n s ⋅ n 0 ) = − w sin 2 (θ ) + const 2 2
(1-16)
where w is the anchoring strength, n s is the director at the surface; and θ is the angle −4 −7 between ns and n 0 . The typical value of w is in the order of 10 ~ 10 J/ m 2 [13].
When an electric field is applied to a nematic liquid crystal cell, the spatial molecular configuration can be determined by minimizing the free energy of the system:
F
=
Fel + Fefield + Fsurf
=
1 3 d x{K1 (∇ ⋅ n) 2 + K 2 [n ⋅ (∇ × n)]2 + K 3 [n × (∇ × n)]2 } 2∫
∫
(
r
)
2
3 + d x[ − Δε E ⋅ n 0 ] +
1 dS[ w sin 2 (θ )] 2∫
(1-17)
13
n0 θ
ns
Fig. 1.10. The direction n0 of a nematic on a solid surface, specified by the polar angle θ 0 and azimuthal angle φ0 .
1.4. Modeling of Director Configuration of Liquid Crystals The basic concept of director configuration modeling is to find the director configuration that minimizes the total free energy of the system. The total free energy includes the bulk term, which is described as the Frank-Oseen strain free energy, the surface term, which is surface free energy, and the term related to the external electric field. For the surface term, both the two cases are discussed: infinite surface anchoring and finite surface anchoring.
1.4.1 Director Configuration in Case of Infinite Surface Anchoring When the surface anchoring energy is strong enough to be treated as infinite, the free energy density equation for a liquid crystal material in an electric field, based on the Frank-Oseen strain free energy density, is given by [7]:
1 1 ) ) ) ) ) f = {k11 (∇ • n ) 2 + k 22 [n • (∇ × n ) − q0 ] 2 + k 33 [n × (∇ × n )]2 } ± D • E 2 2
(1-18)
14
where nˆ is the director and q 0 = 2π / p , p is the natural pitch of the material; and k11 , k 22 ,
k 33 are the splay, twist, and bend elastic constants, respectively. In the ±
1 D • E term, D is the 2
electric flux density and E is the electric field; “ + ” is for the case of constant electric flux density and “−” is for the case of constant electric field. In our application we usually consider the applied voltage as a constant, so we concentrate on the latter constant voltage condition. Assuming the director only changes along the cell normal, defined as the z-axis, a onedimensional condition, we use the coordinate system shown in Fig.1.11.
z n
nz x
ny nx y
Fig.1.11. Co-odinator system of director
It is reasonable to assume there is no free charge in the liquid crystal; from Maxwell’s Equation ∇ • D = 0 we have
dD z = 0 . Also, considering the constant voltage, dz ∫ Edz = V
(1-19)
15
Dz
So from
∫
We obtain
Dz =
ε 0 [ε // n z + ε ⊥ (1 − n z 2 )] 2
dz = V
(1-20)
ε 0V ∫
d 0
(1-21)
dz
ε // n z 2 + ε ⊥ (1 − n z 2 )
This means the normal component of D is constant throughout the cell. The free energy can now be written as:
1 2 2 2 f = {k11 (n& z ) 2 + k 22 [−n x n& y + n y n& x − q0 ]2 + k 33 [n z (n& x + n& y ) + (n x n& x + n y n& y ) 2 ]} 2 2
+
Dz 2 2 2ε 0 [ε // n z + ε ⊥ (1 − n z )]
with use n& representing
(1-22)
d n . The total free energy is given by integrating (1-22) over the dz
volume:
Ft = A ∫ 0d fdz
(1-23)
The free energy function is
F = A ∫ 0d { f − λ D
Dz
− λ n (n x + n y + n z )}dz ≡ A ∫ 0d f ′dz (1-24) 2
ε 0 [ε // n z + ε ⊥ (1 − n z )] 2
2
2
2
where λD is the LaGrange multiplier for the constraint of constant voltage, and λ n is another LaGrange multiplier for the constraint nˆ = 1 . Now the problem becomes the need to find the director configuration nˆ = nˆ ( z ) which will minimize the total free energy function in (1-24) and meets the boundary conditions. The stationary condition leads to the Euler-Lagrange equations. Firstly
16
δf ′ δf 1 = − λD = 0 , resulting in: 2 δDz δDz ε 0 [ε // n z + ε ⊥ (1 − n z 2 )] λ D = Dz
(1-25)
Considering (1-22) and (1-25) we obtain:
1 2 2 2 f ′ = {k11 (n& z ) 2 + k 22 [−n x n& y + n y n& x − q0 ] 2 + k 33 [n z (n& x + n& y ) + (n x n& x + n y n& y ) 2 ]} 2 2
Dz 2 2 2 − λ n (n x + n y + n z ) − 2 2 2ε 0 [ε // n z + ε ⊥ (1 − n z )]
(1-26)
Another Euler-Lagrange equation is:
δf ′ ∂f ′ d ∂f ′ ≡ − ( )=0 , δni ∂ni dz ∂n& i
i = x, y, z
(1-27)
We need to solve (1-27) to find the director configuration of the equilibrium state. If we choose a spherical co-ordinate system in which the parameters θ and φ are used, the constraint nˆ = 1 can be automatically satisfied, but when the director is along the z direction, φ can be any value, and this leads to confusion. In our modeling we instead use the parameters
nx , n y , nz . Instead of solving (1-27), we use the relaxation method based on the dynamic equations of the director to find the director configuration of the equilibrium state:
γ
∂ni ∂f ′ d ∂f ′ δf ′ = −[ − ( )] , i = x, y, z =− ∂ni dz ∂n& i ∂t δn i
(1-28)
where γ is a viscosity coefficient. Discretizing these equations gives:
Δ ni = −
Δt δf ′ γ δ ni
(1-29)
17
In detail, we have:
Δnx =
Δt
γ
{ k 22 [ 2 ( − n x n& y + n y n& x − q 0 ) n& y + ( − n x n&& y + n y n&&x ) n y ] +
2 2 2 k 33 [ n z n&&x + 2 n z n& z n& x + ( n x n&&x + n& x + n y n&&y + n& y ) n x ] + 2λ n n x }
Δn y =
Δt
γ
{− k 22 [ 2 ( − n x n& y + n y n& x − q0 ) n& x − ( − n x n&&y + n y n&&x ) n x ] +
2 2 2 − k 33 [ n z n&&y + 2 n z n& z n& y + ( n x n&&x + n& x + n y n&&y + n& y ) n y ] + 2 λ n n y }
Δnz =
Δt
γ
(1-30)
2 2 { k 11 n&&z − k 33 n z ( n& x + n& y ) −
(1-31)
D z ( ε // − ε ⊥ ) n z + 2λnnz} ε 0 [( ε // − ε ⊥ ) n z 2 + ε ⊥ ] 2 2
(1-32) At each time step, ni is updated by ni + Δni , and D z is also upated as in (1-21). We can neglect the λ n term in f ′ expression (1-26), if we re-normalize n at each time step of k +1 the relaxation. Then, the director ni at the time step k+1 is:
nik +1 = nik −
nik +1 =
Δt δ f ′ ( ) γ δn i
nik +1 n k +1
(1-33)
(1-34)
The iteration repeats until the director converges to the equilibrium state. Fig.1.12 is the calculated director configuration of a 90° twist liquid crystal cell (with planar boundary conditions). Fig.1.12(a) is under no voltage and Fig.1.12(b) is under 5V voltage. The following parameters used are: thickness d=5.0 μm, pretilt angle θ p =0°, elastic constant:
k11 =5.5(pN), k 22 =14.0, k 33 =28.0, ε // =8.1, ε ⊥ =3.3
18
Fig 1.13 is a calculated director configuration of a chiral-doped liquid crystal cell with homeotropic boundary conditions. Fig.1.13 (a) is under 0v voltage and Fig.1.13 (b) is under 10V voltage. The parameters used are: thickness d=5.0 μm, pretilt angle θ p = 90° (homeotropic boundary), elastic constant: k11 = 14.9 (PN), k 22 =7.9 , k 33 =15.2, ε // =3.3 , ε ⊥ =8.1, the d/p ratio is 1.
19
1.0 nx
0.9
Component of Director
0.8 0.7 0.6 0.5 0.4 ny
0.3 0.2 0.1
nz 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Thickness
(a)
nz 1.0 0.9
Director Component
0.8 0.7 0.6 0.5 0.4
ny
nx 0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Thickness
(b) Fig 1.12 (a) Director configuration of 90° twist cell without applied voltage. (b) Director configuration of 90º twist cell under applied voltage of 5V.
20
1.0 nz
0.8
0.4 0.2 ny
0.0 -0.2 -0.4 -0.6 nx
-0.8 -1.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Thickness
(a)
1.0 0.8 ny
0.6
Director Component
Director Component
0.6
0.4 0.2 nz
0.0 -0.2 -0.4 nx -0.6 -0.8 -1.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized thickness
(b) Fig.1.13 Director configuration of a chiral-doped homeotropically aligned cell. (a) no voltage applied; (b) 10V applied voltage.
21
1.4.2 Director Configuration in Case of Finite Surface Anchoring When the anchoring strength of the surface alignment is not infinitely strong, besides considering the free energy in the bulk, which is described by (1-17), we also need to consider the surface anchoring energy. According to the description of Rapini and Papoular [12], the surface anchoring energy is:
1 ) ) f a (nˆ ) = Wa [1 − (n • n0 ) 2 ] 2
(1-35)
)
Where Wa is the surface anchoring strength; n0 is the easy direction for the surface director; and
) ) n is the actual surface director which deviates from n0 . The surface anchoring energy is minimized when the director is aligned along the easy direction. The total free energy is:
1 ) ) ′ − ∫ Wa (n • n0 ) 2 ds ≡ ∫ f ′dv + ∫ gds F = ∫ f dv 2
(1-36)
Note that the integration of the second term is over the surface. According to the Euler-Lagrange equation, the equilibrium state satisfies:
δf ′ ∂f ′ d ∂f ′ ≡ − ( ) = 0 ( for the bulk ) δni ∂ni dz ∂n& i And
(1-37)
∂g ∂f ′ − = 0 (for the surface z=0) ∂ni ∂n& i
(1-38)
∂g ∂f ′ + = 0 (for the surface z=d) ∂ni ∂n& i
(1-39)
Where i is the index for x,y,z . The latter two equations are the so-called torque balance equations. Considering the dynamics in the relaxation method, we obtain:
−γ
∂ni δf ′ ∂f ′ d ∂f ′ ≡ − ( ) (for the bulk) = ∂t δni ∂ni dz ∂n& i
(1-40)
22
And
−γ s
∂ni ∂g ∂f ′ = − ∂t ∂ni ∂n& i
−γ s
∂ni ∂g ∂f ′ = + ∂t ∂ni ∂n& i
(for the surface z=0)
(1-41)
(for the surface z=d)
(1-42)
Where γ s is the viscosity constant of the surface.
1.0
ny
0.8
Director Component
0.6 nz 0.4 0.2 0.0 nx -0.2 -0.4 -0.6 -0.8 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Thickness
Fig.1.14. Director configuration of a chiral-doped homeotropic-alignment cell with finite surface anchoring (no applied voltage).
Fig. 1.14 shows the calculated director configurations of a chiral-doped liquid crystal cell −6 2 with homeotropic surface alignment. The anchoring strength is 8.0 ×10 J / m . The other
parameters are: d/p = 0.6 , thickness d=5.0 μm, k11 = 14.9 (pN), k 22 =7.9 , k 33 =15.2. Because the surface anchoring strength is finite, the bulk twisting strength, which tends to tilt the director away from the cell normal, is relatively stronger; the n z is decreased with the increase of surface
23
anchoring energy. Fig.1.15 plots the n z component of the director through cells with different surface anchoring strengths. The other parameters are the same as those parameters used in Fig.1.14.
1.0
z component of director--nz
0.9 0.8
Black: Blue: Red: Green:
0.7 0.6 0.5
Wa = 5.0e-6 Wa = 7.0e-6 Wa = 8.0e-6 Wa = 1.0e-5
0.4 0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Normalized Thickness
Fig.1.15. Z-component of the director of chiral-doped homeotropic-alignment cells at different surface anchoring strengths.
CHAPTER 2
Light Propagation in Stratified Materials
2.1 Introduction In classical electromagnetic theory, the state of light is described by two vectors, the electric
r
r
field E and the magnetic induction B . Light propagation is described by Maxwell's equations [14].
r r ∂B =0 ∇× E + ∂t r r ∂D r = j ∇× H − ∂t r ∇⋅D = ρ r ∇⋅B = 0
r
r
(2-1)
r
where D the electric flux density, H the magnetic field strength, j the current density, and ρ is the charge density. The properties of the medium through which the light is propagating are also necessary to determine the field distribution:
r r j = σE , r r D = εE , r 1 r H = B,
(2-2)
μ
where σ is the conductivity, ε the dielectric constant, and μ is the magnetic permeability. σ ,
r
r
ε and μ are tensors reflecting the properties of the materials and may depend on E and B . In a dielectric medium, σ is negligible in most situations.
24
25
When discussing light propagation in dielectric media, we usually assume there is no source in the media, i.e. no free charge ( ρ = 0 ) and no free current ( j = 0 ). In a homogeneous and isotropic dielectric medium, the following equations are deduced from Maxwell's equations:
r r ∂2E ∇ E − εμ 2 = 0, ∂t r r ∂2H 2 ∇ H − εμ 2 = 0. ∂t 2
(2.3)
These are the standard wave equations with monochromatic plane wave solutions: r r r r E = E0ei (ωt − k ⋅ r ) , r r r r H = H 0ei (ωt − k ⋅ r ) ,
(2.4)
r
r
where k is the wave vector and ω is the angular frequency. The relationship between k and ω is:
r nω k=k = , c
(2.5)
r
r
where n ≡ εμ is the refractive index. Both the electric field E and the magnetic field H are
r r
r r
perpendicular to the propagation direction ( k ⋅ E0 = k ⋅ H 0 = 0 ), and are perpendicular to each other. The phase velocity of light is described by:
v=
c
c = . εμ n
(2.6)
26
The dielectric constant ε = ε (ω ) depends on the frequency of the electro-magnetic wave. Therefore, electro-magnetic waves of a certain frequency propagate in a medium at their own phase velocity, a phenomenon referred to as dispersion in optics. When considering light propagation in liquid crystal devices, in most cases these materials can be treated as stratified anisotropic materials. The basic idea is to divide a liquid crystal device into many layers; if layer number is large enough, each layer is assumed to be optically uniform. Two widely accepted methods, the Jones matrix method and Berreman's 4x4 method [15] are introduced here.
2.2 Jones Matrix Method In the Jones matrix method, the polarization state of light is described by a Jones vector
r ⎡Ex ⎤ E=⎢ ⎥ ⎣E y ⎦
(2-7)
where E x , E y are the complex components of the electric field in the x, y directions, respectively. The light is assumed to propagate along the z direction. As the amplitude and the phase difference between the two components are consdered, so the vector is re-written as:
r ⎡Ex ⎤ ⎡ Ex ⎤ E=⎢ ⎥=⎢ iφ ⎥ ⎣ E y ⎦ ⎣⎢ E y e ⎦⎥
(2-8)
where φ = 2π( ne − no )d/λ is the phase difference of the o-light and the e-light. The light intensity is given by: 2
I = Ex + E y
2
(2-9)
When light passes through a uniform birefrigent film, a Jones matrix, J, is used to describe the optics of the system:
27
r r E out = JEin r
(2-10)
r
where Ein , E out are the incident light and the transmitted light, respectively. For a non-absorbing birefrigent film with a phase shift ϕ and in which the direction of the fast axis (along which direction the refraction index is the smallest) makes an angle θ with respect to x-axis, the Jones matrix is :
J = R (θ ) J ϕ R(θ ) −1 = R (θ ) J ϕ R(−θ )
(2-11)
⎡1 0 ⎤ ⎡cos(θ ) − sin(θ )⎤ and R(θ ) = ⎢ ⎥ . We may treat a nematic liquid crystal iϕ ⎥ ⎣0 e ⎦ ⎣ sin(θ ) cos(θ ) ⎦
Where J ϕ = ⎢
cell as many birefrigent layers, so the total device can be represented by one Jones matrix J total
J total = J m • J m −1 ... • J 2 • J 1
(2-12)
z n nz ny
Ey
nx
x
Ex
y Fig.2.1 Local coordinate system E x - E y - z in the Jones matrix method.
28
The Jones vector of the output light can then be calculated:
r ⎡ E xout ⎤ ⎡ E xin ⎤ E out = ⎢ out ⎥ = J total • ⎢ in ⎥ ⎣⎢ E y ⎦⎥ ⎣⎢ E y ⎥⎦
(2-13)
In numerical computation, when the liquid crystal has a twist structure, it is convenient to assume a local co-ordinate system, in which the E x axis is held to be the same as the in-plane component of the director, and the E y axis is held in the x-y plane (Fig.2.1). It is important to mention that the E y axis is perpendicular to the E x -z plane, so E y is orthogonal to the director. Suppose the director in the k layer is nˆ ( k ) = (n x( k ) , n (yk ) , n z( k ) ) , the electric field in the local electric field co-ordinate system of the k layer isthen:
r ( k ) ⎡ E x( k ) ⎤ E out = ⎢ ( k ) ⎥ ⎣⎢ E y ⎦⎥ When
light
passes
through
the
(2-14)
k+1
layer
in
which
the
director
is
nˆ ( k +1) = (n x( k +1) , n (yk +1) , n z( k +1) ) , the electric field in the local co-ordinate system of the k+1 layer is:
r ( k +1) ⎡ E x( k +1) ⎤ ⎡e iϕ E out = ⎢ ( k +1) ⎥ = ⎢ ⎣⎢ E y ⎦⎥ ⎣ 0
(k ) 0⎤ ⎡ cos(θ ) sin(θ ) ⎤ ⎡ E x ⎤ ⎥⎢ ⎥ ⎢ (k ) ⎥ 1⎦ ⎣− sin(θ ) cos(θ )⎦ ⎣⎢ E y ⎦⎥
(2-15)
where ϕ is the phase difference of the e-wave and the o-wave:
ϕ=
2π
λ
(neff − no )d
(2-16)
and neff is the effective refractive index of the e-wave in the k+1 layer: 2
neff =
no ne 2
2 2
no [1 − (n z( k +1) ) 2 ] + ne (n z( k +1) ) 2
(2-17)
29
The Jones matrix method is a 2-by-2 matrix method; it is easy to be programmed and applied. When considering the light propagating from one layer to another, reflection is not considered; normally the Jones matrix method is not used to calculate reflectance.
2.3 Berreman's 4-by-4 Matrix Method Berreman’s 4-by-4 matrix method was initially applied to liquid crystals in 1972 [15]. It is based on the solution of Maxell’s equations without any significant approximations; it is an accurate way to describe the optical properties for any medium as long as the medium can be treated as a multi-layer structure and in each layer the optic axis is uniform. An isotropic media layer is also treated the same way. Berreman’s 4-by-4 method has been extensively used in the liquid crystal display (LCD) modeling, for the calculation of transmission, reflection, spectrum and chromaticity, contrast-viewing angle and other optical properties. It is used in most commercialized LCD modeling software. In this method, a Cartesian-coordinate system is used in which the liquid crystal cell is in the x-y plane. The dielectric tensor is assumed to be only a function of z. Considering an obliquely
r
r
→ →
incident light on a sample with a wave-vector k= ( k x , k y , k z ), the electric field E = E0 ei (ωt − k ⋅ r ) can be written as :
r r r i ( ωt − k x x − k y y ) i ( ωt − k x x − k y y ) E = E0 e ikZ z e = E ′( z )e
(2-18)
Similarly the magnetic field is described by:
r r r i (ω t − k x x − k y y ) i (ω t − k x x − k y y ) H = H 0 e ik Z z e = H ′( z ) e In order to make the fields dimensionless, we change variables such that:
r r r ε ij r E r H r k e= , h= , k′ = , ε ij′ = E0 H0 ε0 k0
(2-19)
30
After considering Maxwell’s equations:
r r ∇ × H = i ωε E
(2-20)
r r ∇ × E = −iωμH
(2-21)
We obtain:
r r 1 ∂X = iA X k 0 ∂z ⎡ ex ⎤ r ⎢e y ⎥ where X = ⎢ ⎥ ⎢ hx ⎥ ⎢ ⎥ ⎣⎢hy ⎦⎥
(2-22)
And A is the differential propagation matrix.
⎡ ε ′zx k ′x ⎢ ε ′zz ⎢ ⎢ ε ′zx k ′y ⎢ ε ′zz ⎢ A=⎢ ε′ ε′ ⎞ ⎛ ⎢⎜⎜ k x′ k ′y + ε ′yx − yz zx ⎟⎟ ε ′zz ⎠ ⎢⎝ ⎢ 2 ⎢ ⎛⎜ k ′ 2 − ε ′ + ε ′zx ⎞⎟ xx ⎢ ⎜ y ε ′zz ⎟⎠ ⎣ ⎝
ε ′zy k ′x ε ′zz ε ′zy k ′y ε ′zz
− k x′ k ′y
⎛ ε′ 2 ⎞ ⎜ − k ′ 2 + ε ′ − zy ⎟ yy ⎜ x ε ′zz ⎟⎠ ⎝ ε′ ε′ ⎞ ⎛ ⎜⎜ − k ′y k x′ − ε ′xy + zx zy ⎟⎟ ε ′zz ⎠ ⎝
2 ⎤ k x′ −1⎥ ε ′zz ε ′zz ⎥ 2 k ′y k x′ k ′y ⎥ − +1 ⎥ ε ′zz ε ′zz ⎥ ε ′yz k ′y − ε ′yz k x′ ⎥ ⎥ ε ′zz ε ′zz ⎥ ⎥ ε ′xz k ′y ε ′xz k x′ ⎥ − ε ′zz ε ′zz ⎥⎦
(2-23)
−1 To simplify the equation, A is diagonalized to : A ′ = S A S ; with 4 eigenvalues λ1 , λ 2
, λ3 , λ 4 , which represent the forward and backward propagating ordinary and extra-ordinary waves explicitly written as:
λ1 = (ε 0 − m 2 )1 / 2
λ2 =
n x mt 2
nx + n y
2
(2-24)
(
+ ε t − m 2ε t ε f / ε o ε e
)
1/ 2
(2-25)
31
λ3 = - λ1 λ4 =
(2-26)
n x mt 2
nx + n y
2
(
− ε t − m 2ε t ε f / ε o ε e
)
1/ 2
(2-27)
The four associated eigenvector will be given as:
ψ 1,3
ψ 2, 4
where
− ny ⎤ ⎡ ⎢n − mn / λ ⎥ 1 z 1, 3 ⎥ ⎢ x = sin(ar cos(n z )) ⎢ mn z − λ1,3 n x ⎥ ⎥ ⎢ ⎢⎣ − ε o n y / λ1,3 ⎥⎦
mn z λ 2, 4 ⎤ ⎡ m2 ( 1 )n x − − ⎢ εo ε o ⎥⎥ ⎢ 1 ny = ⎥ ⎢ sin(ar cos(n z )) ⎢ ⎥ − λ 2, 4 n y ⎥ ⎢ λ 2, 4 n x − mn z ⎥⎦ ⎢⎣
εt =
1 nz
2
εo
εf =
+
2
(2-29)
(2-30)
εe
ε o n y 2 + ε t nx 2 1 − nz
mt = m(1 −
m=
1 − nz
(2-28)
2
εt )ctg (ar cos(n z )) εo
kx = sin(θ i ) k0
(2-31)
(2-32)
(2-33)
and where θ i is the angle of incidence in vacuum. A special case should to be mentioned is when
n z =1; then the four eigenvalues and the four eigenvectors cannot be used. They are instead defined as:
32
λ1 = (ε 0 − m 2 )1 / 2
λ 2 = ε o (1 −
m2
εe
(2-34)
)
(2-35)
λ3 = - λ1
(2-36)
λ4 = - λ2
(2-37)
ψ 1,3
⎡ 0 ⎤ ⎢ 1 ⎥ =⎢ ⎥ ⎢λ1,3 ⎥ ⎢ ⎥ ⎣ 0 ⎦
(2-38)
ψ 2, 4
⎡− λ 2, 4 / ε o ⎤ ⎢ ⎥ 0 ⎥ =⎢ ⎢ ⎥ 0 ⎢ ⎥ 1 ⎣ ⎦
(2-39)
−1 The matrices S and S are given by:
S = (ψ 1 ,ψ 2 ,ψ 3 ,ψ 4 )
(2-40)
S −1 = N −1 S T M
(2-41)
T where S is the transpose of S and
0 ⎡0 0 ⎢0 0 − 1 M =⎢ ⎢0 − 1 0 ⎢ 0 ⎣1 0
1⎤ 0⎥⎥ 0⎥ ⎥ 0⎦
⎡ψ 1T Mψ 1 ⎤ 0 0 0 ⎢ ⎥ T 0 ψ 2 Mψ 2 0 0 ⎢ ⎥ N= T ⎢ 0 ⎥ 0 ψ 3 Mψ 3 0 ⎢ ⎥ 0 0 ψ 4 T Mψ 4 ⎦⎥ ⎣⎢ 0
(2-42)
(2-43)
33
Now, (2-43) becomes:
r r 1 ∂Y = iA ′Y k 0 ∂z r
(2-44)
r
Where Y = S −1 X . This equation can be solved analytically by: r r r Y ( z + Δz ) = e ik0 ΔzΛ Y ( z )
(2-45)
r r r X ( z + Δz ) = ( S e ik0 ΔzΛ S −1 ) X ( z )
So
r
e ik0 ΔzΛ
where
⎡ e ik 0 Δzλ1 ⎤ ⎢ ik0 Δzλ2 ⎥ e = ⎢ ik Δzλ ⎥ ⎢e 0 3 ⎥ ⎢ ik0 Δzλ4 ⎥ ⎢⎣e ⎥⎦
(2-46)
(2-47)
In the numerical computation, we divide the sample into n slabs, each of which has a thickness Δz, and Δz is small enough for the dielectric tensor in each slab to be treated as constant. The fields that exit the surface are given by:
r r X out = b X in
(2-48)
r
r
b = ( S e ik0 ΔzΛ S −1 ) n * ...... * ( S e ik0 ΔzΛ S −1 )1
(2-49)
r
We select a coordinate system in which the wave vector k is in the x-z plane, so k y = 0 .
)
Considering a liquid crystal material whose director is n = ( n x , n y , n z ) , then the dielectric tensor is given by:
ε ij′ = no 2δ ij + (ne 2 − no 2 )ni n j
(2-50)
Here no , ne are the ordinary and extrordinary refraction indices if the material is a pure liquid crystal.
34
In order to meet the boundary conditions, considering the reflection of light, the field at the input surface is:
r r r X in = X i + X r r
(2-51)
r
where X i is the incident field and X r is the reflected field at the input surface. The transmited light is described by :
r r r Xt = bXi + bXr
(2-52)
The wave vector of the incident, reflected and transmitted waves are related as:
r r r ) ) k r = k i − 2(k i ⋅ N ) N r
)
(2-53)
r
where N is the surface normal and k t = k i if the sample is surrounded by the same isotropic medium at the input and the output surfaces. The magnetic field components can be written in terms of the electric field, since
We have
r r r ∂B = −iωμH ∇× E = − ∂t
(2-54)
r r r k × E = ωμH
(2-55)
r ) r ωμ r k ×e = h = zh Z0
Or
where Z 0 =
Similarly
E E0 , Z = H0 H
and z =
(2-56)
Z Z0
) v er −k ×h = z
From (2-56) and (2-57), we can solve for hx and h y in terms of e x and e y . In general,
(2-57)
35
⎡ 1 e ⎡ x⎤ ⎢ 0 ⎢e ⎥ ⎢ k k ⎢ y ⎥ = ⎢− x y ⎢ hx ⎥ ⎢ zk z ⎢ ⎥ ⎢1 − k 2 y ⎣⎢h y ⎦⎥ ⎢ ⎢⎣ zk z
0 0⎤ 0 0⎥⎥ ⎡e x ⎤ ⎢e ⎥ 0 0⎥ ⎢ y ⎥ ⎥⎢ 0 ⎥ ⎥⎢ ⎥ 0 0⎥ ⎣ 0 ⎦ ⎥⎦
0 1 2 1− kx − zk z kxky zk z
(2-58)
Given an incident light field, the transmitted and reflected light fields are:
⎡ eix ⎤ ⎡ erx ⎤ ⎡0⎤ ⎡eix ⎤ ⎡ etx ⎤ ⎡etx ⎤ ⎢e ⎥ ⎢e ⎥ ⎢ ⎥ ⎢e ⎥ ⎢ e ⎥ ⎢e ⎥ ⎢ iy ⎥ = α ⎢ iy ⎥ , ⎢ ty ⎥ = β ⎢ ty ⎥ and ⎢ ry ⎥ = γ ⎢ 0 ⎥ ⎢hix ⎥ ⎢hrx ⎥ ⎢erx ⎥ ⎢ 0 ⎥ ⎢ htx ⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣hiy ⎥⎦ ⎢⎣hry ⎥⎦ ⎢⎣ery ⎥⎦ ⎣ 0 ⎦ ⎢⎣hty ⎥⎦ ⎣0⎦
where
⎡ 1 ⎢ 0 ⎢ ⎢ k ix k iy α = ⎢− z i k iz ⎢ 2 ⎢ 1 − k iy ⎢ z i k iz ⎣
0 1 2 1 − k ix − z i k iz k ix k iy
⎡ 1 ⎢ 0 ⎢ ⎢ k tx k ty β = ⎢− z t k tz ⎢ 2 ⎢ 1 − k ty ⎢ z t k tz ⎣
0 1 2 1 − k tx − z t k tz k tx k ty
⎡0 ⎢0 ⎢ ⎢ γ = ⎢0 ⎢ ⎢0 ⎢⎣ From (2-58) we get:
0 0
z t k tz 1 0
0 − 0
z i k iz
k rx k ry
z r k rz 2 1 − k ry z r k rz
(2-59)
0 0⎤ 0 0⎥⎥ 0 0⎥ ⎥ ⎥ 0 0⎥ ⎥ ⎦
(2-60)
0 0⎤ 0 0⎥⎥ 0 0⎥ ⎥ ⎥ 0 0⎥ ⎥ ⎦
(2-61)
0 ⎤ ⎥ 1 2 ⎥ 1 − k rx ⎥ − z r k rz ⎥ ⎥ k rx k ry ⎥ z r k rz ⎥⎦
(2-62)
36
⎡0⎤ ⎡eix ⎤ ⎡etx ⎤ ⎢0⎥ ⎢e ⎥ ⎢e ⎥ ty ⎥ iy ⎥ ⎢ ⎢ β + bγ ⎢ ⎥ = bα ⎢erx ⎥ ⎢0⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0⎦ ⎣0⎦ ⎣⎢ery ⎦⎥
Noting
⎡ etx ⎤ ⎡etx ⎤ ⎢e ⎥ ⎢e ⎥ ty ty β⎢ ⎥ = β⎢ ⎥ ⎢erx ⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ery ⎥⎦ ⎣0⎦
and
(2-63)
⎡ etx ⎤ ⎡0⎤ ⎢e ⎥ ⎢0⎥ ty ⎥ ⎢ γ =γ ⎢ ⎥, ⎢erx ⎥ ⎢erx ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ery ⎥⎦ ⎢⎣ery ⎥⎦
We have
⎡ etx ⎤ ⎡eix ⎤ ⎢e ⎥ ⎢e ⎥ ty ⎥ ⎢ = b α ⎢ iy ⎥ (β − b γ ) ⎢erx ⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ery ⎥⎦ ⎣0⎦
(2-64)
So,
⎡ etx ⎤ ⎡eix ⎤ ⎢e ⎥ ⎢ ⎥ ⎢ ty ⎥ = ( β − b γ ) −1 b α ⎢eiy ⎥ ⎢erx ⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎣0⎦ ⎣⎢ery ⎦⎥
(2-65)
Now, from (2-65), we can calculate both the trasmitted and the reflected electric field, and the magnetic field will also be calculated. The light intensities of both the transmitted and reflected light will be known. Fig.2.2 shows the transmittances of circularly-polarized light from two cholesteric liquid crystal cells. For the cell whose twist sense is identical to the handedness of the circularlypolarized light, there is a reflection band with a band width of ( ne - no )p and a reflection wavelength center at ( ne + no )p/2. The parameters used here are: thickness d = 6.0 μm, pretilt angle θ p =0°, d/p ratio = 15, elastic constant: k11 = 15.5 (PN), k 22 =14.0 , k 33 =28.0,
no =1.5, ne =1.7. The entrance and exit media are air (n=1).
37
100 90
Transmittance ( % )
80 70 60 50 40 30 20 10 0 500
550
600
650
700
750
800
Wavelength (nm )
Fig. 2.2. Spectral response of clock-wise circularly-polarized light from two cholesteric cells. Black: twist sense of the cell is clockwise; Red: twist sense of the cell is counter-clockwise
2.4 Light Propagation in Periodic Media
2.4.1 Introduction to Grating Gratings are periodic media that have been widely used in optical applications. A grating can fall into one of two categories: an amplitude grating, which spatially modulates the intensity of light, and a phase grating, which spatially modulate the phase of light. The latter can be realized by periodically modulating the thickness or the refractive index of a medium, as in holographic polymer dispersed liquid crystals (HPDLC), which are the focus of our research. In a periodic medium, the refractive index exhibits a translational symmetry:
38
r r r n( r ) = n( r + a ),
(2-66)
r
where a is a constant vector. For a one dimensional phase grating, the refractive index satisfies
n( z ) = n( z + mΛ) , where m is an integer and Λ is the period of the grating, or pitch. The refractive index can be expanded in a Fourier series: ∞ ⎛ 2 m πz ⎞ n( z ) = n0 + ∑nm cos⎜ ⎟. ⎝ Λ ⎠ m =1
(2-67)
z
x θ
kd
θ
ki
θ θ
2ksin(θ)
Λ
Fig. 2.3. Diffraction of a monochromatic plane wave by an optically thick grating.
r r r ki = kd = k . The component of momentum perpendicular to the grating vector k is conserved.
The simplest case is a sinusoidal grating, where only n0 and the first Fourier component are non-zero:
39
⎛ 2πz ⎞ n( z ) = n0 + n1 cos⎜ ⎟. ⎝ Λ ⎠
(2-68)
Consider the monochromatic wave diffracted from a periodic medium: the index modulation is lumped to an array of planes separated by equal distance, as illustrated in Fig. 2.3. Assuming that the number of planes is infinite, the reflections from these planes are specular. The path difference for rays reflected from adjacent planes is 2Λ sin θ , where θ is the angle between the incident beam and the grating planes. The interference is constructive when the phase difference for neighboring reflected rays is an integer multiple of 2π , which leads to Bragg's law:
2Λ sin θ = N
λ
(2-69)
n0
where N is an integer, and n0 is the average refractive index of the medium. The angle θ satisfying (2-69) is defined as the Bragg angle.
r
r
In k − space, the grating period is represented by the grating vector k , where k ≡ k = 2π/Λ
r
r
r
r
is known as the grating wave number. Bragg's law becomes ki + k r = Nk , and Nk is a vector in the (one-dimensional) reciprocal lattice. The component of the momentum perpendicular to the
r
grating vector k is conserved. In a thin grating, as depicted in Fig. 2.4, the transverse dimension of the periodic medium is relatively small compared to the beam size and/or the wavelength. Due to the finite size of the grating planes, the diffraction from each plane should be considered in addition to the specular reflection. Constructive interference also occurs in directions other than the specular reflection direction. The condition for constructive interference is:
Λ sin θ + Λ sin θ ′ = N (λ/n).
(2-70)
40
A dimensionless parameter (Klein-Cook parameter), Q ≡ 2πλL/n0 Λ2 is introduced to qualitatively define thick or thin gratings, where L is the thickness of the grating. When Q >> 1 , the grating is classified as a thick grating, and when Q << 1 , the grating is classified as a thin grating.
z θ
θ'
x Λ
Fig. 2.4. Diffraction of a monochromatic plane wave by an optically thin grating. θ may not be equal to θ ′ .
2.4.2 Coupled Wave Theory [16] Coupled wave theory was introduced by Kogelnik in 1969 to treat thick dielectric gratings ( Q >> 1 ) [17]. In this approach, the refraction index profile was assumed to be sinusoidal and the dielectric medium is isotropic. The light propagating in the periodic medium is assumed to be polarized light and the diffracted light is coupled with the incident beam. Coupled wave theory is known for its simplicity and versatility to theoretically simulate grating diffraction phenomena. Let us consider the light propagation in a one-dimensional periodic medium with a sinusoidal index modulation described by (2-68). Assume: (1) plane waves have uniform amplitudes; (2) the grating extends laterally to infinity ( Q >> 1 , thick grating); and (3) the electric fields of both the
41
incident light and the diffracted light are parallel to the grating planes and perpendicular to the grating vector, as in Fig. 2.5. In other words, only the transverse electric (TE) mode is discussed. Based on these three assumptions, the electric field of the incident and diffracted beams can be written as:
[
]
[
r r r r E = A1 exp i (ωt − k1 ⋅ r ) + A2 exp i (ωt − k 2 ⋅ r ) r
]
(2-71)
r
where k1 and k 2 are the wave vectors, A1 and A2 are the amplitudes of the electric fields, and
ω is the frequency of the beams. The wave vectors and the frequency of the light are connected r
r
by: | k1 |=| k 2 |= n0ω/c . When the x − z plane is set as the plane of incidence, the wave vectors
r
r
can then be written in Cartesian coordinates: k1 = (α1 ,0, β1 ) and k 2 = (α 2 ,0, β 2 ) . Therefore, the electric field is:
E = A1 exp[i (ωt − α1 x − β1 z )] + A2 exp[i (ωt − α 2 x − β 2 z )].
(a) Transmission grating
(2-72)
(b) Reflection grating z
y
x
L
θ2
θ1
0
0
L
θ1 θ2
Fig. 2.5. Diffraction by optically thick gratings. (a) Transmission grating; (b) Reflection grating.
42
The electric field satisfies the wave equation:
(∇ 2 +
ω2 c
2
n 2 ) E = 0,
(2-73)
where n is the refractive index of the medium.
2.4.3. Coupled Wave Theory for Transmission Gratings For a transmission grating with infinite dimension in the z direction, the amplitudes of the electric fields are functions of x only. In a medium with a sinusoidal index modulation, as described in (2-68), the wave equation of the electric field can be represented by:
⎡⎛ d 2 d ⎞ ⎤ ⎢⎜⎜ 2 − 2iα j ⎟⎟ A j ⎥ exp i (ωt − α j x − β j z ) ∑ dx ⎠ ⎦ j =1,2 ⎣⎝ dx
= − −
ω2 c2
ω2 c2
[
]
[
]
2n0 n1 cos Kz ∑ A j exp i (ωt − α j x − β j z )
(2-74)
j =1,2
[
]
n12 ∑ A j exp i (ωt − α j x − β j z ) . j =1,2
−1 −3 The n12 term is neglected because n1 is usually of the order of 10 to 10 . Assuming the
energy interchange between the transmitted wave and the reflected wave is slow (i.e., A1 and A2 are slowly varying functions), the second order derivatives of A1 and A2 are neglected. (2-74) becomes:
− 2 iα 1
dA1 dA exp(− iα1 x − iβ1 z ) − 2iα 2 2 exp(− iα 2 x − iβ 2 z ) dx dx
ω2
n0 n1 [exp(− iKz ) + exp(iKz )][ A1 exp(− iα1 x − iβ1 z ) c2 + A2 exp(− iα 2 x − iβ 2 z )].
= −
(2-75)
By multiplying (2-75) with exp ( −iα j x − iβ j z ) , with j = 1 or 2 , and integrating over z , two first order differential equations are obtained:
43
dA1 dx dA2 dx
= − iκ 12 A2 e −iΔαx , (2-76)
= − iκ 21 A1e
iΔαx
,
where:
κ 12 =
πn1
λ cosθ1
, κ 21 =
Δα = α 1 − α 2 =
2π
λ
πn1 , λ cos θ 2
(2-77)
n0 (cos θ 2 − cos θ1 ),
and the following condition is satisfied:
β 2 = β1 ± K .
(a) Δα = 0, phase matched
(2-78)
(b) Δα ≠ 0, phase mismatch
z
Δα
z k2 k1
k2
K x
K x
k1
Fig. 2.6. K-space of transmission gratings (a) Ideal phase match Δα = 0 ; (b) not ideal phase match Δα ≠ 0 .
The phase mismatch Δα determines the coupling and energy exchange between the beams. When cos θ 2 = cos θ1 , Δα = 0 , there is an ideal phase match, as shown in Fig. 2.6. The trivial solution
θ 2 = θ1
corresponds
to
the
transmitted
beam;
the
non-trivial
solution
θ 2 = −θ1 corresponds to the diffracted beam. The Bragg condition is satisfied when Δα = 0 :
44
⎛ λ ⎞ ⎟⎟ ≡ θ B , ⎝ 2n0 Λ ⎠
θ1 = −θ 2 = ± arcsin⎜⎜
(2-79)
where θ B is known as the Bragg angle, and the wave equation becomes:
dA1 = −iκA2 , dx
κ=
where:
dA2 = −iκA1 , dx
πn1 . λ cos θ B
(2-80)
(2-81)
The solutions to (2-80) are:
A1 ( x ) =
A2 ( x ) =
A1 (0) cos κx − iA2 (0)sin κx,
A2 (0) cos κx − iA1 (0)sin κx,
where A1 (0 ) and A2 (0) are the electric field amplitudes of the incident beam and the diffracted beam at the incident surface. For a single incident beam ( A2 (0 ) = 0 ), the solutions become:
A1 ( x ) = A1 (0 ) cos κx,
A2 ( x ) = −iA1 (0 )sin κx.
(2-82)
Energy is conserved in this solution. The diffraction efficiency of the grating is:
η=
I diffracted I incident
=
A2 (L ) A1 (0 )
2
2
= sin 2 κL,
(2-83)
where L is the thickness of the cell. For the ideal gratings in which all of our assumptions are satisfied, the diffraction efficiency reaches its first maximum when κL = π/2 . When κL > π/2 (over-modulation), the diffraction efficiency decreases.
45
z
Δα
z
Δα k2
k2
K
K K x
x k1
k1
(b)
(a)
Fig. 2.7. Phase mismatch in transmission gratings due to (a) angular deviation; and (b) wavelength deviation.
When the incident angle deviates slightly from the Bragg angle, θ1 = −θ B + Δθ , the diffraction angle determined by (2-78) is:
θ 2 = θ B + Δθ .
(2-84)
The deviation from the Bragg angle results in a phase mismatch given by:
Δα = −2kΔθ sin θ B = − KΔθ ,
(2-85)
where k = 2πn0 /λ , as illustrated in Fig. 2.7(a). For the mismatched case, solutions to the coupled wave equation (2-76) are:
A1 ( x ) =
⎛ Δα A1 (0 )exp⎜ − i 2 ⎝
⎞ ⎡ Δα ⎤ x ⎟ ⎢i sin sx + cos sx ⎥, ⎠⎣ 2s ⎦
A2 ( x ) =
⎛ Δα ⎞ κ 21 x⎟ sin sx, − iA1 (0 )exp⎜ i ⎝ 2 ⎠ s 2
where
⎛ Δα ⎞ s2 = κ 2 + ⎜ ⎟ . ⎝ 2 ⎠
(2-86)
46
The corresponding diffraction efficiency is: 2 1/2 ⎫ ⎧ ⎡ ⎤ ⎪ α Δ ⎪ ⎛ ⎞ 2 = κ + L 1 ⎜ ⎟ sin ⎢ ⎥ ⎬. ⎨ 2 2 ⎝ 2κ ⎠ ⎦⎥ ⎪ A1 (0) cosθ1 ⎛ Δα ⎞ ⎢ 2 ⎪ ⎣ ⎭ ⎩ κ +⎜ ⎟ ⎝ 2 ⎠
A2 (L ) cosθ 2 2
η =
κ2
(2-87)
The diffraction efficiency decreases with an increase of phase mismatch, as illustrated in Fig. 2.8. The angular dependence of the diffraction efficiency can be derived from (2-87). The diffraction efficiency drops to 1/2 when the deviation of the incident angle from the Bragg angle is:
Δθ = Δθ1/2 =
2κ κΛ = . K π
(2-88)
1
Diffraction Efficiency
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -10
-8
-6
-4
-2
0
2
4
6
8
10
Phase Mismatch
Fig. 2.8. The diffraction efficiency decreases with the increase of phase mismatch. Red: kL=2π/3, Black: kL=π/2. Phase Mismatch: Δα/2k.
A deviation in wavelength, Δλ , can also lead to a phase mismatch, as shown in Fig. 2.7 (b). The phase mismatch is given by:
47
Δα = −
π
Δλ . Λ n0 Λ cos θ B
(2-89)
The diffraction efficiency decreases to half the maximum value when the wavelength of incident beam changes by:
Δλ 1 = 2
2κΛ
π
n0 Λ cos θ B .
(2-90)
Therefore, Δλ 1 can represent the spectral bandwidth. Fig. 2.9 plots the diffraction efficiency as a 2
function of the phase mismatch for different cell thicknesses. The above discussion is valid for TE waves. For more general cases, the electric field is a vector field, and the dielectric constant is a tensor field. The periodic medium is described by
( r) r
ε ′ = ε + ε 1 cos k ⋅ r
(2-91)
where ε is the average dielectric tensor, and ε1 is the tensor representing the amplitude of the periodic dielectric modulation. Following the same procedures, the coupled wave equations can be derived, which have the same form as those of the TE wave (2-76), but with different coupling constants:
κ12 = κ 21 =
π
r r p1 ⋅ ε1 p2 ,
2n1λ cosθ1ε 0
π 2n2λ cosθ 2ε 0
(2-92)
r r p2 ⋅ ε 1 p1 ;
and Δα is given by:
Δα = α 2 − α 1 ± K x .
(2-93)
r K x is the x component of the grating vector k ; and n1 and n2 are the refractive indices r
[(
r r
associated with p1 exp i ωt − k1 ⋅ r
)] and ps
2
[(
)]
r r exp i ωt − k 2 ⋅ r , respectively.
48
1
Diffraction Efficiency
0.9 0.8 0.7
kL=1.0
0.6
kL=1.5
0.5
kl=2.0
0.4
kL=2.5
0.3 0.2 0.1 0 -1
-0.5
0
0.5
1
Phase Mismatch
Fig. 2.9. Diffraction efficiency as a function of phase mismatch (Δβ/2k) for reflection gratings with different thicknesses kL.
The diffraction efficiency is obtained following the same approach for TE waves. In the case of TM waves, the coupling constant is given by κ p = κcosΘ , where Θ is the angle between the electric fields of the incident beam and the diffracted beam [17]. The diffraction efficiency of the TM wave has the same form as the TE wave, but with a smaller coupling constant.
2.4.4. Coupled Wave Theory for Reflection Gratings For reflection gratings, the boundary conditions require that α1 = α 2 (i.e., θ1 = −θ 2 ), and that the amplitude of both the incident and reflected beams be functions of z only. The coupled wave equation for reflection gratings is deduced by following a similar approach to that of transmission gratings:
49
dA 1 dz dA 2 dz
− i κ A 2 e − iΔ β z
=
(2-94)
i κ A1 e
=
iΔ β z
where:
Δβ = β 2 − β 1 ± K ,
(2-95)
and:
κ=
πn1 . λ sin θ1
(2-96)
The diffraction efficiency of a reflection grating is derived by solving (2-94). In the case of a perfect phase match ( Δβ = 0 ), the diffraction efficiency is:
η = tanh 2 κL.
(2-97)
Similar to the case of transmission gratings, the coupling constant of the TM wave in a reflection grating is given by κ p = κcosΘ , where Θ is the angle between the electric fields of the incident beam and the diffracted beam. The diffraction efficiency of the TM wave is always less than that of the TE wave for reflection gratings. Deviation from the Bragg condition results in a phase mismatch ( Δβ ≠ 0 ) given by:
Δβ =
2π 4π − n0 sin θ1 , Λ λ
(2-98)
when θ1 > 0 , the diffraction efficiency of a reflection grating with a phase mismatch is: 2
κ sinh 2 sL 2 ⎛ Δβ ⎞ 2
η=
,
(2-99)
s 2 cosh sL + ⎜ ⎟ sinh 2 sL 2 ⎝ ⎠ where:
2
⎛ Δβ ⎞ s = κ −⎜ ⎟ . ⎝ 2 ⎠ 2
2
Diffraction efficiency as a function of phase mismatch is illustrated in Fig. 2.10.
(2-100)
50
1 Diffraction Efficiency
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
1.5
2
2.5
3
3.5
4
kL
Fig. 2.10. Diffraction efficiency of a reflection grating as a function of κL for an ideally matched phase Δβ=0.
From (2-97) and (2-99), we notice that over modulation will not occur in reflection gratings. Fig. 2.11 models a reflective HPDLC grating. The period of the grating is 200 nm and the thickness of the grating is 5 μm. In 4x4 modeling, the liquid crystal layer and the polymer layer are assumed to be pure, and their refractive indices are 1.58 and 1.52, respectively. In the coupled wave theory calculation, the refractive indices are assumed to have a sine function modulation as described in (2-68) with n0 = 1.55 , n1 = 0.03 .
51
Modeling of Reflection Grating 0.7
Reflection
0.6 0.5 0.4
4x4 modeling coupled wave theory
0.3 0.2 0.1 0 500
550
600
650
700
750
800
Wavelength ( nm )
Fig. 2.11. Modeling of reflection grating at normal incidence. Berreman’s 4x4 method and coupled wave theory calculation are compared.
2.5 Summary In this chapter, we have investigated light propagation in stratified materials. Based on Maxwell’s equations, both the Jones matrix and Berreman’s 4x4 method are discussed and modeling was given to explain the grating effect of a cholesteric and HPDLC cells. Furthermore, coupled wave theory was discussed with a focus on explaining diffraction by one-dimensional transmission and reflection mode phase gratings. The effects of various factors on diffraction efficiency were mathematically investigated. While only isotropic gratings are discussed in detail, the anisotropic gratings can also be treated with a similar method. The results of this chapter can be applied to the HPDLC gratings, which can be approximated as phase gratings.
CHAPTER 3
Holographic Polymer Dispersed Liquid Crystal
3.1 Introduction to Holography In conventional photography, only the distribution of light intensity of the object is recorded and reconstructed. As nature of light and its application are further researched, the idea of recording the complete wave field, both the phase and the amplitude of light, is inspired. Reconstructing the same wave front from a hologram reproduces the image of the object. Since most recording media respond only to the intensity of light, the phase information is transformed into intensity during a holographic recording by the interference of monochromatic coherent light. So, in definition, a hologram is the recording of the interference pattern of multiple beams where the recording medium is positioned. The interference of coherent light rays can be described by the superposition of the electric
r
N
fields: E =
r
∑E e
− iω t
j
. The holographic recording time is usually much longer than the period of
j =1
the
light.
(
The
average
light
intensity
over
a
period
is
)
r r r r r r 1 N N r r r r I (r ) ∝ E (r ) ⋅ E * (r ) = ∑∑El (r )Em (r ), where I (r ) is the spatial distribution of the light 2 l =1 m =1
r r
r r
r r
intensity, E (r ) the spatial distribution of the electric field, and El (r ) and Em (r ) are the electric field components of the different recording beams. The first holograms were demonstrated by Gabor in 1948 [18,19]. In Gabor's in-line holography setup, the interference pattern was generated by both the transmitted beam passing through the object and the scattered light, as shown in Fig. 3.1. Gabor’s holography setup is only 52
53
applicable to objects with high transmittance and the hologram is usually weak; two superimposed, out-of-focus twin images could be observed.
virtual hologram real image image
object recording medium
point source
(b)
(a)
Fig. 3.1 Optical setup for recording (a) and reconstructing (b) in-line hologram.
object beam
real image
virtual image
hologram
recording medium
reference beam (a)
reference beam (b)
Fig. 3.2. The optical setup for recording (a) and reconstructing (b) off-line hologram.
54
Off-axis holography was introduced by Leith and Upatnieks in 1962 [20], as illustrated in Fig. 3.2. In off-axis holography, the reference beam is separated from the illumination beam and is directed to the photographic plate at an angle with the object beam. When reconstructed, the real image and the virtual image are both located off the direct transmitted beam and are widely separated from each other. The image quality of the hologram can be improved by adjusting the intensity of both beams to increase the contrast ratio in the interference pattern.
3.2 Introduction to Holographic Polymer Dispersed Liquid Crystals Holographic polymer dispersed liquid crystals (HPDLCs) are the successors of traditional polymer dispersed liquid crystals (PDLC) that are comprised of a rigid polymer matrix, and randomly dispersed liquid crystal droplets inside the polymer matrix. Usually HPDLCs have a stratified structure with alternating layers of liquid crystal rich and polymer rich planes, formed by the holographic exposure of a mixture of liquid crystals and photo-polymerizable monomers. HPDLCs are normally treated as switchable phase gratings. The first switchable grating based on a PDLC was introduced by Margerum and co-workers at Hughes Research Labs in 1988 [ 21]. A periodically modulated distribution of the liquid crystal droplet density in the polymer network was generated under a masked UV exposure of the UV-curable PDLC mixture, the periodic index modulation in the PDLC film functioned as a transmission grating that could be deactivated by the realignment of the liquid crystal molecules in the LC droplets under the external electrical field. The first switchable PDLC transmission grating formed by holographic exposure was developed by Sutherland and co-workers in 1993 [22]. The prepolymer mixture was cured in the visible wavelength range. The first HPDLC reflection grating was reported by Tanaka and coworkers [23]. HPDLCs have found applications in displays [24, 25, 26] and optical telecommunication [27] with their switchability and their relatively simple fabrication process.
55
To fabricate a conventional HPDLC cell, two ITO glass substrates with anti-reflection (AR) coatings are assembled with uniforly spread spacers to control the cell gap. The recording medium, or PDLC mixture is sandwiched between these ITO glass substrates. The cell is exposed to the interference pattern generated by two monochromatic coherent laser beams generally carrying equal powers and polarizations. The light induces a phase separation during the photo polymerization of the monomers in the mixture and causes the liquid crystals to be phase separated into randomly oriented droplets in this process. On the other hand, as the lights create regions of complete constructive and destructive interference, the photo polymerization in the bright (constructive) area is faster than in the dark (destructive) region; the liquid crystals diffuse to the dark region and the monomers to the bright region. Finally, a spatially periodic distribution of polymer and liquid crystal (LC) droplets is generated. The average refractive index of the LC droplets, thus the refractive index difference between the LC rich region and polymer rich region, can be changed with a realignment of the liquid crystal molecules by an external electric field. In a normal mode HPDLC, refractive indices of the polymer and the LC droplet are unmatched without an external field, while in the reverse mode, the refractive indices of the polymer and the LC droplets are matched without an external field. In either case, an external electric field can switch the HPDLC from a holographic state to a non-holographic state and Vice Versa. The interference of two collimated laser beams is utilized to generate the interference pattern for the holographic exposure. The interference pattern of the two coherent monochromatic beams has the form:
[(
)
r r r r I (r ) = I1 + I 2 + 2 I1 I 2 cosθ cos k1 − k2 ⋅ r + φ0
]
(3-1)
where I1 and I 2 are the beam intensities; θ is the angle between the polarization direction of the
r
r
recording beams; k1 and k 2 are the wave vectors of the recording beams; and φ0 is the phase
56
difference of the two beams. The contrast ratio of the interference pattern can be described by the fringe constant:
V=
2 cosθ I 2 I 2 I1 + I 2
(3-2)
When I1 = I 2 and both recording beams are s-polarized, the fringe constant V is maximized. Based on the propagation direction of the diffracted light, conventional HPDLCs can be categorized into two types: (1) reflective HPDLCs, as shown in Fig. 3.3(a), where the diffracted light propagates to the same side of the incidence, and (2) transmissive HPDLCs, where the diffracted light propagates to the opposite side of the transmitted light, as shown in Fig. 3.3(b).
Fig. 3.3. Holographic configurations for (a) reflective mode and (b) transmissive mode HPDLCs.
If the recording beams are in the same plane as the two writing beams, as illustrated in Fig. 3.3, the interference pattern can be presented by: I = I 0 [1 + V cos(Λx )], where Λ =
λ 2n sin θ
is
the spatial period (pitch) of the interference pattern, or the pitch of the HPDLC, and the x axis is perpendicular to the HPDLC layers.
57
3.3 Transmission Mode HPDLCs The conventional transmission mode HPDLCs have grating planes oriented perpendicular to the cell surfaces. The interference pattern is generated by two coherent laser beams incident from the same side of the cell, as illustrated in Fig. 3.3(b); a stratified structure of polymer rich and liquid crystal rich layers is formed by the exposure. The pitch of the grating is determined by:
Λ=
λw
2nw sin θ w
(3-3)
where θ w is the incident angle inside the HPDLC, λw is the recording beam wavelength, and nw is
the
refractive
λr
index
at
= 2nr Λ cosθ ri nr =
λw . The wavelength reflected by the grating is: λw nr cosθ ri λ n sin θ r = 2nr Λ sin θ r nr = w r nw cosθ wi nw sin θ w
where nr is the refractive index of the HPDLC at the reading beam wavelength λr , θ ri is the reading beam incident angle inside the HPDLC, and θ r is the angle between the incident (or diffracted) beam and the grating planes. The diffraction properties highly depend on the difference between the refractive index of the polymer rich layer and the liquid crystal rich layer. Fig. 3.4 is a schematic illustration of the operation principle of transmission mode HPDLCs. In the field-off state, light is strongly diffracted by the grating; while in the field-on state, light passes through as the grating is deactivated by the electric field. The polarization dependence of the diffraction efficiency of transmission gratings was first revealed by Sutherland and co-workers. Their research showed that the coupling coefficient for the TE wave (s-polarization) and the TM wave (p-polarization) are different and the index modulation of the TM wave was larger than that of the TE wave [22], which indicated that the orientation of LC droplets has a preferred direction perpendicular to the grating planes.
58
V
V index matching layer
polymer rich
LC rich
(a)
AR coating (b)
Fig. 3.4. Transmission HPDLCs in the (a) on state; (b) off state.
Some HPDLC transmission gratings with liquid crystals highly aligned along specific directions were further developed by Vardanyan and co-workers [28]. Polymer scaffoldings formed in an incomplete phase separation interconnect the neighboring polymer rich layers and provide a strong alignment for the liquid crystal molecules in the liquid crystal rich layers. An effective polymer field is used to mathematically explain the effect of this alignment, which is responsible for the electro-optic response and optical anisotropy of the HPDLCs. The dielectric anisotropy of the LC droplets in transmission mode HPDLCs was investigated by Jazbinsek and co-workers [29]. Ellipsoid shaped droplets in the LC rich plane was found through SEM. The index modulation is revealed to be higher along the grating vector direction, and the dielectric anisotropy higher in short pitch HPDLCs, by fitting the measurement data of diffraction efficiency to the coupled wave theory.
59
(a)
s
p
G2
PR G1 s
p
Fig. 3.5. Polarization independent electro-optical device based on stacking two polarization sensitive transmission mode HPDLCs (G1 and G2) and a polarization rotator (PR). Courtesy of Boiko and co-workers [30].
A polarization insensitive HPDLC device was developed by Boiko and co-workers [30] using two identical transmission HPDLCs with a polarization rotator in between, as illustrated in Fig. 3.5. The polarization rotator fabricated from reactive mesogen film can rotate the polarization of certain wavelength of light by π/2 , thus the s-polarized light traveling through the first HPDLC is transformed to p-polarized light by the polarization rotator and is then diffracted by the second HPDLC, while the p-polarized light diffracted by the first HPDLC is transformed to s-polarized light and then passed through the second HPDLC. The contrast ratio of the diffraction efficiencies of p-polarization and s-polarization of the single transmission HPDLC in this device is high (~30) at 1550 nm. The diffraction efficiency of the stacked device is as high as 98% for non-polarized light. An SEM photograph of a transmission mode HPDLC operating at ~ 1500 nm is illustrated in Fig. 3.6.
60
Fig..3.6. The SEM photograph of a transmission mode HPDLC operating at 1500 nm. The top substrate is pealed before the SEM image is captured.
3.4 Reflection Mode HPDLC Reflection mode HPDLCs are recorded by laser beams incident from different sides of the cell, as illustrated in Fig. 3.3(a). When the two recording beams are symmetric, the periodic interference pattern is along the cell normal direction and the resulting grating planes are parallel to the cell surfaces. Unlike the transmission mode HPDLCs, the LC droplets in reflective HPDLCs are assumed to be randomly oriented in the plane parallel to the cell surface, since normally no polarization dependence of reflection efficiency is identified in reflective HPDLCs. Consequently, the average refractive index of the LC droplets is given by:
nLC =
2no2 + ne2 3
(3-4)
61
Fig. 3.7. The SEM photograph of a reflective HPDLC operating at ~1500 nm. The image is of the cross section of the cell.
However, as the result of an incomplete phase separation, there are still liquid crystals in the polymer rich layer, and the polymer networks extends into the LC rich layer. Therefore, the average refractive index of both the polymer rich region, n p , and the LC rich region, nLC , are in the range defined by nLC and n p (the refractive index of the polymer). The index modulation
n1 ≡ nLC − n p is less than nLC − n p , and is typically in the order of 10−2 . For liquid crystals with a positive dielectric anisotropy ( Δε > 0 ), when no ≈ n p and the LC molecules are aligned parallel to the applied electric field, the HPDLC cell will appear transparent to the reading light. Fig. 3.7 is a SEM photograph of the polymer network of a reflective HPDLC operating at around 1500 nm. The SEM image was taken after the liquid crystals were washed away. The stratified polymer network and cavities that used to be occupied by the liquid crystal are clearly identified.
62
3.5 Variable-Wavelength HPDLC So far most HPDLC related electro-optical devices function as an electrically switchable grating with two states; the grating state (or switch-off state) without an external electric field and transparent state (or switch-on state) with an applied external electric field. Recently, a variablewavelength switchable Bragg grating formed in polymer-dispersed liquid crystals was presented by the Display and Photonics Laboratory of Brown University [31]. This innovative device can switch between two distinctly different reflecting states.
A
blended monomer system was prepared by mixing Ebecryl 4866 with Ebecryl 8301 (both from UCB Radcure) in a ratio of 2:1. This was then mixed with the liquid crystal BL038 (EM Industries) with a weight ratio of 50:36:14 for the monomers: liquid crystal : photoinitiator solutions, respectively. This was homogenized and mixed with a Tergitor Min-Foam 1X surfactant from Union Carbide(3 wt%). The mixture was sensitized with a Rose Bengal and NPhenylglycine in 1-vinyl-2-pyrrolidone photoinitiator so that the polymerization could be carried out in the visible wavelength range with an Ar + laser. A cell was formed by drop filling this mixture between ITO-coated glasses. 5 μm spacers were used to control the cell gap. In the zero field state (Fig. 3.8(a)), the average index of the liquid crystal layer nl 2 (E) is greater than that of the polymer layer nl1 and a reflection peak is observed since n l 1 < nl 2 ( E ) . As the field is increased as shown in Fig. 3.8(b), the partial alignment of the liquid crystal droplet reaches a condition where n l 1 ~ nl 2 ( E ) , and no reflection is observed because the layers are index matched and optically homogeneous. At higher electric fields, as shown in Fig. 3.8 (c), the liquid crystal becomes highly aligned and nl 2 (E) decreases to a value below n l 1 , the sample reflects again at a different wavelength. Fig. 3.9(a) shows a plot of peak wavelength and reflectance as a function of applied voltage. Reflectance is normalized to the zero-voltage value.
63
The wavelength span is from a zero-voltage value of 450 nm to a minimum of 438 nm in the switched state. Fig. 3.9(b) shows the experimentally measured reflectance spectrum at the zerovoltage state, the index-matched state (120 V), and the switched state (220 V).
Fig. 3.8. Schematic illustration of a reflecting variable-wavelength HPDLC. (a) The average index of the liquid crystal layer n l 1 nl 2 (E) is greater than that of the polymer layer nl1 ; (b) As the field is applied, n l 1 ~ nl 2 ( E ) , and the index grating is erased and the sample is optically homogeneous; (c) The further increased field generates the highly aligned state where nl 2 (E) decreases to a value below n l 1 . The darker layers correspond to layers of high index of refraction. Courtesy of C. C. Bowley et. al. [31].
64
(a)
(b) Fig. 3.9. (a) Reflectance and peak reflected wavelength as a function of applied voltage for variable wavelength HPDLC with a 5 μm cell gap. (b) Experimentally (points) and modeled (curves) reflectance spectra of variable wavelength HPDLC measured at 0, 120, and 220 V. Curves were fit by varying the effective index of the liquid crystal droplet rich planes nl 2 (E) using a sinusoidal index profile and coupled wave theory. Courtesy of C. C. Bowley et. al. [31].
65
3.6 HPDLC Materials The HPDLC constituent materials usually consist of liquid crystals, photo-polymerizable monomers/oligomers, and a suitable photoinitiator for the exposure wavelength. Surfactants may be added to improve the interaction of the polymer network and the liquid crystal, in order to improve the electro-optical performance and the diffraction efficiency.
3.6.1 UV Mixtures The first HPDLCs were fabricated using UV curable mixtures [22], adapted and developed from the PDLC materials based on UV light-induce polymerization, and are suitable for holographic recording using a 351 nm Ar + ion laser. A series of formulations developed from PDLC mixtures consists of PN393, which is a mixture of low functionality acrylate monomers and photoinitiators, and one of the TL series of liquid crystals (TL203, TL205, and TL213) [32]. All these materials are developed for PDLC applications by EMD Chemicals; the mixture of PN393 and TL liquid crystal is designed for low intensity UV light curing to form PDLC with cross-linked polymer networks. Transmission HPDLCs based on this mixture exhibit a polarization dependence, and the polymer network is mechanically weak due to the low functionality of the PN393. A substitute material for PN393 was developed by De Sarkar and co-workers [33]. Their formula consists of 80% 2-ethylhexyl acrylate (EHA), 15% Ebecryl 8301, a hexafunctional aliphatic urethane acrylate oligomer (EB8301), 5% trimethylolpropane triacrylate (TMPTA), and 2% UV photoinitiator DAROCUR 4265 (Ciba Specialties), all in mass ratio. The new formula is termed MD393 and has a functionality of ~ 1.85. The HPDLCs fabricated using the MD393 and TL203 (mass ratio 1:1) have high diffraction efficiencies, high mechanical stability, and low switching voltages.
66
The electro-optical performance of HPDLC transmission gratings can be improved by adding fluorinated acrylate, according to the research result of de Sarkar and Qi [34]. When a fraction of the monomer mixture MD393 is substituted by the fluorinated monomer, 2,2,2-trifluoroethyl acrylate or 1,1,1,3,3,3-hexafluoroisopropyl acrylate, the diffraction efficiency increases, the switching voltage decreases, and the switching time rises with the increase of fluorinated acrylate concentration.
3.6.2 Visible Mixtures The first HPDLC mixture cured by a visible laser was developed by Sutherland and coworkers [22]. This HPDLC mixture contains the monomer dipentaerythritol pentaacrylate (DPHA), 10 − 30% liquid crystal E7, 10% of the cross-linking monomer N-vinylpyrrolidone −4 (NVP), 10 moles of photoinitiator Rose Bengal (RB), and a small weight ratio of co-initiator
N-phenylglycine (NPG). Rose Bengal has an absorption band around 500~600 nm, with an absorption peak at 559 nm [35]. Under light excitation, Rose Bengal transfers electrons to NPG generating NPG radicals, which initiate the free radical polymerization. The polymerization process can be described in the following equations: RB + hv Æ
RB*
RB* + NPG Æ • RB* + • NPG
• NPG + M Æ • M NVP serves as a solvent for the Rose Bengal and NPG, helping the LC to dissolvie in the monomer, and also functions as a chain terminator. The mixture of Rose Bengal, NPG, and NVP can be prepared separately as the photo initiator solution (P.I. solution) before being added to the monomer-LC mixture. Sutherland's mixture allows for the use of many convenient holographic-
67
quality lasers, at wavelengths including 488 nm, 514.5 nm and 532 nm. One drawback of this mixture is that the switching voltage of the HPDLCs fabricated with this mixture is very high. In Sutherland's formula, a surfactant, vinyl neononanoate (VN), can be added to the mixture to reduce the switching voltage, and the liquid crystal E7 can be substituted by BL038 with a higher birefringence ( Δn = 0.2720 at 589 nm). The new formula consists of BL038, DPHA, and a P.I. solution consisting of 86% NVP, 10% NPG, and 4% Rose Bengal. The mass ratio of the P.I. solution is 10 ~ 15% of the whole mixture. The optimized ratio of the liquid crystal BL038 and monomer DPHA needs to be determined experimentally depending on the working wavelength of the HPDLC, and diffraction angle, and other parameters. A HPDLC mixture for visible light curing consisting of the LC BL038, the P.I. solution, and aliphatic urethane resin oligomers Ebecryl 8301 (hexa-functional) and Ebecryl 4866 (trifunctional) was developed by Bowley and co-workers [36]. Both Ebecryl 8301 and Ebecryl 4866 are made from UCB Radcure as the monomer blend. When the functionality of the monomer blend is ~4.5 with a mass ratio ~ 1:1 of the two monomers, a maximum reflectance of 70% was achieved.
3.7. Summary The fundamental concepts and operational principles of HPDLC were discussed in this chapter, including the fabrication and operation of both transmission mode and reflective HPDLCs. The wavelength variable HPDLC was discussed in detail. The HPDLC materials for both visible and UV curing were introduced.
CHAPTER 4
Liquid Crystal Fabry-Perot
4.1 Introduction In this chapter, we will discuss the fabrication and characterization of liquid crystal FabryPerot products for application in both spectral imaging and optical telecommunications.
4.2 Introduction to Fabry-Perot Interferometer The Fabry-Perot interferometer was designed by C. Fabry and A. Perot in 1899 [37]. The device contains two partially reflecting plane surfaces between which rays of light from multiple reflections create interference patterns. We will begin with a discussion of multiple-beam fringes with a plane parallel plate, to illustrate the principle of a Fabry-Perot interferometer. Fig. 4.1 is a diagram of a plate with refractive index n immersed in a boundary medium with refractive index of n' . The thickness of the plate is d. Suppose the reflection coefficients of light reflected by the two boundary are r and r ' , and the transmission coefficient of light passing through the two boundaries are t and t ' , the complex amplitudes of the waves reflected from the plate are [38] :
rA(i ) , tt ' r ' A(i ) e iδ , tt ' r '3 A (i ) e i 2δ , …… tt ' r '( 2 p −3) A( i ) e i ( p −1)δ , …… where A( i ) is the amplitude of the incident beam. δ is the phase difference between two neighboring beams that are reflected from the plate
δ=
4πnd cos(θ )
λ0
68
(4-1)
69
where λ0 is the wavelength in vacuum. Similarly, the complex amplitudes of the wave transmitted through the plate are, apart from an unimportant constant phase factor,
tt ' A(i ) , tt ' r '2 A (i ) e iδ , tt ' r '4 A( i ) e i 2δ , …… tt ' r '( 2 p −2) A(i ) e i ( p −1)δ , ……
n'
n
d
n'
Fig. 4.1 Diagram of a plate with a refraction index n immersed in a boundary media with refraction index n' .
For either polarized component, we have
tt ' = T , r = - r ' , and r '2 = r 2 = R,
(4-2)
where T is the transmissivity and R is the reflectivity; they are related by R+T = 1 according to energy conservation. If the first p reflected waves are superposed, the amplitude A( r ) ( p ) of the electric vector of the reflected light is given by the expression:
A( r ) ( p ) = {r + tt ' r ' eiδ (1 + r '2 eiδ + ... + r '2( p − 2 ) ei ( p − 2 )δ )} A( i ) = {r + (
1 − r '2( p −1) ei ( p −1)δ )tt ' r ' eiδ } A( i ) 2 iδ 1 − r' e
(4-3)
70
If the plate is sufficiently long, and as P → ∞ , we have 2
A( r ) = A( r ) ( ∞) = − r '{1−1(−rr' '2+ettiδ') e
iδ
}
A( i )
(4-4)
Considering (4-2), we find: iδ
A( r ) = A( r ) (∞) = − (1−1e− Re) iδ R } A( i )
(4-5)
So that the intensity of the reflected light is: δ
δ )R (i ) = I ( r ) = A( r ) A( r )* = 1+( 2R−22−cos I 2 R cos δ
4 R sin 2 ( ) 2 (1− R ) 2 + 4 R sin
2δ
I (i )
(4-6)
2
In a similar way, we obtain
A( t ) = A( t ) (∞ ) = 1− rtt'2' e iδ A( i )
(4-7)
(i ) T A( t ) = 1− Re iδ A
(4-8)
and using (4-2), we have
The corresponding intensity of the transmitted light is:
I
(t )
=A A (t )
( t )*
=
T2 2 1+ R − 2 R cos δ
I
(i )
=
T2 2
(1− R ) + 4 R sin
2δ
I (i )
(4-9)
2
(4-6) and (4-9) are known as Airy’s formulae. When a parameter F, is defined as:
F=
4R (1 − R ) 2
(4-10)
The intensity distributions of the reflected and transmitted patterns are given by:
I (r ) F sin 2 (δ / 2) = I ( i ) 1 + F sin 2 (δ / 2)
(4-11)
71
1 I (t ) = (i ) I 1 + F sin 2 (δ / 2)
(4-12)
Evidently the two patterns are complementary, in the sense that
I ( r ) I (t ) + =1 I (i ) I (i )
(4-13)
Equation (4-13) shows energy is conserved when the medium has no absorption. When
1 2
(t ) (i ) (t ) (i ) δ=2mπ, I / I has a maximum value; when δ = 2π (m + ) , I / I has a minimized value.
( m is an integer ). When R is big, the pattern in the transmitted light consists of narrow bright fringes on an almost completely dark background and, similarly, the pattern in the reflected light becomes one of narrow dark fringes on an otherwise nearly uniform bright background. The sharpness of the fringes is conveniently measured by their half-intensity width. The ratio of the separation of the adjacent fringes and the half-width is defined as the finesse, Ғ, of the fringes. For the fringe of an integral order m, the points where the intensity is half its maximum value are at
δ = 2πm ± and
ε
(4-14)
2
1 1 = 2 1 + F sin (ε / 2) 2
(4-15)
When F is sufficiently large, ε is so small that we may assume sin(ε/4)=ε/4 in (4-15); the half-
4 F
ε=
width is obtained to be:
(4-16)
The finesse is then Ғ=
2π
ε
=
π F 2
=
π R 1− R
(4-17)
72
(t ) (i ) Fig. 4.2 shows the behavior of I / I as a function of the phase difference δ for various values
of finesse Ғ.
Transmission Performance of different Finesse Number
Transmission
1 0.8
Finesse=3
0.6
Finesse=5
0.4
Finesse=10
0.2
Finesse=20
0 5.8
6.3
6.8
7.3
7.8
8.3
8.8
Phase Shift
(t ) (i ) Fig. 4.2 Behavior of I / I as a function of the phase difference δ for various values of finesse
Ғ. Unit of the phase difference δ is π.
The so far discussed multiple beam interference fringes from a plane parallel plate can also be applied to the air-gap Fabry-Perot interferometer when the incident light is at near normal incidence. An air-gap Fabry-Perot interferometer consists of two glass or quartz plates P1, P2 (Fig. 4.3) with plane surfaces. The inner surfaces are coated with partially transparent films of high reflectivity, and are parallel, so that they enclose a plane parallel plate of air with fixed separation, d, decided by a spacer. This form of the interferometer is often refered to as FabryPerot etalon.
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d
P1
P2
Fig. 4.3. Fabry-Perot interferometer
When the incident light is not collimated, the intensity of light I(θ,λ) transmitted through an ideal Fabry-Perot etalon (one with no defects) is given by
I (θ , λ ) = I o (λ )
4π nd cos(θ ) 1 with δ = 2 2 1 + (2 F / π ) sin (δ / 2) λ
(4-18)
where λ is the light’s wavelength, Io(λ) is the intensity in the center of each fringe, d is the plate separation, n is the index of refraction of the material between the etalon plates. (for an air-gap, n=1.0) and F is the finesse of the etalon defined in (4-17). From now on, we will use F to represent the finesse instead of Ғ. When the light source is monochromatic, the Fabry-Perot allows transmittance of light at only specific incidence angles. Imaging the output of the FabryPerot produces a series of circular fringes such as those shown in Fig. 4.4. As the wavelength of the light decreases, the diameters of the rings increase until they eventually occupy the space left vacant by the next adjacent external ring. As this happens, a new ring appears in the center of the pattern to replace the old one.
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Fig. 4.4. Image of the Fabry-Perot interference pattern with monochromatic incident light.
Ideally the finesse F is only a function of reflectivity R, the reflective finesse, if the etalon plates are perfect with no surface defects and the two etalon-plates are perfectly parallel. Fig. 4.5 shows the relation between the reflective finesse with the reflectivity. In reality, even the best etalon will possess defects that limit the theoretically expected performance. The actual finesse will usually be lower than the reflective finesse. If the gap is specified as d ± λ / 50, we can assume the gap will be within these limits for most of its area, say 95%. At any point the most probable value of the gap thickness is d and the number of places where it departs from this by a large amount will be very small. The total area where the gap departs from d by an amount between
dA( x) =
x
bA
π
and
x+dx
can
be
conveniently
expressed
as
the
Gaussian
formula:
exp(−b 2 x 2 )dx , where b is a “figure of merit” which increases as the gap becomes
more and more uniform. Thus different parts of the gap will contribute to the total intensity at slightly different orders of interference and the effect reveals itself as a convolution of the theoretical Airy profile with a Gaussian curve. If we assume that near a maximum the Airy profile is approximately the same shape as a Gaussian curve, and remember that the convolution
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of two Gaussians is another Gaussian with a half-intensity width which is the Pythagorean sum of the component half-intensity widths, then we can determine the reflection coefficient that the plates should have, provided that we know the flatness of them. If they are flat to λ / 50, then the best possible finesse is F=50. If the reflecting layers have a reflection such that the reflection finesse would be 50, then the effective finesse would be 50 / 2 , or about 35. Three types of defects that contribute to this reduction are spherical defects, surface irregularities, and parallelism defects, as shown in Fig. 4.6.
Reflective Finesse Vs Reflectivity
Reflective Finesse
350 300 250 200 150 100 50 0 20
30
40
50
60
70
80
90
100
Reflectivity ( % )
Fig. 4.5 Relation of the reflective finesse with the reflectivity.
(a)
(b)
(c)
Fig. 4.6. Spherical defects (a), surface irregularities (b), and parallelism defects (c).
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Effective Finesse
Effective Finesse VS Defect Finesse 90 80 70 60 50 40 30 20 10 0
FR=50 FR=100 FR=150
20
30
40
50
60
70
80
90
100
Defect Finesse
Fig. 4.7. Effective finesse changes with the defect finesse. FR represents the reflective finesse.
The related defect finesse are Fds , Fdg , and Fdp . The total defect finesse Fd is decided by:
1 1 1 1 = 2+ + 2 2 2 Fd Fds Fdg Fdp The actual effective finesse Fe is decided by:
1 1 1 = 2+ 2 2 Fe Fd FR Fig. 4.7 Plots how the effective finesse changes with the defect finesse for different reflective finesse. To use the Fabry-Perot etalon as a filter, it is customary to restrict the incidence angle of the light to match that of the innermost ring, or spectral element. The resulting field-of-view (FOV) is given by
FOV =
8δλ
λ
(4-19)
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where δλ is the spectral resolution of the etalon. The Free Spectral Range (FSR) is defined as the wavelength difference of the two neighboring constructive interference peaks, or transmission peaks, and is characterized predominantly by the gap of the etalon. The condition for the constructive interference is determined by:
2nd
λ
=m
(4-20)
Where n is the refraction index, d is the gap, λ is the wavelength and m is the order number. Thus the FSR between the m-th order and the m+1-th order is given by:
FSR =
2nd m(m + 1)
(4-21)
Normally m is a large number, so it becomes:
FSR = Δλ =
λ2 2nd
(4-22)
For the high-resolution etalon, the free spectral range is limited by the aforementioned effective finesse. In order to expand the free spectral range, two etalons can be used in series, the one with a larger gap, known as the resolving etalon, defines the spectral resolution of the system. The etalon with the smallest gap, known as the suppression etalon, suppresses some of the orders of the resolving etalon. The result is a system with the spectral resolution of the resolving etalon and a FSR larger or equal to that of the suppression etalon. The latter depends on the ratio of the FSR of the two etalons with respect to each other. Suppose that the ratio of the FSR of the suppression etalon to the resolving etalon can be expressed as ratio of integers A/B, where A and B do not have a common divisor. The FSR of the twin-etalon system is then given by
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FSR = A × FSRresol = B × FSRsup
(4-23)
Fig. 4.8. Modeling of twin etalon system with the gaps of 3 micron and 12 micron.
Where FSRresol and FSRsup are the free spectral ranges of the resolving and suppression etalons, respectively. If B = 1, the suppression etalon defines the FSR of the system, otherwise the FSR of the system will be larger than that of the suppression etalon by a factor of B. In the twin-etalon system, the FSR of the resolving etalon has now been expanded by the factor A, and so has the number of available spectral resolution element. The factor A will henceforth be called
79
the FSR expansion factor. Fig. 4.8 models twin etalon system, one with the gap of 3 micron and the other with a gap of 12 micron.
4.3 Introduction to Liquid Crystal Fabry-Perot (LCFP) Tunable Filter Tunable filters with a wide tunable range that cover the whole C-band, or L-band, or both, have found wide applications in fiber optical communication systems, mainly in three domains: tunable lasers, wavelength division multiplexing systems (WDM), and channel monitoring. Due to the growing demand in bit-rates and the number of channels in a WDM system, tunable narrow band pass filters are required [39] . So far, many efforts have been made to make tunable filters based on the Fabry-Perot principle [40,41,42]. The tunable optical filters using a LiNbO3 torsional actuator [40] need a high driving voltages up to 500 volts and have a narrow tunable range. The tunable filter [42] reported by M. Iodice is a temperature-tuned silicon etalon filter with a narrow passband; however it possesses a narrow free spectral range and the tuning speed was not reported. Employing a liquid crystal material as a cavity medium in a Fabry-Perot etalon has many merits such as low driving voltage, low insertion loss, and wide tuning range. Tunable LCFPs were first proposed by Maeda in 1990 [43] and Patel in 1992 [44], both from Bellcore. Hirabayashi et. al. from NTT [45] and Bao et. al. from Colorado University [46] have also worked on LCFP. The fabrication of a LCFP device is almost identical to the fabrication of a homogeneously aligned nematic liquid crystal cell, or an ECB cell. The etalon plates are coated with a conductive ITO layers, and then high reflection multi-layer dielectric coatings. A polyimide layer with a thickness of about 50~100 nm is spin-coated on one side and is then uniformly rubbed to generate the alignment layer for the liquid crystals. The etalon plates are then assembled with spacers to
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control the thickness of Fabry-Perot cavity. Finally, the liquid crystal material is vacuum-filled into the cavity. Fig. 4.9 shows the structure of a liquid crystal Fabry-Perot etalon.
Etalon Plate High Reflection dielectric coating
Liquid crystal
ITO layer
Fig. 4.9 Structure of liquid crystal Fabry-Perot.
The LCFP is polarization dependent. The refractive index for light with a polarization direction parallel to the rubbing direction, or e-component, is ne , and the refractive index for light with a polarization direction perpendicular to the rubbing direction, or o-component, is no . LCFPs are tunable only for e-components because ne can be tuned with an applied voltage, thus a polarizer is necessary to be placed parallel to the alignment direction of the liquid crystal to allow the extraordinary mode of light to pass through and block the ordinary mode. When voltage is applied, the refractive index of the e-component changes along the normal direction of the cell with a change of the director configuration inside the nematic cell, as was discussed in Chapter 1; the average refractive index of the e-component inside the cavity is:
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ne =
1 d ne ( z )dz d ∫0
(4-24)
Fig. 4.10 shows how the average refractive index changes with an applied voltage. The model is based on the modeling of director configuration described in chapter 1. The parameters used for the model are: ne =1.79 , no =1.53, K11 = 15.5, K 33 = 28.0, ε // = 15.5 , ε ⊥ = 5.2. The pretilt angle was assumed to be 0°.
Average Refraction Index
Average Refraction Index Vs Voltage 1.85 1.8 1.75 1.7 1.65 1.6 1.55 1.5 0
2
4
6
8
10
Applied Voltage (v)
Fig. 4.10. The average refractive index changes with the applied voltages.
For certain applications, it is desirable to have a tunable filter without a polarization dependence, such as in fiber optic telecommunication where the polarization state of the optical signal may be unknown. Several methods have been discussed to fabricate a polarizationinsensitive Fabry-Perot device [50]. Fig. 4.11 shows a solution to achieve the polarization
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independent optical filtering and tuning. The incident light (polarization state unknown) is separated into two linearly polarized beams by a polarization beam-splitter. Each beam passes through a LCFP with the liquid crystal alignment direction parallel to the polarization direction, and is wavelength-filtered by the tunable LCFP filer. Finally, a second polarization beam splitter (not shown in the figure) will combine the two filtered beams back into one beam.
P o la riz a tio n B e am sp litte r
• M irro r
•
LCFP
• • •• • • •• • • •• ••
•
LCFP Fig. 4.11. Combination of polarization beam splitter and two LCFPs with alignment directions perpendicular to each other, to achieve the polarization-independent wavelength filtering.
Fig. 4.12 shows an another solution for a polarization independent optical filtering and tuning system. Inside the Fabry-Perot cavity, there are two thickness identical liquid crystal layers with the alignment directions perpendicular to each other. Because of the symmetry, light with any polarization state has an identical optical path length of ne d + no d + n g t , where n g is the refractive index of the glass substrate inside the cavity. While in this structure, wavelength filtering is tunable by changing ne through an applied voltage; the tunable range is decreased by
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a factor of
ne d , compared with LCFP with a single liquid crystal layer. The structure ne d + no d + ng t
described in Fig. 4.12 is much more compact than that in Fig. 4.11; however, as the glass thickness t is much larger than the liquid crystal thickness d, the tunable range is very small. The twisted nematic Fabry-Perot interferrometer (TN-FPI) was proposed by Patel and Lee in 1991 [47]. Here the liquid crystal alignment directions at the two opposite etalon plates are perpendicular to each other; thus, the liquid crystal layer has a 90° twist through the whole cavity, as shown in Fig. 4.13. A TN-FPI works in a high driving voltage region, where, in the middle of the cavity, the liquid crystals are homeotropicly aligned due to the electric field, the two residual homogeneous liquid crystal layers close to the substrate surfaces will compensate each other and makes the wavelength-filtering tunable and polarization independent.
t •• • • ••• ••• •••• d
d
Fig. 4.12. Two LC layers inside a Fabry-Perot cavity are used to achieve a polarization independent wavelength filtering and tuning.
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Etalon Plate High Reflection dielectric coating
Twisted Liquid crystal •
•
•
•
ITO layer
Polyimide alignment layer
Fig. 4.13. Tunable and polarization independent twisted nematic Fabry-Perot etalon.
4.4 Fabrication and Testing of LCFP Tunable Filter
4.4.1. Single LCFP System Tunable LCFP interferometers have been proven to be competitive in spectroscopy, LIDAR and IR imaging [48,49,61,62]. The advantages of a tunable liquid crystal Fabry-Perot filter are: high resolution, wide free spectral range and tunable range, fast response time, solid state (no moving parts), low driving voltage, and large aperture. We fabricated a LCFP (#1608) for applications in spectral imaging. The etalon plates used had diameters of 38 mm and surface flatness of 1/100 at 632.8 nm. A reflective coating with a 90% reflectivity from 700-1000 nm was deposited on each of the etalon plates [62]. The etalon was filled with a 10-micron thick layer of nematic liquid crystal and gapped using small spherical fused silica spacers. Final parallelism alignment was made using a UVB cured adhesive while monitoring the fringe pattern.
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Transmission curves as a function of wavelength were obtained as shown in Fig. 4.14 through Fig. 4.16. To make these measurements, light from a monochromator is collimated and passed through a polarizer before reaching the LCFP that is being tested. The beam is then focussed onto a photodiode detector and monitored by a computer. The spectral resolution of this set-up is about 0.2 nm, and the beam is collimated such that only the innermost order of the LCFP is sampled. The polarizer’s aim is to remove the polarization component of the beam that the LCFP cannot tune. To obtain a transmission curve, the monochromator is scanned in wavelength while a specific voltage is applied across the LCFP’s gap. The monochromator is then scanned in wavelength once again with the LCFP out of the beam. Dividing the former by the latter, the transmission of the LCFP as a function of wavelength is obtained. The LCFP (#1608) achieved a finesse of 9-12 depending on wavelength and applied voltage. Peak transmission of polarized light ranged from 40-70%. The testing was performed on a full working aperture, which is 1.2 inch in diameter, 80% of 1.5 inch. The finesse and free spectral range (FSR) is summarized in Table 4.1. The electrooptical response measured at the wavelength 805 nm is shown in Fig. 4.17. Fig. 4.18 shows an image of LCFP #1608 installed in the housing with an electrical connector. From Table 4.1, the trend is apparent that the finesse increases with the wavelength. This is because, while the surface quality of the etalon plate is not wavelength dependent in the unit of nm, it is normally judged by λ/m, where λ is the working wavelength and m is a number, the defect finesse can be estimated by: Fd = m. This means, suppose the defect finesse at the 500 nm is 10, the defect finesse at the 1000 nm is ~ 20. Thus the total finesse increases with wavelength supposing the reflectivity (or reflective finesse) is kept unchanged.
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We have also successfully fabricated a liquid crystal Fabry-Perot for optical telecommunication applications. Table 4.2 shows the result of a tunable LCFP filter in the NIR range (C-band) designed for a tunable laser application. We have successfully modified the highreflection coating design to cover the working range from 1520 nm to 1570 nm. We have also optimized the liquid crystal material and alignment polyimide to minimize the transmission loss. The cavity gap is controlled by 10 μm spacers.
Finesse and Free Spectral Range of LCFP #1608 Voltage Applied: --> Finesse at ~ 700 nm FSR at ~700nm Finesse at ~ 820 nm FSR at ~820nm Finesse at ~ 1000 nm FSR at ~1000nm
1.5 V 9.7 16.84 nm 10.7 18.75nm 12.4 26.23nm
3.5 V 10.7 18.46nm 10.7 20.14nm 11.8 27.48nm
9V 9.3 19.14nm 9.45 20.66nm 10.7 28.39nm
Table 4.1. Finesse and free spectral range of LCFP # 1608 at different voltages.
Parameter Plate Diameter Center Wavelength Test Area Insertion Loss Free Spectral Range Spectral Resolution Finesse Response time
Measured Value 10mm 1550nm 0.2 square-mm 1.5db 4 THz (37nm) 108 GHz (1.2 nm) 31 20 ms
Table 4.2. Testing result of tunable LCFP for tunable laser in NIR range.
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S pectral R esponse of L C F P # 1608
Transmission
(1.5 V voltage applied )
W avelength (A )
Fig. 4.14. Spectral response of LCFP #1608, measured at 1.5 V. The polarizer is parallel to the director direction of the liquid crystals.
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Spectral Response of LCFP # 1608
Transmission
(3.5 V voltage applied )
Wavelength (A)
Fig. 4.15. Spectral response of LCFP #1608, measured at 3.5 V. The polarizer is parallel to the director direction of the liquid crystals.
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S p ectral R esp o n se o f L C F P # 1 6 0 8
Transmission
(9 V v o ltag e ap p lied )
W av elen g th (A )
Fig. 4.16. Spectral response of LCFP #1608, measured at 9.0 V. The polarizer is parallel to the director direction of the liquid crystals.
90
Voltage Scan of LCFP #1608 ( at 805nm )
Fig. 4.17. Electro-optical response of LCFP #1608, measured at 805 nm.
Fig. 4.18. LCFP #1608 in the housing with electrical connector.
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4.4.2 Twin LCFP System
We fabricated a twin-LCFP system for the application of high-resolution spectral imaging. Two LCFP etalons were constructed. Etalon plates have a diameter of 38 mm and a surface flatness of 1/100 at a wavelength of 632.8 nm. A reflective coating with a 90% reflectivity from 700-1000 nm was deposited on each of the etalon plates. The first etalon (resolving etalon) was filled with a 30-micron thick layer of nematic liquid crystal and gapped using small spherical fused silica spacers. Final parallelism alignment was made using a UV cured adhesive while monitoring the fringe pattern.
Fig. 4.19. Photographs of the single etalon in the housing (right) and the twin etalon imaging filter (left).
As the high reflection coating exhibit a phase shift upon reflection that altered the effective gap, the suppression etalon was constructed with a 6 micron gap rather then a 7.5 micron gap in order to compensate for the phase shift. The etalons are housed in a cylindrical housing (Fig. 4.19) that consists of an inner cylinder that holds the etalon with ruby spacers isolating the etalon from the housing. The second cylinder holds the heating element and thermostat. Using a PID controlled heating system the etalon
92
temperature can be maintained to 0.01 degrees Celsius. The housing has a dovetail flange that allows the etalons to be connected and then rotated with respect to each other around the optical axis. This rotational feature allows for the alignment of the polarization states of the etalons to be
Transmission
co-incident.
W avelength (Angstrom)
Fig. 4.20. Transmission as a function of wavelength for the 30 μm gap LCFP.
Transmission
93
W avelength (A ngstrom )
Fig. 4.21. Transmission as a function of wavelength for the 6 μm gap LCFP.
Fig. 4.20 shows the transmission versus wavelength with an applied potential of 9 Volts, for the LCFP with 30 μm spacer. Fig. 4.21 shows the same testing for the LCFP with 6.μm spacers. Both etalons achieved a finesse of 9-12 depending on wavelength and applied voltage. The peak transmission of polarized light ranged from 40-70% .
4.4.3. Environment Test of LCFP
Two environmental tests were performed at the Utah State University Space Dynamics Laboratory on etalons similar to those described above. The etalons used in this series of testing were of a smaller diameter (25 mm) and had a multilayer dielectric reflector with a 90% reflectivity from 500-700 nm. A gap of 10 microns was used in each etalon. The etalons were
94
tested using the monochrometer described previously and after the shake and thermal-vacuum tests. The intent of this test was to thermally cycle the LCFP between at least 40°C and –10°C with an approximate dwell time of 60 minutes for each of the two temperature extremes and to expose the etalons to a 10 G shake equivalent to a launch on a Pegasus launch vehicle. The temperature data from the F-P Liquid Crystal thermal vacuum test is shown in Fig. 4.22. Initially the temperature cycles did not achieve the required temperature of –10 °C during the cold cycles. The temperature program was modified to meet the –10 °C requirement, resulting in five acceptable thermal cycles. The temperature cycling rate was 5 °C/min, with a dwell time of 130 minutes at each end. The thermal couple used to monitor the etalon temperature was attached to the side of the housing that covered the LCFP. A cursory examination of the two LCFP etalons did not reveal any obvious damage from either the shake test or the thermal cycling; the etalon plates were neither chipped nor cracked, and the electrical wires were still solidly attached to the substrate. Transmission curves as a function of wavelength were obtained for each of the two etalons using the technique described previously. This was done for two voltage settings of the LC-FP etalons. As shown in Figures 4.23 and 4.24, the data sets had to be shifted in wavelength for the peaks to be lined up, which can be attributed to two things. First, uncertainties in the orientation of the LCFP with respect to the beam can cause a shift of the peak location each time an etalon is set-up in the beam. Secondly, uncertainties in the absolute wavelength calibration curves could be a factor.
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Thermal Vacuum Temperature Cycling Results for F-P Liquid Crystal including Preliminary Test Cycles TC #4 attached to the side of F-P LC housing 50
Teamperature C TC#4 attached to side wall of housing
Expanded Final Temperature Cycles Shown in "Final Dwell Cycles" Chart. Acceptable Results
40
30
20
10
0
-10 Preliminary Temperature Cycle Test The F-P LC did not reach the required temperature of -10C and the dwell time required adjustment. Unacceptable test.
-20 6/11/03 15:36
6/11/03 22:48
6/12/03 6:00
6/12/03 13:12
6/12/03 20:24
6/13/03 3:36
6/13/03 10:48
Time (m/d/y hr:min)
Fig. 4.22. Temperature versus time for the thermal vacuum testing of the LCFP
96
Fig. 4.23. Transmission of the LCFP that underwent a Pegasus-level shake test for two different voltage settings (1 and 9 Volt). The offset of the curves in transmission and wavelength is indicated in parenthesis.
97
Fig. 4.24. Transmission of the LCFP that underwent thermal cycling, before and after the thermal cycling for two different voltage settings (1 and 9 Volt). The offset of the curves in transmission and wavelength is indicated in parenthesis.
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4.5 Summary and Conclusions
We have analyzed, fabricated and characterized liquid crystal Fabry-Perot products for application in both spectral imaging and optical telecommunication. Both single-etalon and twinetalon systems were fabricated. A Finesse of more than 10 in the visible wavelength range and a finesse of more than 30 in NIR were achieved for the tunable LCFP product.
CHAPTER 5
Switchable Circle-to-Point Converter
5.1 Introduction
This chapter discusses innovative switchable circular-to-point converter (SCPC) devices based on holographic polymer dispersed liquid crystal (HPDLC) technology, Fabry-Perot interferometers, and the holographic circular-to-point converter (HCPC). We will discuss the concept and design of an innovative SCPC device, and the fabrication and characterization of SCPC devices working at different wavelengths (visible and NIR), and with different channel numbers (single channel, 10 channel, and 32 channel).
5.2 Background: Introduction to HCPC
Fabry-Perot interferometers (FPI) are employed as spectral-resolving elements in various applications, such as in Lidar detection of atomospherical, environmental, and climate changes [48,49] and in telecommunications [43,59]. In a direct detection Doppler Lidar or incoherent Lidar system to measure wind velocities by aerosol and/or molecular backscatter, the Doppler shift resulting in a pulse of narrowband laser light from scattering by aerosols or molecules is measured. A reference spectrum of an outgoing laser beam is measured by the collection of light scattered from the zero-wind background. When the return signal of a backscattered laser light passes through the receiving optics, the Doppler shift can be determined by subtracting the reference spectrum from the return signal. A high resolution Fabry-Perot interferometer is used to detect the wavelength shifts.
99
100
The Fabry-Perot interferometer produces a circular interference spectrum or fringe patterns of equal area rings representing equal wavelength intervals, sharing a common axis, at the infinity focus of an objective lens system. There is a long-established problem of collecting and testing the signal from the circular fringe pattern as it’s difficult or expensive to design detectors with ring-shaped geometries. Different types of image plane detectors have been created which attempt to match the circular pattern. One such image plane detector was reported by Timothy et al. in 1983 [51]. Their device consisted of an S-20 photocathode, three micro-channel plate electron multiplication stages, and an equal-area concentric-ring segmented anode to match the interference ring pattern. Another image plane detector invented by Bissonnette et al. was a multi-element detector of concentric rings of PIN photodiode material [52, 53]. All of these image plane detectors typically suffered from blurring of spot sizes and low quantum efficiency. A different approach for converting the Fabry-Perot fringe pattern itself to fit linear detectors has also been accomplished [54, 55]. A 45° half angle internally reflecting cone segment is used to convert the circular Fabry-Perot interferometer fringe pattern into a linear pattern. McGill and co-workers developed the passive holographic optical element for Lidar detection [56,57,58]. The holographic optical element comprises areas, each of which acts as a separate lens to image the light incident in its area to an image point. Each area contains the recorded hologram of a point source object. The image points can be made to lie in a line in the same focal plane so as to align with a linear array detector. Holographic Circular-to-point converter (HCPC) have been developed [57] that have concentric equal areas to match the circular fringe pattern of a Fabry-Perot interferometer. A HCPC has a high transmission efficiency, and when coupled with a high quantum efficiency solid state detector, provides an efficient photon-collecting detection system for a Fabry-Perot interferrometer. The HCPC, as well as other holographic elements, may
101
be used as part of the detection system in a direct detection Doppler Lidar system or multiple field of view Lidar system. HCPC holographic plate is divided into concentric annuli. Each annulus of the holographic plate functions as a single lens and converges the incident beam to a point focus, as depicted in Fig. 5.1. In order to match the Fabry-Perot fringe pattern, the annuli are designed to intercept equal wavelength intervals. The signals from different wavelengths are spatially discriminated by the HCPC device, whereas multiple detectors are required for McGill's HCPC devices.
Fig. 5.1. The ray trace diagram of the holographic circular-to-point converter (HCPC) developed by McGill and co-workers. All light incident onto a given annulus of the HCPC is redirected to a designated point. The focal points appear in a common focal plane parallel to the HCPC plate and are angularly separated.
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5.3 Principle of Operation of SCPC
The switchable circle-to-point converter (SCPC) is a combination of the HCPC and the HPDLC that allows for an electric controllability, while maintaining the traditional highlights of a Fabry-Perot interferometer: high optical throughput and high spectral resolution. Based on HPDLC technology, the SCPC device is designed to convert the signal from a Fabry-Perot etalon to a focus point or an array of points just like the HCPC; what is more, the conversion can be deactivated by applying a strong enough electric field, or the conversion can be electrically switched on and off [60]. When an Indium Tin Oxide (ITO) conductive layer on one of the SCPC substrates is patterned with circular pixels that match the Fabry-Perot circular interference pattern, individual channel are discriminated by the Fabry-Perot etalon and can be separately selected and switched on and off. Fig. 5.2 is a cross sectional drawing of a 4x2 switch employing two identical SCPC elements in series – each element has 4 ring-channels. Both of the SCPC elements have similar ring pixel patterns geometrically matching the Fabry-Perot interference ring pattern, and both are designed so that each ring pixel converts the energy within a corresponding circular wavelength channel to the same point (D1 or D2) when no voltage are applied. To simplify the explanation, only four ring pixels (or channels) are shown, labeled with 0, 1, 2, and 3. The light source 61 has been wavelength-discriminated by a FPI. When only voltages V1 and V3 are applied to Channel 1 and Channel 3 of the first SCPC, respectively, Channel 0 and Channel 2 are routed to the destination D1, and Channel 1 and Channel 3 are transmitted unimpeded down the optical path. For the second SCPC, if only ring pixel 1 has an applied voltage, Channel 3 is routed to destination D2, while Channel 1 passed through along the optical axis. By applying appropriate voltages to different ring pixels of different SCPC elements, channels can be randomly routed to any destination.
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Fig. 5.2.The cross-section drawing of a 4X2 switch employing two identical SCPC elements.
Fig. 5.3 shows a random optical cross-switch can be built by stacking multiple SCPC units. The collimated light from the Fabry-Perot etalon propagates into the stack of identical SCPC units. Each SCPC unit converts a selected wavelength channel in the circular interference pattern to a point, which can be projected onto a detector or routed to different client destinations through optical fibers.
Fig. 5.3. A random optical cross-switch by stacking multiple SCPC units.
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5.4 Optics Design of SCPC
Two types of SCPC have been designed. The first type of SCPC is a plain transmission HPDLC, whose function is steering the beam. The diffracted (steered) beam is focused by a focal lens to a point, as depicted in Fig. 5.4. In the second type of SCPC, the HPDLC, functioning as a lens, focuses the collimated incident light to a point, as shown in Fig. (5.2).
CH1
CH2
CH3
SCPC
F o c u sin g L en s
Fig. 5.4. The first type of SCPC: the diffracted beam is focused by a focal lens to a point. Left: different channels are routed away by the SCPC and further focused by the focusing lens. Right: when some channels (red and blue in the figure) on the SCPC are switched on, the switched channels pass through the SCPC without being steered to the detector.
5.4.1 First Type (Beam Steering) SCPC
In the first type of SCPC device, the HPDLC functions as a beam steering device with the same diffraction angle θ d everywhere in the HPDLC area (Fig. 5.6). As the SCPC works in the condition of normal incidence, the HPDLC in the SCPC device requires that the Bragg condition
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(a) θd
Diffracted beam
(b)
Glass Λ
Glass
Λ
θt
θr
θw
θdi θw1
Incident reading beam
θw1i θ w2i
θw2
Recording beams
Fig. 5.5. Reading beam configuration (a) and recording beam configuration (b) of the beam steering HPDLC for the first type of SCPC.
be satisfied when the incident beam is normal to the cell surface. The reading beam (wavelength
λr ) and diffraction beam configuration is illustrated in Fig. 5.5. The HPDLC grating is recorded by exposing the cell to two interfering laser beams whose wavelength is λw , as shown in Fig. 5.5 (b). The recording beam incident angles can be determined, provided that the refractive index of the HPDLC mixture at the recording wavelength λw , nw , and the average refractive index of the HPDLC at the reading wavelength λr , nr is known. The diffraction angle inside the HPDLC θ di is determined by Fresnel's law:
⎛ sin θ d ⎝ nr
θ di = arcsin⎜⎜
⎞ ⎟⎟. ⎠
(5-1)
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Since the incident beam is perpendicular to the cell surface, the grating plane tilt angle θ t is the same as the angle between the grating plane and the incident reading beam θ r , therefore,
θ di = 2θ t = 2θ r . Bragg's law requires that 2Λ sin θ r 2Λ sin θ w
=
λr
=
λw
nr nw
, (5-2)
,
where θ w is the angle between the recording beam and the grating plane. Therefore,
⎛
θ w = arcsin⎜⎜ sin θ r ⎝
λw nr ⎞ ⎟. λr nw ⎟⎠
(5-3)
The recording beam incident angles inside the HPDLCs θ w1i and θ w 2i , which are defined in Fig. (5.6b), are then given by
θ w1i = θ t + θ w , θ w 2i = θ t − θ w .
(5-4)
The recording beam incident angles in air are finally determined:
θ w1 = arcsin(nw sin θ w1i ), θ w 2 = arcsin(nw sin θ w 2i ).
(5-5)
o If we set the diffraction angle of the HPDLC to be θ = 79 , a large angle in order to
minimize the distance between neighboring SCPC devices when they are stacked together, and assuming that nw = nr = 1.5 , the recording beam incident angles can be derived: θ w1 = 43o 59′ and θ w 2 = 20 o51′ .
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5.4.2 Second Type (Focusing) SCPC
In the second type of SCPC device, the HPDLC in the SCPC device is designed to focus the light to a point, functioning like a holographic lens that was demonstrated by Ritcher and coworkers in 1974 [63]. We constructed an interference pattern using a point source and a plane wave for fabricating the HPDLC with a built-in focus for the second type SCPC. The holography setup is illustrated in Fig. 5.6.
HPDLC cell Mirror Iris Beam Splitter
Shutter Beam Expander Focal Lens
532 nm Laser
Fig. 5.6. The holography setup for fabricating the second type of SCPC.
The point source is generated by adding a focal lens in the optical path of one recording beam. The focal length of the lens is F , and it is placed at a distance 2 F from the sample cell. The recording and reading optics of the HPDLC across the center of the HPDLC area are simulated. The recording beam configuration near the sample is illustrated in Fig. 5.7.
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D
θ w 1A
θw 2 A
A
B
C
θ w 1B
F
θ w 2B
θw 1C
θw 2C
Fig. 5.7. Recording beam profile across the HPDLC area using the setup shown in Fig. 5.6.
5.4.3 Astigmatism in Second Type (Focusing) SCPC
For the SCPC working at a wavelength (such as 1540 nm) different from the recording wavelength (such as 532 nm), astigmatism are explained in our following simulation and calculation, and also confirmed in our testing results. In our modeling, we still set the diffraction angle at the center of the HPDLC to be
θ = 79o and nw = nr = 1.5 ; the recording beam incident angles at the center of the HPDLC are: θ w1 = 43o59′ and θ w 2 = 20 o51′ , as calculated in the previous section. The diameter of the effective HPDLC area is D = 2.54 cm. For the collimated recording beam, the incident angle is identical at different locations of the HPDLC: θ w 2 A = θ w 2 B = θ w 2C = 20 o51′ . The incident angles of the diverging recording beam at different locations are calculated for different F . The incident angles at spot A and spot C are presented in Table 5.1, wherethe values are defined by:
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θ rbA :
the Bragg matched incident angle for 1540 nm at A;
angle for 1540 nm at C; diffraction angle at C;
θ dbA :
θ dnA :
θ rbC :
the Bragg matched incident
the Bragg diffraction angle at point A;
θ dbC :
the Bragg
the diffraction angle with 1540 nm normal incidence at A;
θ dnC :
the diffraction angle with 1540 nm normal incidence at C; TIR: total internal reflection; d: minimum distance between neighboring electrodes determined by the diffracted beam configuration. With the incident angles of the recording beams determined, we calculate the Bragg matched incident angle and diffraction angles for the 1540 nm reading beam on different locations across the HPDLC area. Along the line ABC as shown in Fig. 5.7, the normal incident light at point B, center of the SCPC, is no doubt Bragg-matched; however, as the incident location moves away from the center B, the normal incidence is no more Bragg matched. At points A and C which show the largest deviations from normal incidence, the Bragg matched incident angles θ rbA and θ rbC , and the corresponding diffraction angles θ dbA and θ dbC , are calculated and listed in Table 5.1. Considering the angular dependence of the diffraction efficiency discussed in Chapter 2, our calculated result indicates: (1) The diffraction efficiency for normal incident of 1540 nm light decreases with the increase of the distance from the incident spot to the center of the HPDLC. (2) The deviation of the Bragg matched incidence from the normal incidence decreases with increasing F .
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F (mm) θw2A (°) θw2C (°) 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000
48.43 46.11 45.30 44.88 44.63 44.46 44.34 44.24 44.17 44.11 44.07 44.03 43.99 43.97 43.94 43.92 43.90 43.88 43.87 43.85 43.84 43.83 43.82 43.81 43.80 43.79 43.79 43.78 43.77 43.77
37.84 40.84 41.78 42.24 42.52 42.70 42.83 42.93 43.00 43.06 43.11 43.15 43.18 43.21 43.24 43.26 43.28 43.30 43.31 43.33 43.34 43.35 43.36 43.37 43.38 43.39 43.40 43.40 43.41 43.42
θrbA (°) θdbA (°) θrbC (°) -4.03 -2.11 -1.43 -1.08 -0.87 -0.73 -0.63 -0.55 -0.49 -0.44 -0.4 -0.37 -0.34 -0.31 -0.29 -0.28 -0.26 -0.25 -0.23 -0.22 -0.21 -0.2 -0.19 -0.18 -0.18 -0.17 -0.16 -0.16 -0.15 -0.15
TIR TIR TIR TIR TIR TIR TIR TIR TIR 83.21 82.73 82.36 82.05 81.8 81.59 81.4 81.25 81.11 80.99 80.88 80.78 80.69 80.61 80.54 80.47 80.41 80.36 80.31 80.26 80.21
4.912 2.335 1.531 1.138 0.906 0.753 0.644 0.562 0.499 0.449 0.407 0.373 0.344 0.319 0.298 0.279 0.263 0.248 0.235 0.223 0.212 0.203 0.194 0.186 0.178 0.171 0.165 0.159 0.154 0.149
θdbC (°) θdnA (°) θdnC (°) d (mm) 57.88 66.83 70.25 72.11 73.3 74.13 74.75 75.22 75.6 75.91 76.17 76.38 76.57 76.73 76.87 76.99 77.11 77.2 77.29 77.37 77.45 77.51 77.58 77.63 77.68 77.73 77.78 77.82 77.86 77.89
50.90 62.51 66.98 69.44 71.03 72.14 72.98 73.62 74.14 87.12 85.71 84.83 84.18 83.67 83.26 82.93 82.64 82.39 82.18 81.99 81.83 81.68 81.55 81.43 81.32 81.22 81.13 81.04 80.96 80.89
TIR TIR TIR TIR TIR TIR TIR TIR TIR 74.57 74.92 75.22 75.48 75.71 75.91 76.08 76.24 76.38 76.51 76.62 76.72 76.82 76.91 76.99 77.06 77.13 77.2 77.26 77.32 77.37
N/A N/A N/A N/A N/A N/A N/A N/A N/A 4.801 4.801 4.801 4.801 4.801 4.801 4.801 4.831 4.994 5.136 5.26 5.371 5.47 5.558 5.639 5.712 5.778 5.839 5.895 5.947 5.995
Table 5.1. Converging recording beam incident angles, Bragg reading and diffraction angles, diffraction angles with normal incident reading, and minimum distance between neighboring SCPC units.
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For points A and C, the diffraction angles θ dnA and θ dnC for normal incident reading at 1540 nm are also calculated and listed in Table 5.1. These angles are not Bragg matched. It is important to notice that when F is less than 1000 mm, the diffraction beam at the left edge, A, will be totally reflected at the glass-air interface and trapped in the HPDLC until finally escaping from the edge. This total internal reflection (TIR) effect further decreases the diffraction efficiency at the edge of the HPDLC area. To avoid the TIR effect, the focal length of the lens should be larger than 1000 mm. When the diffraction angles are known, the converging properties of the diffracted beam in the x − y plane can be derived from Bragg's law. Fig. 5.8 shows the converging properties of the diffracted beams across the center of the cell in the y direction. The focus is not ideal in the
x − y plane. With the increase of F , the waist of the diffracted beam (the thinnest width of the diffracted beam) decreases, and the position of the beam waist moves away from the HPDLC. To obtain a relatively good focus, a lens with a large focal length is preferred in the recording setup. Since the lens is placed at 2 F from the HPDLC during exposure, a large F brings inconvenience to the fabrication process. Two focal lenses in series can be used to generate the same diverging recording beam profile at the position of the HPDLC. In the simulation above, only the locations across the center of the HPDLC area are considered. A qualitative simulation of various locations over the entire HPDLC area using ZEMAX, an optical simulation and design software, reveals that the focusing of the diffracted beam is also astigmatic, which matches well with our calculation and analysis. The convergence of the diffracted beam in the x − y plane is faster than that in the z direction. Therefore the linear dimension of the focus “point” is larger than that of the beam waist calculated in the simulation above.
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y F = 1000 A
B
C x
y F = 2000 A
B
C x
y F = 3000 A
B
C x
Fig. 5.8. The diffraction beam profile of 1 inch HPDLCs fabricated using lenses with various focal length F .
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5.5 Fabrication and Characterization of SCPC Working in Visible Wavelengths
5.5.1 Single Channel SCPC
For SCPCs working in visible wavelengths, we began sample fabrication with the nonpixelated HPDLC for the second type of SCPC devices, or single channel SCPC devices with built-in convergence, using ITO glass substrates. A UV-curing mixture consisting of 50% PN393 and 50% liquid crystal TL 205 was prepared. The HPDLC mixture was sandwiched between two substrates with the cell gap controlled by 15 μ m fiber spacers. The cell was exposed to a 351 nm Ar+ ion UV laser for 90 seconds in holographic setup as illustrated in Fig. 5.6. The intensity of a single recording beam was ~ 600 mW/cm 2 .
Z e ro O rd e r
F o c a l P o in t
L e ft: N o v o lta g e a p p lie d ; R ig h t: V o lta g e a p p lie d . ( 1 0 0 V A C sq u a re -w a v e 1 k H Z )
Fig. 5.9. The left panel : the switch-off state of the SCPC (no voltage applied); the right panel : the switch-on state (voltage applied). In each panel, the holographic focal point is the point on the right side, and the “pass-through” light is on the left.
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Fig. 5.9 shows digital images of a HPDLC sample working with a red laser (632.8 nm), showing the switching of the focal point of the SCPC. The image on the left panel is the switchoff state of the SCPC when no voltage is applied, and the image on the right panel is the switchon state when a 100 V square-wave AC 1 KHz signal is applied. On each panel, the large light spot is the pass-through light, or zero order, and the left fine point is the focal diffracted light. The diffraction angle away from the transmission beam is 45°. The holographic focal point was sampled with a detector, and the contrast ratio of the switch-off versus switch-on state is 50:1. The overall efficiency of the hologram was 30%. The loss of the efficiency comes from the scattering of the materials, and reflection from the surfaces as no AR coating is used on the glass substrates. We also developed a formula for green-laser (532 nm) curing. The materials are various ratios of Ebecryl 8301 and 4866 as pre-polymer bases to control the effective functionality of the polymer. These are mixed with the nematic liquid crystal BL038 ( no =1.527, ne =1.799) from EM Industries. Rose Bengal and N-Phenylglycine were selected as the photoinitiator and coinitiator that are sensitive to the visible light. The solution of photoinitiator was prepared with 4.0 wt.% Rose Bengal and 10.0 wt.% N-Phenylglycine in N-Vinyl Pyrrolidinone. The surfactant Sorbitan Mono-Oleate was also added to the mixture, which is known to reduce the surface interaction strength between the liquid crystal and polymer. The weight ratio of Oligomer: liquid crystal: initiator solution: surfactant was 45: 32.4: 12.6: 10. A droplet of the pre-polymer mixture was sandwiched between two AR-coated ITO glass substrates with 5-micron spacing thickness controlled with glass fiber spacers. The SCPC sample is exposed to a 5W green laser at a wavelength of 532 nm for 30 seconds. Fig. 5.10 shows two images of a single channel SCPC device in operation fabricated with green-laser curing. The beam size of the reading laser (532 nm) is about one-half inch, shown as
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the large spots in both left panel and right panel. The SCPC converts the incident light into a fine point (the small spots in both images) that is 45° off axis from the center of the SCPC. The focusing effect of the SCPC is very good, with a focus length of 10 cm, matching the writing condition. The right image in Fig. 5.10 shows the SCPC switched by an AC field of 160 Volts, 1k Hz. A Melles Griot laser power meter was utilized to measure the intensity of the incident beam (Iin) and the transmitted beam (I0th), and an OPHIR infrared power meter was employed to measure the diffracted beam intensity (I1st). The transmittance was calculated as I0th/ Iin, and the diffraction efficiency was calculated as I1st/ Iin. The testing result are detailed in Table 5.2. The contrast ratio of switch-off to switch-on is larger than 40.
Transmittance I0th/ Iin
Diffraction Efficiency I1st/ Iin
Switch On
95%
5%
Switch Off
5%
80%
Table 5.2 Testing result of SCPC.
Fig. 5.10. Switching of a SCPC working at 532 nm.
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5.5.2 10-channel SCPC
The SCPC is designed to work in tandem with a Fabry-Perot etalon to convert the circular interference fringe pattern into points or a point array. Thus the ring pixels of the ITO pattern on one SCPC substrate must match the Fabry-Perot ring. Fig. 5.11 shows an ITO ring pattern for a 10-channel SCPC.
Fig.5.11. A schematic description of CAD design of a 10-pixel ITO pattern in SCPC
The fabrication of a multiple channel SCPC is similar to the fabrication of one-pixel SCPC, because the holographic pattern in all the pixels are written at the same time. A 10-pixel SCPC working for 532 nm was fabricated. Fig. 5.12 shows the switching of the center pixel of the 10pixel SCPC that focuses 10 channels to one point. On both the left and right images, the left light spot (bigger spot) is the light that pass through the SCPC, the right spot (smaller) is the focusing point. In the left image, the center pixel is switched on with a voltage of 150 Volts. In the right image, no voltage is applied. Because all of the 10 pixels focus on one point, there is still light on the focusing point when the center pixel is switched on.
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Fig. 5.12. Switching of the center pixel of 10-pixel type-II SCPC.
Fig. 5.13. Switching of one non-center pixel of 10-pixel type-I SCPC
We also demonstrate the 10-pixel switching of the beam-steering SCPC (type I) that directs all 10 channels to different directions without focusing. An additional focusing lens (not shown in the figure) after the SCPC is needed to focus the selected channels to one point. Fig. 5.13 shows the switching of one ring-pixel of the beam-steering SCPC. The left point shows the light that is
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steered by the SCPC, the right point is the light passing through. The left point and the right point have the same size. When the ring-pixel is switched on with a voltage of 150 volts, all of the signal from the selected ring-pixel pass through without being steered to the left point, so a black ring appears in a bright background that is the other 9 channels (including the center pixel). Fig. 5.14 shows the switching of the center pixel of a beam-steering SCPC.
Fig.5.14 Switch on the center pixel of a beam-steering 10-channel SCPC.
5.6 Fabrication and Characterization of SCPC Working in NIR Wavelengths
5.6.1 Material Optimization for Big-Area SCPC Working in NIR
The HPDLC materials can be divided into two categories: UV curable materials and visible curable materials. Both materials have been used to demonstrate the SCPC concept working in the visible wavelength range. When fabricating the SCPC with a large area (minimum 2.54 cm diameter) working in the 1550 nm wavelength range, a 532 nm laser instead of a UV laser is used
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for the holographic recording, due to the lack of sufficient power in the UV laser, and the high efficiency of visible curable materials. Material optimization was initiated from several base formulas with different monomers and surfactants. The liquid crystal in all these formulas is BL038, a liquid crystal mixture from EMD Chemicals that has a high optical birefringence (0.27). The contents of the HPDLC mixtures are listed in Table 5.3. The monomer mixture in Formula 1 consists of the urethane acrylate monomers Ebecryl 8301 and Ebecryl 4866, both of which are produced by UCB Chemicals. Sorbitan mono-oleate, S-271 by Chem Services, is utilized as a surfactant to reduce the switching voltage. The monomer mixture in Formula 2 consists of Ebecryl 8301 and Trimethylolpropane tris (3-mercaptoporpionate)
(TT3) from Sigma-Aldrich.
A fluorinated acrylate monomer,
1,1,1,3,3,3 – Hexafluoroisopropyl Acrylate (HFIPA) is utilized as a surfactant in Formula 2. The monomer in Formula 3 is Dipentaerythritol Pentaacrylate (DPHA), a tetra-functional monomer. A mono-functional monomer, Vinyl Neononanoate (VN), serves as the surfactant in this formula. Photoinitiator solutions are required in all of these formulas to enable the photo polymerization and for resolving the liquid crystal in the monomers. The photo initiator solutions consist of Rose Bengal (photoinitiator), N-Phenylglycine (NPG, co-initiator) and 1-Vinyl-2-Pyrrolidinone (NVP, solvent and chain terminator), all available from Sigma- Aldrich. The HPDLCs fabricated using these materials were characterized and several issues arose. As the operation wavelength of the HPDLC is ~1500 nm, a 15-micron cell gap is necessary to achieve sufficient diffraction efficiency.
The switching voltage necessary to completely
deactivate the HPDLC is high (> 250V), and the samples may experience a dielectric breakdown when a high voltage is applied across the cell. The surfactant in Formula 1 introduces ions into the HPDLC and substantially increases the conductance of the HPDLC. The HPDLCs fabricated using Formula 1 material are easy to heat up when a voltage is applied and experience thermal
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switching instead of electrical switching. Significant effort was made to search for an appropriate HPDLC formula that solved these issues. When the liquid crystal BL011 is used to replace the liquid crystal BL038 in all three formulas, a reduction in the switching voltage was observed; however, the diffraction efficiency also decreased substantially. Signs of thermal switching were also observed in some of the samples. It is clear the BL011 formulas were not suitable for the device. Materials based on the three original formulas but with varied component ratios were further investigated.
Formula Ebecryl 8301 1
Ebecryl 4866
BL038
P.I.Solution 1
S-271
22.50%
32.40%
12.60%
10.00%
TT (3)
BL038
P.I.Solution 2
HFIPA
36%
4%
34%
16%
10%
DPHA
BL 038
P.I.Solution 1
VN
47%
38%
10%
5%
22.50%
Formula Ebecryl 8301 2
Formula 3
P.I.Solution 1 P.I.Solution 2
Rose Bengal 4% 3%
NPG 10% 7%
NVP 86% 90%
Table 5.3. Components of the HPDLC mixtures initially investigated.
After a series of experiments, a HPDLC mixture that is suitable for the HPDLC operating at ~ 1500nm based on Formula 3 is developed. The contents of the new formula (Formula-SCPC) are presented in Table 5.4.
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Formula-SCPC DPHA
BL038
P.I.Solution 1
VN
35.92%
43.36%
10.45%
10.27%
Table 5.4. Material contents of Formula-SCPC.
5.6.2 Fabrication and Characterization of Single Channel SCPC Working in NIR
As we have discussed in the optical design of the SCPC, the type-II (focusing) SCPC working at 1550 nm has strong astigmatism, which is also confirmed in testing of our type-II samples working at 1550 nm. The focusing in the x − y plane was faster than in the z direction, which is in agreement with the results of our qualitative simulation. Our research focuses on typeI (beam-steering) for SCPC devices working at 1550 nm.
Beam Splitter
Sample
θw1 θw2
Mirror
Shutter
Iris
Beam Expander
532 nm Laser
Fig. 5.15. Holographic recording setup for fabricating the SCPCs working in the 1550 nm range.
We fabricated the single channel HPDLC using ITO-coated glass substrates with AR coating. The “Formula-SCPC” HPDLC mixture was sandwiched between two substrates. The cell gap was controlled by 15 μ m fiber spacers. The cell was exposed in the holographic setup illustrated in Fig. 5.15 for 90 seconds. The recording beam incident angles at the HPDLC were:
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θ w1 = 43o59′ and θ w 2 = 20 o51′ to enable a large diffraction angle of θ = 79o . The intensity of a single recording beam was 600 mW/cm 2 . We characterized the electro-optical properties of the HPDLCs fabricated for the SCPC device. All of the HPLDCs showed strong polarization dependence. The diffraction efficiency for s-polarized light was much less than that of the p-polarized light and was therefore neglected, only p-polarized light was tested. A 1 KHz square wave signal was used to address the HPDLCs. The transmittance and diffraction efficiency as a function of voltage are presented in Fig. 5.16. The switching voltage was substantially decreased and the diffraction efficiency was 55%.
Electro-Optical Response of SCPC 0.6
Efficiency
0.5 0.4 Transmission Diffraction
0.3 0.2 0.1 0 0
20
40
60
80
100
120
140
160
Voltage ( v )
Fig. 5.16.Transmittance and diffraction efficiency as a function of voltage.
Compared with the high diffraction efficiency (80%) of the SCPC working at 532 nm, a number of factors lower the diffraction efficiency of the SCPC operating at 1550 nm: (1) The Pitch of grating of the HPDLC is much larger; (2) The AR coating is optimized for 532 nm, not 1550 nm;
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(3) The Sharp diffraction angle further degrades the AR coating performance;
5.6.3. Fabrication and Characterization of 32-channel SCPC Working in NIR
5.6.3.1 Fabrication Process
The 32-channel SCPC units were fabricated with AR ITO-coated substrates with thickness of 0.5 mm, thinner than the previously used glass substrates with a thickness of 1.0 mm. The purpose of using the thin glass is to minimize the size of the total device when multiple SCPC units must be stacked together. To hold the cells of the thin substrates properly during exposure, a special sample holder was designed and fabricated. The holographic recording setup for fabricating the SCPC units is illustrated in Fig. 5.15. The two recording beam incident angles were set to be θ w1 =41°25′, and θ w 2 =18°25′, for the normal o incidence reading beam to be diffracted to a large angle of θ = 79 . The Formula-SCPC material
was utilized. 15-micron fiber spacers controlled the cell gap. The cells were pressed at 4.5 psi by a balloon for 12 minutes to ensure cell gap homogeneity prior to the holographic exposure. The exposure time was 5 minutes and the total output power of the laser was set to 3W during the exposure. After exposure, the cell was cured in a 3W laser beam to polymerize the monomers outside the HPDLC area. The edges of the cells were secured with 5-minute epoxy.
5.6.3.2 Switching of the 32-channel SCPC
The switching properties of the SCPC were investigated. The driving signal was a 1kHz square wave generated by a HP function generator and a Trek amplifier. Voltage was applied on each of the 32-channels of the SCPC unit one channel at a time, and the switching of each single ring pixel was observed. Fig. 5.17 demonstrates the switching of several channels of a SCPC sample.
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Fig. 5.17. Switching of independent channels in the SCPC unit. The photos, from left to right, show the deactivation of the central pixel, the 5th pixel (count from the center), and the outmost pixel (32th), respectively.
The electro-optical performance of the central pixel of this SCPC unit is shown in Fig. 5.18. The transmittance of some channels is summarized in Table 5.5. The diffraction efficiency was lower than that of the HPDLC fabricated using unpatterned glass substrates. This is attributed to the scattering from the etched lines under laser exposure and the nonuniformity caused by the thin glass, which further degrade the interference pattern of the recording beams and lower the grating quality.
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unit (%)
Electro-optical Response of SCPC 100 90 80 70 60 50 40 30 20 10 0
Transmission Diffraction
0
20
40
60
80
100
120
140
160
Applied Voltage ( v )
Fig. 5.18. The normalized transmittance and diffraction efficiency of the center channel of a SCPC unit as a function of voltage.
Channel Number 1 2 3 13 16 19 24 26 28
Ton (%) 45.7 50 52 56.7 57.5 58.7 62.3 64.6 67.6
Toff (%) 62.2 69.1 66.9 73.3 78.7 81.6 76.3 78.3 82.7
Table 5.5. The transmittance of some channels of a SCPC unit.
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5.6.3.3 Wavelength Dependence
The SCPC device was designed to operate in the wavelength range of 1530- 1560 nm. Therefore, the wavelength dependence of the device performance was a critical factor. An Agilent 81689A tunable laser was utilized to characterize the wavelength dependence of the 32channel SCPC devices. The measurement setup is illustrated in Fig. 5.19. The incident light was perpendicular to the SCPC cell surface. The intensity of the incident light, the transmitted light, and the diffracted light was measured using a Melles Griot universal power meter at various wavelengths. The zero field transmittance and diffraction efficiency of the SCPC device were calculated and the results are presented in Fig. 5.20.
Photo Detector
Collimation Lens I1st
Iris
Agilent 81689A Tunable Laser
Iin
I0th
Fig. 5.19. Optical setup for measuring the wavelength dependence of the SCPC units.
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Fig. 5.20. Transmittance and diffraction efficiency as a function of incident wavelength of the switch-off state of a SCPC sample: JL101404B.
The SCPC samples have strong polarization dependence; the diffraction efficiency of spolarized light is substantially smaller than that of p-polarized light and therefore can be neglected. The incident light in the measurement was p-polarized. The diffraction efficiency and transmittance were calculated in reference to the incident light intensity. The wavelength of the incident light was increased from 1525 nm to 1575 nm, in increment steps of 5 nm. The transmittance of the SCPC shows no substantial change with the wavelength; however, the diffraction efficiency decreased substantially when the incident wavelength was greater than 1560 nm. The decrease is attributed to the increase of the diffraction angle with an increase in the reading beam wavelength. When the reading beam wavelength is increased from 1528 nm to 1560 nm, the diffraction angle changes from 75°46′ to 83°37′, which is very close to total internal reflection. The performance of the anti-reflection coating degrades substantially when the incident angle is close to TIR. When the incident wavelength is 1575 nm, the incident light is
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totally reflected at the glass-air interface. The diffraction efficiency is fitted with the coupled wave theory for a transmission grating, 2 1/2 ⎫ ⎧ ⎡ ⎪ ⎛ Δα ⎞ ⎤ ⎪ κL ⎢1 + ⎜ = ⎟ ⎥ ⎬. 2 2 sin ⎨ ⎝ 2κ ⎠ ⎦⎥ ⎪ A1 (0) cosθ1 ⎛ Δα ⎞ ⎢ 2 ⎪ ⎣ ⎩ ⎭ κ +⎜ ⎟ ⎝ 2 ⎠
A2 (L ) cosθ 2 2
η =
κ2
where k = πn1 cos(2θ B ) / λ cos(θ B ) ,
Δα = −
2
(5-6)
π
Δλ is the phase mismatch generated Λ n0 Λ cosθ B
from the deviation in wavelength. As the recording beam incident angles are: θ w1 =43°59′ and
θ w 2 = 20 o51′ as shown in Fig. 5.6(b), the writing angle θ w inside the HPDLC is determined by: 2 θ w = arc(sin( θ w1 )/1.5) - arcsin(sin( θ w 2 )/1.5); θ w = 6.9264°. Considering a writing wavelength of λ=532 nm and refractive index n ~ 1.5, the period of the grating Λ is 2nΛ sin θ w = λ and calculated to be Λ = 1.478 μm. For a reading beam with normal incidence on the cell surface, as the designed exit angle in air is 79°, the Bragg angle is
2 θ B =arcsin((sin(79°))/1.5),
θ B = 20.44o .(see Fig. 5.2(a)). The wavelength that meets the Bragg condition is: λB = 2nΛ sin θ B =1.548 μm. As the wavelength varies from λB , the diffracted angle shifts to meet the condition of constructive interference: Λ sin θ + Λ sin θ ′ = λ/n , as the grating can be treated as a thin grating (defined in Chapter 2). When the diffracted light passes through the glass-air boundary, the intensity of the refracted light is [64]:
I // = (
sin 2θ di sin 2θ d )2 sin (θ di +θ d ) cos 2 (θ d − θ di ) 2
(5-7)
Considering the coupled wave theory and the light refraction equation (5-7), we fit the modeling result with the measurement result, as shown in Fig. 5.21. The model does not consider
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the effect of scattering, which may explain the discrepancy between the model and the
a
experimental result.
Wavelength Dependence of Diffraction Efficiency
Diffraction Efficiency
0.25
0.2
0.15 Modeling Measurement 0.1
0.05
0 1520
1530
1540
1550
1560
1570
Wavelength
Fig. 5.21. Fit of the model based on coupled wave theory and the refraction principle, with the experimental result, for the wavelength dependence of the diffraction efficiency.
5.6.3.4 Angular Dependence
The dependence of the transmittance and diffraction efficiency on the reading beam incident angle was characterized to evaluate the holographic recording optical setup. A 1540 nm fiber laser generated the incident beam. The transmitted beam intensity and diffracted beam intensity were measured using a Melles Griot universal power meter and an OPHIR infrared power meter with various reading beam incident angles. The transmission as a function of incident angle is presented in Fig. 5.22, and the diffraction efficiency as a function of incident angle of the SCPC is presented in Fig. 5.23. Both Fig.s 5.22 and 5.23 are based on the test result of SCPC sample
130
JL101404B. The diffraction efficiency reached its maximum at normal incidence, while the transmittance minimized at an incident angle of 1°. This result proved the choice and control of recording beam incident angle was optimized for normal incidence reading of the designated wavelength. When the reading beam incident angle was greater than 1°, the diffraction efficiency decreased much faster than the increase of the transmittance. The abruptness of the decrease in diffraction efficiency was due to the increase of the diffraction angle with the increase of the incident angle. The designed diffraction angle in the device was 79° in air and the diffraction angle inside the SCPC was close to the total reflection angle at the glass-air interface. The efficiency of the anti-reflection coating on the glass substrates decreased substantially when the diffraction angle was close to the total internal reflection angle and the diffracted light would be totally reflected when the total internal reflection angle was reached or exceeded.
Transmission
Transmission Vs Incident Angle 0.7 0.68 0.66 0.64 0.62 0.6 0.58 0.56 0.54 0.52 0.5 -8 -7
-6 -5 -4
-3 -2 -1
0
1
2
3
4
5
6
7
8
Incident Angle
Fig. 5.22. The transmission as a function of incident angle of the SCPC.
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On the other hand, the performance of the anti-reflection coating hardly changed for the transmitted light. The reason that the transmittance changes with the incidence angle can be mostly attributed to the change of diffraction efficiency with the change of incident angle. Considering coupled wave theory, the diffraction efficiency η for transmission gratings can be fit using the transmission data: 2 1/ 2 ⎫ ⎧ ⎡ ⎪ ⎛ Δα ⎞ ⎤ ⎪ sin ⎨κ L ⎢1 + ⎜ η= 2 ⎟ ⎥ ⎬, 2 2 κ ⎝ ⎠ ⎦⎥ ⎪ κ + ( Δ2α ) ⎪⎩ ⎣⎢ ⎭
κ2
where k = πn1 cos(2θ B ) / λ cos(θ B ) is the coupling constant, and Δα = −4πn0 Δθ sin θ B / λ is the phase mismatch. Here λ is the reading beam wavelength, θ B =20.44° is the Bragg angle,
Diffraction Vs Incident Angle 0.16 0.14 Diffraction
0.12 0.1 0.08 0.06 0.04 0.02 0 -8
-6
-4
-2
0
2
Incident Angle
Fig. 5.23.The diffraction efficiency as a function of incident angle of the SCPC.
4
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n0 =1.6 is the average refractive index of the HPDLC, n1 is the index modulation of the grating, and Δθ is the deviation from Bragg angle. The corresponding transmittance is given by 1-η, provided that all other losses are neglected. The measurement was for p-polarized light only; therefore, the coupling constant κ was calculated for p-polarized light and the fitting was only valid for p-polarized light. The best-fit curve is presented in Fig. 5.24. The index modulation of the HPDLC is n1 = 0.02 ± 0.002.
Normalized Transmission
Transmission Vs Incident Angle 1 0.95 0.9 Modeling result Measurement
0.85 0.8 0.75 0.7 -8
-6
-4
-2
0
2
4
6
8
Incident Angle
Fig. 5.24. Normalized transmittance are fit to the formula for a transmission grating derived by coupled wave theory. Measurement results are of SCPC sample JL101404B.
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5.7 Summary and Conclusions
We have demonstrated the concept and design of an innovative SCPC device, and have fabricated and tested SCPC devices working at different wavelengths (visible and NIR), and with different channel numbers (single channel, 10 channel, and 32 channel). Two types of SCPC devices are analyzed with more focus on the second type, a beam-steering SCPC. The high diffraction efficiency of up to 80% in the visible, and 60% in the NIR was achieved. The wavelength and angular dependence were also investigated.
This research illustrates the
potential for making electrically tunable optical devices such as random optical switches and spectral imaging detectors.
CHAPTER 6
Lasing of Dye-Doped HPDLC
6.1 Introduction
In this chapter, we will discuss the materials, fabrication and characterization of lasing emission in dye doped HPDLCs. Lasing from different modes of HPDLCs will be studied and both the switching and tunability of the lasing function will be demonstrated. Lasing from twodimensional HPDLC based photonic band gap (PBG) materials will also be demonstrated. Finally, lasing from polarization modulated gratings will be discussed.
6.2 Introduction to Dye
Typically, dye molecules are more or less rod-like. Usually the major component of the transition moment of the molecule is along the long molecular axis (positive dye) or short axis (negative dye) [65]. Positive dyes, as in Fig. 6.1(a), absorb the component of unpolarized light in the long axis of the molecules. In terms of absorbance A, A// > A⊥ . In negative dyes (see Fig. 6.1(b)), A// < A⊥ and absorption occurs orthogonal to the molecular axis. Since the molecule is rotating, light is absorbed in any direction orthogonal to the axis. The dichroic ratio D and is defined by:
D=
A// A⊥
(6-1)
Clearly, for a positive dye, D > 1 and for a negative dye, D< 1. The most widely used dichroic dyes in guest-host LCDs fall basically into two classes from a chemical structure point of view – azo dyes and anthraquinone dyes. Fig. 6.2 describes the
134
135
molecular structure of two kinds of dyes. In Fig. 6.2(a), A represents an acceptor, such as − NO3 and D represents a donor such as − NH 2 . Without A and D, the dye absorbs light in the UV band, while with different acceptors and different donors, the azo dyes will absorb light in a different wavelength band. Many azo dyes have been found to absorb light in the visible wavelength band. Fig. 6.2(b) shows a basic anthraquinone dye. Without any substitute, the dye absorbs UV light. To make a dye that absorbs visible light, substitutes are introduced at 1,4,5 or 8.
(a)
(b)
Fig..6.1. Absorption of positive dye (a) and negative dye (b).
Usually azo dyes have higher solubility in liquid crystals than anthraquinone dyes because azo dye molecules are more rod-like than anthraquinone dye molecules. To increase the solubility of anthraquinone dyes in liquid crystals, usually donors such as − NH 2 and acceptors like
− NO2 or − CN are introduced @2,3,6 or 7. A dichroic mixture is basically a homogeneous mixture of dye(s) in a liquid crystal host. The various physical properties of dichroic mixtures depend upon the physical properties of the dyes, the host, and the combination. For example the color of the mixture is mainly dependent on the
136
dyes, while the dielectric anisotropy, elastic constants and refractive indices are basically those of the liquid crystal. Viscosity is dependent on both. Some of the properties such as absorbance and percent transmittances are also dependent on alignment, cell gap, etc. The addition of a dye may slightly modify the physical properties of the liquid crystal mixture such as its operable temperature range, dielectric and optical anisotropy, etc. 2
3
1 N N
A
D
4
O
O
5
8 7
(a)
6
(b)
Fig. 6.2 Two basic kinds of dyes (a) azo dye (b) anthraquinone dye
)
The director of dyes in the host liquid crystal coincides with the director of the host, n .
)
However, the direction of each dye molecule deviates from the director n due to thermal fluctuations. The impact of the thermal fluctuation may be different in the dye and liquid crystal molecules depending on their molecular lengths and geometry, which are shown in Fig. 6.3. The liquid crystal (Lm) and dye (Dm) molecules makes an angle θ and φ respectively with the
)
director n . The order parameters of the liquid crystal molecule ( S L ) and dye molecule ( S D ) and the transition moment of the dye absorption ( S T ) as determined from the distribution of their long molecular axes are given by:
137
SL =
SD =
ST =
3 cos 2 θ − 1
(6-2)
2 3 cos 2 φ − 1
(6-3)
2 3 cos 2 θ T − 1
(6-4)
2
Where θ and φ are the angles made by the long molecular axes of the liquid crystal and dye molecules, respectively, with the director of the liquid crystal (n). θ T is the angle between the transition moment and the director. If we assume that the direction of the transition moment, T, of the dye deviates from its long molecular axis, Dm, at an angle β: the absorbance, A, of the incident polarized light whose electric field vector, E, makes an angle, ψ, with the director, n, can be given by [65]:
⎡⎛ s ⎞ ⎤ ⎛ 1− SD ⎞ ⎛ SD ⎞ 2 2 A( β ,ψ ) = kcd ⎢⎜ D ⎟ sin 2 β + ⎜ ⎟ + ⎜ ⎟ 2 − 3 sin β cos ψ ⎥ ⎝ 3 ⎠ ⎝ 2 ⎠ ⎣⎝ 2 ⎠ ⎦
(
)
(6-5)
Where k is the magnitude of the transition moment and c and d are, respectively, the concentration of the dye and the thickness of the liquid crystal layer. The dichroic ratio is expressed as the ratio of the absorbance at ψ= 0 and ψ=π/2:
A// A( β ;ψ = 0) 2 + 4S D − 6 S D sin 2 β D= = = π A⊥ 2 − 2S D + 3S D sin 2 β A( β ;ψ = ) 2
(6-6)
It is interesting to note that (6-5) can be written as
A( β ,ψ ) = A⊥ + ( A// − A⊥ ) cos 2 (ψ ) The order parameter, S T , of the transition moment is determined experimentally as:
(6-7)
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ST =
A// − A⊥ D −1 = A// + 2 A⊥ D + 2
(6-8)
S D (2 − 3 sin 2 β ) 2
(6-9)
From (6-6) and (6-8) we get:
ST =
For elongated pleochroic dyes, β is extremely small, so S T = S D . The later discussion will use S to represent both the dye order parameter and the order parameter of the transition moment when we assume β=0. D irecto r D m : A x is o f D ye M o lecu le φ
β
T : T ran sitio n M o m en t L m : L on g A x is o f liq uid crystal M o lecu le
θ
ψ
P : E lectric V ector of P o larized L igh t
Fig. 6.3. Dye molecules inside liquid crystals
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6.3 Introduction to Laser
The word "laser" is an acronym for Light Amplification by Stimulated Emission of Radiation. A traditional laser is composed of a pumping source, an active laser, or gain, medium and a resonant optical cavity. The gain medium serves to transfer the external energy from the pump source into the laser beam. The simplest laser model to understand is the two energy-level system, with a ground state energy and an excited state of energy , where >, as shown in Fig. 6.4(a). Assume there is a group of N atoms; the number of these atoms in the ground state is defined as N1, while the number in the excited state is N2 , satisfying N1 + N2 = N. The energy difference between the two states is given by ΔE = E2 − E1 = hν 0 , Where h is Planck's constant, and ν 0 determines the characteristic frequency of light that interacts with the atoms. In a thermal equilibrium state, the ratio of the number of atoms in each state is given by a Boltzmann distribution[131, 132]:
N2 − ( E2 − E1 ) = exp[ ], N1 kT
(6-10)
where T is the temperature (in unit of kelvin) of the group of atoms, and k is Boltzmann's constant. There are three possible interactions between a system of atoms and a light that we must consider: Spontaneous emission, absorption, and stimulated emission. If an atom is in an excited state, it may spontaneously decay to the ground state at a rate proportional to N2, thereby emitting a photon of frequency ν 0 . Here, the photons are emitted stochastically and there is no fixed phase relationship between the photons emitted from a group of excited atoms; in other words, this spontaneous emission is incoherent. In the absence of other processes, the number of atoms in the excited state at time t, is given by:
140
N 2 (t ) = N 2 (0) exp(
−t
τ 21
),
(6-11)
where N2(0) is the number of excited atoms at time t=0, and τ21 is the lifetime of the transition between the two states. This emission is defined as a spontaneous emission.
E3
E2
R (radiationless transition)
E2
hν 0 E1
P (pump transition)
L (laser transition)
E1 (a) two-level system
(b) three-level system
Fig. 6.4. (a) Two-level energy system of laser medium. (b) three-level energy system.
If light (photons) of frequency ν 0 pass through the group of atoms, there exists a defined probability of atoms in the ground state absorbing a photon and being excited to the higher energy state. When an atom in the excited state interacts with a photon of frequency ν 0 , the atom may decay, emitting another photon with the same phase and frequency as the incident photon. This process is known as stimulated emission. Critically, stimulated emission is defined by the fact that the induced photon has the same frequency, phase, and polarization as the inducing photon. In other words, the two photons are coherent. It is this property that allows for optical amplification, and the production of a laser system.
141
In the operation of a laser, all three light-matter interactions (spontaneous emission, absorption, and stimulated emission) occur. Initially, atoms are energized from the ground state to the excited state by a process referred to as pumping. If the ground state has a higher population density than the excited state (N1 > N2), the process of absorption is dominant and there is a net attenuation of photons. If the populations of the two states are the same (N1 = N2), the rate of absorption of light exactly balances the rate of emission; the medium is optically transparent. If the higher energy state has a greater population density than the lower energy state (N1 < N2), then the emission processes dominate, and the radiation field within the system undergoes a net increase in intensity. In order to produce a faster rate of stimulated emission than absorption, a population inversion is required: N2/N1 > 1. In a two-level system, the lower energy state contains a larger population than the higher energy state, as described by equation (6-10), a population inversion (N2/N1 > 1) can never exist for a system in thermal equilibrium. To achieve the necessary population inversion, the system must be pushed into a non-equilibrated state. At minimum, a three-level system, as shown in Fig. 6.4(b), is required. Consider a group of N atoms with three energy states, E1, E2 and E3, and E1 < E2 < E3. The population densities of each state are N1, N2 and N3, respectively. Initially, the system of atoms is at thermal equilibrium and the majority of the atoms will be in the ground state: i.e. N1 ≈ N, N2 ≈ N3 ≈ 0. When the atoms are subjected to light of a frequency ν31, where E3 - E1 = hν31, the process of optical absorption will excite the atoms from the ground state to level 3, such that N3 > 0. The energy transition E1 → E3 is referred to as the pump transition. In an optical medium suitable for laser operation, it is required that these excited atoms quickly decay to level 2. The energy released in this transition may be emitted as a photon
142
(spontaneous emission), or, in practice, the 3→2 transition (labeled R in Fig. 6.4(b)) is usually radiationless, with the energy being transferred to a vibrational motion (heat) of the host material surrounding the atoms. An atom in level 2 may decay by spontaneous emission to the ground state, releasing a photon of frequency ν21 (given by E2 - E1 = hν21), which is shown as a laser transition in Fig. 6.4(b). If the lifetime of this transition, τ21 is much longer than the lifetime of the radiationless 3→2 transition τ32 (if τ21 >> τ32), the population of E3 will essentially be zero (N3 ≈ 0) and a population of excited state atoms will accumulate in level 2. If over half the N atoms can be accumulated in this state, then the population inversion condition (N2 > N1) is met, and optical amplification at the frequency ν21 can be obtained. Though the first type of laser to be discovered (based on a ruby laser medium, by Theodore Maiman in 1960) was a three-level system, in practice, most lasers are four-level systems, as depicted in Fig. 6.5. Here, the pumping transition P excites the atoms in the ground state (level 1) into the pump band (level 4). The atoms in the upper level, E4, and lower laser level, E2, decay through fast, non-radiative transitions into E3 and E1, respectively, leading to negligible population densities in the states E4 and E2: N2 ≈ 0 and N4 ≈ 0. The laser transition occurs in the energy transfer from E3 to E2. Since the lifetime of the laser transition L is long compared to that of transitions R1 and R2 (τ32 >> τ43 and τ32 >> τ21), any appreciable population accumulating in level 3 will form a population inversion with respect to level 2. Thus, the optical amplification and laser operation occurs at a frequency of ν32 (E3 - E2 = hν32). As the light generated by stimulated emission is equivalent to the input signal in terms of wavelength, phase, and polarization, this gives laser light its characteristic coherence and allows
143
it to maintain the uniform polarization and monochromaticity established by the optical cavity design, as was discussed in the chapter on Fabry-Perot.
E4 R 1 (radiationless transition) E3 P (pump transition) L (laser transition)
E2 R 2 (radiationless transition)
E1
Fig. 6.5. A four-level laser energy diagram.
The lasing threshold is the lowest excitation level at which the laser's output is dominated by stimulated emission rather than by spontaneous emission. Below the threshold, the laser's output power rises slowly with increasing excitation. Above this threshold, the slope of power vs. excitation is orders of magnitude greater. The linewidth of the laser's emission also becomes orders of magnitude smaller above the threshold. It is above the threshold that the laser is considered to be lasing. The lasing threshold is reached when the optical gain of the laser medium is exactly balanced by the sum of all of the losses experienced by the radiation field in one round trip through the laser's optical cavity. This can be expressed, assuming a steady-state operation, as [133]:
144
R1 R2 exp(2 g th l ) exp(−2αl ) = 1
(6-12)
where R1 and R2 are the mirror reflectivities, l is the length of the gain medium, g th is the gain parameter and exp(2 g th l ) is the round trip threshold power gain, while α is loss parameter and exp(−2αl) is the round trip power loss. The optical loss is a near constant for any particular laser (α = α0), especially close to threshold. Under this assumption the threshold condition can be rearranged as:
g th = α 0 −
1 ln( R1 R2 ) 2l
(6-13)
6.4 Introduction to Dye Laser
Organic dyes are widely known for their ability to generate laser emission over a wide wavelength range because of the special electronic energy levels of the dye molecules [66]. In a dye molecule, each electronic level of the molecule is associated with a set of vibrational and rotational energy levels spaced closely together compared with the electronic level spacing. Optical pumping by external radiation brings the molecule from one of the vibrational-rotational levels of the ground electronic states to one of the vibrational-rotational levels of an excited state. The excited dye molecule tends to decay very quickly to the lowest-lying vibrational-rotational level of the excited state, which serves as the upper laser level. The decay process is non-radiative and typical lifetimes are in the picosecond range. The lasing emission occurs when the dye molecule returns to one of the vibrational-rotational levels of the ground state. As a result, the emission spectrum of a dye molecule has a broad curve and is normally shifted from the absorption spectrum.
145
At threshold, the conditions for dye lasing at frequency ω L are that the gain G(ω ) be equal + − to the effective cavity loss L. Supposing N and N are the density of the excited and ground
state molecules, respectively, and σ e (ω ) and σ a (ω ) are the induced emission and absorption cross sections, respectively, these conditions may be written as [134]:
G (ω L ) = σ e (ω L ) N + − σ a (ω L ) N − = L
(6-14 )
dG dω
(6-15 )
ωL
=0
Intensity
absorption
fluorescence Lasing emission
Wavelength
Fig. 6.6. Absorption, fluorescene and lasing of dye.
The bars indicate a thermodynamic average over the vibrational sublevels of the electronic levels. It is convenient to express the gain in terms of the fluorescence spectrum K (ω ) , giving [135]:
G (ω ) = ( N + − e h[(ω −μ ) / kT ] N − )(
πc 2 ) K (ω ) ωn
(6-16)
146
where n is the refractive index of the host medium, and μ the chemical potential difference arising from the general result
σ a (ω ) h[(ω − μ ) / kT ] . In analyzing equation (6-16) at threshold, =e σ e (ω )
we find the dye lasing threshold frequency ω L increases with cavity loss L and decreases with + − dye concentration (N = N + N ).
Fig. 6.6 depicts the absorption and fluorescence spectra along, as well as the lasing emission. Experimental results [134] agree with the theoretical analysis that the dye lasing threshold frequency ω L increases with cavity loss L and decreases with dye concentration (N = N + N ). +
−
This will also be shown in our experimental results. As the fluorescence spectrum of the dye shifts towards longer wavelengths as compared with the absorption spectrum, lasing tends to occur on the longer wavelength side of the peak of the fluorescence spectrum. While the fluorescence spectrum “pulls” the lasing wavelength closer to the fluorescence peak with lower values, the absorption spectrum tends to force the lasing wavelength farther away from the fluorescence peak. Generally, for an isotropic dye molecule, the photoexcitation is insensitive to the polarization state of the excitation light. While for an anisotropic dye molecule, the photoexcitation highly depends on the polarization state of the pumping light, due to the dichroism of the dye molecules [67]. For a positive dye, the photoexcitation of the dye molecules by a linearly polarized pumping light source with polarization parallel to the dipole moment of the dye molecules is larger than that perpendicular to the dipole moment. If all of the dye molecules are oriented in the same direction, the photo-excitation of the sample is polarization dependent. If all the dye molecules are randomly distributed in a sample, they function as an isotropic medium and the photoexcitation is polarization independent.
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Laser emission tuning within the emission band of the dye is accomplished by discriminating against most of the frequencies, i.e., by making the cavity loss larger than the gain for most frequencies. The traditional method of accomplishing this makes use of the “Littrow arrangement” sketched in Fig. 6.7, which shows that tuning the center peak of the laser is achieved by rotating a diffraction grating, which reflects radiation of wavelength λ only in those directions satisfying the Bragg condition: 2dsin(θ)=mλ, m=1,2,3,…
(6-17)
where d is the spacing between the lines of the grating. Wavelengths not satisfying (6-10) are not fed back along the cavity axis and consequently have large losses. Thus, the bandwidth of the laser radiation is greatly reduced, and tuning is accomplished by rotation of the grating. Basically, the Littrow arrangement for the wavelength tuning of a dye laser is a mechanical tuning method.
Flashlamp
θ
Output Dye
Diffraction Grating
M irror
Fig. 6.7. “Littrow arrangement” tunes the center peak of a laser by rotating the grating.
148
Fig. 6.8 shows the molecular structures of the laser dyes Pyrromethene 580 (P580) (a) and DCM (b). While the dye P580 has a lasing emission wavelength ranging from 545 nm to 585 nm at the excitation of a Nd:YAG(532) laser, the wavelength range of lasing emission of the DCM dye covers 600 nm~ 655nm. We have used both P580 and DCM in our dye-doped HPDLC systems that will be discussed later in this chapter.
H3C(CH2)3
(CH2)3CH3
+
N
N BF F (a)
N O
NC
CN
(b) Fig. 6.8. Molecular structure of the lasing dye Pyrromethene 580(a) and DCM(b).
149
6.5 Introduction to Photonic Band Gap Materials
Photonic band gap (PBG) materials, with considerable promise for the emerging generation of nano- and mesoscale optoelectronic components [68, 69], have recently been utilized for high technology applications. One-, two- and three-dimensional PBG materials have been studied for more than a decade. Refractive index modulation at periodicities comparable to optical wavelengths influences the behavior of photons in a manner akin to the influence of the crystal lattice on the behavior of semiconductors [70]. Constructive and destructive interference of a propagation wave leads to the enhancement and depletion of the density of states of sustainable optical energies (frequencies) within a material. A well-defined density of states exists in these materials and propagation of specific energies or photons may be completely, or partially prohibited by the physical structure of these photonic devices. Complete suppression of the density of optical states, where the propagation of photons with a specific energy is prohibited, depends on the structure (symmetry-defined space group), composition, and refractive index contrast of the PBG [71]. These characteristics are the basis for next generation optical waveguides, sensor platforms, lasers, and display devices. Various fabrication techniques are being used to fabricate PBG materials, such as advanced lithographic techniques [72,73,74], layer-by-layer chemical vapor deposition [75], colloidal crystal
growth
[76],
self-assembly
of block
co-polymers
[77,78],
and
two-photon
microfabrication [79]. Synonymous with the electronic band gap properties of semiconductors, photonic crystals and PBG materials exhibit interesting properties at or near their band gaps. Dowling, Bowden and co-workers performed early work in this field. They developed a theoretical framework whereby lasing could be achieved in a periodic structure composed of materials of different dielectric
150
constants [80]. Lasing has since been achieved in a wide variety of photonic crystals composed of various organic and in-organic materials exhibiting band gap structures.
6.6 Introduction to Lasing in Liquid Crystal Materials
A variety of liquid crystal materials have been used for both lasing and amplified spontaneous emission. The structure of liquid crystal materials is easily controlled through surface treatment techniques, holographic methods, additive material such as chiral dopants, and confining geometries. PBG structures have been created in cholesteric [81-86] and ferroelectric liquid crystals [87-88], liquid crystal elastomers [89, 90], and polymer dispersed [91, 92] and holographic-polymer dispersed liquid crystals (HPDLCs) [93-95]. With the existence of a band gap in these structures, it is possible to achieve lasing with the use of the proper laser dyes and pumping sources. The most familiar medium for lasing from liquid crystal based photonic crystals is the cholesteric liquid crystal. Cholesteric liquid crystals are chiral nematics, where the handness of the constituent molecules causes the orientation of the local nematic director to vary linearly with position along the helix axis, which is perpendicular to the director. The spatial period of the structure is defined as the pitch, which is determined by the twisting power and the concentration of the chiral constituents. As a consequence of the birefringence of the liquid crystal and the periodicity of the helical cholesteric structure, light propagation along the helix axis is forbidden in a range of wavelengths, incident light is strongly reflected when the wavelength of light lies in this band and has the same helicity as the cholesteric. The edges of this reflection band are at wavelengths of no p and ne p , where no and ne are the ordinary and extraordinary refractive indices of the liquid crystal, and p is the pitch [96].
151
Because of the existence of the selective band, cholesteric liquid crystals are 1D photonic bandgap materials with a bandgap structure, and allow for the possibility of lasing without the external mirrors that usually forms a laser cavity. When a fluorescent dye is dissolved in the cholesteric host, the propagation of one normal mode of the emitted light is forbidden if the peak of the fluorescent emission of the dye is in the selective reflection band of the cholesteric. Lasing in dye-doped cholesterics was proposed as early as the 1970s by L.S. Golderberg and J.M. Schnur [97]. Their proposed lasing medium comprises of a mixture of a strongly fluorescent dye, 7-diethyl-4-methyl coumarin, and a cholesteric liquid crystal solution of 40 percent cholesteryl oleyl carbonate, 30 percent cholesteryl chloride and 30 percent cholesteryl monanoate. The dye is pumped with light at its absorbing wavelength of 340 nm, and emits light in a band centered at a wavelength of about 450 nm. Other observations of lasing in dye-doped cholesteric liquid crystals were made approximately thirty years ago [98]. Yablonovitch theoretically described the lasing action from these structures, and other photonic crystals, through the use of distributed feedback theory [99]. Kopp and co-workers have investigated lasing at the band edge of cholesteric liquid crystal materials [100,101]. Chanishvili and co-workers and have done extensive work toward creating tunable cholesteric lasing sources in both the ultra-violet and visible regimes [102-104]. A.Munoz et al. [84] further reported UV lasing in cholesteric liquid crystals without dye material, where the cholesteric liquid crystal act as both the active material and a distributed cavity host. Their material comprises the cholesteric mixture BL061 from EM Industries with right-handed helicity and the right-handed chiral dopant 4-(2-methylbututyl)-4-cyanobiphenyl) (CB-15). Lasing at different wavelengths in the near UV are observed by changing the ratio of the
152
chiral dopant, thus shifting the edge of the reflection band in the range of 385 nm – 405 nm, when samples are under picosecond excitation at 355 nm. Tuning of the lasing of a dye-doped cholesteric material always corresponds to tuning of the pitch of the cholesteric. Traditionally, thermal tuning uses the temperature dependence of the cholesteric liquid crystal or the chiral dopant; changing the temperature changes the pitch of the cholesteric and in turn the edge of the forbidden band, thus the lasing wavelength can be thermally tuned. Other different methods have been exploited to achieve the tunability of lasing in cholesteric systems, such as applying a mechanical stress [105] or an electric field [125]. A photo-induced tuning of cholesteric lasing was achieved by A. Chanishvili et al. [106, 107], based on the photo-induced transformation of the chiral dopant that allows the selective reflection band to be shifted. The chiral dopant ZLI-811 undergoes a phototransformation when irradiated at UV wavelengths below 300 nm. This transformation is a photo-Fries rearrangement. The tunable range is up to 30 nm; however, the tuning speed is very slow.
6.7 Introduction to Dye-Lasing in HPDLC
Holographic formation of PBG materials has enabled a wide arrange of device applications by leveraging existing liquid crystal and display technologies. Holographic formed PBG materials allow for a simple, rapid fabrication process by the exposure of a prepolymer mixture to an interference pattern. More recently, attention has been given to lasing in holographic-polymer dispersed liquid crystals. Bunning and coworkers were the first to observe laser emission from a HPDLC [108, 109]. As their first trial fabrication of gratings with Coumarin 485 (C485) added to the pre-polymerized syrup was unsuccessful because of the photodegradation of C485 and the overlap of the C485 fluorescence with the absorption spectrum of Rose Bengal, they incorporated C485 in the reflection grating by simply washing a C485-butyl acetate solution over a formed
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HPDLC sample after the removal of the top glass substrate [108]. When photo excited with the tripled-output (355 nm) of a Nd:YAG, the lasing emission occurs at the band edge of the reflection notch that was designed to be within the fluorescence spectrum of C485. Subsequently, Bunning and coworker’s work has been followed by Matsui and coworkers [110,111], Luchetta and coworkers [93,112], and several others [94, 114]. A HPDLC is formed by the exposure of a prepolymer mixture of liquid crystal, monomer and photoinitiator to an interference pattern, which has been discussed in previous chapters. To create a 1D grating structure, two incident beams compose an interference pattern. The photonic band gap arises due to the index of refraction mismatch between the liquid crystal and polymer layers. The index of refraction of the liquid crystal planes is some average of the ordinary and extraordinary indices of the material. This average is mismatched from the index of the polymer by Δn ≈ 0.1. When photons of certain energies encounter this index mismatch, they observe it as a mirror like boundary and may reflect off it. These reflections form the photonic band gap; specific wavelengths of light will not be able to pass through the material. Holographic-polymer dispersed liquid crystals (HPDLCs) have also been fabricated into twoand three-dimensional structures [115], through the use of more than two beams to create the complicated interference pattern. Using a complex setup of multiple beams, it is possible to create any Bravais lattice structure as a HPDLC PBG material [116], as well as a wide array of quasicrystal structures [117,118]. Nearly any imaginable 3D periodic structure can be fabricated using this holographic technique. The HPDLC represents an organic PBG material, which exhibits several advantages, including low-cost, rapid, simple fabrication techniques, and disadvantages, low index contrasts between dielectric materials, over inorganic PBG materials [108].
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We have investigated the dye-lasing of HPDLC with different structure and different materials, which will be discussed from section 6.6.
6.8 Dye Lasing from HPDLC of Different Modes: Materials, Fabrications and Results
6.8.1 Lasing of single reflective dye-doped HPDLC
In order to demonstrate lasing action in dye-doped HPDLCs, a number of HPDLC cells were produced in reflection mode. A prepolymer mixture of liquid crystal, monomer and photoinitiator were used. The mixture consisted of 40 % of the liquid crystal BL038 (ne = 1.799, no = 1.527, Δε = +16.4, Merck), 55 % monomer (1:1 Ebecryl 8301 and Ebecryl 4866, Ciba Specialty Chemicals) and 5% photoinitiator containing 4 % Rose Bengal, 10% n-phenylglycine and 86 % 1-vinyl-2pyrrolidone (Sigma-Aldrich) to sensitize the mixture to visible light. To this mixture was added 0.3% of laser dye Pyrromethene 580 (1,3,5,7,8-pentamethyl-2,6-di-n-butylpyrromethenedifluoroborate), available from Exciton, Inc. The mixture was thoroughly mixed using a stir bar for several hours to ensure the laser dye completely entered the solution homogeneously. Glass spacers were employed in order to control the sample thickness. The samples were pressed using a balloon press for 5 minutes resulting in a homogenous distribution of the mixture. Initial samples were fabricated using the interference of two coherent beams. The angles of incidence for the two beams were determined through the use of the Bragg condition. The pitch of the grating was selected such that the reflection band, or photonic band gap, was within the range of observed lasing from PR580, which is 545 nm – 590 nm. The pitch, or periodicity, of the HPDLC, Λ is: Λ=
λw 2n sin(θ )
(6-18)
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where λw is the wavelength of the laser used to write the HPDLC grating, θ is the angle of incidence for the writing beams, n is the average index of refraction of the prepolymer mixture, in the case of our prepolymer mixture n ≈ 1.55. Once fabricated, the PBG, or reflection band will occur at a wavelength λr = 2nΛ. An angle of incidence for the writing beams of 52° will result in a pitch Λ = 187.1 nm and a reflected band near λr = 580 nm, ideal for the laser dye Pyrromethene 580.
Fig. 6.9. Lasing emission from a reflection mode HPDLC (solid line) and transmission spectra of the same sample (dotted line).
A Brilliant model frequency doubled Nd:YAG laser operating at 532 nm was used to pump our HPDLCs. While the laser has a repetition rate of 10 Hz and a maximum output of 200 mJ per pulse, the laser output was attenuated down to approximately 6 mJ per pulse in our experiments. A laser line filter was placed between the lasing sample and the fiber spectrometer to block all light below 540 nm and eliminate the pump laser in the measured spectra. The solid
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line in Fig. 6.9 shows a strong laser line peaking at 556.7 nm with a full width at half maximum (FWHM) of ~6.6 nm. The transmission spectrum of the sample is also shown as a dotted line in the figure. The thickness of the sample is 20 µm. It is clear that lasing occurs at the edge of the band gap, or reflection notch of the grating. The other notch in the transmission spectrum is due to the absorption by the Rose Bengal used in the photoinitiator, and the absorption of laser dye. In order to switch the sample, an external voltage of 1 kHz square wave was applied to the sample. The voltages measured ranged from 0 V to 300 V. Lasing emission at 100 V increments is shown in Fig. 6.10. The peak of lasing is located at 556.7 nm with a FWHM of 6.6 nm at a voltage of 0 V applied to the sample; peak at 555.1 nm with a FWHM of 8.8 nm, at a voltage of 100 V; peak at 555.1 nm with a FWHM of 9.7 nm, at 200 V; and peak at 555.1 nm with a FWHM of 10.0 nm, at 300 V. Higher voltages were not attempted as they would have shorted the HPDLC and destroyed the grating.
Fig. 6.10. Switching of the dye lasing emission from a reflection mode HPDLC
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6.8.2 Lasing of transmissive Dye-doped HPDLC
Transmission mode HPDLCs were made with a small doping concentration of the laser dye Pyrromethene 580. Prepolymer mixtures consisting of the materials identical to the recipe used for reflective dye-doped HPDLC were also used for these gratings, with 27.5 % Ebecryl 4866, 27.5 % Ebecryl 8301, 40 % BL038 liquid crystal and 5 % photoinitiator, all in weight percentage. The photoinitiator contained 4 % Rose Bengal, 10% n-phenylglycine and 86 % 1-vinyl-2pyrrolidone. Three different mixtures were made, each with a different doping level of laser dye: 0.5, 1.0 and 2.0 %. The mixtures were mixed with a stir bar for more than 2 hours to ensure homogeneity.
Pum p L aser Focal L ens
C y lin d ric a l L ens L a s in g E m is s io n D y e-d o p e d HPDLC
Fig. 6.11 Two lens were used to generate the vertical line across the HPDLC grating in order to increase the area of the gain medium being pumped
The fabrication laser setup consisted of two beams from a Verdi frequency doubled Nd:YAG laser (532 nm), each with a beam power of ~1 W. The beams were incident on opposite sides of the sample with a half angle between the two beams of 46°. This angle is designed to generate a
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pitch length of ~369 nm within the HPDLC, corresponding to a band gap around 1145 nm. A second order gap exists at half this wavelength, ~572 nm. We expect lasing to occur on the blue edge of the gap, or in the range of 550 nm – 565 nm. Samples were fabricated using 20 µm spacers to maintain sample thickness. Exposures in the laser setup were carried out for 1 minute. In the laser pumping setup, two lenses were placed between the pump laser and the lasing sample. The first lens was a focal lens used to focus the pump beam down to a spot and then expand the beam, the second was cylindrical lens used to transform the beam into a vertical line across the HPDLC grating, increasing the area of the gain medium being pumped. The setup is shown in Fig. 6.11. Lasing was measured from the edge of the sample, parallel to the grating vector of the HPDLC. A fiber spectrometer with a resolution of ~2 nm was used for lasing intensity measurement. Unlike the reflective HPDLC in the previous section, lasing from the dye-doped transmissive HPDLC is dependent on the polarization state of pump beam. For the convenience of discussion, here we define the S-component of the pumping laser as the polarized component with polarization direction parallel to the grating direction or lasing emission direction, the Pcomponent with polarization direction perpendicular to the grating direction. Measurements were first made of the HPDLC sample with 0.5% dye concentration. Fig. 6.12 shows the emission as the pump beam polarization is changed from linear s-polarized to linear p-polarized by rotating a λ/2 waveplate 45° between the polarizer and the pumping laser. The polarizer is set with its direction parallel to the grating direction of the HPDLC. The pumping light polarization dependence of the lasing emission was also observed for the dye-doped HPDLC samples with different concentrations (1% and 2%) of laser dye, which are shown in Fig. 6.13 and Fig. 6.14.
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The phenomenon of the pumping light polarization dependence of the lasing emission can be explained by the anisotropic alignment of he dye molecules inside the HPDLC, as we have discussed in previous sections. Fig. 6.15 further illustrates the microstructure of the dye-doped transmissive HPDLC system. While alternating liquid crystal-rich layers and polymer-rich layer are generated during the polymerization process under the laser interference , the dye molecules accumulate in the liquid crystal layer much more than in polymer layer and are aligned along the direction of the liquid crystals, which, on average, is perpendicular to the grating direction. Thus a p-polarized pump laser will generate much more lasing emission.
Lasing Vs Polarization State of Pumping Light (0.5% Dye Concentration) 16000 Lasing Intensity (Arb. Units)
14000 12000 10000
S-Polarized
8000
P-Polarized
6000 4000 2000 0 545
547
549
551
553
555
Wavelength (nm)
Fig. 6.12. Lasing emission of the sample with 0.5% Dye concentration as the pump beam polarization is changed from s-polarized to p-polarization.
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Lasing Intensity (Arb. Units)
Lasing Vs Polarization State of Pumping Light (1% Dye Concentration) 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 548
S-polarized P-polarized
549
550
551
552
553
Wavelength (nm)
Fig. 6.13. Lasing emission of the sample with 1% Dye concentration as the pump beam polarization is changed from s-polarized to p-polarization.
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Lasing Intensity (Arb. Units)
Lasing Vs Polarization State of Pumping Light (2% dye concentration) 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 548
S-polarized P-polarized
550
552
554
556
558
Wavelength (nm)
Fig. 6.14. Lasing emission of the sample with 2% Dye concentration as the pump beam polarization is changed from s-polarization to p-polarization.
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P-polarized Pump Laser S-polarized Pump Laser
Polymer layer
LC layer
: Dye
: LC
Fig. 6.15. Dye molecules are distributed in the liquid crystal layers and are aligned with the liquid crystal in the surface, and are perpendicular to the grating direction.
Measurements were also made to investigate the threshold nature of these samples, one of the key features of a lasing system. The spectrum was acquired at increasing pump energies with incident p-polarization, and the peak intensity was recorded for each spectrum. Fig. 6.16 and Fig. 6.17 show the intensities of lasing emission increase with the pumping energies, for dye-doped HPDLC samples with dye concentration of 0.5% and 1%. It is apparent that the FWHM of the lasing peak at around 551 nm is ~2 nm, at the minimum resolution of our spectrometer. The emission peak grows significantly with increasing pump power. When the pumping energy reaches the higher level, the lasing intensity is saturated. When the peak intensity is plotted as a function of the pump energy, a threshold can be seen in the neighborhood of 18 µJ for the sample with a dye concentration of 0.5%. Fig. 6.18 shows the emission intensity of the lasing peak at
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various pump energies for the sample with a dye concentration of 0.5%. The threshold is determined to be ~18 µJ.
Lasing Emission (0.5% dye concentration) 70000
Lasing Intensity (Arb. Units)
60000 9 Micro-J
50000
15.5 Micro-J
40000
21 Micro-J
30000
32 Micro-J 44 Micro-J
20000
70 Micro-J
10000 0 546
548
550
552
554
Wavelength ( nm )
Fig. 6.16. Lasing emission at various pump energies in a sample with 0.5% dye concentration.
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Lasing Emission (1% dye concentration)
Lasing Intensity (Arb. Units)
25000 32 Micro-J
20000
44 Micro-J 66 Micro-J
15000
77 Micro-J 92 Micro-J
10000
120 Micro-J 145 Micro-J
5000 0 548
200 Micro-J
549
550
551
552
553
Wavelength ( nm )
Fig. 6.17. Lasing emission at various pump energies in a sample with 1% dye concentration.
Fig. 6.18. Peak emission intensity at various pump energies. A threshold is seen at ~18 µJ. Sample has dye concentration of 0.5%.
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Lasing at Various Voltages 9000 8000
Lasing Intensity (Arb. Units)
7000 6000
0v
5000
50 v
4000
100 v 200 v
3000 2000 1000 0 548
550
552
554
Wavelength (nm)
Fig. 6.19. Effect of electric fields on lasing in a transmission mode HPDLC. Energy of Pumping laser is 20 μJ.
The electro-optic response was observed for these lasing systems. An electric field was applied across the sample doped with a 1.0 % laser dye. The applied voltage ranged from 0 V to 200 V. The measured spectra at 0 V, 100 V and 200 V are shown in Fig. 6.19. We observe the emission falls off by a factor of two as a voltage of 200 V is applied across the sample, as compared to the zero voltage case. The grating is essentially being switched off by the electric field, allowing the refractive index of the liquid crystal droplets to be index matched
166
to that of the polymer. Further work is needed to optimize the materials, perfectly matching the ordinary index of the liquid crystal and the index of the polymer.
6.8.3 Multiple-Method for Lasing Tuning
In order to tune the wavelength peak of the laser emission, several modes of operation are proposed, as shown in Fig. 6.20. While Figures 6.20(a) and (b) show the basic principle of an HPDLC as discussed before, Fig. 6.20(c) shows a three panel stack with a broad band incident white light, ΔλW, and three reflection bands λB1, λB2, and λB3, whose peak wavelengths are dictated by Bragg’s law, λB=2d for normal incident light, where d is the plane thickness. This is a very attractive approach to create a tunable laser element since the given Bragg reflection band can be turned on or off electrically. Fig. 6.20(d) shows the structure of a chirped HPDLC, where the reflected Bragg peak, λ(xi), is a function of the spatial position in the sample, x. A chirped HPDLC is obtained by creating the interference pattern with diverging beams thereby spatially changing the thickness of the Bragg planes, d, within the sample as a function of position. Fig. 6.20(e) shows a single electrically tuned HPDLC, where the peak wavelength, λ(Vi), can be tuned as a function of applied voltage.
While in Fig. 6.10 we have already shown the electrical tuning of the lasing peak, though the tuning range is only ~ 1 nm, new configurations and materials are necessary to enable a larger modulation of the refraction index of HPDLC, in order to expand the tunable range.
6.8.3.1 Lasing Tuning in Stack of HPDLCs
As a possibility for creating a tunable laser source, the stacked HPDLC configuration was fabricated as shown in Fig. 6.20(c).
167
Fig. 6.20. Various modes of operation to tune the wavelength peak of the lasing.
Lasing Switching of HPDLC stack 3500
Lasing Emission (Arb. Units)
3000 2500 0V
2000
150 V 1500
250 V
1000 500 0 540
560
580
600
620
640
Wavelength (nm)
Fig. 6.21. Stacked grating configuration for tunable lasing. The grating with the smaller pitch, lower reflection band is in a zero voltage state, while the larger pitch grating has a field applied across it to switch off lasing
168
Two HPDLCs doped with different laser dyes, at different pitches, were fabricated separately using the procedures mentioned above and then sandwiched together. One cell contained the laser dye Pyrromethene 580 and the other was doped with DCM; each cell was 10 µm thick. These stacked cells were then placed in front of the pump beam and the lasing emission was observed with the spectrometer. Two laser emission lines were observed. The solid line in Fig. 6.21 shows the lasing emission when no voltage was applied. A field of increasing strength was applied across one of the cells as the other was left in a zero field state, as shown by the dashed and dotted lines in Fig. 6.21.
Transmission of HPDLC Gratings
Transmission (Arb. Units)
120 100 80 Grating A
60
Grating B
40 20 0 540
560
580
600
620
640
Wavelength (nm)
Fig. 6.22. Transmission of the two gratings used in the stack. Grating A is doped with dye P580, and grating B is doped with dye DCM.
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The spectral response of the two gratings of the stack were measured separately, as shown in Fig. 6.22. Grating A is doped with dye P580, and grating B is doped with dye DCM. Comparing Fig.s 6.21 and 6.22, it is clearly shown that the wavelength of the lasing emission of the left peak (in Fig. 6.21) is located at the left edge of the reflection band in grating A, and the wavelength of lasing emission of the right peak (in Fig. 6.21) is located at the left edge of the reflection band in grating B. We have demonstrated that the creation of stacked, dye-doped HPDLCs is a viable option for multiple line lasing devices.
6.8.3.2 Lasing Tuning in Chirped HPDLC
Chirped HPDLCs with reflection bands ranging between 560 nm and 590 nm were fabricated. The following formulation has been used for sample preparation: 55% homogeneous prepolymer mixture composed of a photo-polymerizable urethane acrylate (from Ciba Specialty Chemicals), 45% liquid crystal (BL038 from EM Industries, ne=1.799, no=1.527, Δε=+16.4) and 5% photoinitiator (Rose Bengal, n-phenylglycine, and 1-vinyl-2-pyrrolidone; all available from Sigma-Aldrich). The mixture was placed between two AR-ITO coated glass substrates; glass spacers were employed in order to control the sample thickness and the samples were pressed using a balloon press for 5 minutes resulting in a homogenous distribution of the mixture. Two additional lenses were used in the holographic writing system to generate a chirped grating. These lenses, placed in the path of the split beams, created diverging beams that created the spatially varying grating spacing within the sample. When tested with a spectrometer, a shift in the location of the reflection notch of the transmission spectra was observed when the sample was probed along one direction. Samples with shifts as high as 10 nm were fabricated. More divergent beams and larger samples can be used to create larger shifts of the lasing peak.
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Two lenses with focal lengths of 100 mm were used in our setup to create diverging beams at the cell. The samples are exposed to a 532 nm Verdi laser for 30 seconds at a power of 2 W. An Ocean Optics fiber spectrometer was used to measure the transmission of the formed grating within the samples. A reflection notch varying as a function of position in the sample relative to the spectrometer was observed in all of the samples made with a wavelength range between 560 nm and 590 nm. When measured at two opposite edges of the sample, a chirped HPDLCs has reflection notches with minima in the transmission spectra between 562 nm and 572 nm. This shift, at three spots in the transmission spectra, is seen in the top of Fig. 6.23. These samples were pumped as described above; the lasing spectra were measured at the same three spots and the resulting spectra are shown in the bottom of Fig. 6.23. Laser emission appears at the edge of the band gap, or reflection band, for each point. The lasing peaks are located at 554.4, 555.1, and 557.8 nm for the left, middle and right points, respectively. This corresponds to approximately a 4 nm shift in the lasing wavelength. It is clear these grating structures have spatial tunability. In addition to this spatial tunability, lasing from each spot of these chirped dye-doped HPDLCs are also switchable, as is seen in Fig. 6.24. The fields necessary for switching a chirped HPDLC are comparable to those required for switching a non-chirped configuration. The peak of laser emission is at 555.3 nm with a FWHM of 4.9 nm, at 0 V; the peak of laser emission is at 555.3 nm with a FWHM of 5.2 nm, at 100 V; the peak of laser emission is at 556.5 nm with a FWHM of 7.0 nm, at 200 V,; the peak of laser emission is at 556.5 nm with a FWHM of 7.4 nm, at 300 V.
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Fig. 6.23. Tuning of a chirped HPDLC. Transmission spectra at left (solid), middle (dashed) and right (dotted) points on the sample (top); and lasing emission at left (solid), middle (dashed) and right (dotted) points on the sample (bottom).
Fig. 6.24. Switching of a reflection mode chirped HPDLC
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6.9 Two Dimensional Dye-Doped HPDLC Lasing
2D structures in PDLC materials can be fabricated through multiple-beam holography. For N linear polarized plane wave beams, the irradiance of the interference pattern can be expressed as:
r I (r ) =
N
N
v
∑∑E i =1
j =1
r
j
r r r r* ⋅E i exp[ i ( k j − k i ) r ]
(6-19)
r
r
r
If the reciprocal wave vectors G ji = k j − k i form a reciprocal lattice, then the I (r ) has a period structure with the scale of lattice comparable to the wavelength of light. The ability to manipulate the wave vectors and polarization directions allows for the design of 2D or even 3D interference patterns [119-122]. We will discuss a 4-beam interference in details. The 4 beams are all of the same polar angle θ, evenly distributed at Δφ = 90° intervals around the azimuth, and have the same intensity. For
simplicity, let us assume that 2 beams lie in the X-Z plane and have the polarization directions along the Y direction, while the other 2 beams lie in the Y-Z plane and have a polarization directions along the X direction. The 4 beams can be analytically expressed as:
r r E1 = e y E0 [ei ( kx x+kz z ) e −iωt + e −i ( kx x+kz z ) eiωt ]
,
r r E2 = e y E0 [ei ( − kx x+k z z ) e − iωt + e − i ( − k x x+kz z ) eiωt ]
r r i ( k y+k z ) −i ( k y+k z ) E3 = ex E0 [e y z e −iωt + e y z eiωt ]
,
r r i ( − k y+k z ) −i ( − k y +k z ) E3 = ex E0 [e y z e −iωt + e y z eiωt ] where
,
(6-20)
r r ex and ey are two unit vector along X-axis and Y-axis directions. When the 4 beams meet,
the total electric field is:
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4 r r E = ∑ Ei
(6-21)
i =1
and 4 r r2 r2 r 2 r2 r 2 r r r r E = (∑ Ei ) 2 = E1 + E2 + E3 + E4 + 2 E1 • E2 + 2 E3 • E4
(6-22)
i =1
Here, some terms, such as are perpendicular, or
r r r r E1 • E3 and E2 • E4 , disappear because the polarization directions
v v ex • ey = 0 . The total intensity is:
r r2 r2 r2 r2 r r r r I = E 2 = E1 + E2 + E3 + E4 + 2 E1 • E2 + 2 E3 • E4 As
(6-23)
r2 r2 r2 r2 2 E1 = E2 = E3 = E4 = E0 = I 0
and
r r 2 2 E1 • E2 = 2 E0 [e i ( k x x+ k z z ) e − iωt + e − i ( k x x +k z z ) e iωt ] • [e i ( − k x x+k z z ) e − iωt + e − i ( − k x x+k z z ) e iωt ]
= 2 E0 ei 2 k z z e −i 2ωt + e i 2 kx x + e − i 2 k x x + e − i 2 k z z ) e i 2ωt ] = 4 E0 cos(2k x x) 2
2
(6-24)
Similarly,
r r 2 2 E3 • E4 = 4 E0 cos( 2k y y )
(6-25)
Thus the total intensity is:
I = 4I 0 + 4 I 0 [cos(2k x x) + cos(2k y y)]
(6-26)
Fig. 6.25(a) shows the structure of a 4-beam interference. As the polar angle of the 4 beams are the same, we have
kx = k y =
2π
λ
sin(θ ) , thus (6.26) generates a 2-dimentional square-
lattice with intensity modulation. Fig. 6.25(b) shows the resulting interference pattern as
174
described by (6.26). The lattice structure was confirmed by an SEM image previously taken by the Display and Photonics Laboratory of Brown University, as shown in Fig. 6.26. The designed lattice period was ideally 222 nm.
X Y
Z
(a)
(b)
Fig. 6.25. (a) Setup for creating 4-beam interference pattern and (b) the resulting interference pattern; the bright (dark) regions represent areas of high (low) intensity.
Fig. 6.26. SEM image of a HPDLC lattice generated by 4-beam interference.
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Another 2D configuration made use of 6 incident beams evenly distributed at Δφ = 60° increments around the azimuth, each with the same polar angle and same intensity. The 6-beam interference results in a 2D hexagonal lattice structure. The beam setup and the resulting interference pattern are shown in Fig. 6.27.
Fig. 6.27. (a) Setup for creating a 6-beam interference pattern and (b) the resulting interference pattern; the bright (dark) regions represent areas of high (low) intensity.
While these two configurations may not exhibit complete band gaps, band gaps do exist along certain directions, along the unit vectors comprising the lattice, for instance; for this reason a strong lasing intensity should be observed along these directions when lasing dye materials are doped into the HPDLC. We fabricated both 4- and 6-beam 2D dye-doped HPDLC grating structures using the optical setup discussed above. The polar angles of incidence from the normal of the glass substrate were θ = 10° for the 4- beams setup. From 6.19, we know the pitch of the 4-beam setup is decided by
176
Λ=
π kx
=
π ky
=
λ 2 sin(θ )
=1532 nm, for the writing laser wavelength 532 nm and polar angle
θ=10°.
y
x
Fig. 6.28. (a) Isointensity plot for four-beam fabrication with directions of the band gap and subsequently lasing; and (b) lasing from this structure doped with the laser dyes Pyrromethene 580 (solid line) and DCM (dotted line). Lasing emission is measured along x-direction.
The materials used in the fabrication of the 2-D dye-doped HPDLC are basically the same as the mixtures in fabricating the 1-D samples. The prepolymer mixture consisted of 40 % of the liquid crystal BL038 (ne = 1.799, no = 1.527, Δε = +16.4, Merck), 55 % monomer (1:1 Ebecryl 8301 and Ebecryl 4866, Ciba Specialty Chemicals) and 5% photoinitiator containing Rose Bengal, n-phenylglycine and 1-vinyl-2-pyrrolidone (Sigma-Aldrich) to sensitize the mixture to visible light. To this mixture was added 0.3% of one of the laser dyes Pyrromethene 580 (1,3,5,7,8-pentamethyl-2,6-di-n-butylpyrromethene-difluoroborate)
or
DCM
(4-
Dicyanomethylene-2-methyl-6-p-diethylaminostyryl-4-H-pyran), both available from Exciton,
177
Inc. The mixture was thoroughly mixed using a stir bar for several hours to ensure the laser dye completely entered the solution homogeneously. In the 4-beam structure, a band gap is expected in x-direction, y-direction and diagonal direction, as shown in Fig. 6.28(a). When viewed along these directions, the cross section of the HPDLC sample will appear as a simple 1D grating structure, like that of the 1D gratings fabricated previously. When measured from the x-direction (and the same for y-direction), laser emission peaked at 558 nm with a FWHM of 5 nm (solid line) was observed in the sample doped with dye Pyrromethene 580; lasing emission peaked at 614.3 nm with a FWHM of 12.2 nm (dashed line) in the sample doped with the dye DCM. Both of the lasing emission results are shown in Fig. 6.28(b).
Fig. 6.29. (a) Isointensity plot for six-beam fabrication with directions of band gap and subsequently lasing; and (b) lasing from this structure doped with the laser dyes Pyrromethene 580 (solid line) and DCM (dotted line).
178
We also fabricated a 6-beam 2D dye-doped HPDLC grating structure using the optical setup discussed in Fig. 6.27(a). The polar angles of incidence from the normal of the glass substrate were θ = 27°. The 6-beam structure contains three axes along which lasing is expected, as seen in Fig. 6.29(a). Laser emission of the sample doped with PM580 peaked at 554.5 nm, with a FWHM of 5.1 nm (solid line); Lasing of sample doped with DCM peaked at 613 nm, with a FWHM of 14.8 nm (dashed line), both are shown in Fig. 6.29(b). Comparable lasing was observed along all three axes.
6.10 Lasing of Polarization Grating
In most applications of holography, the means of interference is the intensity interference, as most holographic materials are sensitive to exposure intensity. Recently a few materials have been found to be sensitive to the local polarization direction and intensity, and they have been utilized to record the so-called polarization interference. The first demonstration of tan interference pattern recorded using two orthogonally polarized beams was proposed by Kalichashvili [123], and further configurations were systematically studied by Nikolova and Todorov [124]. The interference of two orthogonally polarized beams gives a uniform intensity, however, the distribution of the polarization direction is spatially modulated. Two special and interesting cases are considered as shown in Fig. 6.30: (1) Two beams are linearly polarized with orthogonal polarization directions (Fig. 6.30(a)); (2) Two beams are circularly polarized with opposite handedness, or one left-handed and the other right handed. (Fig. 6.30(b))
179
For the two circularly polarized beams with different handedness, they can be described in
⎛ 1 ⎞
r
r
⎛ 1 ⎞
the Jones matrix method by: E1 = ⎜⎜ iπ / 2 ⎟⎟ , and E2 = ⎜⎜ −iπ / 2 ⎟⎟ . Assuming there is no difference ⎝e ⎠ ⎝e ⎠ of optical path length when the two beams meet at the point x = 0, then the difference of the optical path (OPD) along the x-axis is decided by: OPD = 2 x sin(θ ) , and the phase mismatch Between the two beams is Δ =
4πx sin(θ )
λ
, thus the total electric field is given by:
r ⎛ 1 ⎞ ⎛ 1 ⎞ iΔ ⎛ e iΔ / 2 (2 cos(Δ / 2) ⎞ ⎛ cos(Δ / 2) ⎞ iΔ / 2 ⎟ = ⎜⎜ ⎟⎟2e E = ⎜⎜ iπ / 2 ⎟⎟ + ⎜⎜ −iπ / 2 ⎟⎟e = ⎜⎜ iΔ / 2 ⎟ e e sin( / 2 ) Δ e 2 cos( / 2 / 2 ) π − Δ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ Here it is interesting that the polarization states along the x-axis are all linearly polarized, with periodic modulation of the polarization directions. Because of the fact that the polarization
⎛a⎞
⎛− a⎞
π
λ
⎟⎟ are the same, the period along x axis is Λ = = . states ⎜⎜ ⎟⎟ and ⎜⎜ Δ / 2 2 sin(θ ) ⎝b⎠ ⎝ − b⎠ Similarly, considering the two linearly polarized beams with equal intensity, the total electric field of interference is given by:
r r r ⎛ 1⎞ E = E1 + E2 ∝ ⎜⎜ iΔ ⎟⎟ ⎝e ⎠ Thus, the polarization is also modulated along the x-direction. As the morphology of the liquid crystal system is easily controlled using surface alignment and holographic techniques, liquid crystal and polymer can be utilized for polarization recordings of polarization grating. The optically induced molecular reorientation in a nematic LC film has been well studied both theoretically and experimentally. A linear photopolymerizable polymer (LPP) was reported by Crawford’s group at Brown University to create a polarization grating on a surface alignment layer for a nematic liquid crystal, which propagated into the bulk nematic
180
[126,127]. Fig. 6.31 shows two microscope images of a cell of a polarization grating. While one substrate of the cell is coated with the polyimide SE-7511 to generate homeotropic alignment, the other substrate is coated with the LPP material from Rolic and is exposed to two interfering laser beams of circular polarization with opposite handedness. The period of the grating is 7.5 μm. In Fig. 6.31(a), the image is taken with the cell between two crossed polarizers when no voltage is applied. The periodic structure of the liquid crystal alignment direction dominated by the substrate with the LPP material coating is clearly shown in the microscope image. When a high voltage is applied, the liquid crystals change to a homeotropic alignment and a black image is shown in Fig. 6.31(b). The differences between a HPDLC grating and a polarization grating are that: (1)
HPDLC grating is composed of two different materials (liquid crystal and polymer) in a periodic structure, while a polarization grating is composed of pure liquid crystal with periodic modulation of the director alignment;
(2)
In fabrication, a HPDLC grating does not need surface alignment layer, while a polarization grating needs the LPP material for the liquid crystal alignment and for recording the periodic modulation of the polarization caused by the interference of the two holographic writing beams.
The polarization-modulated grating can also be treated as a photonic band gap (PBG) structure. When a lasing dye is doped into the system, lasing emission is expected. In order to obtain ideal conditions for a distributed feedback emission, the periodicity, or pitch of the polarization grating, is selected so that an edge of the reflection band gap would lie within the fluorescence band of the laser dye. According to Kogelnick’s coupled wave theory [128,129], the wavelength of the enhanced emission from this Bragg grating structure is decided by: 2 nΛ = kλ ,
181
where n is an effective refractive index, Λ is the pitch of the periodic medium and k is an integer representing the diffraction order of the grating. We fabricated the dye-doped nematic liquid crystal samples with a polarization grating structure as discussed. The LPP material (ROLIC) was spin-coated onto two glass substrates with an anti-reflection coating on one side, a thin layer of index-matched indium tin oxide (ITO) coating on the other. 5 µm fiber spacers were spread on one substrate and then the uniform empty cells were assembled with epoxy. These samples were then exposed to an interference pattern generated by two circularly polarized beams of identical intensity and opposite handedness, from an Ar+ laser operating at 351 nm. Each beam contained a power of approximately 40 mW. The exposure time was about 30 seconds. Under the ultraviolet radiation of these two interfering beams, the LPP molecules polymerize with their molecular axes parallel to the local polarization direction that was periodically modulated as shown in Fig. 6.30(b). The nematic liquid crystal BL038 (ne = 1.7999, no = 1.527, Δε = +16.4, EM Industries) was chosen, for its large birefringence and dielectric anisotropy, as the host solvent for an organic laser dye Pyrromethene 580 (PM580, Exciton, Inc.). The concentration of the dye was ~ 0.5 wt.%. The dye-doped liquid crystal mixture was mixed for approximately one hour using a stir bar to ensure a homogeneous mixture, before being capillary filled into the exposed cells. These samples were pumped using a 532 nm frequency doubled Nd:YAG laser (Quantel) with pulse widths of 9 ns, a repetition rate of 10 Hz and a variable beam intensity. The incident angle of pumping beam on the sample is set at approximately 45°. The emission from the sample was measured from the edge of the grating, as shown in Fig. 6.32(b). Two lenses were used to focus the incident beam on the grating structure; a plano-convex lens focuses the incident beam to a point and a cylindrical lens spreads the beam along a direction parallel to the grating vector of
182
the structure. This setup increases the area of region excited by the pump beam within the grating structure, as is shown in Fig. 6.11. Laser emission from these samples was measured and is shown in Fig. 6.33. The lasing emission peak has a full width half maximum (FWHM) of ~5 nm, compared with that of the FWHM of the fluorescence band (>50 nm).
y (a)
E1 x
2θ
z
E2
x y (b)
E1 2θ z
x E2
x Fig. 6.30. (a) Two linearly polarized beams with orthogonal polarization directions; (b)Two circularly polarized beams with opposite sense of clockwise. [127].
183
(a)
(b)
Fig. 6.31. Microscope images of a cell of polarization grating between two crossed polarizers. (a) no voltage is applied; (b) 20 V voltage is applied.
Fig. 6.32. Writing beam and pump beam for fabrication and lasing emission testing of the polarization gratings.
184
Fig. 6.33. Lasing emission from a liquid crystal polarization holography grating.
Fig. 6.34. Threshold of laser emission for the dye-dope liquid crystal polarization grating.
185
The threshold of these samples was measured and is shown in Fig. 6.34. The threshold pump energy is ~ 225 µJ. Below this threshold point, the lasing emission energy changes slowly relative to the rate of change of pump energy. Above this threshold point, the rate of change increases more drastically. The FWHM of emission drops down to ~5 nm under a 700 µJ pump beam. Also measured was the pump beam polarization dependence of these grating structures. A half-wave plate was placed after the polarizer in front of the pumping laser. The rotation of the polarization was controlled by the rotation of a half wave plate. A 50% increase in lasing emission was observed, as shown in Fig. 6.35, when the incident polarization state was rotated from s- polarization to p-polarization. As we have discussed in the lasing of transmissive HPDLC gratings, dye molecules tend to align themselves along with the liquid crystals. In the structure of the polarization grating, the surface alignment directions of the liquid crystal rotate within the plane of the substrates along with the modulation of the polarization directions. When the pump laser is at normal incidence, both the p-polarized beam and the s-polarized beam are identical to the dye molecules that have a uniform sinusoidal distribution in different in-plane directions. However, when the incident angle of the pump beam is 45°, the electric field of the p-polarized light still lies in plane, while that of s-polarized light has only 50% that lies in plane, the other 50% has the electric field perpendicular to the plane of the liquid crystals and the dye molecules make little contribution to the lasing emission.
186
Fig. 6.35. Lasing emission increases by 50% as the incident polarization is rotated from s- to p-polarization.
The switching and tuning of the lasing of liquid crystal polarization gratings was also measured. When an electric field was applied to a sample pumped with a p-polarization laser, along with the switching off of the polarization grating, the intensity of the lasing emission drops by 60%, along with a red shift of the wavelength of emission by approximately 5 nm, as is seen in Fig. 6.36. The application of an electric field applied to a sample pumped by s-polarized light had no effect on the output intensity of emission, but did red shift the lasing wavelength by 7 nm.
187
Fig. 6.36. Effect of an applied electric field on a liquid crystal polarization grating pumped by p-polarized light.
188
6.11 Summary and Conclusions
We have investigated the materials, fabrication and characterization of lasing emission of dye doped HPDLCs. Lasing from different modes of HPDLCs has been studied, with a lasing peak resolution of ~ 2 nm. Both the switching and tunability of the lasing function were demonstrated. The research illustrates a potential for making electrically tunable lasers. Lasing from twodimensional HPDLC based photonic band gap (PBG) materials was also demonstrated. Finally, lasing from polarization modulated gratings was investigated.
CHAPTER 7
Conclusions and Considerations on Future Work
Several innovative wavelength tunable devices based on liquid crystal technology, especially on Holographic Polymer Dispersed Liquid Crystals (HPDLC) have been developed. Based on the electrically controllable beam steering capability of transmission mode HPDLCs, the concept and design of novel switchable circular to point converter (SCPC) devices have been demonstrated for selecting and routing the wavelength channels discriminated by a Fabry-Perot interferometer, with applications in Lidar detection, spectral imaging and optical telecommunication. SCPC devices working at different wavelengths (visible and NIR) with different channel numbers (single channel, 10-channel, and 32-channel) were fabricated and investigated. Two types of SCPC devices were analyzed with more focus on the second type, a beam-steering SCPC. A high diffraction efficiency of up to 80% in the visible, and 60% in the NIR was achieved. The wavelength dependence and angular dependence were also investigated. A random optical switch was proposed by integrating a Fabry-Perot interferometer with a stack of SCPC units. The research on SCPC devices gives a potential for making electrically tunable optical devices such as random optical switches and spectral imaging detectors. Liquid crystal Fabry-Perot products were analyzed, fabricated and characterized for their application in both spectral imaging and optical telecommunications. Both single-etalon systems and twin-etalon systems were fabricated. Finesse value of more than 10 at the visible wavelength range and finesse value of more than 30 in the NIR are achieved for the tunable LCFP product. The materials, fabrication and characterization of lasing emission of dye doped HPDLCs was also investigated. Lasing from different modes of HPDLCs has been demonstrated, with a lasing 189
190
peak resolution of up to 2 nm, and both the switching and tunability of the lasing function was demonstrated. Lasing from two-dimensional HPDLC based photonic and gap (PBG) materials was also studied. Finally, lasing from polarization modulated gratings was investigated. In the future we will continue all of the research work based on HPDLC. Considering the combination of dye-lasing with SCPC technology, one interesting research topic would be a SCPC based dye-laser with an automatic focusing effect that may route all of the laser signals to the focusing point, with the similar design of focusing SCPC. Based on all of the switchable and wavelength-variable HDPLC devices, SCPC devices, and dye-lasing devices that we have discussed, with more research work on different combinations of materials, we would like to predict and further investigate a wavelength-tuning device by electrical-tuning. In Fig. 3.8 we have discussed some of the previous work of the Display and Photonics Laboratory at Brown University, where they used a polymer material with the refraction index n p between the value of no (ordinary refraction index of liquid crystal) and ne (extraordinary refraction index of liquid crystal), which can be summarized as no < n p < ne , for the optical positive material satisfying no < ne . When a polymer material is chosen with its refraction index satisfying either n p < no < ne or n p > ne > no , then the external field will never change the mismatch condition of the polymer layer and liquid crystal layer inside the HPDLC, however, as the average refraction index of liquid crystal layer, n , is tuned by the voltage, or
n = n (v), the center wavelength of the HPDLC grating, λ, can also be tuned based on the grating equation at the normal incidence: 2( n d lc + n p d p ) = mλ , where d lc and d p are the thickness of the liquid crystal layer, and polymer layer, respectively.
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