WAVE PROPAGATION in NEGATIVE INDEX MATERIALS Dissertation

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WAVE PROPAGATION IN NEGATIVE INDEX MATERIALS

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Doctor of Philosophy in Electrical and Computer Engineering

Rola Aylo, M.S.

Dayton, Ohio

August, 2010

WAVE PROPAGATION IN NEGATIVE INDEX MATERIALS

APPROVED BY:

Partha P. Banerjee, Ph.D Andrew Sarangan, Ph.D

Advisory Committee Chairman Committee member

Professor Associate Professor

Electrical and Computer Engineering Electrical and Computer Engineering

Youssef Raffoul, Ph.D Karolyn Hansen, Ph.D Committee member Committee member Associate Professor Assistant Professor Mathematics Department Biology Department

Guru Subramanyam, Ph.D Committee member Chair, Professor Electrical and Computer Engineering

Malcolm Daniels, Ph.D Tony E. Saliba, Ph.D Associate Dean Dean, Professor School of Engineering School of Engineering

ABSTRACT

WAVE PROPAGATION IN NEGATIVE INDEX MATERIALS

Name: Aylo, Rola University of Dayton

Advisor: Dr. Partha Banerjee

Properties of electromagnetic propagation in materials with negative permittivities and permeabilities were first studied in 1968. In such meta materials, the electric field vector, the magnetic field vector, and the propagation vector form a left hand triad, thus the name left hand materials. Research in this area was practically non-existent, until about 10 years ago, a composite material consisting of periodic metallic rods and split-ring resonators showed left-handed properties. Because the dimension of the constituents of the metamaterial are small compared to the operating wavelength, it is possible to describe the electromagnetic properties of the composite using the concept of effective permittivity and permeability.

In this dissertation, the basic properties of electromagnetic propagation through homogenous left hand materials are first studied. Many of the basic properties of left hand materials are in contrast to those in right hand materials, viz., negative refraction, perfect lensing, and the inverse Doppler effect. Dispersion relations are used to study wave propagation in negative index materials. For the first time to the best of our

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knowledge, we show that a reduced dispersion relation, obtained from the frequency

dependence of the propagation constant by neglecting a linear frequency dependent term,

obeys causality. Causality of the propagation constant enables us to use a novel and

simple operator formalism approach to derive the underlying partial differential equations

for baseband and envelope wave propagation.

Various tools for understanding and characterizing left-handed materials are thereafter presented. The transfer matrix method is used to analyze periodic and random structures composed of positive and negative index materials. By random structures we mean randomness in layer position, index of refraction, and thickness. As an application of alternating periodic negative index and positive index structures, we propose a novel sensor using the zero average gap that only appears in such structures which has different properties from the usual Bragg gap occurring in alternating positive index structures.

Also in this dissertation, we propose a novel negative index material in the visible range based on nanoparticle dispersed liquid crystal cells. The extended Maxwell Garnett theory, which is combination of the regular Maxwell Garnett and Mie scattering theories, is used to find the effective refractive index of the proposed cell. Nanoparticle dispersed liquid crystal cells can also be used as plasmonic sensors. A theoretical study of such sensors is presented. Finally, fabrication and testing of such cells is proposed and initial progress in fabrication is reported. The final assembly and testing of nanoparticle dispersed liquid crystal cells constitute ongoing and future work.

ACKNOWLEDGMENTS

First, I would like to thank my advisor Prof. Partha Banerjee for his continual support and his vast knowledge and expertise in different areas, for without him the completion of my dissertation would have been impossible. Secondly, I would like to thank Prof. Andrew Sarangan for his time and for letting me using his fabrication facility during different stages of my dissertation. Also, I would like to thank Prof. Guru

Subramanyam for his help for suggesting the DNA structures and for giving me some samples for testing. I would like to thank Dr. Karolyn Hansen for her help in the biology questions during my dissertation. Also, I would like to thank Prof. Youssef Raffoul for his caring, both in the personal and academic life, and for his important mathematics course I took under him, which helped me greatly during my dissertation.

Next, I would like to thank Profs. John Loomis and Peter Powers for their help in

my course work and their time, their classes, and their big heart and vast knowledge and

experience, which I benefited from a lot. Also, I would like to thank Drs. Joseph and

Tony Saliba for providing a caring, well understanding environment. Next, I would like

to thank the University of Dayton and the DAGSI scholarship committee, which I greatly

benefited from during my stay at the University of Dayton.

Also, I would like to thank my mom and dad for their continual love and financial support, my sisters Rana and Aline for their encouragement to pursue my studies and taking care of my dog Pipo. Last but not least, I would like to thank my colleague, namely, Dr. Georges Nehmetallah for his valuable scientific debates and long discussions.

TABLE OF CONTENTS

ABSTRACT ...... iii

ACKNOWLEDGMENTS ...... v

LIST OF ILLUSTRATIONS ...... xii

LIST OF TABLES ...... xxi

LIST OF ABBREVIATIONS ...... xxii

LIST OF SYMBOLS ...... xxv

CHAPTER 1 INTRODUCTION ...... 1

1.1 Brief History of left-handed medium and negative refraction ...... 2

1.2 Objectives and novelty of this work...... 5

1.3 Organization of the Dissertation ...... 7

CHAPTER 2 METAMATERIAL THEORY AND APPLICATIONS ...... 10

2.1 Introduction ...... 10

2.2 Maxwell’s equations ...... 10

2.3 Phase and group velocities ...... 14

2.4 Reversal of fundamental optical phenomena in LHM ...... 16

2.4.1 Reversal of Snell’s law ...... 18

2.4.2 Reversal of Doppler effect and Vavilov-Cerenkov radiation ...... 20

2.4.3 Reversal of convergence and divergence in convex and concave lenses ...... 21

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2.4.4 Superlensing ...... 22

2.5 Other applications of metamaterials (cloaking) ...... 24

2.6 Conclusion ...... 26

CHAPTER 3 DISPERSION RELATIONS FOR NEGATIVE INDEX MATERIALS ....29

3.1 Introduction ...... 29

3.2 Complex dielectric permittivity ...... 30

3.3 Effective medium theory and causality ...... 32

3.4 General dispersion relations ...... 36

3.5 Illustrative examples of dispersion relations ...... 39

3.5.1 Resonant permittivity ...... 39

3.5.2 Resonant permittivity and permeability ...... 41

3.5.3 Effective medium comprising particles with resonant permittivity ...... 43

3.6 Baseband and envelope propagation in NIM ...... 44

3.6.1 Baseband propagation with loss ...... 45

3.6.2 Envelope propagation in NIMs with loss ...... 50

3.6.3 Baseband and Envelope Propagation in NIM with gain ...... 51

3.7 Conclusion ...... 55

CHAPTER 4 OPTICAL PROPAGATION IN NEMATIC LIQUID CRYSTALS ...... 57

4.1 Introduction ...... 57

4.2 Mathematical Model ...... 59

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4.3 Negative Refraction ...... 65

4.4 Conclusion ...... 68

CHAPTER 5 MULTILAYER STRUCTURES ...... 69

5.1 Introduction ...... 69

5.2 Transfer Matrix Method ...... 70

5.2.1 Reflectance, transmittance and absorbance ...... 73

5.2.2 Limit of the transmittance ...... 74

5.3 Periodic Model ...... 76

5.3.1 Infinite periodic structures ...... 76

5.3.2 Finite Periodic Structures ...... 78

5.3.3 Anisotropic model ...... 84

5.4 Random Model ...... 87

5.4.1 Randomness in the position of the layers ...... 88

5.4.2 Perturbation in layer thickness ...... 91

5.4.3 Perturbation in refractive index ...... 95

5.5 Applications: Sensors (periodic structures) ...... 97

5.6 LC and NIM defect layers in periodic structures ...... 102

5.7 Conclusion ...... 107

CHAPTER 6 NANOPARTICLE-DISPERSED LCCs ...... 108

6.1 INTRODUCTION ...... 108

6.2 MG, MS and EMG Theories ...... 109

6.2.1 MG theory (static limit) ...... 109

6.2.2 EMG theory (Quasi-static limit) ...... 110

6.3 Tunable Negative Index Binary NDLCC ...... 113

6.4 Numerical results ...... 118

6.5 Plasmonic Sensor Array ...... 126

6.5.1 Electrostatic field sensor ...... 128

6.5.2 Pressure/temperature sensor ...... 133

6.6 Fabrication Procedure of LCC between two Glass Substrates ...... 136

6.6.1 Experiment 1: Testing of a biased twisted LCC ...... 139

6.6.2 Experiment 2: Testing of unbiased parallel LCC ...... 140

6.6.3 Experiment 3: Testing of unbiased twisted LCC ...... 141

6.7 Conclusion ...... 142

CHAPTER 7 CONCLUSION AND FUTURE WORK ...... 144

7.1 Conclusions ...... 144

7.2 Future Work ...... 146

7.3 Preliminary Results of Z-scan on NDLCCs ...... 147

BIBLIOGRAPHY ...... 151

APPENDICES ...... 159

A. MATLAB Code for Hilbert Transform...... 160

B. MATLAB Code for Baseband Propagation...... 161

C. MATLAB Code for Envelope Propagation...... 162

D. MATLAB Code for Beam Propagation in LC...... 163

E. MATLAB Code for TMM ...... 164

F. MATLAB Code for Anisotropic TMM ...... 165

G. MATLAB Code for Mie Scattering Coefficients ...... 170

LIST OF ILLUSTRATIONS

Figure 1.1: Relationship between metamaterials, left handed materials and negative index materials...... 2

Figure 2.1: Electric field E , magnetic field H , propagation vector k and Poynting vector S for propagating electromagnetic waves, (a) RHM (ε >0, µ >0) (b) LHM ( ε <0, µ <0)...... 14

ε µ Figure 2.2: Permittivity-Permeability ( r - r ) space (real parts) for classification of materials and types of supported waves. The first quadrant represent RHM medium and the third quadrant represents LHM...... 17

Figure 2.3: Interface between RHM/LHM with a negative angle of refraction...... 18

Figure 2.4: (a) RHM-RHM-RHM interface and (b) RHM-LHM-RHM interface at λ=0.9 m...... 19

Figure 2.5: Doppler effect in (a) PIM (b) NIM...... 21

Figure 2.6: Beam paths in (a) rectangular, (b) convex, (c) concave lenses made of LHMs...... 22

Figure 2.7: Difference between a conventional and a NIM lens...... 24

Figure 2.8: (a) Ideal cloak (b) with loss...... 27

Figure 2.9: Relative permittivity of the cloaking shell...... 28

Figure 3.1: Different types of effective media: (a) MG, (b) Bruggeman and (c) layered nanostructures...... 33

Figure 3.2: Real and imaginary parts of the relative permittivity for SiC-KBr mixture. .36

ε r ε(∞) = ε Figure 3.3: Real and imaginary parts of , found from Eq. (3.34), with 13 4. 0 , ω = π × × 12 ω = π × × 12 γ = × 12 T 2 25.4 10 rad/s, L 2 46.7 10 rad/s, and 94.0 10 rad/s...... 40

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Figure 3.4: (a) real and (b) imaginary parts of K (ω) from Eq. (3.3); (c) numerically calculated HT of the imaginary part in (b); (d) numerically calculated HT of the real part in (a). Note that (a) is identical to (c) and (b) is identical to (d)...... 41

Figure 3.5: Real and imaginary parts of the relative permittivity (a) and relative permeability (b), found from Eq. (3.35) and (3.36) respectively...... 42

Figure 3.6: (a) real and (b) imaginary parts of K (ω) from Eq. (3.33); (c) numerically calculated HT of the imaginary part in (b); (d) numerically calculated HT of the real part in (a). Note that (a) is identical to (c) and (b) is identical to (d)...... 43

(ω) Figure 3.7: (a) real and (b) imaginary parts of Keff from Eq. (3.37); (c) numerically calculated HT of the imaginary part in (b); (d) numerically calculated HT of the real part in (a). Note that (a) is identical to (c) and (b) is identical to (d)...... 44

Figure 3.8: Normalized dispersion relation (real part shown only)...... 46

Figure 3.9: (a) Real and (b) imaginary parts of the complex propagation constant using Eq. (3.39) and assuming ω =10 14 rad / s...... 47 0

Figure 3.10: (a) Real and (b) imaginary parts of the complex susceptibility using Eq. (3.40) and assuming ω =10 14 rad / s...... 47 0

Figure 3.11: (a) Baseband initial Gaussian pulse (initial width τ=5) in time domain and after propagating a distance Z=10. (b) Energy decay of the propagating pulse...... 49

ω = Figure 3.12: Gaussian pulse envelope in time domain in a NIM for cn 2and initial width τ = 20 . (a) Initial pulse, (b) after propagation by Z=10...... 51

Figure 3.13: (a) Initial ( Z=0) baseband initial pulse in time domain and (b) after propagation a distance Z=10. The initial Gaussian pulse is taken as ψ ( ) = (− 2 τ 2 ) τ = n ,0 T exp T with 5 ...... 54

ω = Figure 3.14: Gaussian pulse envelope in time domain in a NIM for cn 4 and initial width τ = 20 . (a) initial pulse, (b) after propagation by Z=10...... 55

Figure 4.1: NLC orientation: (a) without external field (b) with external field applied ....58

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Figure 4.2: Geometry and axes of a NLC cell...... 59

Figure 4.3: Tilt angle ( θ) for a thickness L=75µm and at V0=1.5V...... 61

Figure 4.4: Calculated walk-off angle versus applied voltage V0...... 61

Figure 4.5: Refractive index profile in the absence of a light beam for a bias V0=1.5 V. .62

Figure 4.6: Initial Gaussian beam at z=0...... 63

Figure 4.7: Final beam shape at z=500µm ...... 63

Figure 4.8: Final beam shape at z=500µm due to diffraction only...... 64

Figure 4.9: Beam shape (a) at z=0 (b) at z=500 µ m with r0=9 µm...... 64

Figure 4.10: Experimental Setup...... 66

( ) Figure 4.11: Variation of the time averaged Poynting vector angle θr,s θi , α with respect α = to optical axis angle with the z-axis at normal incidence ( θi 0) for different wavelength...... 66

θ Figure 4.12: Variation of the time averaged Poynting vector angle s,r with respect to α θ optical axis angle with the z-axis and angle of incidence i . Maximum negative angle of refraction θ max (θ = 0) = − 72.7 ;α = 48 7. ...... 67 r,s i

Figure 4.13: Positive and negative refraction, and predominantly positive refraction, for an angle of incidence of ≈ 5°. For λ=0.632 µm, (a) shows predominantly negative refraction without bias voltage, while (b) shows predominantly positive refraction with 10 V bias...... 68

Figure 5.1: Section of a slab composed of q layers...... 71

Figure 5.2: Schematic structure of N periods. Each period may comprise p layers...... 75

Figure 5.3: Periodic structure composed of 2 materials with refractive index n1 and n2. ..77

Figure 5.4: Transmittance of a periodic PIM/NIM slab with n1=1.5, n2=-3, d1=0.1 m, d2=0.05 m and N=4...... 79

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Figure 5.5: Transmittance of a periodic PIM/PIM slab with n1=1.5, n2=3, d1=0.1 m, d2=0.05 m and N=4...... 80

Figure 5.6: Electric field for a periodic PIM/PIM slab with n1=1.5, n2=3, d1=0.1 m, d2=0.05 m and N=4...... 80

Figure 5.7: (a) Permittivity and permeability of the NIM as given by Eq. (5.27); (b) Transmittance of the alternating periodic layered structure of Figure 5.3 where the length of each layer is d=d1=d2=1cm and for normal incidence...... 82

Figure 5.8: Photonic band structures of the periodic PIM/NIM stack in terms of the frequency and incident angle θ. The black areas represent forbidden gaps (transmittance is less than 0.001)...... 82

Figure 5.9: (a) Transmittance and (b) reflectance of a 14 layers stack with nPIM =1.5, nNIM 11 6 given by Eq. (5.28) where ωpe =1.1543 × 10 rad/s, ω1e = ω1m =2 π × 5×10 rad/s, 11 6 ωpm =1.6324×10 rad/s, γe=2× γm=2 π×6×10 rad/s and dPIM = d NIM = 0.005m. Normal incidence is assumed...... 83

Figure 5.10: (a) Transmission spectrum for the case where the output-polarized field is the same direction as the input along the y axis. (b) Transmission spectrum for the case where the output-polarized field is the same direction as the input along the x axis. The ε = ε = ε = ε = parameters are the following for both cases: 1xx ,6 2xx ,3 1yy ,2 2 yy ,7 ε = ε = = = ε 1zz 2 zz ,4 d1 41 7. nm ,. d 2125 nm , N 15 , and the rotation angle between 1x and ε 2x is zero...... 87

Figure 5.11: Comparison of the average transmittance for 17 layers where the PIM ( nA= 1.5 ) and NIM ( nB= -1) with dA=d B=0.125 µm...... 88

Figure 5.12: Transmittance for a 2 layer stack for different layer thickness dA=d B and for different λ, with nA=1.5 , nB=-1...... 89

Figure 5.13: Photonic band structure for a random stack of 200 layers. The black area represent forbidden gap (Transmittance is less than 0.001). The top figure is for TE polarization and the bottom for TM polarization...... 90

Figure 5.14: Localization length as a function of frequency for a random (position) stack of 100 layers...... 91

Figure 5.15: (a) Tr [T]for the case of d=d1=d2=1cm, PIM is air, NIM index of refraction from Eq. (9), and for increasing values of perturbation parameter ε = 1,0 20 1, 1,2 3 . Inset shows the standard deviation σ (Tr [T]). (b) Localization length as a function of frequency for the periodic stack (solid) and random stack (dashed)...... 93

Figure 5.16: Average transmission for a transparent mode as a function of the disorder strength ε at f=9 GHz. The dashed line is the mixed structure (PIM/NIM) and the solid line is the purely positive index structure (PIM/PIM)...... 94

Figure 5.17: Localization length as a function of frequency a random stack of 200 layers...... 96

Figure 5.18: Photonic band structures of a disordered PIM/NIM stack in terms of the frequency and incident angle θ. The black area represent forbidden gap (Transmittance is less than 0.001)...... 96

Figure 5.19: Variation of transmittance with wavelength for three different values of NIM layer thickness fraction. All other parameters are as in Figure 5.9...... 98

Figure 5.20: Transmittance through the structure when the cell thickness d is scaled by - 10% and +10%. All other parameters are as in Figure 5.9...... 99

Figure 5.21: Variation in the spectral width at T=0.2 with respect to the change in the thickness of the cell for different values of index of refraction for the PIM layer. The duty cycle D=50%. All other parameters are as in Figure 5.9...... 100

Figure 5.22: Transmittance for change in the refractive index of the PIM layer, the thickness of NIM and PIM layer and both a change in the refractive index of the PIM layer with a change in the thickness in both layers. The duty cycle D=50%. All other parameters are as in Figure 5.9...... 101

Figure 5.23: Schematic of a homogeneous periodic anisotropic stack on top of a substrate...... 102

Figure 5.24: Dependence of edge frequencies of the first PBG of p – polarized waves on the incident angle (a) V/V c=2.853 (b) and V/V c=1...... 103

Figure 5.25: Schematic of a LC defect stack...... 104

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Figure 5.26: Transmission characteristics of a 1D PC with a LC as a defect layer of width 11 a/6. Applied voltage is V=1.5 Vc, d1=d2=a/2, n1=1.34, n2=2.5 as in [70]...... 104

Figure 5.27: Schematic of a multi LC defect stack...... 105

Figure 5.28: Transmission characteristics of the multi LC defect stack for different combinations of bias voltage...... 105

Figure 5.29: Figure 5.29: Effect on the transmittance of a NIM defect layer on an optical bandpass filter made of a periodic stack and (a) PIM/NIM defect, (b) NIM defect with varying n, (c) PIM defect with varying n...... 106

Figure 6.1: Schematic of the metamaterials comprising binary nanoparticles randomly distributed in aligned nematic LC...... 116

Figure 6.2: (a) No electric field applied on a NDLCC, (b) Electric field applied on the LC cell, (c) Index profile of the LC due to the electric field (bias), (d) Equivalent setup showing a layered stack with each layer’s index of refraction according to the profile in (c)...... 118

ε ′ ε′′ Figure 6.3: (a, b) 2,1r and (c, d) 2,1r parts of the relative permittivity for the case of LiTaO 3 and Ge nanoparticles...... 119

Figure 6.4: Scattering, extinction, and absorption coefficients for (LiTaO 3, Ge), (a) for m=1 and (b) for m=1 to 10 terms...... 120

′ ′′ θ Figure 6.5: (a) neff (solid blue) and neff (dashed red) versus wavelength at =0 for the ′ case of LiTaO 3 and Ge nanoparticles, (b) neff profile in the LCC due to the electric field (bias voltage) around the cell where λ = 78 .466 µm ...... 121

′ ′′ Figure 6.6: (a) neff (b) neff versus wavelength and angle of incidence for the case of

LiTaO 3 and Ge nanoparticles...... 122

Figure 6.7: Real (a, b) and imaginary (c, d) parts of the relative permittivity for CuCl and Ag for the case of CuCl and Ag nanoparticles...... 123

Figure 6.8: Scattering, extinction, and absorption coefficients for the (CuCl, Ag) NDLCC for (a) for m=1 and (b) for m=1 to 10 terms...... 123

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′ ′′ Figure 6.9: (a) neff (solid blue) and neff (dashed red) versus wavelength for the case of θ ′ CuCl and Ag nanoparticles in the middle of the LCC (at =0 ), (b) neff profile in the LCC due to the electric field (bias voltage) around the cell where λ = .0 38722 µm ...... 125

′ ′′ Figure 6.10: (a) neff (b) neff versus wavelength and angle of incidence for the case of CuCl and Ag nanoparticles...... 125

Figure 6.11: (a) Homogeneous NDLCC containing plasmonic nanoparticles. (b) NDLCA, (c) An applied electric or magnetic field can switch the director axis of the LC from parallel to perpendicular to the cell glass (side view)...... 128

Figure 6.12: Real (up) and imaginary (down) effective index of refraction of LC cell function of FIR frequency region and propagation distance z without nanoparticles with bias voltage (a) V=1 V, (b) V=2V...... 129

Figure 6.13: Real (up) and imaginary (down) effective index of refraction of LC cell function of visible frequency region and propagation distance z without nanoparticles with bias voltage (a) V=1V, (b) V=2V...... 130

Figure 6.14: Real (up) and imaginary (down) effective index of refraction of Ge dispersed LC cell function of frequency and propagation distance z for the following parameters = = × −6 ω = × 12 ε ( ) = γ = ω µ = µ = f 38.0 , r 25.2 10 , p 26 7. 10 , r 0 15 ,8. p / 100 , 1r rh 1 with bias voltage (a) V=1 V, (b) V=2V...... 130

Figure 6.15: Real (up) and imaginary (down) effective index of refraction of Ag dispersed LC cell function of frequency and propagation distance z for the following ε ( ) = = = × −9 ω = π × × 15 γ = π × × 12 parameters, r 0 ,1 f r,1.0 20 10 , p 2 18.2 10 , 2 35.4 10 , µ = µ = 1r rh 1 with bias voltage (a) V=1 V, (b) V=2V...... 131

Figure 6.16: Transmittance of a LC cell for two different electrostatic fields V=1V (blue) and V=2V (red), (a) without nanoparticles at visible frequencies (b) with Ag nanoparticles at visible frequencies, (c) without nanoparticles at FIR frequencies, and (d) with Ge nanoparticles at FIR frequencies...... 132

Figure 6.17: (a) LC cell before applying pressure, (b) LC after applying pressure...... 133

Figure 6.18: Nanodispersed Ge nanoparticles LC cell effective index of refraction before applying pressure ( f=0.38) and after applying pressure ( f=0.7), (Top) real part, (Bottom) imaginary part...... 134 xviii

Figure 6.19: Nanodispersed Ag nanoparticles LC cell effective index of refraction before applying pressure( f=0.1) and after applying pressure ( f=0.3), (Top) real part, (Bottom) imaginary part...... 134

Figure 6.21: Transmittance for the case of Ag nanoparticles dispersed LCC. (red) no pressure is applied ( f=0.1) and cell thickness 8 µm, (blue) after applying pressure and cell thickness 8 µm ( f=0.3), (green) after applying pressure and cell thickness 7 µm ( f=0.3). 136

Figure 6.22: LC cell structure. ITO coated glass substrate is 2.5cm x 2.5 cm, and the distance between the two glass slabs is around: 10 microns...... 137

Figure 6.23: Clean bench class 100-Series 301-Laminar Flow Workstation +Dust Cover(air clean product)...... 138

Figure 6.24: Rubbing machine RM-3I -08+Control Unit (Beam Eng. For Adv Meas. Co.) (left) and spin coater(KW-4AC-4) + vacuum chucks+ vacuum pump (Chemat technology Inc.)...... 138

Figure 6.25: Corning hotplate and stirrer-CLS6(left), E7 LC (middle), and A&D-HR-200 analytical balance (right)...... 139

Figure 6.26: Testing of a biased liquid crystal cell...... 139

Figure 6.27: (a) No bias, transmission is T=6mW, (b) bias voltage =3V, T=80 µW...... 140

Figure 6.28: Lab setup for LCC testing...... 140

Figure 6.29: Testing of unbiased parallel LCC...... 141

Figure 6.30: Transmittance vs. angle between the director axis and the linearly polarized laser output. Theoretical (solid Red line), experimental (diamond blue dots)...... 141

Figure 6.31: Testing of unbiased twisted LCC...... 142

Figure 6.32: Transmittance vs. angle between the director axis and the linearly polarized laser output...... 142

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Figure 7.1: Z-scan apparatus...... 147

Figure 7.2: Normalized transmittance for the TL205+ BaTiO 3 sample (closed aperture)...... 148

Figure 7.3: Normalized Transmittance for the TL205+ BaTiO 3 sample (open aperture)...... 149

Figure 7.4: (a) The lab setup, (b) the motion controller stage...... 150

LIST OF TABLES

Table 2.1 Phase and group velocities for RHM and LHM...... 15

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LIST OF ABBREVIATIONS

LHM Left handed material

PC Photonic crystal

NIM Negative index material

DNM Double negative material

SRR Split ring resonator

IR Infrared

CMM Composite metamaterial

TL Transmission line

CRLH Composite right-left handed

LC Liquid crystal

NLC Nematic liquid crystal

PIM Positive index materials

EMG Extended Maxwell Garnett

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PDE Partial differential equation

TMM Transfer matrix method

NDLCC Nanodispersed liquid crystal cell

EM Electromagnetic

MG Maxwell Garnett

MS Mie scattering

LCC Liquid crystal cell

RHM Right handed material

TE Transverse electric

PML Perfect matching layers

PMC Perfect magnetic conductors

HT Hilbert transform

TM Transverse magnetic

KSE Kuramoto-Shivahinsky equation

KDV Korteweg de Vries

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FT Fourier transform

ITO Indium tin oxide

FEM Finite element method

PBG Photonic band gap

DP Double positive

SNR Signal to noise ratio

SPR Surface plasmon resonance

ORB Omnidirectional reflection band

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LIST OF SYMBOLS

Symbol Definition r Position vector t Time variable ∇• Divergence ∇× Curl ∇ Gradient

~ Time dependent electric field E ~ D Time dependent electric displacement ~ H Time dependent magnetic field ~ B Time dependent magnetic induction

ε~(t) Time dependent permittivity

µ~( ) Time dependent permeability t

ρ Free charge density ~ Time dependent current density J

ℑ ℑ −1 t , t Fourier transform and inverse Fourier transform

ω Angular frequency

xxv f Frequency in Hz

ε(ω) Frequency dependent permittivity

µ(ω) Frequency dependent permeability

ε Permittivity of free space 0

µ Permeability of free space 0

ℜe Real part of a complex number

Im Imaginary part of a complex number E Frequency dependent electric field D Frequency dependent electric displacement H Frequency dependent magnetic field B Frequency dependent magnetic induction k Wave propagation vector

k0 Wave number in space

k ,z,y,x ⊥ Wave vector components on the x, y, z and transverse

directions S Poynting vector S Time-averaged Poynting vector

pˆ Polarization direction unit vector

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v p Phase velocity

λ Wavelength c Speed of light in vacuum n Index of refraction v g Group velocity

ε Relative permittivity r

µ Relative permeability r

ω pe Electric plasma frequency

ω pm Magnetic plasma frequency

η Intrinsic impedance

η Intrinsic impedance of free space 0

η Relative impedance r

θ Incident, reflection, and transmission angles t,r,i

∆ω Doppler frequency shift

Source velocity vs

ω Doppler Doppler frequency f0 Focal length

Radius of curvature of a lens R0

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T,R,A Transmittance, reflectance, and absorbance, respectively

ε ,r φ z, Relative permittivity coefficients of the permittivity tensor in

cylindrical coordinates

ε xx yy, zz, Relative permittivity coefficients of the permittivity tensor in

Cartesian coordinates

µ ,r φ z, Relative permeability coefficients of the permittivity tensor in

cylindrical coordinates

µ xx yy, zz, Relative permeability coefficients of the permittivity tensor in

Cartesian coordinates µ,ε Material permeability and permittivity tensors

ε ' (ω) Real part of ε(ω)

ε '' (ω) Imaginary part of ε(ω)

χ(ω) Linear susceptibility function of frequency

χ~(t) Linear susceptibility function of time

P Principal value of an improper integral

ε Low frequency permittivity s

ε(∞) High frequency permittivity

ε( ) Permittivity at zero frequency 0

τ Relaxation time

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ω The ith plasma frequency pi

ω The ith resonance frequency 0i

γ The ith damping amplitude i

σ Conductivity

Volume fraction fi

ε (ω) The permittivity of the nanospheres i

ε (ω) The permittivity of the host material h

ε (ω ) eff Effective permittivity

µ (ω ) eff Effective permeability

(ω ) n eff Effective index of refraction

(ω) Real part of propagation constant 'k k '' (ω) Imaginary part of propagation constant

χ' (ω) χ '' (ω) Real and imaginary parts of the susceptibility

(ω) Reduced dispersion relation K

K (ω ) Reduced effective dispersion relation eff

~ ( ) Inverse Fourier transform of the dispersion relation K t

B Bulk modulus

ρ Density

xxix

υ Velocity of acoustic waves a

υ Velocity of electromagnetic waves em

ωT and ωL Transverse and longitudinal optical phonon frequencies

u( ),t δ(t) Unit step function and Dirac delta function

Z Space normalizations

K1 and K3 Frank elastic constants for splay and bend respectively

∆ε Static dielectric anisotropy STA

ε ε Extraordinary and ordinary relative permittivities e , o

V Voltage n and n Refractive index of light polarized parallel (extraordinary) or e o normal (ordinary) to the molecular axis

n Optical birefringence

< n > Zero averaged- refractive index

Tr Trace of a matrix

N,d, L Number of layers, thickness of each layer, thickness of

structure, respectively

ς Localization length

ς Decay length d

~ Z Independent normalized identically distributed random r variables

xxx var Variance

Λ 2,1 Eigenvalues

σ (Tr [T ]) Standard deviation of the trace of the transmittance

Sn Sensitivity of a sensor

∆ε Optical dielectric anisotropy OPT

α Electric dipole polarizability i

Particle number density Ni

Q Q, Q, Absorption, extinction, and scattering and cross sections abs i, ext i, sca i,

a , and b Mie coefficients which correspond to the electric-dipole and mi mi

magnetic dipole components of the scattering matrix ( ) ( ) mth order spherical Bessel functions of the first and second jm z , ym z types

(1) ( ) mth order spherical Hankel function of the first type hm z

ψ ( ) χ ( ) mth order Riccati-Bessel functions of first and second type m z , m z

ξ ( ) mth order Riccati-Hankel function of first type m z

f 12 Total filling factor of the spheres

C1 , C2 Concentrations of the spheres types 1 and 2

a0 First neighbors’ distance a Lattice constant

xxxi

Focused and initial beam width wf , w0

xxxii

CHAPTER 1

INTRODUCTION

Metamaterials are composite materials artificially constructed to exhibit exceptional properties not found in nature. Left-handed materials (LHMs) are a subset of metamaterials with an anti-parallel relation between wave propagation vector and the

Poynting vector, which lead to negative refraction. This phenomenon may happen in chiral media, photonic crystals (PCs) or waveguide structures. LHMs have been studied extensively in recent years due to their unique physical properties and novel applications

[1]. Negative refraction of electromagnetic waves is the most interesting physical phenomenon exhibited by the left-handed metamaterial structures. Negative index materials (NIMs) are a subset of LHM which, on top of exhibiting negative refraction, have negative values of the refractive index which may potentially lead to many novel optical devices such as perfect lensing (or super resolution) [2]. Finally, double negative materials (DNM) are a subset of NIM where the real parts of the permittivity and the permeability are both negative which is a sufficient but not necessary condition for NIMs

(see Figure 1.1). Other unusual physical characteristics of NIMs are the reversal of

Doppler shift and backward Cerenkov radiation [3]. In the literature, DNM, NIMs and

LHMs are sometimes used interchangeably.

Metamaterials LHM

NIM DNM

Figure 1.1: Relationship between metamaterials, left handed materials and negative index materials.

1.1 Brief History of NIMs and Negative Refraction

Historically, negative refraction of light and other waves has been first discussed by

Mandelstam [4] in 1945. The dispersion curve of an optical phonon branch in a crystal lattice has been given as an example of such unusual media. The fact that refraction at the interface of one medium with ε>0 and µ>0 and another with ε<0 and µ<0 can not

only be negative, but also reflectionless, has been pointed out by Veselago [3] in 1968.

Also, Veselago has introduced the concept of reversed Doppler shift and backward

Cerenkov radiation beside negative refraction and backward wave propagation.

After nearly 30 years since Veselago’s seminal work, Pendry has proposed

various magnetic resonant structures such as an array of cylinders, and a capacitive array

of sheets wound on cylinders [5]. For each structure, he has shown that the effective

permeability can be described as a resonant model, which exhibits a negative

permeability over a certain frequency range. Additionally, Pendry later has proposed

some variations on the split ring resonator (SRR) design so that it operates at infrared

(IR) frequencies [6]. The first artificial LHM has been designed by Smith [7] using a

combination of wires and SRRs. Smith et al. have reported the experimental

demonstration of left-handed materials by stacking SRRs and thin wire structures as

arrays of 1-D and 2-D structured composite metamaterials (CMMs) [8]. The medium is built by first printing a thin sheet of metal in the shape of circular split rings on one side of the dielectric substrate. The printed metal sheets have then been placed adjacent to a set of relatively large metallic posts [7]. Smith et al . have shown that a metamaterial containing only SRR has a stop band characteristic, while a metamaterial containing only rods has a high-pass characteristic. However, when both are present, a pass band appears near the stop band for the SRRs. From that, Smith et al . conclude that propagation is allowed at those frequencies due to the fact that the permittivity and permeability are both negative [8]. A medium with negative permittivity and permeability has a negative index of refraction and vice versa [3]. The first experimental demonstration in this manner has been performed by Shelby et al. [9]. The authors have placed a prism shaped metamaterial structure in a parallel plate waveguide. They have been able to observe the negative refraction and measure the negative index of refraction of the composite structure.

Negative refraction, and in general, wave propagation in negative index media have also been a controversial subject and have generated an intense debate. Valanju et al. [10] have claimed that dispersion implies positive refraction of group velocity even when the phase velocity is refracted negatively. The perfect lens behavior of left-handed materials has also been criticized. In particular, Garcia [11] has claimed that a perfect lens cannot theoretically exist because it would require infinite energy generation. In

[12] Pendry contradicts the claims of [11] and explains how he takes losses fully into consideration, thus establishing the existence of super resolution in the limit of no absorption. Much of the confusion arises from inconsistent definitions of the group

velocity and Valanju et al [10] have erroneously identified the group velocity as the direction of the interference pattern movement [13].

NIMs have attracted growing attention in both theory and experiment. An alternative representation of NIM is using the transmission line (TL) approach [14].

Using the dual model of conventional TL, a new TL has been constructed using capacitance in series and shunt inductance. A general composite right-left handed

(CRLH) is composed by capacitance and inductance in series and shunt inductance and capacitance. The CRLH is the base of a lot of microwave components including antennas, couplers, and resonators [14]. NIM suffers from high losses and narrow bandwidth. Cloaking is a widely sought useful military application. After the experimental realization of invisible cloak in the microwave regime [15], a high interest in optical transformation, used for explaining cloaking in the optical region of the spectrum, thus invisibility, has been developed [16].

As noted before, LHM are not the only materials that give rise to the negative refraction concept. Negative refraction is also achievable by using 2-D PCs [17, 18].

PCs are typically periodic structures constructed using high dielectric materials [19].

Negative refraction can also be obtained using liquid crystal (LC) cells [20, 21]. The refracted beam is controlled using the electric field and temperature [20]. Kang et al.

[22] have used an external magnetic field to obtain negative refraction in nematic LCs

(NLCs). The authors have used a prism shaped NLC device and have shown that negative refraction is conveniently realized by changing the incidence angle of light and the direction of the applied magnetic field.

1.2 Objectives and Novelty of this Work

The objective of this dissertation is to systematically investigate wave propagation through NIMs, and through a mixture of positive index materials (PIMs) and NIMs.

Understanding wave propagation in NIMs enable us to design and model a possible metamaterial. First, we find a novel and convenient way to model wave propagation in

metamaterials based on the underlying dispersion relation or a new reduced propagation

constant, which describes the variation of the wavenumber on the frequency, and is in

agreement with existing models of resonant permittivities and permeabilities. Secondly,

we demonstrate negative refraction in a pre-fabricated NLC sample with initial in-plane

pre-tilt, and show that one can switch between positive and negative refraction

electronically by applying a bias voltage across the sample. While negative refraction is

a necessary condition for negative index, it is not a sufficient condition. In general, we

are more interested in developing a NIM. Hence, we enunciate a novel way of

incorporating negative index in LC structures, e.g., by doping it with resonant

nanoparticles. It becomes evident that this method of achieving negative index (or net

zero index) involves creating a mixture of a NIM and a PIM. Therefore, as part of this

dissertation, we formulate a novel approach to study random mixtures of PIMs and NIMs,

which can be generalized to study propagation in homogeneous or periodic NIM-PIM

structures. A sensor based on such mixture is also proposed. The extended Maxwell

Garnett (EMG) theory is used to study the nanoparticle doped LC cell (NDLCC).

Specific novel aspects of this dissertation are listed below:

• Concept of causal reduced propagation constant. The propagation constant is

identical to the reduced propagation constant under appropriate limiting

values of the physical parameters as in acoustics. Causality of the propagation

constant enables the use of simple operator formalisms to derive the

underlying partial differential equations (PDEs) for baseband and envelope

wave propagation.

• Class of simplified dispersion relations describing NIMs and their underlying

wave equations (both baseband and envelope) based on the real and imaginary

parts of the propagation constant, which are assumed to be related through

Hilbert transform to ensure causality of the propagation constant. Baseband

and envelope propagation in a gain/loss medium is also studied.

• Extension of transfer matrix method (TMM) to study the transmission and

reflection of random sequences of PIM and NIM layers, which approach a

homogeneous mixture of PIMs and NIMs in the limit. All kinds of

randomness has been studied and compared with PIM. The limit of the

transmission of a PIM/NIM structure has been derived.

• Use of properties of the zero gap in a sensor design.

• Use of a defect slab into a periodic structure. Two cases have been studied,

viz., using LC or NIM as the defect layer.

• NIMS based on NDLCCS. Starting from negative refraction from NLCs with

in-plane pre-tilt, which can be switched with the application of a bias voltage

[21], we develop a new type of negative index media in the infrared and in the

optical regime based on nanodispersed liquid crystal cells (NDLCCs).

• Use of NDLCC as plasmonic sensors for measuring electrostatic fields,

pressure, and temperature.

1.3 Organization of the Dissertation

In Chapter 2, we present a summary of the theoretical background of metamaterials. A review of Maxwell’s equations and electromagnetic (EM) theory is briefly discussed in the time and frequency domains. A summary of the direct consequences of negative permittivity and permeability are discussed in Section 2.3. In Section 2.4, we present an overview of the applications of metamaterials with some numerical simulations. In

Chapter 3 we study causality and dispersion relations in negative index media. In Section

3.2 various complex permittivity models are defined. Section 3.3 presents an example of an effective medium theory. Next, Section 3.4 gives a summary of general dispersion relation and the reduced dispersion relation is introduced. Section 3.5 gives examples of different dispersion relations either from the classical models or suggested ones. Finally in Section 3.6, we study baseband and envelope propagation starting from suggested dispersion relations representing a gain and a lossy medium. In Chapter 4, we discuss liquid crystals and negative refraction. We demonstrate the theoretical model and experimental results of negative refraction using a NLC cell. In Section 4.2, we show the model that governs the change in the director angle of NLCs due to an applied voltage.

Then we model beam propagation in LC. Section 4.3 gives an example of negative refraction using LC cell. In Chapter 5 we discuss multilayer structures of PIM and NIM.

In Section 5.2 a summary of the TMM is presented. In Section 5.2.1 we derive the reflection, transmission and absorption of waves passing through these periodic structures. In Section 5.2.2, for the first time to the best of our knowledge, we employ the

transfer matrix to derive the limit of transmittance for mixed (PIM/NIM) structures when the number of layers tends to infinity. In Section 5.3 we consider periodic structures for different types of materials (i.e dispersive, nondispersive, lossy, etc.). In Section 5.4 we study random system of PIM/NIMs. Randomness of layer thickness, index of refraction and position are studied. In Section 5.5, we study a proposed application of the zero gap. In Section 5.6, we study the effect of introducing defect slab into a periodic structure. Two cases are studied with LC or NIM as the defect layer. In Chapter 6 we study theoretically and experimentally NDLCCs used as metamaterials with tunable properties. These tunable NDLCCs are of great importance due to potential applications in “super-resolution lensing” and as “plasmonic sensors”. In Section 6.2 we briefly discuss the Maxwell-Garnett (MG) theory, and the EMG theory or the Mie scattering

(MS) theory based extension of MG theory. In Section 6.3 we study the feasibility of the fabrication of a metamaterial using binary NDLCC. In Section 6.4 we examine the analytic basis of the plasmonic resonance behavior of a nanodispersed (Ge, Cu, Ag, Au)

Liquid Crystal cell (LCC) under (i) externally applied electrostatic field, and (ii) externally applied pressure/temperature. In Section 6.5, we discuss the fabrication and testing of a NDLCC. Chapter 7 summarizes the dissertation and proposes future work.

Various parts of the dissertation have been published in the literature. These are listed below in chronological order, and are also referenced in the appropriate chapters. It is expected that several concepts developed in this dissertation are going to be useful in an ongoing project titled “ Metamaterial Lens”.

Referred journal publications

1. R. Aylo and P.P. Banerjee, “Modeling propagation in negative index media using causal complex dispersion relations,” J. Opt. Soc. Am. B 27 , 1583-1588 (2010).

2. R. Aylo , P.P. Banerjee and G. Nehmetallah, “Perturbed multilayered structures of positive and negative index materials” J. Opt. Soc. Am. B 27 , 599-604 (2010).

3. P. P. Banerjee, R. Aylo , and G. Nehmetallah, “Baseband envelope propagation in media modeled by a class of complex dispersion relations,” J. Opt. Soc. Am. B 25 , 990-994 (2008).

Conference proceedings

1. R. Aylo , P. P. Banerjee, and G. Nehmetallah, “Tunable metamaterial binary nanoparticle dispersed liquid crystal cells,” to be presented in the SPIE Optics and Photonics San Diego August 1-5 (2010).

2. A. K. Ghosh, P. Verma, P. P. Banerjee, and R. Aylo , “Propagation of a Gaussian beam in a metamaterial based sensor,” to be presented in the SPIE Optics and Photonics San Diego August 1-5 (2010).

3. R. Aylo , P.P. Banerjee, A.K. Ghosh, and P. Verma, “Design of metamaterial based photonic sensors for pressure measurement,” Proc. SPIE 7604 (2010).

4. R. Aylo , P.P. Banerjee, and G. Nehmetallah, “Perturbation of multilayered structures of positive and negative index materials,” Proc. SPIE 7392 (2009).

5. R. Aylo , P.P. Banerjee, and G. Nehmetallah, “Optical propagtion through a homogeneous mixture of positive and negative index materials,” Proc. SPIE 7029 (2008).

6. P. P. Banerjee, G.T. Nehmetallah, R. Aylo , and P. Buranasiri, “Dispersion relations for negative index materials and slow light,” Proc. SPIE 6698 (2007). (Invited Paper) .

CHAPTER 2

METAMATERIAL THEORY AND APPLICATIONS

2.1 Introduction

In this Chapter, we present a summary of the theoretical background of metamaterials. In

Section 2.2, a review of Maxwell’s equations, and EM theory is briefly discussed in the time and frequency domains. A summary of the direct consequences of negative permittivity and permeability is discussed in Section 2.3. In Section 2.4, we present an overview of the applications of metamaterials with some numerical simulations. Section

2.5 concludes the Chapter.

2.2 Maxwell’s equations

Before considering more complex phenomena, it is useful to review the simple concept of

EM wave propagation. Throughout this dissertation, all media are assumed to be linear

(properties do not depend on electric and magnetic fields), homogeneous (properties do not depend on position), dispersive (properties depend on frequency) and isotropic (no polarization effect except otherwise noted). The EM fields are fully described by the

Maxwell’s equations along with the constitutive relations and boundary conditions. The time dependent Maxwell’s equations are

~ ∇ • D()r,t = ρ , (2.1)

~ ∇ • B()r,t = 0 , (2.2)

~ ~ ∂B()r,t ∇ × E()r,t = − , (2.3) ∂t ~ ~ ~ ∂D()r ,t ∇ × H ()()r,t = J r,t + , (2.4) ∂t

~ ~ t ~ D()()()()()r,t = ε~ t ∗ E r,t = ∫ε~ t − t′ E r,t′ td ′, (2.5) −∞

~ ~ t ~ B()()()()()r,t = µ~ t ∗ H r,t = ∫ µ~ t − t′ H r,t′ td ′ , (2.6) −∞

~ ~ ~ ~ where E is the electric field, D is the electric displacement, H is the magnetic field, B ~ is the magnetic induction, ρ is the free charge density and J is the current density. For

wave propagation in optics, media without free charges and conduction current are most ~ relevant i. e. ρ = 0 and J = 0 . Equations (2.1) and (2.4) become

~ ∇ • D()r,t = 0 , (2.7)

~ ~ ∂D()t,r ∇ × H ()t,r = . (2.8) ∂t

The permittivity and permeability are generally defined in the frequency domain.

Using the Fourier transform pair:

∞ ()()ω = ℑ ~ = ~ − jωt f t [ )]t(f ∫ f t e dt , (2.9) −∞

∞ ~ − 1 ω f ()t = ℑ 1 [ (f ω )] = f ()ω e j t dω, (2.10) t π ∫ 2 −∞

ε(ω) = ℑ {ε~( )} µ(ω) = ℑ {µ~( )} ε (ω ) µ(ω ) we have t t , t t , where and are the frequency dependent permittivity and permeability, respectively. In order to properly define wave propagation we also need the constitutive relations in frequency domain. In the case of linear, homogenous, isotropic media, taking the Fourier transform of the constitutive equations for metamaterials of Eqs. (2.5) and (2.6), we get D( ,r ω) = ε(ω)E( ,r ω), (2.11)

B( ,r ω) = µ(ω)H( ,r ω). (2.12)

In the frequency domain, Maxwell’s Eqs. (2.3) and (2.4) can be written as: ∇× E( ,r ω) = − jωµ (ω)H( ,r ω) , (2.13)

∇× H( ,r ω) = jωε (ω)E( ,r ω) . (2.14)

In this dissertation we plan to deal with time-harmonic fields with a time variation

ω ω of the form e j t . Defining the corresponding generic phasor F(r,ω) = ℜe[F(r )e j t ], where F represents any of the physical quantities in Eqs. (2.11) to (2.14), Maxwell’s

equations become ∇• E(r ) = 0 , (2.15) ∇ • H(r ) = 0 , (2.16) ∇× E(r ) = − jωµ (ω)H(r ), (2.17)

∇× H(r ) = jωε (ω)E(r ) . (2.18)

− For plane wave solution of the form e kj .r , and after straight forward algebra,

Maxwell’s curl equations in frequency domain become k × E(r ) = ωµ (ω)H(r ) , (2.19)

k × H(r ) = −ωε (ω)E(r ), (2.20) where k is the wave propagation vector. For ε < ,0 µ < 0 , Eqs. (2.19) and (2.20) show

that the wave vector, the electric field and the magnetic field form a LH system; hence

the name LHM.

The energy flux associated with the EM wave is denoted by the Poynting vector

as S(r ) = E(r )× H(r ). (2.21) From Eq. (2.21) it follows that S , E and H form a right-handed set of vectors. S is independent of the signs and values of ε and µ . Consider an electric field that is

polarized along the pˆ direction. From Maxwell’s equations and the Poynting theorem,

the power flow direction can be written as

= − r.kj E ˆ ep , (2.22)

1 − H = (k × ˆp)e r.kj , (2.23) ωµ

∗ 1 ⇒ S = ℜe{E × H }= pˆ × (k × pˆ ), ωµ

1 k = [()ˆ .p ˆp k − ()ˆ k.p k ]= . (2.24) ωµ ωµ

where S is the time-averaged Poynting vector. This result shows that time-averaged

Poynting vector is in the opposite direction of the wave vector in a LHM. Figure

2.1(a),(b) show the vector orientations for right handed material (RHM) and LHM, respectively. The Poynting vector is always directed away from the source of radiation

[14]. But the wave vector is directed toward the source for LHM and away from the source for RHM. We can say that the phase wave fronts run backwards in LHM.

S S (a) (b)

H H k E E

Figure 2.1: Electric field E , magnetic field H , propagation vector k and Poynting vector S for propagating electromagnetic waves, (a) RHM (ε >0, µ >0) (b) LHM ( ε <0, µ <0).

2.3 Phase and group velocities

Two important quantities in wave propagation are the phase and the group velocities. The phase velocity of the surfaces of constant phase, (or wavefronts) r.k = const is given by:

  ω  k  v = kˆ , kˆ = . (2.25) p k    k  where, k = 2π λ = ε µω = nω c is the wave number, λ is the wavelength in the

medium, c is the velocity of light in vacuum, and n is the index of refraction. Since the

frequency is a positive quantity, we can say that the phase velocity in a LHM ( n<0) is

opposite to the phase velocity in RHM ( n>0).

The group velocity is the propagation of the envelope of the wave packet, is

defined as [23]:

∂ω ∂ω ∂ω v = ∇ ω = aˆ + aˆ + aˆ , (2.26) g k ∂ x ∂ y ∂ z kx k y kz

∇ where k denotes the gradient of ω in the wavevector space. In linear, isotropic and

nondissipative media (with no absorption), the group velocity is equal to the energy flow

velocity associated with the direction of the Poynting vector, which doesn’t depend on

material properties. The group and energy velocities are not equal in the presence of

absorption or in case of anomalous dispersion [13]. For the case of isotropic, low loss

materials the group velocity can be written as [7]: ∂ω = k vg (2.27) ∂k k

Combining observations of the directions of k and S we find that in a LHM the phase and group velocities are of opposite signs and the wave fronts travel toward the source. Table 2.1 summarizes the phase and group velocities homogeneous isotropic dielectric RHM and LHM materials.

Table 2.1: Phase and group velocities for RHM and LHM.

ε > ,0 µ > 0 ε < ,0 µ < 0

S.k > 0 S.k < 0 > > S.vg 0 S.vg 0

> < vg .k 0 vg .k 0

> > v p .k 0 v p .k 0

2.4 Reversal of fundamental optical phenomena in LHM

Let us define the relative permittivity and permeability with respect to those of free space

ε = ε ε µ = µ µ ε µ as r 0 r 0 , respectively. Figure 2.2 shows the r and r space [14].

ε µ~ Materials in the first quadrant with r >0 and >0 allow forward propagating waves and

ε µ are called RHMs. Materials in the second and fourth quadrant have either r <0 or r <0.

EM waves cannot propagate inside them and evanescent waves occur. The angular

ω ω frequencies pe and pm represents the electric and magnetic plasma frequencies

ε µ ε µ respectively. For the third quadrant r <0 and r <0. LHMs with r <0, r <0, which fall in the third quadrant, allow backward propagating waves. Most natural materials

µ ε ≥ occupy the first quadrant with r = 1 and r 1, and rare natural electrical plasma and magnetic plasma occupy the second and fourth quadrants.

Another important quantity of interest is the intrinsic impedance η defined as:

µ η = =η η , (2.28) ε 0 r

µ where, η is the intrinsic impedance of free space, η = r . Consider the case where 0 r ε r

ε µ ε µ we have a nondispersive LHM having r = r =-1. r and r can be written in another

µ e jπ 2 form as ε = exp ( jπ ) and µ = exp ( jπ ). Hence, η = r = = ,1 which is always r r r ε jπ 2 r e positive, which is not the case for the index of refraction. Here, we need to be careful

ε µ when taking the square root of the index of refraction, because r and r are analytic functions whose values are generally complex. The ambiguity in the sign of the square

root can be resolved with a proper analysis of the problem. Since

= ε µ = ( π ) ( π ) = ( π ) = − n r r exp j / 2 exp j / 2 exp j 1 . The important step is that the

ε µ square root of either r or r alone must have a positive imaginary part, which is a

necessity for passive material [1]. Materials with n < 0 are called NIMs, as opposed to normal materials that are called PIMs.

µ~ r ε µ ε µ r <0, r >0 r >0, r >0 Electric plasma Metal at optical freq. Right-Handed Medium (ω < ω ) E pe Evanescent waves H k r r E ε~ ε <0, µ <0 ε µ r r r r >0, r <0 r k H Magnetic plasma LHM or NIM ferrites (ω < ω ) pm r Evanescent waves

Many of the unusual effects of negative index are subject to many researches.

But the most immediate accessible phenomenon from an experimental or computational

point of view is the reversal of wave refraction, as discussed below.

2.4.1 Reversal of Snell’s law

θ = θ As an immediate consequence of Snell’s law n1 sin i n2 sin t , energy is

refracted on the same side of the normal as the incident wave, which means that the

θ < transmission angle is negative t 0as shown in Figure 2.3. Figure 2.4 shows a

simulation in RSOFT with a incident Gaussian wave with a wavelength of 0.9 m and an incident angle (as shown from the arrows) of θi=25° into a 3 layer slab where the first and

ε µ ε µ last medium RHM with ( r = 2, r =1) and the middle slab is RHM with ( r = 1, r =1)

ε µ for case (a) and is LHM with ( r = -1, r =-1) for case (b).

LHM

-θt

RHM

θr θi

Figure 2.3: Interface between RHM/LHM with a negative angle of refraction.

RHM

(a)

RHM

LHM

RHM

(b)

Figure 2.4: (a) RHM-RHM-RHM interface and (b) RHM-LHM-RHM interface at λ=0.9 m.

2.4.2 Reversal of Doppler effect and Vavilov-Cerenkov radiation

Another direct consequence of having the permittivity and permeability negative

simultaneously is the reversal in Doppler shift. In the far field, the field of a source

moving in the z direction and radiating with an angular frequency ω have the following form [14]:

ϕ (ω ) ~ ~ e j t, E()(),t,z H t,z ∝ , (2.29) r

ϕ(ω ) = ω − with , t t kr where k represents the wave number of the medium where the source moves and r is the radial variable in the spherical coordinates. Figure 2.5 shows the source in (a) RHM and in (b) LHM. The position of the source as a function of time is

= z vst . For an observer to the left of the source the phase seen is:

 k   v  ϕ = ωt − kv t = ω1− v t = ω1− s t (2.30) s  ω s   v   p 

The Doppler frequency shift is:

v ∆ω = s . (2.31) v p

∆ω < ∆ω > ω For NIM, 0 , while for PIM 0 . The Doppler frequency Doppler is then

ω = ω − ∆ω Doppler (2.32)

ω where, Doppler is shifted downward for PIM but shifted upward for PIM.

Also, Veselago has shown that Vavilov-Cerenkov radiation is reversed in LH

medium [3]. Vavilov-Cerenkov radiation is the visible EM radiation emitted by liquids

and solids when bombarded with fast-moving electron beams with high velocity [14].

vs z vs z z S S

(a) (b)

Figure 2.5: Doppler effect in (a) PIM (b) NIM.

2.4.3 Reversal of convergence and divergence in convex and concave lenses

Existence of a negative refractive index implies an entirely new form of geometrical optics. A striking example is shown in Figure 2.6 (a), where a slab of NIM focuses the point source, which is not the case in PIMs. The shaded area represents the

LHM. A rectangular lens made of positive index material expectedly diverge the beam.

But it is possible to focus EM waves using rectangular slab lenses made of LHM.

ε = µ = − Moreover, under the condition r r 1 , impedance is matched and therefore no reflection occurs. These conditions are considered “ideal” because they lead to the same impedance and speed of light as free space. Figure 2.6 (b) shows the case when a convex lens of LHM is used. Different from the case in right handed materials, wave diverge instead of converging. Figure 2.6 (c) shows the case of a concave lens made with LHM is used: Instead of a diverging beam, a converging beam is obtained. One advantage of using LHM lenses is they could be more compact. The focal length of a thin convex lens is given by [24]:

R f = 0 , (2.33) 0 2 n −1

where R0 is the radius of curvature of the surface of the lens. For two lenses having the

same focal length and the same refractive index magnitude, the LHM lens has a higher

radius of curvature and therefore is more compact.

nNIM

z z z nPIM

Convex Concave Rectangular lens

Figure 2.6: Beam paths in (a) rectangular, (b) convex, (c) concave lenses made of LHMs.

2.4.4 Superlensing

Pendry [2] has shown that the diffraction limit encountered in conventional lenses could

ε = µ = − be overcome with a flat lens as in Figure 2.6 (a) with r r 1 and thickness d.

The EM waves scattered by an object in air have all the Fourier components

= 2 − 2 − 2 kz k0 kx k y . The function of the lens is to apply phase corrections to each of the

Fourier components of the object wave. The propagating waves, which form the image,

= 2 + 2 < are limited to the transverse wave vector k⊥ kx k y k0 . For larger values of the

= 2 + 2 > = − 2 + 2 − 2 transverse wave vector, k⊥ kx k y k0 , k z j k x k y k0 . These are evanescent waves that decay exponentially with z and no phase correction can restore them to the right amplitude to be useful in the image plane. Hence, the maximum resolution in the

image is given by:

2π 2π 2πc ∆ ≈ λ = = = = λ . (2.34) ⊥,min ω 0 k⊥,max k0

λ < λ Hence, for ⊥ 0 , kz is imaginary (for a wave propagating along z) and the wave evanescent along z and the contents of the image either do not reach the lens or are strongly attenuated when reaching the location of the focus[14].

ε = µ = − Let us assume an s-polarized light entering a slab of LHM of r r 1 , then

µ the relative impedance η = r =1, and all the energy is transmitted into the medium r ε r

= ω 2 −2ε µ − ( 2 + 2 ) where, 'k z c r r k x k y . For evanescent waves we have

2 + 2 > ω 2 −2ε µ = − ( 2 + 2 )−ω 2 −2ε µ k x k y c r r then 'k z j k x k y c r r and the limit of the

− ′ (k 2 +k 2 )−ω2c 2ε µ d transmission through both surfaces of the slab is lim T = e kj zd = e x y r r and ε µ →− s r , r 1 the initially evanescent waves now grow inside the slab as shown in Figure 2.7 [2]. The recovery of the evanescent waves allows the image resolution to be better than the diffraction limit and this material could theoretically produce a perfect image of the source, which led Pendry to call the system a “perfect lens”. A LHM material must necessarily be dispersive as shown [3] and therefore all this would only work properly at certain frequencies. Note that a dispersive medium has to be lossy by the Kramers-

Kronig relations discussed in later chapter [23].

Such a perfect lens can be used in medical imaging, optical imaging and nondestructive detections. The first superlens in the microwave regime was realized in

2004, which demonstrated resolution three times better than the diffraction limit [25].

The first optical superlens was proposed using thin silver film [26].

Conventional Lens NIM Lens Object high Image low Object high Image high

frequency frequency frequency nNIM=-1 frequency

d-d z z 1 d1 d2 d nPIM 1 nPIM

nNIM=-1 Intensity Intensity

z z

Propagating waves d=d 1+d 2

Evanescent waves

Figure 2.7: Difference between a conventional and a NIM lens.

2.5 Other applications of metamaterials (cloaking)

Another important potential application of metamaterials is cloaking. Cloaking requires a material whose permittivity and permeability can be independently controlled. It also requires permittivity and permeability elements with relative magnitudes less than unity, and consequently the bandwidth of a passive cloaking material is limited. Using optical transformation the material parameters for a cylindrical invisible cloak are given by [27]

r − a ε = µ = , (2.35) r r r

r εφ = µφ = , (2.36) r − a

 b 2 r − a ε = µ =   , (2.37) z z  b − a  r in which a ≤ r ≤ b , and a and b are radii of the inner and outer spherical surfaces of the cloak, respectively. Now, in Cartesian coordinates,

ε = ε 2 φ + ε 2 φ xx r cos φ sin , (2.38)

ε = ε = (ε −ε ) φ φ xy yx r φ sin cos , (2.39)

ε = ε 2 φ + ε 2 φ yy r sin φ cos , (2.40)

ε = ε zz z , (2.41) with µ = ε completing the material tensor description, defined as:

ε ε µ µ  xx xy 0   xx xy 0      ε = ε ε µ = µ µ  yx yy 0 ,  yx yy 0  . (2.42)  ε   µ   0 0 zz   0 0 zz 

Figure 2.8(a) shows the electric field distribution and EM power flow for the simulation of an ideal cloak using COMSOL. The incident wave has a frequency f= 2 GHz

transverse-electric (TE) polarized time-harmonic uniform plane wave incident from the

left of the computational domain. The left and right boundaries are set to perfect

matching layers (PML) and the top and bottom are set to perfect magnetic conductor

(PMC). The cloaking shell has an inner radius a=0.2 m and an outer radius b=0.4 m .

Note the zero fields inside the cloak. Any object can be placed in the interior of the thin

shell without influencing the fields anywhere in the domain. In Figure 2.8 (b) a loss is

introduced to the ideal cloak parameters. Figure 2.9 shows the permittivity of the

simulated cloak.

2.6 Conclusion

In summary, in this chapter we have discussed the fundamental properties of propagation in the presence of LHMs. Secondly, the reversal in the basic optics phenomena is summarized, viz., Snell’s law, Doppler effect and Vavilov-Cerenkov radiation. Next the change in geometrical optics has been presented and the concept of the perfect lens has been introduced. Finally cloaking, which is a promising application of metamaterials, has been introduced. It has been shown how small losses can significantly destroy the cloaking effect.

(a)

(b)

Figure 2.8: (a) Ideal cloak (b) with loss.

Figure 2.9: Relative permittivity of the cloaking shell.

CHAPTER 3

DISPERSION RELATIONS FOR NEGATIVE INDEX MATERIALS

3.1 Introduction

In general, metamaterials are made from mixtures of other materials. Thus it is important to understand how the effective medium theory and causality apply to such a mixture.

Moreover, since modeling the wave equation starting from the dispersion relation is easy

[28], thus we study the causality of the propagation constant. In this chapter the causality of the permittivity, permeability, propagation constant and the refractive index are discussed. In Section 3.2 various complex permittivity models are defined. Section 3.3 presents an example of an effective medium theory. Next, Section 3.4 gives a summary of general dispersion relation and the reduced dispersion relation is introduced. Section

3.5 gives example of different dispersion relations either from the classical models or suggested ones. Finally in Section 3.6, we study baseband and envelope propagation starting from suggested dispersion relations representing a gain and a lossy medium.

3.2 Complex dielectric permittivity

The permittivity ε (ω ) can be written from Eq. (2.9) as:

∞ ε ()()ω = ∫ε~ t e− jωt dt , (3.1) −∞

Because of causality, the integration range for t can be redefined from 0 to ∞. Then Eq.

(3.1) becomes

∞ ε ()()ω = ∫ε~ t e− jωt dt . (3.2) 0

From Eq. (3.2) we notice that

∞ ε ∗ ()()()ω = ∫ε~ t e jωt dt = ε −ω , (3.3) 0

Now, writing ε(ω) = ε ' (ω)− jε '' (ω), where, ε ' (ω)is the real part of ε (ω ) and ε '' (ω)is the

imaginary part of ε (ω ), separating ε (ω ) into real and imaginary parts and using Eq.

(3.3) we obtain

ε ′(ω) = ε ′(− ω), (3.4a)

ε ′′(ω) = −ε ′′(− ω) . (3.4b)

Thus ε ' (ω)is an even function of ω and ε '' (ω)is an odd function of ω.

The permittivity and electric susceptibility are connected by the relation [29]

ε(ω) = ε [ + χ(ω)] 0 1 , (3.5) where χ(ω) is the linear susceptibility. The real and imaginary parts of the susceptibility

and the permittivity must be related through the Hilbert transform (HT) in order to satisfy

causality. The HT pair for a function a(ω )can be written as [30]:

∞ 1 Im {a(ω')} a'()()ω = Re {}a ω = P dω' , (3.6) π ∫ ω −ω −∞ '

∞ 1 Re {a(ω')} a '' ()()ω = Im {}a ω = − P dω' . (3.7) π ∫ ω −ω −∞ '

where P is the principal value of the improper integral.

Traditionally, there are three types of permittivity models: Debye, Lorentz, and

Drude, depending on the type of the material and the frequency range.

The Debye model is suitable to describe the dielectric properties of polar fluids that have permanent electric dipole moments. The permittivity is given by [31]:

ε − ε (∞) ε ()()ω = ε ∞ + s , (3.8) 1+ jωτ

ε ε (∞) where s and are the low frequency and high frequency permittivities of the material

and τ is the relaxation time. An example of such a medium is water up to 100 GHz [32].

The Lorentz model is widely used in solid state physics and it gives the frequency

dependence of the permittivity as

ω 2 ε ()()ω = ε ∞ + pi ε , (3.9) ∑ ω 2 −ω 2 + γ ω 0 i 0i j i

where ε (∞) is again the high frequency permittivity of the material, ω is the ith plasma pi

ω th γ th frequency, 0i is the i resonance frequency and i is the i damping amplitude. A

special case of the Lorentz model is the Drude model, used to describe the optical

ω = properties of metals. Setting the resonance frequency to zero ( 0 0 ) in Eq. (3.9) we obtain

ω 2 ε ()()ω = ε ∞ − p ε . (3.10) ω 2 − jωγ 0

The typical conductivity behavior for low frequencies

jσ ε()ω → − , (3.11) ωε 0

σ = ω 2ε γ is apparent in the model, where the conductivity is p 0 / .

3.3 Effective medium theory and causality

Different types of effective media are shown in Figure 3.1. The simplest case of an effective medium is the two-phase MG nanosphere system (Figure 3.1a). Solid nanospheres are embedded in a host material, and they can be insulators, metals or semiconductors, and are assumed to have a size much smaller than the wavelength of the

incident light. In the case of dielectric nanosphere with a volume fraction fi ,

ε (ω)− ε (ω) ε (ω)− ε (ω) eff h = f i h , (3.12) ε ()()ω + ε ω i ε ()()ω + ε ω eff 2 h i 2 h

ε (ω) ε (ω) where i is the permittivity of the nanospheres and h is the permittivity of the host material [30]. The MG approximation of an effective medium can be reformulated as: 1+ 2υ(ω) f ε ()()ω = ε ω i , (3.13) eff h −υ()ω 1 fi where υ(ω )is given by

ε (ω)−ε (ω) υ()ω = i h . (3.14) ε ()()ω + ε ω i 2 h

Two major limitations of the MG approximation are: (a) the interaction between

nanospheres and the scattering of light are neglected (the case where the scattering is not

neglected is discussed in a later Chapter), (b) the MG model does not take into account

either the size or the distance between nanospheres, thus neglecting agglomeration and polydispersion effects. Also, the volume fraction must be relatively low ( fi <<1) for the approximation to be valid. Bruggeman has solved this problem with the assumption that the inclusions are embedded in the effective medium itself and the permittivity of the host material is replaced by Eq. (3.12).

ε (ω) h ε (ω) i

(a) ε (ω) a ε (ω) b

(b) ε (ω) a ε (ω) b

(c)

Figure 3.1: Different types of effective media: (a) MG, (b) Bruggeman and (c) layered nanostructures [30].

The effective dielectric function is derived from

ε (ω)− ε (ω) ε (ω)−ε (ω) f a eff + f b eff = 0, (3.15) a ε ()()ω + ε ω b ε ()()ω + ε ω a 2 eff b 2 eff

ε ε where a and b denote the two components having different permittivity a , b and

volume fractions fa , fb respectively. Eq. (3.15) applies to spherical inclusions. Zeng

[33] has generalized the Bruggeman theory in order to include different shapes of

inclusions. The generalized formula becomes

ε (ω)−ε (ω) ε (ω)−ε (ω) f a eff + f b eff = 0 (3.16) a ε ()()()ω + g[]ε ω − ε ω b ε ()()()ω + g[]ε ω − ε ω eff a eff eff b eff where g is a geometric factor depends on the shape of the inclusions (Figure 3.1(b)). For

spherical inclusions g=1/3 Eq. (3.16) reduces to Eq. (3.15). Solving Eq. (3.16), and

taking only the valid root we get:

−c()()()()()ω + c ω 2 + 4g 1− g ε ω ε ω ε ()ω = a b , (3.17) eff 4()1− g

(ω) = ( − )ε (ω)+ ( − )ε (ω) where c g fa a g fb b .

Finally we discuss the case of layered nanostructures (Figure 3.1(c)). We

consider a two-phase layered nanostructure, built by alternating layers of subwavelength

thicknesses. Two cases have to be considered, TE and transverse magnetic (TM)

polarized light.

For TE polarized light the tangential component of the electric field is continuous

at the boundary between the layers. The average electric displacement is:

(ω) = a (ω)+ b (ω) = [ ε (ω)+ ε (ω)] (ω) DTE fa DTE fbDTE fa a fb b ETE , (3.18)

since ETE is continuous over the structure. The average electric displacement is:

(ω) = ε (ω) (ω) DTE eff ETE . (3.19)

Combining Eqs. (3.18) and (3.19), the effective dielectric function is given by

ε (ω) = ε (ω)+ ε (ω) eff fa a fb b . (3.20)

For TM polarized light the normal component of DTM is continuous at the boundary. The average electric field is

 f f  E ()()()ω = f E a ω + f E b ω = a + b D ()ω (3.21) TM a TM b TM ε ()ω ε ()ω  TM  a b 

where DTM is continuous. The effective permittivity for TM polarized light is given by

1 f f = a + b . (3.22) ε ()ω ε ()ω ε ()ω eff a b

Hence, the refractive index of a layered medium is different for TE- and TM- polarized

light thus anisotropic yielding a form of birefringence.

Now we study the causality of the effective medium permittivity through an

example. Consider a system of SiC particles in a KBr host with a volume fraction f=0.5.

We must use the Bruggeman formula since f is large. The permittivity is given by the

Lorentz model as in Eq. (3.9) with the following parameters:

for KBr:

ε(∞) = 3.2 ε γ = 92 ×10 8 rad / s, γ = 603 ×10 8 rad / s, ω = × 11 0 , 1 2 p1 2.3 10 rad / s,

ω = 1.1 ×10 11 rad / s, ω = 13.2 ×10 11 rad / s, ω = 11.3 ×10 11 rad / s, p2 01 02

for SiC:

ε(∞) = ε ω = × 11 ω = × 11 γ = × 11 7.6 0, p 18 10 rad / s, 0 14 9. 10 rad / s, 14.0 10 rad / s [34].

Figure 3.2 shows that the given (from Lorentz model) and calculated HT real and

imaginary part are in agreement (See Appendix A). Thus, the effective medium theory

satisfies causality.

Figure 3.2: Real and imaginary parts of the relative permittivity for SiC-KBr mixture.

3.4 General dispersion relations

General dispersion relations can be expressed in the form P(ω k, ) = 0 . The corresponding PDE for the wave ψ ( t,z )can be found by substituting ω and k by the

operators:

ω → − j ∂ / ∂ ,t k → j ∂ / ∂ ,z (3.23)

for 1-D propagation along the z direction [28].

The complex permittivity or susceptibility can be used to determine the complex

frequency dependent propagation constant k(ω) = k'(ω)− jk '' (ω). Note that

k 2 (ω) = ω 2µ(ω)ε(ω). Now, using the relation n2 (ω) = ( v/c )2 = [ck (ω)/ ω]2 where c and v denote the velocity of light in vacuum and in the material, respectively and assuming in

µ = µ this Section that 0 , after some straight forward algebra we get,

 ω 2  /1 2 ' ()ω = ± [ε ' + ε 2' + ε 2'' ] k  r r r  , (3.24)  c2 2 

 ω 2  /1 2 '' ()ω = ± [ε ' − ε 2' + ε 2'' ] k  r r r  . (3.25)  c2 2 

Conversely, the real and imaginary parts of the susceptibility (or permittivity) can be derived from the real and imaginary parts of the propagation constant using the relations

 c 2 χ ' ()()ω = ε ' ω −1 = ()k 2' − k 2''   −1, (3.26) r  ω 

 c 2 χ '' ()()ω = ε '' ω = ()2k 'k ''   . (3.27) r  ω 

Now we investigate the condition on k(ω ) in order to have ψ (z,t) real. The inverse FT of Ψ(ω )can be written as:

∞ 1 ψ ()z,t = Ψ()()ω exp {}j[]ωt − k ω z dω π ∫ 2 −∞

∞ 1 = Ψ()()ω exp {}j[]ωt − k ' (ω)z + jk '' ω z dω . (3.28) π ∫ 2 −∞

The complex conjugate Eq. (3.28) becomes

∞ ∗ 1 ψ ()z,t = Ψ* ()()ω exp {}− j[]ωt − k ' ω z exp []− k '' ()ω z dω π ∫ 2 −∞

∞ 1 = Ψ* ()()−ω exp {}j[]ωt + k ' −ω z exp []− k()−ω z dω , (3.29) π ∫ 2 −∞

∗ through the replacement ω → −ω . For ψ (z,t) to be real, ψ (z,t) =ψ (z,t), so that upon comparing Eqs.(3.28) and (3.29),

k ' (−ω) = −k ' (ω), k '' (−ω) = k '' (ω). (3.30)

This means that k' (ω) must be an odd function of ω , while k '' (ω) must be an even function of ω .

It is well known that the permittivity and permeability must satisfy causality. In addition, knowing that the convolution of two causal functions is also a causal function

[35], it also follows that the square of the refractive index, n~2 (t), should also be a causal function. Now we examine the causality of the propagation constant. It has been proven that the inverse FT of the function

ωm[n(ω)−1]m (3.31) is causal [30]. For the case of m = 1, and because causality is insensitive to multiplication by a constant, relation (3.31) can be rewritten as

ω[n(ω)−1] ω K()ω = = k()ω − . (3.32) c c

ε (ω) The assumption in relation (3.31) and Eq. (3.32) is that lim n()ω = lim = 1 and ω→∞ ω→∞ ε 0

µ = µ µ = (ω) r 0 1. Hence, the reduced dispersion relation K must have its real and

~ imaginary parts related by the HT, and K(t) must be causal [36]. As a generalization of

Eq. (3.32) we suggest rewriting the reduced dispersion relation as

K(ω) = ω ε (ω)µ(ω)−ε (∞)µ(∞) , (3.33) where limε (ω) = ε (∞) and limµ(ω) = µ(∞) . ω→∞ ω→∞

We would like to mention that propagation of acoustic waves can be analyzed

similar to electromagnetic waves by invoking the replacements ε → B−1 ; µ → ρ where

B is the bulk modulus and ρ is the density [37]. The velocity of acoustic waves is

υ = ρ υ = εµ a B , in analogy with the velocity of electromagnetic waves em 1 .

Assuming ρ to be a frequency independent constant, and with B → ∞ or B −1 → 0 as ~ ~ ω → ∞ [38], it follows that K(ω) = ω ρ B = k(ω), hence k (t) = K(t)must be causal,

implying that the real and imaginary parts of k(ω )are related through the HT. A simple physical reason for B → ∞ as ω → ∞ is that B is the ratio of stress to strain, and the material cannot respond to fast variations in stress, leading to diminished strain, and hence higher values of the bulk modulus. The comments above are also valid for a nonmagnetic medium with µ constant, and ε (∞) → 0 . Propagation is such media can be readily modeled by starting from k(ω ) and using operator algebra as in Eq. (3.23) to derive the underlying PDE as it is shown in a later section.

3.5 Illustrative examples of dispersion relations

In this Section we give several examples of dispersion relations. We start with the classical permittivity model from Section 3.2 and also give more simple pairs of HTs.

3.5.1 Resonant permittivity

Consider a material such as LiTaO 3 whose permittivity is given as [39]:

 ω 2 − ω 2  ε ()()ω = ε ∞ 1+ L T  , (3.34)  ω 2 − ω 2 + ωγ   T j  where ε(∞)is the high frequency limit of the permittivity, with ε(ω) satisfying Kramers-

Kronig relations, ωT and ωL are the transverse and longitudinal optical phonon

ω 2 = ω 2 ε ( ) ε (∞) frequencies, respectively, and are related by the Sachs-Teller relation L T 0 which can directly be derived from Eq. (3.34), and γ is the damping coefficient (loss

factor). Eq. (3.34) is the Lorentz model written for polatronic materials. For LiTaO 3,

ω = π × × 12 ω = π × × 12 γ = × 12 T 2 25.4 10 rad/s, L 2 46.7 10 rad/s, 94.0 10 rad/s, and

ε(∞) = ε µ = µ 13 4. 0 . Since the material is nonmagnetic we take 0 . The real and

imaginary parts of the relative permittivity are shown in Figure 3.3. Note that over the

ω < ω < ω region T L , the real part of the permittivity is negative. The plots of the real and imaginary parts of the dispersion relation, as given by Eq. (3.33), are shown in Figure

3.4. It is clear that the real and imaginary parts of K (ω) are related by the HT.

Figure 3.3: Real and imaginary parts of ε , found from Eq. (3.34), with ε(∞) = 13 4. ε , r 0 ω = π × × 12 ω = π × × 12 γ = × 12 T 2 25.4 10 rad/s, L 2 46.7 10 rad/s, and 94.0 10 rad/s.

3.5.2 Resonant permittivity and permeability

Consider a hypothetical Lorentz type metamaterial with both dispersive permittivity and permeability, and given as in Eq. (3.9)

 ω 2  ε ()ω = 1+ pe ε , (3.35)  ω 2 − ω 2 + γ ω  0  1e j e 

 ω 2  µ()ω = 1+ pm µ , (3.36)  ω 2 − ω 2 + γ ω  0  1m j m 

ω 11 ω ω = 10 with the following parameters: pe = 1.1543 × 10 rad/s , 1e = 1m 9.42 ×10 rad/s ,

ω 11 γ γ 8 pm = 1.3024 ×10 rad/s , e = 2× m = 3.769 ×10 rad/s . The real and imaginary parts of

the relative permittivity are shown in Figure 3.5.

Figure 3.5: Real and imaginary parts of the relative permittivity (a) and relative permeability (b), found from Eq. (3.35) and (3.36) respectively.

The plot of the dispersion relation K (ω) is shown in Figure 3.6. As is clear from Figure

3.6, the real and imaginary parts of K (ω) calculated from Eq. (3.33) are related by the

ω ω HT. Note that for < 1e, m , the permeability and permittivity are positive implying that

the propagation constant should also be positive and for

ω < ω < ω 2 + ω 2 = × 11 1e,m 1e,m pe ,m 49.1 10 rad / s the permeability and permittivity are negative, giving a negative propagation constant. This is evident from Figure 3.6.

3.5.3 Effective medium comprising particles with resonant permittivity

In this sub-section we discuss the causality of the propagation constant in an effective medium comprising nonmagnetic particles with resonant permittivity in a nonmagnetic dispersionless background (or host) medium. The MG rule shows that a mixture comprising a Lorentz material in a dispersionless background medium is also a Lorentz material [31]. In this case the reduced propagation constant can be written as

(ω) = ω [(ε − ε (∞))]µ Keff eff eff 0 . (3.37)

ε (∞) ε (ω) As in Section 3.4, the value of eff is subtracted from eff . Let us assume a host

ε = ε µ = µ = medium with permittivity h 5.1 0 , and permeability h 0 with f1 1.0 . Also

ε (ω ) ω ω γ ε (∞) 1 is given by Eq. (3.34) with the same values for T , L , and as in Figure

3.3. ε (ω) is found using the MG formula in Eq. (3.13). The value of ε (∞) has been eff eff analytically found from Eq. (3.13) and substituted in Eq. (3.37). Figure 3.7 shows (a)

(ω) real and (b) imaginary parts of K eff from Eq. (3.37). Once again, it is clear that the

(ω) real and imaginary parts of K eff are related by the HT.

(ω) Figure 3.7: (a) real and (b) imaginary parts of K eff from Eq. (3.37); (c) numerically calculated HT of the imaginary part in (b); (d) numerically calculated HT of the real part in (a). Note that (a) is identical to (c) and (b) is identical to (d).

3.6 Baseband and envelope propagation in NIM

From Eq. (3.33) if we assume the case where ε (∞) = 0 (or B(∞) → ∞ ), then

K (ω) = k(ω) . As stated earlier, it has been shown in Ref. [40] that modeling of wave propagation (both baseband and envelope) can be conveniently done starting from the dispersion relation k = k(ω ) and by replacing the k and ω by the respective operators

k → j∂ / ∂ ,z ω → − j∂ / ∂t . Based on these assumptions, in what follows, we provide

examples of dispersion relations that exhibit either loss or gain, and where the phase and

group velocities may be contradirected. Consider the following dispersion relation where the real and imaginary part of k(ω ) is a HT pair [41]:

ω  (ω /ω ) 1  k()()()ω = k ' ω − jk '' ω ≡ − 0  0 + j  . (3.38) + ()ω ω 2 + ()ω ω 2  v 1 / 0 1 / 0 

~ −1 −ω ( ) ≡ ℑ [ (ω)]∝ − 0t ( ) Note that k t t k je u t , is a causal function as expected. Note that for

ω ≡ ω ω > ' ≡ ' (ω ) < n / 0 1, kn k / 0 / v 0 , and the phase velocity is negative while the group

ω >> ' → − ω velocity is positive, characteristic of a NIM. Indeed, for n 1, kn /1 n ,

'' ≡ '' (ω ) → kn k / 0 / v 0 which is the simplest type of dispersion relation that models a NIM, as in [42]. The variation of the real part with frequency is plotted in Figure 3.8.

3.6.1 Baseband propagation with loss

Since the HT is unique up to a constant [41], we can construct a second dispersion relation from Eq. (3.38) of the form

ω  (ω /ω ) (ω /ω )2  k ()()()()ω = −k ω − j 1. ≡ k ' ω − jk '' ω ≡  0  0 − j 0  , (3.39) L L L + ()ω ω 2 + ()ω ω 2  v 1 / 0 1 / 0 

~ −ω ( ) = [ 0t ( )−δ ( )] which again yields a causal function kL t ej u t t . The real and imaginary

(ω) part of k L are shown in Figure 3.9. Using Eq. (3.26), the real and imaginary parts of

the effective susceptibility corresponding to Eq. (3.29) can be shown to be

 2  − ()ω ω 2   ()ω ω  χ ()()()ω = χ ' ω − χ '' ω ∝ c 1 / 0 − − 2 / 0 L L j L    1 j  . (3.40)  v  []+ ()ω ω 2 2  []+ ()ω ω 2 2   1 / 0   1 / 0 

Figure 3.8: Normalized dispersion relation (real part shown only).

Again, since the HT is unique up to a constant, it can be verified that the real and

imaginary parts of χ(ω) are indeed related through the HT, thereby implying causality of

χ~(t). The variation of the real part and imaginary part with frequency is plotted in

Figure 3.10. In this case the phase velocity is always positive, while the group velocity

ω = changes sign at n = 1. For simplicity, we take v c .

Figure 3.9: (a) Real and (b) imaginary parts of the complex propagation constant using Eq. (3.39) and assuming ω =10 14 rad / s. 0

Figure 3.10: (a) Real and (b) imaginary parts of the complex susceptibility using Eq. (3.40) and assuming ω =10 14 rad / s. 0

We derive the PDE for baseband propagation, starting from the dispersion relation

in Eq. (3.38). Using the substitutions defined in Eq. (3.23) to Eq. (3.39) we get:

∂ψ 1 ∂ 3ψ 1 ∂ψ 1 ∂ 2ψ − + − = 0 . (3.41) ∂ ω 2 ∂ ∂ 2 ∂ ω ∂ 2 z 0 z t v t v 0 t

ω Upon performing space and time normalizations as T = ω t, Z = 0 z , we get 0 v

∂ψ ∂3ψ ∂ψ ∂2ψ n − n + n − n = ,0 ψ (Z,T) =ψ ( tz ), . (3.42) ∂Z ∂Z∂T 2 ∂T ∂T 2 n

ω ω << For 0 1, the dispersion relation Eq. (3.38) can be approximated as

(ω ) ≈ ω −ω 3 − (ω 2 −ω 4 ) knL n n n j n n . (3.43)

(ω ) Note the presence of attenuation terms, arising from the imaginary part of knL n . The leading term is proportional to the square of the frequency, which is observed during sound wave propagation through fluids [43]. The above dispersion relation yields the following baseband wave propagation

∂ψ ∂ψ ∂3ψ ∂2ψ ∂4ψ n + n + n − n − n = 0. (3.44) ∂Z ∂T ∂T 3 ∂T 2 ∂T 4

Note that the first two terms in Eq. (3.43) is the familiar KdV type dispersion relation,

and hence, the first three terms on the left-hand side of Eq. (3.44) is the linear part of the

Korteweg de vries (KdV) equation [28]. The additional terms represent the contribution

of losses, such as through second and fourth order diffusion terms. Eq. (3.44) is the well-

known Kuramoto-Shivashinsky equation (KSE) extensively used to model wave

propagation in dispersive-dissipative media [44].

Eq. (3.42) can be solved using the Fourier transform (FT) method. Upon taking the FT in time, straightforward algebra, and inverse Fourier transform, it can be shown that

 2  −1 −1  −ω − jω  ψ ()()Z,T = ℑ {}()Ψ Z,ω = ℑ Ψ ,0 ω exp  n n Z, (3.45) n t n n t n n + ω 2   1 n 

Ψ = ℑ {ψ ( )} where n t n ,0 T is the temporal spectrum of the initial pulse [40]. Figure 3.11

(a) shows the baseband pulse after propagation of a normalized distance Z=10. Figure

3.11 (b) shows the energy decay of the pulse during propagation, due to the presence of

diffusion (See Appendix B).

Figure 3.11: (a) Baseband initial Gaussian pulse (initial width τ=5) in time domain and after propagating a distance Z=10. (b) Energy decay of the propagating pulse.

3.6.2 Envelope propagation in NIMs with loss

For envelope solution in NIMs we start from the dispersion relation Eq. (3.38), and use

Eq. (3.33) to obtain after normalization in space and time:

∂ψ ∂3ψ ∂ψ n − n − n +ψ = 0. (3.46) ∂Z ∂Z∂T 2 ∂T n

ψ = ℜ {ψ [ (ω − )]} Now setting n e en exp j cn T kcn Z where

ω k −ω ω = c ,k = c = cn , (3.47) cn ω cn ω +ω 2 0 0 v 1 cn

in Eq. (3.46), we get after some algebra:

∂ψ ∂3ψ ∂2ψ ∂2ψ ∂ψ a en +b en + jc en + jd en + e en + fψ = 0 , (3.48) ∂Z ∂Z∂T 2 ∂T∂Z ∂T 2 ∂T en where

= −( +ω 2 ) = = ω = − = ω + = − a 1 cn , b ,1 c 2 cn ,d kcn ,e 2kcn cn ,1 f 1. (3.49)

As in the previous example, we solve Eq. (3.48) using the FT method. It can be shown that for an initial condition ψen ,0( T) , the solution ψen (Z,T) can be expressed as

 2  − −    dω − je ω − f   ψ ()()Z,T = ℑ 1{}()Ψ Z,ω = ℑ 1 Ψ ,0 ω exp −  n n Z  . (3.50) en t en n t en n  ω 2 − ω −     b n jc n a  

ω = Plots for cn 2 are shown in Figure 3.12, and it is clear from careful monitoring of the peaks in the inset that the group velocity is, indeed, positive. The decay of the pulse during propagation is due to attenuation, as expected. We have checked from our

ω = simulations that for cn 5.0 , the group velocity is negative, as discussed earlier. In the

ω →∞ limit cn , the attenuation goes to zero, and the pulse acquires a large group velocity, so that its profile for arbitrary Z is essentially centered at T = 0 , as expected.

ω = Figure 3.12: Gaussian pulse envelope in time domain in a NIM for cn 2and initial width τ = 20 . (a) Initial pulse, (b) after propagation by Z=10.

3.6.3 Baseband and Envelope Propagation in NIM with gain

From k (ω )in Eq. (3.38) above, we can construct a second dispersion relation by adding a constant term to the imaginary part of Eq. (3.38). Again, since the HT is unique up to a

~ constant, the causality of k (t) is not affected, and k (ω ) in Eq. (3.38) above can be modified to

 ω  − (ω /ω ) − (ω /ω )2  k ()ω = k(ω) +1j =  0  0 − j 0  , (3.51) G  + ()ω ω 2 + ()ω ω 2   v 1 / 0 1 / 0 

where the subscript G stands for gain.

µ = −µ Now, assuming a negative material ( 0 for simplicity), the real and

imaginary parts of the relative permittivity can be expressed in terms of the real and

imaginary parts of the propagation constant as in Eq. (3.26) and Eq. (3.27) but with

µ = −µ 0 .

 c 2 ε ' ()ω −1 = −()k 2' − k 2''   , (3.52) r ω 

 c 2 ε '' ()ω = −()2k 'k ''   . (3.53) r  ω 

The frequency dependent electric permittivity corresponding to Eq. (3.51) becomes

 2  ()ω ω 2 −   − ()ω ω  ε ()ω = ()ε ' − ε '' ε = c ε / 0 1 − 2 / 0 r j r 0   0   j . (3.54)  v  []+ ()ω ω 2 2  []+ ()ω ω 2 2   1 / 0   1 / 0 

It can be analytically checked that the real and imaginary parts of the permittivity are indeed also related by the HT. Examples of such active metamaterials are nanowire composites [45].

Starting from Eq. (3.51), we can derive the underlying PDEs for baseband and envelope wave propagation. Using the operator formalism for frequency and wave number stated above, we get

∂ψ 1 ∂3ψ  ω  1  ∂ψ  ω  1  ∂ 2ψ − −  0   +  0   = 0 . (3.55) ∂ ω 2 ∂ 2∂  ω  ∂  ω 2  ∂ 2 z 0 t z  v  0  t  v  0  t

= ω = (ω ) Now using the coordinate transformations T 0t, Z 0 /v z , we can readily derive

the normalized PDE for baseband propagation:

∂ψ ∂ 3ψ ∂ψ ∂ 2ψ n − n − n + n = 0 . (3.56) ∂Z ∂T 2∂Z ∂T ∂T 2

Eq. (3.56) can be solved using the FT method. Upon taking the FT in time we get

 ω − jω 2   Ψ ()()Z,ω = Ψ ,0 ω exp  j n n Z n n n n  +ω2    1 n  

  ω    ω 2  = Ψ (),0 ω exp  j n Z • exp  n Z  , (3.57) n n  +ω 2   +ω 2   1 n   1 n 

Ψ ( ω ) = ℑ {ψ ( )} where n ,0 n t n ,0 T is the temporal spectrum of the initial pulse. Then

ψ (Z,T ) is obtained by taking the inverse FT of Ψ (Z,ω ). From Figure 3.13 we notice n n n that the Gaussian pulse, plotted as a function of time, gains in amplitude as it propagates, due to the fact that the imaginary part of the propagation constant is negative for positive values of the angular frequency, which, in turn, implies a wave of the form

− jkz −k z e ∝ e i . This is also directly evident from Eq. (3.57), which shows a real

exponential with positive argument for positive values of angular frequency and

propagation. The medium is, therefore, an amplifying medium or a medium with gain,

instead of an attenuating or absorbing medium. It is also clear from this example that

identical real parts of dispersion can support imaginary parts (without violating the HT

condition) which gives rise to either attenuation or gain. Note also that the first

exponential in Eq. (3.57) suggests that for positive frequencies, the wavenumber is

negative, implying negative phase velocity. Although the above analysis is for baseband

propagation thus far, note that the negative value for the imaginary part of the

propagation constant correspondingly implies a negative sign of the imaginary part of the

permittivity, as shown by Eq. (3.54).

To further demonstrate the effect of the negative value of the imaginary part of

the permittivity, we now simulate envelope propagation in NIMs using the dispersion

ψ = ℜ {ψ [ (ω − )]} relation given in Eq. (3.51). In Eq. (3.56) we set n e en exp j cn T kcn Z ,

ω k −ω 2 where ω = c ,k = c = cn , and after some algebra we get cn ω cn ω + ω 2 0 0 v 1 cn

∂ψ ∂ψ ∂ 2ψ ∂ 2ψ ∂ 3ψ a e + b e + c e + d e + e e + fψ = 0 , (3.58) ∂Z ∂T ∂T 2 ∂T∂Z ∂T 2∂Z e

= + ω 2 = − + ω − ω = + = − ω = − where a 1 cn , b 1 2 j cn 2kcn cn , c 1 jk cn , d 2 j cn , e 1 and

= − − ω − ω 2 − ω 2 f jk cn j cn cn j cn kcn . Eq. (3.58) can again be solved using a FT method. It

ψ ( ) ψ ( ) can be shown that for an initial condition en ,0 T , the solution en Z,T can be expressed as

2 − −   − f − jb ω + cω   ψ ()()Z,T = ℑ 1{}()Ψ Z,ω = ℑ 1 Ψ ,0 ω exp  Z . (3.59) en t en n t en n  − ω 2 + ω    a e jd  

ω = Results for cn 4are shown in Figure 3.14, which shows a positive group velocity and a gain of the signal during propagation, as expected (See Appendix C).

(a) (b)

Figure 3.13: (a) Initial ( Z=0) baseband initial pulse in time domain and (b) after propagation a distance Z=10. The initial Gaussian pulse is taken as ψ ( ) = (− 2 τ 2 ) τ = n 0 T, exp T with 5 .

ω = Figure 3.14: Gaussian pulse envelope in time domain in a NIM for cn 4 and initial width τ = 20 . (a) initial pulse, (b) after propagation by Z=10.

3.7 Conclusion

In this Chapter, we have first introduced the concept of causality for permittivity and permeability, both for a homogeneous as well as an effective medium comprising binary constituents. We have also introduced the concept of a reduced propagation constant, obtained by starting from a causal permittivity and permeability, and have shown that by neglecting a linear frequency dependent term from the dispersion of the propagation constant, obeys causality. The propagation constant is identical to the reduced propagation constant under appropriate limiting values of the physical parameters. We have found dependence on frequency of the reduced propagation constant for (a) a

nonmagnetic material where the permittivity is given by the Lorentz model, (b) a material where the permittivity and permeability are Lorentz-type, and (c) an effective medium comprising a nonmagnetic material with Lorentz-type permittivity in a dispersionless background, where the effective permittivity is given by the MG rule. In each case, the reduced propagation constant has real and imaginary parts related through the HT, implying causality. Causality of the propagation constant enables the use of simple operator formalisms to derive the underlying PDEs for baseband and envelope wave propagation. Two illustrative examples have been given to model propagation in NIMs, one with loss and one with gain.

CHAPTER 4

OPTICAL PROPAGATION IN NEMATIC LIQUID CRYSTALS

4.1 Introduction

The term LCs, or mesogens , refers to materials which exhibit intermediate phases between the isotropic liquid and crystalline solid states. A LC has a fluid phase in the sense that a liquid crystal “flows” and can take the shape of its container. It contains rod- like molecules which exhibit orientational alignment without positional order [46].

Although LC combines the properties of a solid and an isotropic liquid, they have very specific electro-optical properties [47]. LCs are found in many consumer electronic devices.

Depending on the constituents, concentration, substituent, and finally the temperature, LCs exist in many so called mesophases: nematic, cholesteric, smectic, and ferroelectric. Nematics are the most widely studied LCs. A unique property of a NLC is its ability to change optical characteristics under the action of an external field. The incident light on the NLC can also modify the electric permittivity tensor, leading to reorientational nonlinearity [46]. Figure 4.1 shows the orientation of the LC with applied external voltage. NLCs are known to exhibit enormous optical

nonlinearities, owing to their large refractive index anisotropy, coupled with the

optically-induced collective molecular reorientation. Negative refraction at optical

frequencies can be realized using LCs because of its optical anisotropy [20,21].

(b) (a)

Figure 4.1: NLC orientation: (a) without external field (b) with external field applied.

In this Chapter, we examine the basics of optical propagation in LCs. In the spirit of the title of the disseration, we give an example of a LC arrangement whereby one can demonstrate effective negative refraction by channeling a light beam through the LC material upon introducing the beam through the side of the LC- indium tin oxide (ITO) sandwich. In Section 4.2, we show the model that governs the change in the director angle of NLCs due to an applied voltage. Then we model optical propagation in LC, which is coupled with the refractive index which is dependent on the director orientation.

Section 4.3 gives an example of negative refraction using LC cell, achieved by utilizing the voltage dependent anisotropy of the LC sandwich. Although this arrangement shows negative refraction, it is not a consequence of negative index. True negative index configurations using nanoparticles in a LC host is discussed later in the dissertation. In this case, the negative index can be achieved by engineering the material to have negative

effective permittivity and permeability over a range of frequencies. Section 4.4

concludes the Chapter.

4.2 Mathematical Model

Due to the one-dimensional orientational ordering in NLCs, the molecules have an average orientation defined by a vector called the director [48]. Figure 4.2 shows the 2-D geometry of a NLC cell. A static electric field is applied over the cell as the glass is coated with ITO. The structure being invariant in the z direction, the static electric field has a component along the x axis only.

x V θ

Figure 4.2: Geometry and axes of a NLC cell.

In what follows we show the effect of biasing, along with a predetermined pre-tilt,

on optical propagation in a NLC. With reference to the sample geometry sketched in

Figure 4.2, let us consider an x-polarized beam propagating in the z direction and injected

into a glass cell filled with NLC. We assume that the molecules have a small (viz., 2°)

pretilt in the x-z plane. The bias induced orientation is governed by the Euler-Lagrange

equation [49]:

2θ  −   θ 2  2 ()2 θ + 2 θ d + K3 K1 θ d + 1 ε dV θ = K1 cos K3 sin 2  sin 2   a   sin 2 ,0 (4.1) dx  2   dx  2  dx  59

where θ is the angle between the molecular director and the propagation vector, K1 and K3

∆ε = ε − ε are the Frank elastic constants for splay and bend respectively, STA e o is the

ε ε dielectric anisotropy where e and o are the extraordinary and ordinary (relative)

permittivities. The potential distribution is given by:

d 2V dθ dV ()ε sin 2 θ + ε cos 2 θ + ∆ε sin 2θ = .0 (4.2) e o dx 2 STA dx dx

The two equations are solved using a commercial software (COMSOL) which uses finite

( ) = (− ) = element methods (FEMs) with boundary conditions V L 2/ V0 , V L 2/ 0,

θ(−L 2/ ) =θ(L 2/ ) = 2π /180 (corresponding to 2° pre-tilt), to derive the NLC orientation

across its thickness L. Also pertinent to optical propagation in NLCs is the refractive

index. In what follows, ne and no refer to the refractive index of light polarized parallel

(extraordinary) or normal (ordinary) to the molecular axis, respectively, due to optical anisotropy. For the nematic E7 type at room temperature the material parameters are

= × −11 = × −11 ε = ε = ,1.5 n K1 2.1 10 N , K3 95.1 10 N, e 19 ,6. o ne=1.6954 and o =1.5038. For a thickness L=75 µm and a V0=1.5V the tilt angle is shown in Figure 4.3. Due to the optical anisotropy in NLCs, the orientation of the Poynting vector S deviates from the wave vector k . The difference in these two vectors can be defined as the walk off-angle. The walk-off angle can be expressed in terms of θ and the optical

− birefringence n= ne no as:

 ∆n 2 sin (2θ )  δ ()θ = arctan  . (4.3)  ∆ 2 + 2 + ∆ 2 ()θ   n 2no n cos 2 

Tilt angle for an applied voltage of V 0 = 1 .5 V 6 0

5 0

4 0

3 0 (degrees) θ θ θ θ 2 0

1 0

0 -4 0 -3 0 -2 0 -1 0 0 1 0 2 0 3 0 4 0 x

Figure 4.3: Tilt angle ( θ) for a thickness L=75µm and at V0=1.5V.

Figure 4.4 displays the calculated walk-off versus applied voltage V0. The refractive index is given by:

1 cos 2 (θ ) sin 2 (θ ) = + . (4.4) 2 ()θ 2 2 n no ne

Calculated maximum (on-axis) walk-off vesrsus cell bia s 7

(degrees) 3 δ δ δ δ

0 0 1 2 3 4 5 V (V ) 0

Figure 4.4: Calculated walk-off angle versus applied voltage V0.

For the sample described above, the calculated refractive index profile in the absence of a

light beam is shown in Figure 4.5 for a bias V0=1.5 V.

1 .7 5

1 .7

1 .6 5 n

1 .6

1 .5 5

1 .5 -4 0 -3 0 -2 0 -1 0 0 1 0 2 0 3 0 4 0 x

Figure 4.5: Refractive index profile in the absence of a light beam for a bias V0=1.5 V.

In order to study the beam propagation in the LC cell, we have to solve the

evolution equation for the optical envelope Ee . Assuming that light is linearly polarized along x, and neglecting the reorientation of the director due to the optical field, the

evolution equation is given by [50]:

∂E  ∂ 2 ∂ 2  e +  +  = 2 jk 0n  2 2 Ee 0 (4.5) ∂z  ∂x ∂y 

For the initial distribution of the optical field, a circular-symmetric Gaussian

 2 + 2  = = − x y = µ beam is considered: Ee (z )0 exp  2  with a value of r0 2 m. Figures 4.6  r0 

and 4.7 show the initial and the final beam shapes at z=0 and z=500µm respectively. As

expected the beam diffracts while propagating but with less diffraction due to focusing caused by the parabolic index profile.

Figure 4.6: Initial Gaussian beam at z=0.

Figure 4.7: Final beam shape at z=500µm .

Figure 4.8 shows the beam shape while taking into account diffraction only. As expected

the beam diffraction is more than the one shown in Figure 4.7. Figure 4.9 shows the

= µ beam propagation for an initial width of r0 9 m. As noticed from the simulation the

= µ beam does not change its shape for r0 9 m; hence this is a mode of propagation in the

“guided index” medium for the chosen parameters of the NLC (See Appendix D).

Figure 4.8: Final beam shape at z=500µm due to diffraction only.

(a) (b)

Figure 4.9: Beam shape (a) at z=0 (b) at z=500 µ m with r0=9 µm.

4.3 Negative Refraction

The concept of negative index in NLCs arises due to voltage induced anisotropic refraction. This has previously been reported by Zhao [20]. To test this concept experimentally, we use a LC cell which has been fabricated at ARFL/MXPJ as in [21].

The LC cell of thickness d=100 m is assembled using two glass substrates with

conductive ITO layers. The substrates are coated with a polyimide film rubbed to align the director n uniformly at 45° to the z axis in the plane xz. The experimental setup is shown in Figure 4.10. The He-Ne laser beam ( λ=633 nm ) passes through a polarizer and

is focused by a lens ( f=5.5 cm ) to pass through the middle of the cell filled with E7. The angle of refraction is given by [21]:

2n sin θ + cn n bn 2n2 − n2 sin 2 θ θ = tan −1 i i o e o e i i (4.6) r,s 2 2 − 2 2 θ 2bn o ne bn o ne ni sin i

= −2 2 α + −2 2 α = α −2 − −2 α where b n0 cos ne sin , c sin 2 (ne n0 ) , is the angle between the

θ director n and the z-axis, n i is the refractive index of the incident medium, i is the incident angle. Eq. (4.6) is derived by writing the index ellipsoid coordinates and finding the components of the wave vector kt . For a typical liquid crystal material like E7, we may assume n =1α = 45 0 ,n = 52.1 , n = 74.1 ,λ = 633 nm . From Eq. (4.6) we can i 0 e

<θ <θ ≈ deduce that the range of incident angles 0 i ic 12 5. , where negative refraction

θ max = − occurs [21]. The maximum negative angle of refraction is r,S 72.7 , when the

θ = α = incidence angle is i 0 , =48.7°, and λ 633 nm . Figure 4.11 shows the angle of refraction obtained for different wavelengths as a function of α .

Entry Glass Plate

He-Ne ααα n Laser x z Filter P y LC

Figure 4.10: Experimental Setup.

( ) Figure 4.11: Variation of the time averaged Poynting vector angle θr,s θi , α with respect α = to optical axis angle with the z-axis at normal incidence ( θi 0) for different wavelength.

Figure 4.12 shows the variation of the time averaged Poynting vector angle

( ) α θr,s θi ,α with respect to optical axis angle with the z-axis and angle of incidence θi .

Maximum negative angle of refraction remains the same as in Figure 4.11 where

θ max = − α = θ = r,s 72.7 , at 48 7. i 0 . Hence our assumption of normal incidence is valid.

< < θ < θ ≈ Also, in Figure 4.12 we have negative refraction θr,s 0 only when 0 i ic 12 5. .

θ Figure 4.12: Variation of the time averaged Poynting vector angle s,r with respect to α θ optical axis angle with the z-axis and angle of incidence i . Maximum negative angle θmax (θ = ) = − α = of refraction s,r i 0 72.7 ; 48 7. .

Figure 4.13 (a) shows the path of the incident beam, the positive and negative refracted beams when no voltage is applied. Figure 4.13 (b) shows a predominantly positive refraction when the bias voltage is turned on. When the voltage is applied n is parallel to the y axis (as in Figure 4.10), the refractive index of the LC is no and the angle of refraction is found from Snell’s law. The experimental results are reproducible at other wavelengths, viz., the different emissions from an Ar laser (532nm, 514nm, 488nm).

(a) (b)

Figure 4.13: Positive and negative refraction, and p redominantly positive refraction, for an angle of incidence of ≈ 5°. For λ=0.632 µm, (a) shows predominantly negative refraction without bias voltage, while (b) shows predominantly positive refraction with 10 V bias.

4.4 Conclusion

In this Chapter we have modeled the reorientation angle in a LC cell due to an applied external voltage. The model is used in later chapters. We have summarized beam propagation in LCs. And finally, an experimental setup using a NLC cell with initial in - plane pre-tilt has been described and negative refraction has been observed. Although negative refraction has many applications, we are more interested in a negative index material. Thus, in the remaining chapters we consider features, fabrication and testing of negative index materials.

CHAPTER 5

MULTILAYER STRUCTURES

5.1 Introduction

Propagation in NIMs is of current interest due to its potential applications in superlensing. Photonic crystals, which are widely used for both light control and beam manipulation, may offer many new possibilities being combined with metamaterials.

Work has been done on photonic bandgap like structures that contain alternating layers of

PIMs and NIMs [51,52]. Also it has been demonstrated that stacking alternating layers of positive and negative index media leads to a type of photonic band gap (PBG) corresponding to a zero averaged- refractive index (zero < n > )[51]. The zero < n > gap

was verified experimentally in [53]. Most of the previous work on metamaterials have

focused on periodic structures. It is important to study disorder in these types of

structures because non-uniformities may inevitably occur in the fabrication of the

microstructures or nanostructures, which may be crucial for smaller operating

wavelengths. In general there are three kinds of randomness, viz., randomness in the

thickness of the layers, randomness in the refractive index of the layers, and random

stacking (randomness in the position of each layer in the stack). In this Chapter we

consider periodic as well as random systems.

This Chapter is organized as follows. In Section 5.2 a summary of the transfer

matrix method is presented. In Section 5.2.1 we derive the reflection, transmission and

absorption of waves passing through these periodic structures. In Section 5.2.2, for the

first time to the best of our knowledge, we employ the transfer matrix to derive the limit

of transmittance for mixed (PIM/NIM) structures when the number of layers tends to

infinity. In Section 5.3 we consider periodic structures for different types of materials

(i.e dispersive, nondispersive, lossy…). In Section 5.4 we study random systems of

PIM/NIMs. All the types of randomness, to be made more specific later, are studied. In

Section 5.5, we propose an application of the zero gap. In Section 5.6, we study the

effect of introducing defect slab into a periodic structure. Two cases were studied with

LC or NIM as the defect layer. Section 5.7 concludes the Chapter.

5.2 Transfer Matrix Method

The standard approach used for analyzing stacks or layers of various refractive indices is

the TMM. We first summarize the derivation of the transfer matrix from first principles

using Maxwell’s equations and the constitutive equations [24]. We rewrite for

convenience from Chapter 2 Eqs. (2.11), (2.12), (2.17), and (2.18): ∇ × E(r,ω) = − jωµ (ω)H(r,ω), (5.1)

∇× H(r,ω) = jωε (ω)E(r,ω), (5.2)

D(r,ω) = ε(ω)E(r,ω), (5.3)

B(r,ω) = µ(ω)H(r,ω). (5.4)

Consider an incident wave from vacuum at an angle θ onto a periodical layered

structure containing NIMs and PIMs, as shown in Figure 5.1.

d1 d2 dq

Incident Substrate . . . n0 n1 n2 nq ns θθθ0

Boundary 1 2 3 4 5 …. q-1 q q+1

Figure 5.1: Section of a slab composed of q layers.

ω Suppose wave vectors k( ) lie in the y-z plane. For the TE case, assume that the = = electric field E is in the x direction and Ey Ez .0 Using Eq. (5.1), it follows that

= H x 0 . Also,

∂H ∂H z − y = jωε ()ω E , (5.5a) ∂ ∂ x y z

∂E x − jωµ ()ω H = ,0 (5.5b) ∂z y

∂E x − jωµ ()ω H = 0 . (5.5c) ∂ z y

Eliminating H y , H z we get from Eqs. (5.5a), (5.5b), and (5.5c)

∂2 E ∂2 E x + x + ε ()()ω µ ω k 2 E = 0, (5.6) ∂y2 ∂z2 r r 0 x

dµ(ω ) dε (ω) where we assumed that = 0 , and = 0 (true within each layer). Substituting dz dz

ε (ω)µ (ω) = ( ) (− α ) α 2 = r r 2 θ Ex Ex z exp jk 0 y , in Eq. (5.6) with 2 sin , we get k0

d 2 E x + k 2ε ()()ω µ ω cos 2 θ E = .0 (5.7) dz 2 0 r r x

= ( ) (− α ) In a similar way, we can show if H y H y z exp jk 0 y that

d 2 H y + k 2ε ()()ω µ ω cos 2 θ H = .0 (5.8) dz 2 0 r r y

The solution to Eqs. (5.7), (5.8) along with Eq. (5.5c), can be found to be:

( ) = (β )+ (β ) Ex z Acos z Bsin z , (5.9a)

β H ()z = − []Bcos ()()βz − Asin βz , (5.9b) y ωµ ()()ω µ ω j 0 r

β = ε (ω) µ (ω) (θ ) where k0 r r cos . We write the linear system (Eqs. (5.9a) and (5.9b)) in

a matrix notation. The electric component and magnetic component in z and z+ z are related via a transfer matrix:

 E (z + ∆z)  E (z) x = x  ()+ ∆  M  (), (5.10) H y z z  H y z 

where

jωµ µ (ω ) E ()()()z + ∆z = cos β∆z E z − 0 r sin ()()β∆z H z , (5.11a) x x β y

β H ()z + ∆z = sin ()()()()β∆z E z + cos β∆z H z . (5.11b) y ωµ µ ()ω x y j 0 r

Replacing Eqs. (5.11a) and (5.11b) in Eq. (5.10), the transfer matrix M becomes

 ()β∆ j ()β∆  =  cos z sin z  M  P , (5.12)  jP sin ()()β∆z cos β∆z 

cos θ µ where P = , η =η η , ∆z is the layer thickness d , , and η = 0 is the η 0 r i 0 ε 0

characteristic impedance of vacuum. The matrix connecting the incident end and the exit

n ()()ω = ω end is M n ∏ M j . We can rewrite Eq. (5.10) as j=1

  j  B  q  cos δ sin δ  1  = ∏ r r  , (5.13)    Pr   η  C  r=1  δ δ  /1 s    jP r sin r cos r 

= = η δ = β∆ where B E1 Eq+1 ,C H1 Eq+1 , s is the impedance of the substrate and r z .

Note that the treatment for a transverse magnetic TM wave is similar to that for a TE wave and can be derived using the following replacements E → H , H → E and µ →ε

1 ε → µ . Note that for TM polarization, P = [23]. η cos θ

5.2.1 Reflectance, transmittance and absorbance

The net irradiance at the exit of the assembly (entering the substrate) is given by

1 I = Re (1 η * )E E * , and that at the entrance to the assembly (immediately inside layer q 2 s q q

1 * * 1) is given by I = Re (BC )E E [54]. Let the incident irradiance be denoted by Ii, then 1 2 q q

1 I = ()1 − R I = Re (BC * )E E * . Hence 1 i 2 q q

Re (BC * )E E * I = q q , (5.14) i 2()1− R

where

*  B −η C  B −η C  R =  0  0  . (5.15)  +η  +η   B 0C  B 0C 

The transmittance T into the substrate is defined as:

I Re (1 η )(1 − R) 4 Re (1 η ) T = q = s = s . (5.16) ()* (+η )(+η )* I i Re BC B 0C B 0C

The absorbance A in the multilayered structure is related to R and T by R +T + A =1.

Hence the absorbance is written as:

 Re (1 η ) 4 Re (BC * −1 η ) A = ()1 − R 1 − s = s . (5.17)  ()*  (+η )(+η )*  Re BC  B 0C B 0C

5.2.2 Limit of the transmittance

For a wave propagating in a one dimensional periodic stratified medium in the propagation direction (shown in Figure 5.2), the unimodular matrix in Eq. (5.13) can be written as [55]

N m m  m S − S − m S  M N = 11 12 = 11 N N 1 12 N , (5.18) p    −  m21 m22   m21 SN m22 SN SN−1 

 j  p  cos δ sin δ  where = r r , N is the number of p-layer periods comprising the M p ∏ Pr  r=1  δ δ   jP r sin r cos r 

sin (Nφ) 1 1 q-layered structure ( Np = q ), S = , Tr (M ) = cos ()φ = ()m + m , where N sin φ 2 p 2 11 22 ( ) Tr M p is the trace of M p . One can easily prove this result by induction by using

= (φ) − = det( M p ) 1 and the recurrence relation: 2cos SN SN−1 SN+1 . As the number of

[ ( )]2 > = periods N tends to infinity, if Tr M p 4 , then the limit of transmittancelim T 0 , N→∞

implying that we have a stop band.

1st period Nth period

d1 d2 dp

Incident Substrate ...... n0 n1 n2 np n1 n2 . . . np ns

θθθ0

Figure 5.2: Schematic structure of N periods. Each period may comprise p layers.

[ ( )]2 < For Tr M p 4 , from Eq. (5.16), replacing B and C with their respective values, the

transmittance T can be written as

4(1 η )Re (1 η ) T = 0 s . (5.19) ()η []− + ()η + + ()η − ()η 2 1 0 m11 S N S N −1 1 s m12 S N m21 S N 1 s m22 S N 1 s S N −1

= − + η = + ( − ) η where B m11 SN SN−1 m12 SN s , C m21 SN m22 SN SN−1 s where the mij s are

obtained from Eq. (5.18) and B, C are from Eq. (5.13). In the limit as N → ∞,

sin (Nφ ) → 1 [56]. Then

sin Nφ 1 S = → , (5.20) N sin φ sin φ

and

sin (N −1)φ sin (Nφ)cos φ − sin φ cos (Nφ) cos φ S − = = → . (5.21) N 1 sin φ sin φ sin φ

Replacing Eq. (5.20) and Eq. (5.21) into Eq. (5.19) we get

4(1 η )Re (1 η )sin 2 φ T = 0 s . (5.22) ()η ()− φ + ()η + + ()η − ()η φ 2 1 0 m11 cos 1 s m12 m21 1 s m22 1 s cos

5.3 Periodic Model

In this Section, we apply the TMM method explained in Section 5.2 to simple cases of alternating NIM/PIM layers for the nondispersive and dispersive cases, in order to validate our approach against known results.

5.3.1 Infinite periodic structures

In this subsection, we derive the conditions that lead to bandgaps. Let us consider a periodic infinite structure as shown in Figure 5.3. Each period is composed of 2 layers.

η = Each layer has a thickness di , refractive index ni , and impedance i , where i 2,1 .

Assume that the incident angle θ = 0. From Eq. 5.12 with

cos θ 1 β = k ε (ω ) µ (ω ) cos (θ ) = k ε (ω ) µ (ω ) = k and P = = , the transfer i 0 ri ri 0 ri ri i i η η i i

matrix for one period becomes:

 cos (k d ) jη sin (k d )  cos (k d ) jη sin (k d ) M = M M =  1 1 1 1 1  • 2 2 2 2 2  . (5.23) 1 2  η ()()   η ()()   j 1 sin k1d1 cos k1d1   j 2 sin k2d 2 cos k2d 2 

( + ) = − jKd ( ) = + We impose the periodicity constraint E z d e E z , where d d1 d 2 is the thickness of one period. The diagonal elements of the matrix M = M M are: 1 2

η m = cos ()()k d cos k d − 1 sin ()()k d sin k d , (5.24a) 11 1 1 2 2 η 1 1 2 2 2

η m = cos ()()k d cos k d − 2 sin ()()k d sin k d . (5.24b) 22 1 1 2 2 η 1 1 2 2 1

d1 d2

Incident Substrate . . . n0 n1 n2 n1 n2 ns

θθθ0=0

Boundary 1 2 …. q-1 q q+1

Figure 5.3: Periodic structure composed of 2 materials with refractive index n1 and n2.

From Eqs. (5.24a) and (5.24b), the trace of M is found to be

η η  Tr [][][]M = Tr M M = m + m = 2 cos ()()k d cos k d −  2 + 1 sin ()()k d sin k d . 1 2 11 22 1 1 2 2  η η  1 1 2 2  1 2 

Since 1 2Tr [M ] = cos (Kd ) [51], it follows that

1 1 η 2 +η 2  cos ()Kd = cos ()k d + k d + cos ()k d − k d −  1 2  sin ()()k d sin k d 2 1 1 2 2 2 1 1 2 2  2η η  1 1 2 2  1 2 

η 2 +η 2  = cos ()()()k d + k d + sin k d sin k d −  1 2 sin ()()k d sin k d 1 1 2 2 1 1 2 2  η η  1 1 2 2  2 1 2 

 η 2 +η 2  = cos ()()()k d + k d + sin k d sin k d 1− 1 2 . (5.25) 1 1 2 2 1 1 2 2  η η   2 1 2 

+ = π If k1d1 k 2d 2 m , where m is an integer,

 η 2 +η 2  cos ()()()()Kd = −1 m + sin k d sin mπ − k d 1− 1 2  1 1 1 1  η η   2 1 2 

η 2 +η 2  = ()()()−1 m + −1 m sin 2 k d  1 2 −1. (5.26) 1 1  2η η   1 2 

( ) ≥ ( ≠ ′π ) ′ It follows then that cos Kd 1. Now, if k1d1 m where m is an integer, then

cos (Kd ) >1, thus implying a spectral gap. This is the ordinary Bragg condition. From

Eq. ( 5.25) we notice that if we allow layer 1 or 2 to have a negative index of refraction,

+ = then we could have the case when k1d1 k2d 2 0 which also leads to Eq. (5.26 ). This

+ = condition is equivalent to n1d1 n2 d 2 0 , which leads to a new type of gaps called zero

< n > gap, which is only present in structures containing negative index materials. The main difference between the two gaps is that, the zero < n > gap is invariant with scaling,

i.e., the position of the gap remains unchanged if d1 , d 2 are both scaled by the same factor. Some of the advantages of the zero < n > gap are discussed in a later Section.

5.3.2 Finite Periodic Structures

We now give different examples of periodic structures containing NIMs in some cases.

Example 1: non dispersive case

We consider a system composed of alternating NIM and PIM layers as shown in Figure

5.3. Medium 1 has a refractive index of n1=1.5 and medium 2 has a refractive index of

n2=-3. The thickness of the PIM and the NIM layers are d1=0.1 m, d2=0.05 m with N=4

(8 layers). The transmittance computed using the TMM with MATLAB as in Section 5.2

is compared in Figure 5.4 with simulation using the FEM (using COMSOL). Note that,

+ = ≠ although k1d1 k 2 d 2 0 , we do not do not see the gap at all frequencies (i.e, T 0).

This is because the above condition is for infinite number of layers. As the number of layers is increased the results is equal to N → ∞(See Appendix E).

Figure 5.4: Transmittance of a periodic PIM/NIM slab with n1=1.5, n2=-3, d1=0.1 m, d2=0.05 m and N=4.

Example 2: (non dispersive)

We now consider an example similar to example 1 but the second layer with n2=+3. The

thickness of the layers are d1=0.1 m and d2=0.05 m with N=4 (8 layers). The results are

shown in Figure 5.5. In this case, no zero gap is expected. Figure 5.6 shows the

electric field inside the slab.

Figure 5.5: Transmittance of a periodic PIM/PIM slab with n1=1.5, n2=3, d1=0.1 m, d2=0.05 m and N=4.

Figure 5.6: Electric field for a periodic PIM/PIM slab with n1=1.5, n2=3, d1=0.1 m, d2=0.05 m and N=4.

Example 3: dispersive, nonlossy medium

= = = In this example we consider an alternating stack ( p ;2 d1 d 2 1cm ) of PIM (air) and

a dispersive NIM with the permittivity and permeability given by:

52 10 2 32 ε ()f =1+ + , µ ()f = 1+ , (5.27) r 9.0 2 − f 2 11 5. 2 − f 2 r .0 902 2 − f 2

where f is the frequency measured in GHz [52]. Figure 5.7 (a) shows the values of the permittivity and permeability as functions of frequency. Figure 5.7(b) shows the transmittance for normal incidence for different number of periods, namely N = (4,25, ∞ ).

As N increases it reaches the limit where N → ∞ computed through Eq. (5.22). The

zero < n > bandgap is between 2-3 GHz while the Bragg gap (where the stack is

PIM/PIM) is between 6.5-7.5 GHz. We note that other secondary Bragg bandgaps also exist, for instance, a very narrow bandgap around 1.38 GHz (where the stack is

PIM/NIM), and other around 9.8-10.5 GHz, 11-15 GHz (where the stack is PIM/PIM).

In our investigation, we restrict ourselves to the zero < n > and the first Bragg bandgap.

Figure 5.8 shows the photonic bandgaps for the periodic stack in terms of the frequency

and the incident angle for (a) TE and (b) TM modes. The black regions represent

transmittance below 0.001. As seen from Figure 5.8, the zero < n > band gap is relatively invariant w.r.t. the incident angle and the optical polarization.

Figure 5.8: Photonic band structures of the periodic PIM/NIM stack in terms of the frequency and incident angle θ. The black areas represent forbidden gaps (transmittance is less than 0.001).

Example 4: dispersive, lossy media

Let us reconsider the hypothetical Lorentz type metamaterial from Eq. (3.35) with both dispersive permittivity and permeability. We can assume that the permittivity and the permeability can be written as [57]:

ω 2 ω 2 ε ()ω = 1 + pe ,µ ()ω = 1 + pm , (5.28) r ω 2 − ω 2 + γ ω r ω 2 − ω 2 − γ ω e1 j e 1m i m

ω ω ω ω γ γ where pe , pm , e , m , e , m are constants measured in rad/s. Figure 5.9 (a) shows the

transmittance and Figure 5.9 (b) shows the reflectance, when ε and µ vary with frequency

as in Eq. (5.28), which agrees with the result found in [ 58 ].

(a) (b)

5.3.3 Anisotropic model

The study of birefringent stratified layered media is very important in a lot of applications, especially as narrow-band birefringent filters, polarizers, multistage electrooptic modulators, and in our case tunable NIMs. Thus, the most logical material that exhibits such phenomena is liquid crystals. We follow the method treated by the

following authors [59-61] to study the general theory of EM propagation in such media.

We note that the transmission spectrum is now expected to be a function of the

orientation of the optical axis of the different layers and polarization of incident wave.

For a monochromatic plane wave in homogeneous birefringent layered

anisotropic media, the EM radiation consists of four waves. Mode coupling takes place

at the different interfaces, resulting in waves with different polarizations due to the

anisotropy of the layers. Hence a 4x4 transfer matrix approach is needed in this case

[62]. The dielectric tensor in rectangular coordinates can be written as:

ε  1 0 0  ε =  ε  −1 A 0 2 0  A , (5.29)  0 0 ε   3 

ε ε ε where, 1 , 2 ,and 3 are the principal dielectric constants and A is the coordinate rotation matrix in 3D and given by [63]:

cos ψ cos φ − cos θ sin φ sin ψ − sin ψ cos φ − cos θ sin φ sin ψ sin θ sin φ  =  ψ φ + θ φ ψ − ψ φ + θ φ ψ − θ φ A cos cos cos sin sin sin sin cos cos cos sin cos  , (5.30)  sin θ sin ψ sin θ sin ψ cos θ  where, θ ,φ , and ψ are the Euler angles of orientations of the crystal axes with respect to

(xyz ) coordinate system. The z component γ of the propagation vector can be found by

× ( × )+ω 2 µε = solving the following momentum space wave equation k k E E ,0 or equivalently the determinant of the following matrix [62]

ω 2 µε − β 2 − γ 2 ω 2 µε +αβ ω 2 µε +αγ xx xy xz ω 2 µε +αβ ω 2 µε −α 2 − γ 2 ω 2 µε + βγ = yx yy yz 0 , ( 5.31) ω 2 µε +αγ ω 2 µε + βγ ω 2 µε −α 2 − β 2 zx zy zz

where, α ,β are the transverse components of the propagation vector. Since the whole birefringent layered medium is homogeneous in the xy plane α and β remains the same

throughout the layered medium. The above determinant in Eq. (5.31) gives a quartic

γ γ equation in . This leads into 4 roots σ =1,2 3,, 4 and hence 4 wave vectors: kσ =αxˆ + βyˆ +γ σ zˆ which lie in the same plane of incidence which remains the same through the whole layered medium. The polarization of these waves are given by

 (ω 2 µε − α 2 − γ 2 )(ω 2 µε − α 2 − β 2 )− (ω 2 µε + βγ )2   yy σ zz yz σ  = (ω 2 µε + βγ )(ω 2 µε + αγ )− (ω 2 µε + αβ )(ω 2 µε − α 2 − β 2 ) p Nσ  yz σ zx σ xy zz  , (5.32) (ω 2 µε + αβ )(ω 2 µε + βγ )− (ω 2 µε + αγ )(ω 2 µε − α 2 − γ 2 )  xy yz σ xz σ yy σ 

where, Nσ is a normalization factor such that p • p =1. The electric field can be written

4 = ( ) ( )[]()ω − − − ( )(− ) as: E ∑ Aσ n pσ n exp i t αx βy γσ n z zn , where, n is the layer number. σ =1

Imposing continuity on the transverse components of the electric and magnetic fields at

the interfaces we get

( − ) ( )  A1 n 1   A1 n      A ()n − 1 A ()n  2  = D −1 ()()()n −1 D n P n  2 , (5.33)  ()−   () A3 n 1 A3 n  ()−   () A4 n 1  A4 n 

where

[ γ ( ) ] exp i 1 n tn 0 0 0    0 exp []iγ ()n t 0 0 P()n =  2 n  , (5.34)  0 0 exp []iγ ()n t 0   3 n  0 0 0 exp []iγ ()n t  4 n  ( ) ( ) ( ) ( ) ˆ p.x 1 n ˆ p.x 2 n ˆ p.x 3 n ˆ p.x 4 n    ˆ q.y ()()()()n ˆ q.y n ˆ q.y n ˆ q.y n D()n =  1 2 3 4  , (5.35)  ()()()()  ˆ p.y 1 n ˆ p.y 2 n ˆ p.y 3 n ˆ p.y 4 n  ()()()()   ˆ q.x 1 n ˆ q.x 2 n ˆ q.x 3 n ˆ q.x 4 n 

c and, qσ ()n = kσ ()()n × pσ n (may not be unit vectors), t = z − z − n, =1,2,..., N.. If ω n n n 1

= −1 ( − ) ( ) ( ) we let Tn− ,1 n D n 1 D n P n , then the overall matrix equation for the whole layered

= = + = structure is: T T0,1T1,2 ...TN s, s, N 1 t, N+1 0. Assume the light is incident from the left

side of the layered structure, and let As A, p B, s B, p , and Cs C, p be the incident, reflected, and transmitted electric field amplitudes, respectively. The transfer matrix can be written as:

 As  M 11 M 12 M 13 M 14 Cs       B M M M M 0  s  =  21 22 23 24  . (5.36) A  M M M M C   p   31 32 33 34  p  B  M M M M 0  p   41 42 43 44   T

The transmission coefficients can be found from Eq. (5.36) as:

 C   C   C   C  t =  s  t, =  p  t, =  s  t, =  p  . ss  A  sp  A  ps  A  pp  A   s  A =0  s  A =0  p  =  p  = p p As 0 As 0

As an illustrative example of anisotropic layered media, we take

ε = ,6 ε = ,3 ε = ,2 ε = ,7 ε = ε = ,4 d = 41 7. nm ,d 125 nm , N =15 . We 1xx 2xx 1yy 2 yy 1zz 2zz 1 2 assume that the principal axes of the second layer are parallel to the ones of the first

layer, the polarization is conserved. Figure 5.10(a) shows the transmission spectrum for

the case where the output-polarized field is the same direction as the input and along the

y axis, and Figure 5.10 (b) shows the transmission spectrum for the case where the

output-polarized field is the same direction as the input but now along the x axis (See

Appendix F).

5.4 Random Model

Most of the previous work on metamaterials has focused on the study of the properties of periodic structures. Traditional random systems made of double-positive (DP) have also been extensively studied [64]. In this Section, we extend our formalism to the analysis of randomly placed PIM/NIM layers. The idea is to extend our approach to a homogeneous

mixture of PIM/NIM materials with a given ratio of PIM to NIM. We consider all types

of disorder and emphasizes the main difference between PIM/NIM mixtures and

PIM/PIM mixtures.

5.4.1 Randomness in the position of the layers

The number of interfaces in the system plays an important role for the wave propagation in PIM-NIM. In the periodical system discussed so far, A and B layers are alternatively

and periodically stacked. However, in our random model, A and B layers are randomly stacked. Thus, the total number of the interfaces between A and B layers is always less

than that in a periodic sample. From the simulation of a PIM/NIM random stack with

nA=1.5 and n B=-1 we can conclude that the last resonance peak (near 100% transmittance) with increasing wavelength is independent of the PIM-NIM ratio (see

Figure 5.11), and increases with layer thickness (see Figures 5.12); also we get less transmittance for equal ratio as shown in Figure 5.11.

Figure 5.11: Comparison of the average transmittance for 17 layers where the PIM ( nA= 1.5 ) and NIM ( nB= -1) with dA=d B=0.125 µm.

Figure 5.12: Transmittance for a 2 layer stack for different layer thickness dA=d B and for different λ, with nA=1.5 , nB=-1.

The localization length is an important parameter often used to characterize the

transmission through random layers. Approximately speaking, it is similar to the decay

length of the EM wave through the structure. Here we study the localization behavior in

a randomly perturbed stack. The localization length is found from the following formula

[64]:

ς = −L ln/ T (5.37)

For a periodic structure the decay length is given by

ς = − d L / ln( T) (5.38)

where L is the multilayered structure length. In other words, localization length or decay

length is smaller for lower values of the transmission, as expected.

We change randomly the order of PIM and NIM layers in the stack keeping the

ratio to be 1:1, since minimum transmission has been previously predicted for equal ratio

of PIM to NIM. Figure 5.13 shows the photonic band structure for different angles of

incidence of a random stack of 200 layers. The top figure is for TE polarization and the

bottom one is for TM polarization. We note here that the zero < n > bandgap does not appear for a small number of layers (<50). Also, we note that the Bragg gap disappears as expected when we are studying structures that have randomness in the position(s) of the layers. The localization length is computed as in Eq. 5.37. Figure 5.14 shows the localization length of a stack of 50 cells (100 layers). The denominator in Eq. 5.38 is

3 averaged over 10 realizations. Existence of the zero < n > bandgap between 2-3 Ghz is again clear from this figure. The reason for an apparent extra gap around 9-10 Ghz is under investigation.

Figure 5.14: Localization length as a function of frequency for a random (position) stack of 100 layers.

5.4.2 Perturbation in layer thickness

In this Section we introduce an example of perturbation to the thickness of each layer.

We assume that the material is composed of successive layers of PIM and dispersive

NIM material. Further, we assume that a PIM/NIM sandwich or “cell” has constant thickness ( 2d ) but there is randomness in the location of the boundary between the PIM

= + − = and NIM. The composite transfer matrix of this cell is Pr M r M r r 1,..., p 2/ where

 π π   2 ()± ε ~  η  2 ()± ε ~   cos  nr d 1 Z r  j r sin  nr d 1 Z r  ±  λ   λ  M =   . (5.39) r  j  2π ~   2π ~   sin  n d()1± ε Z  cos  n d()1± ε Z  η λ r r λ r r   r     

~ In Eq. (5.39) εdZr is the additional random thickness for the first (+) and second (-)

layer comprising each “cell” respectively, while maintaining a fixed overall cell ~ thickness; and Zr are independent normalized (to d ) identically distributed random

~ ~ 1 ~ ~ ~ variables with probability density functions f ~ (Z )with Z r = ∫ Z f ~ ()Z dZ = ,0 and Z r r r Z r r r −1

~ 1 ~ ~ 2 ~ ~ var []Z = ∫ (Z − Z ) f ~ ()Z dZ = .1 ε is a (dimensionless) perturbation parameter. r r r Z r r r −1

The transfer matrix of a “macrocell” which we define as a collection of p 2/ cells can be

p 2/ = written as T ∏ Pr . Note that several macrocells may comprise the entire structure, r=1 i.e., Np = q , where N is an integer. Since the random variables are independent, let us define the average transfer matrix of each cell as

 P P  = =  1,1 2,1  Pr P  . (5.40) P 1,2 P 2,2 

 p 2/   p 2/  [] = = = []p 2/ = Λp 2/ + Λp 2/ Λ Then Tr T Tr ∏ Pr  Tr ∏ Pr  Tr P 1 2 , where 2,1 are the  r=1   r=1 

eigenvalues of P . Figure 5.15(a) shows Tr [T] for the case of d=d1=d2=1cm, PIM is air,

NIM index of refraction is from Eq. (5.27), and for increasing values of perturbation

1 1 1 parameter ε = ,0 , , and with p 2/ = 10 . As mentioned earlier, for values of the 20 2 3 trace greater than 2, one expects a bandgap. The mean in Eq. (5.37) is taken as the average over 10 3 realizations. We notice that if the perturbation parameter is small (

1 ε < ) on average there is no major shift of the bandgaps and that signifies the 2 robustness of the multilayered structure due to imprecision in fabrication. The inset in

Figure 5.15 (a) shows the standard deviation σ (Tr [T]) . The standard deviation is

contained to values less than unity, signifying relatively small changes in the average

bandgap. Figure 5.15(b) shows the difference between the localization length in the random case and the decay length in the periodic case of a stack of 20 cells. We notice that in PIM/PIM bandgap region (around 6.5-7.5 GHz), the localization length is much larger than the decay length. In the PIM/NIM (zero < n > ) region (around 6.5-7.5 GHz), the localization length is only slightly larger than the decay length; this is because in mixed structures that contain PIM and NIM we have an average cancelation of phase between the PIM and NIM layers in both the periodic and random cases. This result agrees with those obtained by Ref. [64] for a non-dispersive stack of single negative materials ( ε or µ <0).

For completeness, we enumerate the methodology to study the effect of disorder on transmission at a certain frequency in PIM/PIM and PIM/NIM structures through an illustrative example. For instance, Figure 5.16 shows the average transmission for a

“transparent” mode (high transmittance) as a function of the disorder strength ε for a

PIM/PIM and PIM/NIM structures composed of 50 layers each. For the PIM/PIM

structure the indices of refraction are 1 and 1.53 respectively, and for the PIM/NIM

structure the indices of refraction are 1 and -1.53, respectively. We note that the mixed

structure (PIM/NIM) is more sensitive to disorder than the ordinary structure (PIM/PIM)

in that the average transmittance of the former is smaller than that of the latter. So, in the

development of devices, such as bandpass filters that uses the properties of PIM/NIM and

light localization, the strength of disorder should be carefully selected. The analysis can

be readily extended to other refractive index pairs, frequencies and values of the

transmission.

5.4.3 Perturbation in refractive index

ξ Now we assume that the refractive index of NIM have random fluctuations r , where

ξ the r ’s are independent random variables distributed uniformly on [-0.4,0.4]. The refractive index of NIM layers becomes:

 2 2  2  () =  + 5 + 10  + 3  + ξ n f 1 2 2 2 2 1 2 2  r (5.41)  9.0 − f 11 5. − f  .0 902 − f 

The randomness in the refractive index can be attributed to possible frequency

independent randomness in the composition of the material [65]. The localization

length is computed as in Eq. 5.37(a). Figure 5.17 shows the localization length of a stack

of 100 cells (200 layers). The denominator in Eq. 5.37(a) is averaged over 10 3 realizations. We notice that randomness in dispersive left handed materials does not introduce new localized states (outside the bandgap) as in conventional random positive index layered structures. This is apparent from Figure 5.17 where the left edge of the zero < n > bandgap is unchanged, regardless of whether the stack is periodic or random.

Hence, dispersive left handed materials do not change Anderson localization in 1D disordered structures, which agrees with the result shown in [65] for non-dispersive left handed materials. Figure 5.18 shows the photonic band structure for the disordered stack.

We can conclude that the omnidirectional gap is not affected by disorder in the refractive index. So the structure is robust against fabrication irregularities and hence can be used as omnidirectional reflectors. Finally, we note that this perturbation in index of refraction discussed in this Section is different from the perturbation discussed in the previous

Section due to the fact that in this Section the optical path length of PIM layers are kept

constant where as in the previous section both layers’ optical path lengths change. Other types of randomness in refractive index can be analyzed in a similar way.

Figure 5.17: Localization length as a function of frequency a random stack of 200 layers.

1.57

0.78

1.57 TM

Incidence angle (radians) angle Incidence 0.78

0 2 3 4 5 6 7 8 9 10 Frequency (GHz)

Figure 5.18: Photonic band structures of a disordered PIM/NIM stack in terms of the frequency and incident angle θ. The black area represent forbidden gap (Transmittance is less than 0.001).

5.5 Applications: Sensors (periodic structures)

In this Section, one possible application of the zero < n > gap is proposed [66]. There are many performance parameters for a sensor. These include sensitivity, signal to noise ratio

(SNR) and operating range. These parameters should be as high as possible for a sensor.

Detection limit is the minimum variation in the refractive index or thickness of the layer which can be detected by the sensor. Conventionally, the sensitivity ( Sn) of sensor such as a surface plasmon resonance (SPR) sensor, is defined as the rate of shift in the frequency ( δf ) due to a variation in the refractive index ( δn ) of the layers [67]. For our

proposed sensor we define a change in the spectral width of the bandgap δw at a reference transmittance level corresponding to a variation δd in the thickness of the layers respectively as another figure of merit:

δω S = . (5.42) d δd

As an example we assume a multilayer structure composed of 7 periods of NIM/PIM.

The index of refraction of PIM is taken to be 1.5. The permittivity and permeability of

the NIM layer are as in Section 5.3.2 example 4 [57]. The real parts of εr and µr are negative for frequencies f<18.5 GHz. Assume that the thickness of each cell composed

of a NIM layer and a PIM layer is d= d PIM + d NIM = 0.01m. Figure 5.9 shows the

transmittance and reflectance for a multilayer structure composed of 14 layers (7 cells)

where d PIM = d NIM = 0.005m. Normal incidence is assumed in all simulations in this

Section. The zero- < n > bandgap is approximately between 12.25-15.5 GHz, while a neighboring Bragg bandgap is approximately between 8-10 Ghz. Figure 5.19 shows the variation of the transmittance with frequency for three values of the “duty cycle” of the

NIM layer (D=d NIM /d). For different duty cycles, the zero- < n > bandgap occurs around different center frequencies with all other parameters kept unchanged. For D=0.5, the center frequency is 13.75 GHz. For smaller duty cycle (D=0.3), the center frequency for the zero- < n > bandgap is lower, viz., at 10.2 GHz; while for a higher duty cycle

(D=0.7), the center frequency is 16.32 GHz. This is because for lower (higher) frequency, the refractive index of the NIM material, calculated from the permittivity and permeability relations, is more (less) negative; hence, a smaller (larger) layer thickness dNIM is needed to attain zero- < n > . We remark that other bandgaps seen in Figure 5.19 for D=0.3, 0.7 correspond to Bragg gaps. This shows that for a given spectral range of the source and the detector, a suitable duty cycle can be selected.

Figure 5.19: Variation of transmittance with wavelength for three different values of NIM layer thickness fraction. All other parameters are as in Figure 5.9.

Figure 5.20 shows the transmittance for different scale changes in the cell thickness. As discussed earlier in Section 5.3.1 the center frequency of the Bragg gap is shifted with the thickness change but the center frequency for the zero < n > gap remains

invariant with scaling. However, as clear from the figure, the spectral width of the zero

< n > gap changes with the scaling. Interestingly, for the chosen dispersion relation, the

width of the Bragg gap does not appreciably change with scaling. A decrease (increase)

in the cell thickness is seen to broaden (narrow) the width of the zero < n > gap at a reference transmittance value picked for convenience below the side lobes.

Figure 5.20: Transmittance through the structure when the cell thickness d is scaled by - 10% and +10%. All other parameters are as in Figure 5.9.

Figure 5.21 shows the change in the spectral width of the transmittance at a reference value T=0.2 with the scaling in length of the multilayer structure from – 10% to + 10% for different values of refractive index for the (nondispersive) PIM layer. We notice that a

higher refractive index for the PIM layer gives us a higher δω , and thus a higher

sensitivity Sd .

A common issue with sensors is their cross-sensitivity [68]. This pertains to change in sensor response from more than one stimulus (viz., temperature and pressure/strain for fiber-optic sensors). The use of both the Bragg gap and the zero < n > gap could alleviate this problem. Figure 5.22 shows the effect of change in the refractive index and/or the thickness of the multilayer on the center frequencies (and bandwidths) of the Bragg gap and the zero < n > gap. For a duty cycle of 50%, a small change in the refractive index of the PIM relative to that of the NIM (possibly occurring from

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temperature change) shifts the center frequencies of both zero < n > gap (which only

appears in PIM/NIM structures) and the Bragg gap; but a change in the thickness of the

cell (possibly occurring from pressure/strain) only shifts the center frequency of the

Bragg gap. Note that any thickness change due to temperature would in general change

the center frequencies of both gaps. Thus, change in cell thickness due to pressure only

can be uniquely determined. Therefore, by employing this unique feature of the two

types of bandgaps, it should be possible to segregate the effect of temperature change

from the change in pressure/strain, hence eliminating cross-sensitivity. Measurement of

center frequencies and/or bandwidth can be effectively performed by using spectrum

analyzers or network analyzers, depending on the frequency of operation [69].

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5.6 LC and NIM defect layers in periodic structures

In this section we investigate the tunability of the omnidirectional reflection band (ORB) in 1-D PC consisting of alternating dielectric and NLC layers by an external electric field as shown in Figure 5.23 below [70-72].

x Dielectric LC z

. . . ne, . . . Air=1 n Air=1 E θθθ n0 p-pol k E H s-pol H d1 d2

Figure 5.23: Schematic of a homogeneous periodic anisotropic stack on top of a substrate .

The transmission characteristics of this 1D PC structure is computed using Ref. [73] which takes into account the variation of the dielectric constant of each LC layer in the longitudinal z direction. This transmission depends on the dielectric function, incident

wave vector, and the orientation of optical axis. The dielectric tensor for a NLC is given

by [46]

ε 0 ε  n2 + ∆ε cos 2 α 0 ∆ε cos α sin α  xx xz  o  ε =  ε  = 2  0 yy 0   0 no 0  , ( 5.43) ε ε   ∆ε α α 2 + ∆ε 2 α   xz 0 zz   cos sin 0 no cos 

∆ε = 2 − 2 where, OPT ne no is the optical dielectric anisotropy, ne and no are the extraordinary

and ordinary refractive indices, respectively. The tilt angle α is the angle between the principal axis of nematic liquid crystal molecules as explained in Chapter 4.

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For this example we use NLC 5CB with the frank elastic constant

= = ∆ε = = = = K1 4.7 pN , K3 10 2. pN , STA 16 , ne 75.1 , n0 53.1 , Vc 1.18 V (critical voltage) and n=2.35 for the ZnS dielectric d = d = a 2/ . (20 periods) As we can see for p- 1 2 polarized light in Figure 5.24(a), we notice an ORB when V=2.85 Vc. The ORB is not

present when V= Vc as shown in Figure 5.24 (b). The frequencies for the edges of the gap were taken when the transmission was less than 0.01.

(a)

(b)

Figure 5.24 : Dependence of edge frequencies of the first PBG of p – polarized waves on the incident angle (a) V/V c=2.853 (b) and V/V c=1.

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Now we consider a 1D PC stack with a LC layer acting as a defect shown in

Figure 5.25. We consider 12 periods of alternating dielectric layers with refractive indices n1=1.34, n2=2.5, respectively, and a thickness d1=d2=a/2 ( a~1 µm) and the

thickness of the LC defect layer is 11 a/6. Therefore the defect mode vary with the applied voltage. Figure 5.26 shows the transmission spectra of the normally incident s- and p- polarized waves when the voltage is V=1.5 Vc. The tuning of the bandgap is useful in the implementation of tunable optical waveguides and filters.

d1 d2 LC d3

n Air e, Air n0

n1 n2

Figure 5.25: Schematic of a LC defect stack.

Figure 5.26: Transmission characteristics of a 1D PC with a LC as a defect layer of width 11 a/6 . Applied voltage is V=1.5 Vc, d1=d2=a/2, n1=1.34, n2=2.5 as in [70]. 104

Now we propose a structure with more than 1 defect layer. Such a structure gives us more tunability without the need of a tunable voltage source because each defect layer is independently controlled by a fixed amplitude bias voltage. As an example we consider a structure consisting of 3 defect LC layers as shown in Figure 5.27. The

-5 thickness of each defect layer is d3=10 m. Figure 5.28 shows the different results for all possible combinations where one, two, or all the bias voltages on the LC defect layers are on.

d1 d2 d LC LC 3 LC

n n n Air e, e, e, Air n0 n0 n0

n1 n2

Figure 5.27: Schematic of a multi LC defect stack.

Figure 5.28: Transmission characteristics of the multi LC defect stack for different combinations of bias voltage.

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As a final example, we consider the effect of an NIM defect layer for different value of

λ the index of refraction of the defect layer n, d3= 0 2/ , n1=1.38, n2=2.6, respectively, and

λ a thickness n1d1= n2d2= 0 4/ in a 10 period stack. We notice that if the defect layer has a negative index we get a broader bandgap than in the PIM case as shown in Figure

5.29(a). For a larger value of n we get a sharper bandpass filter and a broad bandgap wich can be used in optical filter in the NIM case as shown in Figure 5.29(b). In Figure

5.29(c) we notice that in the PIM case we don’t get a noticeable change in the bandpass filter but we get broadening of the bandgap for smaller values of n.

(a) (b)

(c) Figure 5.29: Effect on the transmittance of a NIM defect layer on an optical bandpass filter made of a periodic stack and (a) PIM/NIM defect, (b) NIM defect with varying n, (c) PIM defect with varying n.

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In the future the defect layer could be made from nanoparticles dispersed LC cells

(NDLCC) for more tunability. NDLCC can be used as a NIM as it is shown in the next

Chapter.

5.7 Conclusion

We have investigated the transmission properties of EM/optical wave through periodic and random stacks of PIMs and NIMs using a matrix-based multilayered approach.

Randomness in layer position, thickness, refractive index has been studied. This randomness mimic a homogeneous material composed of PIM and NIM materials. The limit of the transmittance for mixed (PIM/NIM) structures when the number of layers tends to infinity has been derived. Also we have considered periodic structures of different types of materials (i.e dispersive, Nondispersive, lossy…). We also discussed how the zero gap could be used as a sensor. Finally, we studied the effect of introducing defect slab into a periodic structure. Two cases were studied with LC or

NIM as the defect layer.

In the next Chapter we derive the theory, describe the fabrication techniques, and the testing methods of nanoparticles dispersed LC as a candidate for an NIM.

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CHAPTER 6

NANOPARTICLE-DISPERSED LCCs

6.1 INTRODUCTION

Metamaterials with tunable properties are of great importance due to potential applications in “super-resolution lensing” and “plasmonic sensors”. In Section 6.2 we briefly discuss the Maxwell-Garnett (MG) theory and the extended Maxwell-Garnett

(EMG) theory. EMG theory is the Mie scattering (MS) theory based extension of the

MG theory. In Section 6.3 we study the feasibility of the fabrication of a metamaterial using binary nanoparticle-dispersed liquid crystal cell (NDLCC). Depending on the angle between the director axis of the LCC and the incident beam, and the types, radii, and volume filling fractions of the nanoparticles, a negative index of refraction cell can be obtained in a certain range of frequencies. The effective index of refraction of the

NDLCC is calculated using EMG. The scattering, extinction, and absorption of such a

NDLCC are found. Also, the influence of the various parameters to obtain such a NIM

has been investigated. In Section 6.4 we examine the analytic basis of the plasmonic

resonance behavior of a nanodispersed (Ge, Cu, Ag, Au depending on the range of

frequencies) LCC under (i) externally applied electrostatic field, and (ii) externally

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applied pressure/temperature. In Section 6.5, we discuss the fabrication and testing of a

NDLCC. Section 6.6 concludes the Chapter.

6.2 MG, MS and EMG Theories

6.2.1 MG theory (static limit)

ε The relative effective dielectric constant eff,r of small spherical particles of radii r1 , r2 ...,

ε embedded in a host of relative dielectric constant rh is given by the Claussius-Mosotti equation [74]

ε − ε f r,eff rh = i α , (6.1) ε + ε ∑ 3 i r,eff 2 rh i ri

4π where α is the particle dipole polarizability, f = N r 3 is the volume fraction of the i i 3 i i

embedded particles, and Ni is the particle number density. Let us define the unitless

ωr quantities x = ε µ i , where µ is the relative permeability of the host, and r hi rh rh c rh i

ωr are the radii for polaritonic and plasmonic nanoparticles, and x = ε µ i , where ε i ri ri c ri

µ , ri are the relative permittivities and permeabilities of the nano-particles spheres.

According to MG theory, if xhi and xi are <<1 ( static limit case ), the electrostatic value of

the polarizability is defined as [75,76]

ε −ε α = ri rh r3 , (6.2) i ε + ε i ri 2 rh

ε th where ri is the relative dielectric constant of the i type material. For low volume

fraction fi, the MG formula is written as

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ε − ε P ε − ε r,eff rh = f ri rh , (6.3) ε + ε ∑ i ε + ε r,eff 2 rh i=1 ri 2 rh which is a generalization of Eq. (3.12).

Explicitly, we can rewrite Eq. (6.3) as:

 P ε − ε  + ri rh 1 ∑ 2 fi   = ε + 2ε  ε = ε  i 1 ri rh  . (6.4) r,eff rh P ε − ε  1− f ri rh   ∑ i ε + ε   i=1 ri 2 rh 

For high volume fraction fi we use the Bruggeman approximation, known as the effective

ε ≈ ε medium approximation (EMA) where we assume eff,r rh . With this, Eq. (6.3) becomes

P ε − ε f ri r,eff = .0 (6.5) ∑ i ε + ε i=1 ri 2 r,eff

After some algebra, Eq. (6.5) can be written as

  P  P  ()ε + 2ε f = .0 (6.6) ∑ ∏ rk r,eff i  i=1 k=1  k≠i 

6.2.2 EMG theory (Quasi-static limit)

Now if we assume that xhi <<1, but not necessary xi <<1 (quasi-static limit), then the

α static polarization factor i in Eq. (6.2) does not reflect the polarization property of single

particles and we have to rely on the EMG formulas or Mie theory based extension of

MG. Recall that the Mie results for the extinction, scattering and absorption cross

sections are [77,78]

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∞ = 2 ()()+ + Qext ,i ∑ 2m 1 Re ami bmi , (6.7) xhi m=1

∞ = 2 ()+ ( 2 + 2 ) Qsca ,i ∑ 2m 1 ami bmi , (6.8) xhi m=1

= − Qabs ,i Qext ,i Qsca ,i . (6.9)

where ami and bmi are the Mie coefficients which correspond to the electric-dipole and magnetic dipole components of the scattering matrix respectively, and are given by ′ ′ j ()()x [ψ x ] ε − j ()()x [ψ x ] ε ()ω = m i m hi ri m hi m i rh ami ′ ′ , (6.10) ()()[]ξ ε − ()1 ()()[]ψ ε jm xi m xhi ri hm xhi m xi rh

′ ′ j ()()x [ψ x ] µ − j ()()x [ψ x ] µ ()ω = m i m hi ri m hi m i rh bmi ′ ′ , (6.11) ()()[]ξ µ − ()1 ()()[]ψ µ jm xi m xhi ri hm xhi m xi rh

π π th where, j ()z = J + ()z , y ()z = Y + ()z , ( Jm, Ym) are the m order spherical m 2z m 5.0 m 2z m 5.0

(1) ( ) = ( ) + ( ) Bessel functions of the first and second types respectively, and hm z jm z jy m z , is the mth order spherical Hankel function of the first type. Also the functions

πz πz th ψ ()()z = zj z = J + ()z , χ ()()z = −zy z = − Y + ()z , are called the m order m m 2 m 5.0 m m 2 m 5.0

Riccati-Bessel functions of first and second type respectively, and

π ()1 z ()1 th ξ ()z = zh ()z = H + ()()()z =ψ z − jχ z , is the m order Riccati-Hankel function m m 2 m 5.0 m m of first type. If we only need the 1 st order the above functions can be found to be

sin z cos z j ()z = − , (6.12a) 1 z 2 z

cos z sin z y ()z = − − , (6.12b) 1 z 2 z

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()  sin z cos z   cos z sin z  h 1 ()z =  −  − j +  , (6.12c) 1  z 2 z   z 2 z 

sin z ψ ()z = − cos z , (6.12d) 1 z

cos z χ ()z = + sin z , (6.12e) 1 z

 sin z   cos z  ξ ()z =  − cos z − j + sin z , (6.12f) 1  z   z 

cos z sin z ψ ′()z = − + sin z , (6.12g) 1 z z 2

sin z cos z χ ′()z = − − + cos z , (6.12h) 1 z z 2

 cos z sin z   sin z cos z  ξ′()z =  − + sin z + j + − cos z . (6.12i) 1  z z 2   z z 2 

µ ≈ µ ≈ th Note that in our calculation we assume h i 1. Also, for the m order derivatives of the Reccati-Bessel and Riccati-Hankel functions we derived the following Equations:

π π ′ 1 1 z ψ ()z = J + ()z + []J − ()()z − J + z , (6.13) m 2 2z m 5.0 2 2 m 5.0 m 5.1

 π π  ξ′ ()()=ψ ′ + 1 () + 1 z []()()− m z m z j Ym+ 5.0 z Ym− 5.0 z Ym+ 5.1 z  . (6.14) 2 2z 2 2 

α So in Eq. (6.2) we replace i from the electric dipole polarizability that follows from the

Mie theory, which is defined as [75,76]:

3r 3 α = i i j 3 a1i , (6.15) 2xhi

For single nanospheres, Eq. (6.2) becomes:

112

ε − ε 3 f r,eff rh = j i a . (6.16) ε + ε 3 1i r,eff 2 rh 2 xhi

After some straightforward algebra we get

 f  + i  1 3 j a1i   x3  ε = ε hi , (6.17) r,eff rh  3 f  1− j i a  2 x3 1i  hi  which is the EMG formula. In a similar way we can find the effective permeability to be

 f  + i  1 3 j b1i   x3  µ = µ hi , (6.18) r,eff rh  3 f  1− j i b  2 x3 1i  hi 

6.3 Tunable Negative Index Binary NDLCC

A tunable NIM leads to new optical phenomena such as negative refraction [3], superlensing [2-7 ], and can lead to numerous possible applications such as sensors

[3,66], which we study in Section 6.4. Using a tunable material to construct such structure such as LC was reported in literature [79]. A structure constructed with silver film and LC interface shows a shift in the surface plasmon resonance frequency using the change in refractive index due to a voltage [80]. Another design has used nanostrip pairs of silver separated by a LC layer [81].

Nematic LCs are anisotropic materials, and the physical properties of the system vary with the average alignment with the director. LCs have a large optical anisotropy and are sensitive to temperature and external electromagnetic field which make them a good candidate for dynamic metamaterials [80-82].

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In this Section we explore and characterize novel materials such as NDLCC for use in low loss NIM applications such as a metamaterial lens in optical or near optical regime. The most important advantages of our approach are: ease of obtaining/designing

NIMs in optical range, tunability depending on the electric, magnetic, or optical fields applied to the LCC, ease of fabrication at low cost, and finally, the ability to incorporate a large variety of nanoparticles due to the fluid nature of LCs.

ε µ The eff,r and eff,r of the NDLCCs can be found according to the EMG or the quasi-static limit which is like the regular MG theory with the incorporation of MS theory as explained in the previous Section [39,83,84]. Unlike the static limit, the effective medium parameters in the quasi-static limit are dispersive even for a medium composed of non-dispersive components. The EMG method can be used to calculate all the optical

= ε µ properties of the composite medium. In particular, the effective index neff eff,r eff,r can be used in the Snell’s law to find refraction of the incident beam. Note that the EMG effective constants obtained are complex even in the case of non-absorbing spheres embedded in a non-absorbing host. In this case, the imaginary part is due to the extinction of the propagation beam due to scattering.

For a binary composite comprising two different kinds of nano-spheres the effective-medium parameters can be obtained from the condition that the average extinction of random unit cells compared with that of the surrounding medium is zero

[39] and the effective permittivity and permeability for m=1 are written as:

2 ε − ε + 3 j (2x3 )a f (2ε + ε ) C rh r,eff hi 1i 12 rh r,eff = 0 , (6.19) ∑ i ε + ε + ()3 ()ε − ε i=1 rh 2 r,eff 3 j xhi a1i f12 rh r,eff

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2 µ − µ + 3 j (2x3 )b f (2µ + µ ) C rh r,eff hi 1i 12 rh r,eff = 0 , (6.20) ∑ i µ + µ + ()3 ()µ − µ i=1 rh 2 r,eff 3 j xhi b1i f12 rh r,eff

= + where f12 f1 f 2 is the total filling factor of the spheres, C1 and C2=1-C1 are the

concentrations of the spheres types 1 and 2, respectively. Eqs. (6.19) and (6.20) can be

simplified to minimize the computation time into two quadratic equations:

ε 2 + ε + = p1 eff p2 eff p3 ,0 (6.21a)

µ 2 + µ + = q1 eff q2 eff q3 ,0 (6.21b)

where

= − ( + )( + )( + ) p1 2 C1 C 2 1 A1 1 B1 , (6.22a)

C [2(1− 2A )(1+ B ) − (1+ A )(1− 2B )]  p = ε  1 1 1 1 1  , (6.22b) 2 h + [](− )(+ )− (+ )(− )  C2 2 1 2B1 1 A1 1 B1 1 2A1 

= ε 2 ( + )( − )( − ) p3 h C1 C2 1 2A1 1 2B1 , (6.22c)

= − ( + )( + )( + ) q1 2 C1 C 2 1 A2 1 B2 , (6.22d)

C [2(1− 2A )(1+ B ) − (1+ A )(1− 2B )]  q = µ  1 2 2 2 2  , (6.22e) 2 h + [](− )(+ )− (+ )(− )  C2 2 1 2B2 1 A2 1 B2 1 2A2 

= µ 2 ( + )( − )( − ) q3 h C1 C2 1 2A2 1 2B2 , (6.22f)

= − 3 j = − 3 j = − 3 j = − 3 j and A1 3 a11 f12 , B1 3 a12 f12 , A2 3 b11 f12 ,b2 3 b12 f12 . 2xh1 2xh2 2xh1 2xh2

The filling factors of the nanoparticles are obtained from their lattice structures.

For an fcc crystal (4 atoms/unit cell) with period a consisting of spheres with radius r = a0/2, where a0 is the first neighbours distance. The period, or the lattice constant in the following, is the length of the side of the conventional unit cell of a cubic lattice, so

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= = a0 a 2 2 and r a 2 4. In a similar way, for a diamond crystal (8 atoms/unit cell),

r = a 3 8. Hence, for a maximum packing the volume filling fraction becomes

Volume of each sphere × # of atoms in unit cell f = Volume of a cube

3  4 3πr 3 × 4 4 3π ()a 2 4/ × 4  = = 74 %: fcc lattice , (6.23) =  a 3 a3  3 4 3πr 3 × 8 4 3π ()a 3 8/ × 8 = = 34 % : diamond lattice .  a3 a 3

ε = ε′ − ε′′ Materials are characterized by their relative permittivity ri ri j ri and

µ = µ ′ − µ ′′ relative permeability ri ri j ri . Consider the case of binary dispersed nanoparticles

ε ε µ µ scatterers of electric permittivity r1, r ,2 and permeability r1 , r 2 , respectively in a host

ε µ medium of electric permittivity rh and permeability rh as shown in Figure 6.1. In the

NDLCs, we have a medium with three different regions: the LC host material (region h), the plasmonic nanoparticle material (region 1) and the polaritonic nanoparticles (region

2). Since both materials are nonmagnetic, their relative permeabilities are equal to 1.

Plasmonic Polaritonic r1 r2 ε µ ε µ ( r2 , r2) ( r1, r1)

Host ( εrh ,µrh ) µr1 ≅µr2 ≅ µrh

Figure 6.1: Schematic of the metamaterials comprising binary nanoparticles randomly distributed in aligned nematic LC.

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Good candidates of polaritonic nanoparticles materials are: LiTaO 3 in the far infrared regime (FIR), viz., 4.25x10 12 Hz [39], TlBr, TlCl, SiC in the mid infrared regime

12 12 (MIR) viz., 50x10 Hz [85], and Cu 2O, CuCl in the visible regime, viz., 520x10 Hz,

775x10 12 Hz respectively [86]. Their permittivities vary according to Drude-Lorentz

model defined in Chapter 3 and rewritten here as

 ω 2 − ω 2  ε = ε ()∞ 1+ L T  , (far infrared) (6.24) r1 r  ω 2 − ω 2 + ωγ   T j 1 

γ ε = ε ()∞ + A 1 , (visible) (6.25) r1 r ω −ω + γ 0 j 1

ω =ω ε ( ) ε (∞) ε ( ) where L T r 0 r according to the Sachs-Teller relation, r 0 is the static

ε (∞) ω ω dielectric constant , r is the high frequency limit of the permittivity, T and L are

the transverse and longitudinal optical phonon frequencies γ1 is the damping coefficient,

ω the constant A is proportional to the exciton oscillation strength, 0 is the exciton- polariton resonance frequency (CuCl possesses a Z3 exciton line at 386 .93 nm) [86].

A good candidate for the plasmonic nanoparticles materials are: Ge, Ag, Au, and

Cu, where their permittivities vary according to Drude model:

 ω 2  ε = ε ()0 1− p , (6.26) r2 r  ω 2 − ωγ   j 2  where, ωp is the plasma frequency which is proportional to the square root of the impurity density of the semiconductor, and γ2 is the damping factor defined above [87].

As shown in Figure 6.2(a), if the incident light on the LCC is a linearly polarized

light as an extraordinary wave, the permittivity is given by [46]

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ε ε ε = e o , (6.27) rh ε 2 θ +ε 2 θ e cos o sin

ε ε where, e, and o are the permittivities for light polarized parallel and perpendicular to

the director axis nˆ respectively, and θ is the angle between the director axis and the optical wave vector k of the incident wave. Note that for a biased LCC (shown in Figure

6.2(b)) θ varies according to the profile in Figures 6.2 (c) and (d), and the permittivity of

the LC host is not constant throughout the cell. The angle θ is found as explained in

Chapter 4.

(a) (b) (c) (d)

6.4 Numerical results

Different types of nanoparticles are chosen in order to give negative refractive index in the desired operating wavelength range. In the infrared range LiTaO 3 and Ge can be used as polaritonic and plasmonic nanoparticles, respectively [39]. As an example, in the far

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ε ′ infrared region of the spectrum, Figures 6.3(a) and (b) show the real ( 2,1r ) and Figures

ε ′′ 6.3(c) and (d) show the imaginary ( 2,1r ) parts of the relative permittivity for the case of

LiTaO 3 and Ge nanoparticles, respectively computed using Eq. (6.24) and Eq. (6.26), for

ε ( ) = ε (∞) = ω = ω = π × × 12 the following parameters, r 0 15 ,8. r 13 ,4. p T 2 25.4 10 rad / s,

γ ≈ ω ω = ×π × × 12 2,1 p /1000 ., L 2 46.7 10 rad / s . Figure 6.4 shows the scattering, absorption, and extinction coefficients as found from Eqs (6.7), (6.8) and (6.9). Figure

6.4 (a) shows the approximation using the first term ( m=1) only and Figure 6.4 (b) uses

the first 10 terms ( m=1to 10). We notice that difference is small. So, for simplicity, in

finding the effective medium, we use only the first term.

(a) (b)

ε ′ ε′′ Figure 6.3: (a, b) 2,1r and (c, d) 2,1r parts of the relative permittivity for the case of LiTaO 3 and Ge nanoparticles.

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(a) (b)

Figure 6.4: Scattering, extinction, and absorption coefficients for (LiTaO 3, Ge), (a) for m=1 and (b) for m=1 to 10 terms.

′ ′′ Figure 6.5(a) shows the real ( neff ) and imaginary ( neff ) parts of the effective index of

= µ = µ refraction computed from Eqs. (6.21) and (6.22) where, r1 8.2 m, r2 25.2 m,

µ = µ = µ = ε = ε = = = = = − 1r r2 rh 1, ⊥ 96.1 , || ,4 f1 44.0 f, 2 34.0 , C1 56.0 C, 2 1 C1 . The angle between the incident field and the director axis in the middle of the cell is θ=0 .

′ θ Figure 6.5(b) shows neff profile in the LCC due to the electric field (bias voltage) where is not constant throughout the cell where λ = 78 .466 µm . Figures 6.6 (a) and (b), show

′ ′′ neff and neff parts of the refractive index as a function of wavelength and of the angle of incidence θ. We notice that the maximum negative index of refraction occurs at θ=0.

Also, we noticed that increasing the filling factor increases the negative index of refraction.

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(a)

(b)

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(a)

(b) ′ ′′ Figure 6.6: (a) neff (b) neff versus wavelength and angle of incidence for the case of

LiTaO 3 and Ge nanoparticles.

For the optical region of the spectrum we use Au, Cu, or Ag instead of Ge as

plasmonic nanoparticles, since their plasma frequency is in the visible region, and we use

Cu 2O, or CuCl instead of LiTaO 3 as for the polaritonic nanoparticles. The polaritonic material chosen for this example is CuCl due to its large exciton oscillation strength A

ε′ parameter. Figures 6.7(a) and (b) show the real ( r 2,1 ) and Figures 6.7(c) and (d) show

ε′′ the imaginary ( r 2,1 ) parts of the relative permittivity for the case of CuCl and Ag

nanoparticles, respectively computed using Eq. (6.24) and Eq. (6.25), for the following

parameters, A = 632 , ω = 2π × 18.2 ×10 15 rad ,s/ ω = 2π × .0 775 ×10 15 rad ,s/ p 0

ε (∞) = ε ( ) = γ = π × × 10 γ = π × × 12 r 59.5 , r 0 1 , 1 2 2.1 10 rad s/ , 2 2 35.4 10 rad s/ . Figure

6.7 shows the scattering, absorption, and extinction coefficients as found from Eqs (6.7),

(6.8) and (6.9). Figure 6.7 (a) shows the approximation using the first term ( m=1) only

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and Figure 6.7 (b) uses the first 10 terms ( m=1 to 10). We notice that difference is small.

So, for simplicity, in finding the effective medium, we use only the first term.

(a) (b)

(c) (d) Figure 6.7: Real (a, b) and imaginary (c, d) parts of the relative permittivity for CuCl and Ag for the case of CuCl and Ag nanoparticles.

(a) (b) Figure 6.8: Scattering, extinction, and absorption coefficients for the (CuCl, Ag) NDLCC for (a) for m=1 and (b) for m=1 to 10 terms.

′ ′′ Figure 6.9(a) shows the real ( neff ) and imaginary ( neff ) parts of the effective index of refraction computed from Eqs. (6.21) and (6.22) where, r = 35 nm , r = 10 nm , 1 2

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µ = µ = µ = ε = ε = = = = = − 1r r2 r3 1, o 96.1 , e ,4 f1 f,4.0 2 4.0 , C1 C,5.0 2 1 C1 . The

angle between the incident field and the director axis in the middle of the cell is θ=0 .

′ θ Figure 6.9(b) shows neff profile in the LCC due to the electric field (bias voltage) where

λ = µ ′ is not constant throughout the cell at .0 38722 m . Figures 6.9 (a) and (b), show neff

′′ and neff parts of the refractive index as a function of wavelength and of the angle of

incidence θ. We notice that the maximum negative index of refraction occurs at θ=0.

Also, we noticed that increasing the filling factor increases the negative index of refraction.

(a)

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(b)

(a)

(b)

′ ′′ Figure 6.10: (a) neff (b) neff versus wavelength and angle of incidence for the case of CuCl and Ag nanoparticles.

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Finally, the scattering, extinction, and absorption coefficients for the two cases of

(LiTaO 3, Ge) and for the (CuCl, Ag) NDLCC are shown in Figures 6.9(a) and (b), respectively (See Appendix G).

6.5 Plasmonic Sensor Array

In this section we develop 2-D sensor arrays based on a planar grid made of LC substrates containing dispersed metallic (such as Ge, copper, Ag) nanoparticles, with external electrodes that may be connected to addressable voltage sources, in order to quantify and analyze the applied stimuli. Two specific stimuli are addressed in this

Section- (a) an applied pressure/temperature field, and (b) an arbitrary electrostatic field wherein the sensor array is placed or embedded. The application (a) is based on the idea that when the LC substrate is subjected to (say) a normal external pressure, the modified dispersed nanoparticles concentration generate a plasmon excitation whose resonance characteristics is sensitive to the applied pressure. This change the effective refractive index, thus changing the transmittance of the sensor. In this manner, by probing the resonances across the array, we may ascertain optically the pressure distribution. Same thing apply to temperature. The application (b) is based on a similar or related concept that in response to an external electrostatic field, the index of refraction of the LC cell vary in the lateral direction (propagation direction) and thus varying the transmittance of the cell. The inclusion of nanoparticles generate enhanced resonance of the effective refractive index that enhance the detection of the electrostatic field change through the variation of the transmittance.

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There are several approaches in the literature dealing with the use of plasmonics in a variety of sensor applications [88-93]. Many of these utilize periodic arrays of apertures with subwavelength dimensions and submicron periodicity [88]. Others use fiber-optic SPR sensors of different chemical, physical, and biochemical parameters [89].

Localized surface plasmon resonances of metallic nanoparticles such as gold, silver, copper or other suitable nanoshells with appropriate sensory coatings have also been employed for sensing of chemical and biological molecules [94-97]. There are reported operations that use nanoparticles, in particular those dispersed, for instance, in cholesteric

LCs [98,99]. However, there does not appear to be any significant amount of work where by such nanodispersed LCs have been developed into grids or arrays for the purpose of sensing pressure or electrostatic fields. The advantages of using plasmonic nanoparticles embedded in LCs host as sensors are:

• The ability to accurately measure plasmonic resonances, thereby obtaining data

pertinent to localized pressure or static field information through the measure of

transmittance;

• The ability to increase or decrease plasmon-enhanced resonances by altering the

nanoparticle volume filling fraction, or the average particle size;

Figure 6.11 (a) shows the schematic of a homogeneous or planar structure aligned nematic LC cell containing plasmonic nanoparticle samples. Figure 6.11(b) shows an array of the same structure. Figure 6.11(c) shows a side view of a biased LC cell.

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+ + + + + + + + + + E

------(a) (b) (c) Figure 6.11: (a) Homogeneous NDLCC containing plasmonic nanoparticles. (b) NDLCA, (c) An applied electric or magnetic field can switch the director axis of the LC from parallel to perpendicular to the cell glass (side view).

Several important theoretical and modeling issues are addressed in this Section.

The first one is how to relate pressure, temperature, electrostatic field to the dispersed nanoparticles which generate a plasmon excitation whose resonance characteristics change the effective refractive index, thus changing the transmittance of the sensor. The second one is how to model the mixture of nanoparticles embedded in LCs and how to measure the effective index of refraction and hence transmittance. For a layered stack of materials we used the TMM for analysis of the reflectance and transmittance as a function of the wavelength of the incident light as shown in Chapter 5. This approach can be extended to a large number of layers, with the aim of approaching a homogenous mixture.

6.5.1 Electrostatic field sensor

Let us consider the schematic of Figure 6.11(c) where we have plasmonic nanoparticles dispersed in a LC cell. The electrostatic field is equivalent to the voltage bias around the cell. Since the index of refraction profile of the LC changes (see Section 6.3.1) along the propagation direction as shown in Figure 6.2(c) we can consider the setup to be

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equivalent to a layered stack as shown in Figure 6.2(d). In the presence of nanoparticles, these stacks have different refractive indices for the host in which the nanoparticles are embedded. This leads to different effective refractive indices for each layer. The overall transmission is thus determined by the change in these effective refractive indices, which is also therefore a function of the applied voltage. If an electromagnetic pulse is applied on the LC with center frequency around the plasmonic resonance of the nanoparticles, the effective index of refraction due to the plasmonic resonance varies significantly in that region and hence resulting in a noticeable change in transmittance for two different electrostatic fields unlike the case when there are no nanoparticles. Figures 6.12 (a), (b) show the effective indices of refraction for two electrostatic fields V 1=1 V and V 2=2 V without nanoparticles respectively. Figures 6.13 is similar to Figures 6.12 but with a different range of frequencies. Figures 6.14 is the same as Figures 6.12 with Ge nanoparticles are included, respectively. Figures 6.15 is similar to Figures 6.13 but with

Ag instead of Ge.

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(a) (b) Figure 6.13: Real (up) and imaginary (down) effective index of refraction of LC cell function of visible frequency region and propagation distance z without nanoparticles with bias voltage (a) V=1V, (b) V=2V.

(a) (b) Figure 6.14: Real (up) and imaginary (down) effective index of refraction of Ge dispersed LC cell function of frequency and propagation distance z for the following parameters = = × −6 ω = × 12 ε ( ) = γ = ω µ = µ = f 38.0 , r 25.2 10 , p 26 7. 10 , r 0 15 ,8. p / 100 , 1r rh 1 with bias voltage (a) V=1 V, (b) V=2V.

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(a) (b) Figure 6.15: Real (up) and imaginary (down) effective index of refraction of Ag dispersed LC cell function of frequency and propagation distance z for the following ε ( ) = = = × −9 ω = π × × 15 γ = π × × 12 parameters, r 0 ,1 f r,1.0 20 10 , p 2 18.2 10 , 2 35.4 10 , µ = µ = 1r rh 1 with bias voltage (a) V=1 V, (b) V=2V.

The transmittance is found using the TMM explained in Chapter 5. Figure

6.16(a) shows the transmittances of a LC cell without nanoparticles at visible frequencies for two different electrostatic fields V=1V and V=2V. Figure 6.16(b) shows the transmittances of a LC cell with Ag nanoparticles at visible frequencies for two different electrostatic fields V=1V and V=2V. Figure 6.16(c) shows the transmittances of a LC cell without nanoparticles at FIR frequencies for two different electrostatic fields V=1V and V=2V. Figure 6.16(d) shows the transmittances of a LC cell with Ge nanoparticles at

FIR frequencies for two different electrostatic fields V=1V and V=2V. The quasiperiodic behavior of Figures 6.16 (a,c) is due to the fact that the optical path length of the LC in terms of wavelength changes with frequency. However, the variation in Figures

6.16(b,d) is due to the existence of a zero transmission bandgap, similar to what is observed in layered positive index – negative index media [100]. Careful examination of

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the graphs (Figures 6.16(a,b)) reveal that without nanoparticles, the change in transmission due to a voltage change from 1V to 2V (corresponding to an electrostatic

ω field change from 125 kV/m to 250 kV/m) at a frequency of 1.0 p for Ag yields a change in transmission of 3%; whereas, with nanoparticles, the change is 19%, which is a factor of over 6 times the case without nanoparticles. Thus, with a careful choice of the operating frequency, one can achieve marked improvement in the sensitivity of response

(transmission) to the electrostatic field due to enhancement introduced by the plasmonic resonance of the nanoparticles.

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6.5.2 Pressure/temperature sensor

Consider the schematic of Figure 6.17 where we again have plasmonic nanoparticles dispersed in a LC cell. Figure 6.17 (a) shows the schematic before applying pressure and

Figure 6.17(b) shows the schematic after we apply pressure. Note that pressure/temperature result in a change in the filling factor of the nanoparticles as well the cell thickness. Both of these parameters change the transmittance of the LC cell. (see

Section 6.5).

(a) (b) Figure 6.17: (a) LC cell before applying pressure, (b) LC after applying pressure.

Figure 6.18 shows the real and the imaginary parts of the effective index of refraction for a LC cell with Ge nanoparticles with two different volume filling fractions f=0.38 and f=0.7 simulating the scenario without and with applied pressure, respectively.

Figure 6.19 is similar to Figure 6.18 but for the case of Ag nanoparticles with two

different volume filling fractions f=0.1 (no pressure) and f=0.3 (after applying pressure).

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15 f=0.38 f=0.7

) 10 eff

Real(n 5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ωωω ωωω Normalized frequency, / p

15 f=0.38 f=0.7

) 10 eff

Iamg(n 5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Normalized frequency, ωωω/ωωω p

Figure 6.18: Nanodispersed Ge nanoparticles LC cell effective index of refraction before applying pressure ( f=0.38) and after applying pressure ( f=0.7), (Top) real part, (Bottom) imaginary part.

5 f=0.1 4 f=0.3 )

eff 3

2 Real(n 1

0 0 0.2 0.4 0.6 0.8 1 ωωω ωωω Normalized frequency, / p

5 f=0.1 4 f=0.3 )

eff 3

2 Iamg(n 1

0 0 0.2 0.4 0.6 0.8 1 Normalized frequency, ωωω/ωωω p

Figure 6.19: Nanodispersed Ag nanoparticles LC cell effective index of refraction before applying pressure( f=0.1) and after applying pressure ( f=0.3), (Top) real part, (Bottom) imaginary part.

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Figure 6.20 shows the difference in transmittance for the Ge nanoparticle dispersed LC cell of initial thickness 80 µm. The red curve indicates the transmittance with no pressure applied. The blue curve is the transmittance when the volume filling fraction only has been assumed to have changed due to applied pressure. Finally, the green curve shows the transmittance when the volume filling fraction and the LC thickness has changed due to applied pressure. The LC cell is assumed to have compressed to 70 µm due to applied pressure. Figure 6.21 is the same as Figure 6.20, but for Ag nanoparticles.

Figure 6.20: Transmittance for the case of Ge nanoparticles dispersed LCC. (red) no pressure is applied ( f=0.38) and cell thickness 80 µm, (blue) after applying pressure and cell thickness 80 µm ( f=0.7), (green) after applying pressure and cell thickness 70 µm (f=0.7). As in the case of electrostatic field, monitoring of the transmittance at a band edge can be used to determine the change in pressure/temperature. For instance, at a frequency

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ω of 2.0 p for Ag, the transmittance without applied pressure is 50%, while the same with applied pressure is almost 0% due to the shift in the transmission spectrum.

6.6 Fabrication Procedure of LCC between two Glass Substrates

In this Section we explain the main steps to fabricate a nanodispersed LC cell. The glass substrate is placed on a stage with a vacuum hole, and fixed on the stage with vacuum pressure and placed in the spin coater where an alignment material PI-2555 (0.1%-1% by weight of PI 2555 in solution of N-Methyl-2-Pyrrolidone) coats the glass substrate. The glass substrate is then put in the vacuum oven till the PI-2555 dries on the substrate. The

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coated alignment material is rubbed by the rubbing machine, in order to produce an alignment layer on the substrate. Next an appropriate spacer is put on one of the substrates. The slide sandwich is inserted on a heated plate and held inclined, while the

NDLC is inserted using a dropper at the top of the sandwich. The NDLC fills the space between the two slides under gravity and its viscosity and surface tension keeps it contained within the sandwich. (See Figure 6.22).

Later, in subsequent work we plan to mix the nanoparticles with a non polar

organic solvent like hexane or heptane, stir, and then lightly sonicate the suspension. We

plan to thereafter mix the resulting nanoparticle suspension with the LC. Finally we plan

to evaporate the hexane/heptane and ultrasonically treat the nanparticle dispersed LC for

approximately 5 min.

In this dissertation we have only fabricated parallel and twisted LCCs without

nanoparticles. The reason behind this is to master the fabrication of LCCs. In later

research, we plan to add nanoparticles to the LCC as indicated above. The experiments

in the following subsections describe the testing of the LCCs.

Glass Substrate

ITO Polyamide PI-2555 E7 LC with binary nanoparticles

Figure 6.22: LC cell structure. ITO coated glass substrate is 2.5cm x 2.5 cm, and the distance between the two glass slabs is around: 10 microns.

Some pictures of the LC facility setup are shown in Figure 6.23 to Figure 6.25.

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Figure 6.23: Clean bench class 100-Series 301-Laminar Flow Workstation + Dust Cover(air clean product).

Figure 6.24: Rubbing machine RM-3I -08+Control Unit (Beam Eng. For Adv Meas. Co.) (left) and spin coater(KW-4AC-4) + vacuum chucks + vacuum pump (Chemat technology Inc.).

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Figure 6.25: Corning hotplate and stirrer -CLS6(left), E7 LC (middle), and A&D -HR-200 analytical balance (right) .

6.6.1 Experiment 1: Testing of a biased twisted LCC

As a first experiment we have fabricated a twisted LCC. Then we have biased it and put it between 2 linear crossed polarizers as shown in the schematic of Figure 6.26. The cell in this case acts like a switch as expected. When there is no bias (V=0), the transmission is

T=6mW (see Figure 6.27(a)), when there is bias (V=3V) the transmission is T=80 µW

(see Figure 6.27(b)). The lab setup is shown in Figure 6.28.

Figure 6.26: Testing of a biased liquid crystal cell.

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(a) (b) Figure 6.27: (a) No bias, transmission is T=6mW, (b) bias voltage =3V, T=80 µW.

Figure 6.28: Lab setup for LCC testing.

6.6.2 Experiment 2: Testing of unbiased parallel LCC

As a second experiment we have fabricated a parallel (non-twisted) LCC shown in Figure

6.29. Then we put the LC cell between 2 linear crossed polarizers without biasing. The sample is rotated until the director axis of the sample is 45 0 with the vertical linearly

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polarized (VLP) output of the laser beam. This give us the maximum transmission as expecte d according to the formula:

T∞ sin ()2φ , where φ is the angle between the director axis and the linearly polarized laser output.

Figure 6.30 shows the comparison between the theoretical and the experimental resul

Figure 6.29: Testing of unbiased parallel LCC.

Figure 6.30: Transmittance vs. angle between the director axis and the linearly polarized laser output. Theoretical (solid Red line), experimental (diamond blue dots).

6.6.3 Experiment 3: Testing of u nbiased twisted LCC

As a third experiment we have fabricated a twisted LCC. Then we put the sample between 2 linear crossed polarizers without biasing as shown in the schematic of Figure

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6.31. In this case no matter how the sample is rotated we get within sample a similar transmission which verifies the theory. The result is shown in Figure

6.32.

Figure 6.31: Testing of unbiased twisted LCC.

Figure 6.32: Transmittance vs. angle between the director axis and the linearly polarized laser output.

6.7 Conclusion

In this Chapter we have laid the grounds of possi ble cheap, tunable metamaterials great importance due to potential applications in super -resolution lensing and sensors.

We have studied the feasibility of the fabrication of a b inary NDLCC that exhibits negative index of refraction in the infrared and optical region of the spectrum. The

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effective index of refraction is calculated according to the quasi-static limit of the MG mixing rule EMG, and Mie theory. The scattering, extinction, and absorption of such

NDLC cells were also found. The tunability of these cells is also studied. This tunability is due to the radii of the nanoparticles, filling fractions and the angle between the director axis of the LCC and the incident beam which varies according to electric field biasing the

LCC. We notice that the maximum negative index of refraction occurs at θ=0. Also, we noticed that the negative index of refraction increase with increasing filling factors.

Next, in this Chapter we have developed 2-D sensor based on NDLCC with external electrodes as a pressure/temperature and electrostatic field sensor. For the pressure/temperature sensor the change in concentration of the nanoparticles change the effective refractive index, thus changing the transmittance of the sensor. For the electrostatic field the inclusion of nanoparticles generate enhanced resonance of the effective refractive index that enhance the detection of the electrostatic field change through the variation of the transmittance.

Also, in this Chapter we have outlined the fabrication procedure of LCC between two glass substrates and how to incorporate nanoparticles. Also, we have built a laboratory LC processing setup from scratch, fabricated parallel and twisted LCC samples, and tested the fabricated samples.

Finally, we want to point out that the only drawback of these NDLCC is the high absorption. We hope that by carefully choosing the right materials we can reduce the absorption and hence get better transmission through these cells.

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CHAPTER 7

CONCLUSION AND FUTURE WORK

7.1 Conclusions

In this dissertation, we have attempted to understand the properties of EM propagation through metamaterials in the simplest possible way, and in the process, design novel easily to fabricate and cheap structures that can exhibit metamaterial behavior and be used for potential applications as sensors.

Specifically, in Chapter 2, we have discussed the fundamental properties of propagation in the presence of NIMs. Secondly, the reversal in the basic optics phenomena is summarized, viz., Snell’s law, Doppler effect and Vavilov-Cerenkov radiation. Next, the concept of the super resolution or superlensing has been introduced.

Finally cloaking, which is a promising application of metamaterials, has been introduced through transformation optics. Cloaking or invisibility in the microwave or optical regime has many military as well as civilian applications.

In Chapter 3, we have discussed the concept of causality for permittivity and permeability, both for a homogeneous as well as an effective medium comprising binary constituents. We have also introduced for the first time the concept of a reduced propagation constant which obeys causality. The propagation constant is identical to the

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reduced propagation constant under appropriate limiting values of the physical parameters. We have found dependence on frequency of the reduced propagation constant for (a) a nonmagnetic material where the permittivity is given by the Lorentz model, (b) a material where the permittivity and permeability are Lorentz-type, and (c) an effective medium comprising a nonmagnetic material with Lorentz-type permittivity in a dispersionless background, where the effective permittivity is given by the MG rule. In each case, the reduced propagation constant has real and imaginary parts related through the HT, implying causality. Causality of the propagation constant enables the use of simple operator formalisms to derive the underlying partial differential equations (PDEs) for baseband and envelope wave propagation. Two illustrative examples have been given to model propagation in NIMs, one with loss and one with gain.

In Chapter 4, we have modeled the reorientation angle in a LC cell due to an applied external electrostatic field. We have also studied beam propagation in LCs.

Also, a setup using a NLC cell with initial in-plane pre-tilt has been described and negative refraction has been observed theoretically and experimentally.

In Chapter 5, we have investigated the transmission properties of EM/optical wave through periodic and random stacks of PIMs and NIMs using a matrix-based multilayered approach. Randomness in layer position, thickness, and refractive index has been studied. This randomness mimic a homogeneous material composed of PIM and

NIM materials. The limit of the transmittance for mixed (PIM/NIM) structures when the number of layers tends to infinity has been derived. Also we have considered periodic structures of different types of materials (i.e dispersive, Nondispersive, lossy…). We also discussed how the zero gap could be used as a sensor. Finally, we studied the

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effect of introducing defect slab into a periodic structure. Two cases were studied with

LC or NIM as the defect layer.

In Chapter 6 we have derived the theory, describe the fabrication techniques, and the testing methods of nanoparticles dispersed LC as a candidate for an NIM. Also, we have laid the grounds of possible cheap, tunable metamaterials of great importance due to potential applications in super-resolution lensing and sensors. Next, we have studied the feasibility of the fabrication of a binary NDLCC that exhibits negative index of refraction in the infrared and optical region of the spectrum. The effective index of refraction has been calculated according to the quasi-static limit of the MG mixing rule EMG, and Mie theory. The scattering, extinction, and absorption of such NDLC cells is also found. The tunability of these cells is also studied. This tunability is due to the radii of the nanoparticles, filling fractions and the angle between the director axis of the LCC and the incident beam which varies according to electric field biasing the LCC. We notice that the maximum negative index of refraction occurs at θ=0. Also, we have noticed that the negative index of refraction increases with increasing filling factors. Finally, in this

Chapter we have also developed the theory of using NDLCC as plasmonic sensors for measuring electrostatic fields, pressure, and temperature.

7.2 Future Work

As a continuation of this work, we plan to attempt the following in the near future:

• Install an argon environment setup for nanoparticles treatment.

• Fabricate the NDLCs according to the steps laid out in Chapter 6, Section 6.5.

• Measure the NDLCC transmittance and reflectance using the Cary 500

spectrophotometer to determine the magnitude of the refractive index.

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• Develop a phase measurement technique for the refractive index using

polarization and walk-off interferometers.[101,102]

• Characterize NDLCCs for metamaterial lensing by using the Z-scan technique

which is commonly used for characterizing induced lensing [103-104], but has

been adapted to characterize linear lenses as well [105].

7.3 Preliminary Results of Z-scan on NDLCCs

Before ending, we report here on the first Z-scan measurement of NDLCCs where the LC sample is doped with ferroelectric BaTiO 3 nanoparticles which has possible applications in photorefractive two beam coupling. These samples have been obtained from AFRL for characterization purposes. The Z-scan apparatus shown in Figure 7.1 is used. In this technique a polarized Gaussian laser beam, propagating in the z direction, is focused to a narrow waist by using a convex lens. The sample is moved along the z direction through

the focal point and the transmitted intensity is measured in the far field using a

photodetector behind a small iris, as a function of the z position. As the sample moves along the beam focus, focusing and/or defocusing by the sample modifies the wave front phase, thereby modifying the detected beam intensity.

Figure 7.1: Z-scan apparatus.

147

In nonlinear induced lensing characterization, the value of the nonlinear refracted index is obtained by measuring the variation of the on-axis transmittance with scan length about the focus of the external lens. In linear lens characterization, the on-axis transmittance variation gives information about the strength of the sample lens. Open aperture Z-scans are also used to quantify nonlinear absorption [103-104].

A continuous-wave (514 nm) laser beam is focused with a lens of focal length 5 cm. The beam radius at the sample is found to be 10.5 µm and the Rayleigh range ( z0) is

0.673 mm. The power illuminating the NDLCC TL205 +BaTiO 3 sample is 30 mW. The iris diameter for closed aperture measurement was 3 mm. The motorized translation range is around 10 Rayleigh ranges from each side of the focal plane. The data are collected automatically from the power meter using a GPIB to USB converter to a PC where we developed an in-house MATLAB program as an interface to capture the intensity plot as shown in Figure 7.2.

Figure 7.2: Normalized transmittance for the TL205+ BaTiO 3 sample (closed aperture).

148

Also, we have used the open aperture z-scan configuration in order to measure possible nonlinear absorption. For open aperture, a pulsed laser beam (repetition rate: fr=10 MHz, average power: P average =200mw, pulse width: τ=100 fs ) has been used. The

sample, once again, is TL205 +BaTiO 3. The obtained result is shown in Figure 7.3.

Figure 7.3: Normalized Transmittance for the TL205+ BaTiO 3 sample (open aperture).

The laboratory setup is shown in Figures 7.4 (a) and (b). The characteristic features akin to nonlinear induced refraction and absorption are observed with the nanoparticle doped samples, unlike the LC samples that are not doped. The reason for this Z-scan behavior is currently under investigation.

149

Photodetector Iris Powermeter

Motorized stage

LC sample Convex lens

(a)

Motion Controller

(b)

Figure 7.4: (a) The lab setup, (b) the motion controller stage.

150

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APPENDICES

159

APPENDIX A

MATLAB Code for Hilbert Transform

% k causal clear all; clc; w=linspace(-5,5,2^10)*10^15; w0=10^14; v=3*10^8; c=3*10^8; a=w/w0; b=w0/v; kr=b.*(a-a.^3); ki=b.*(a.^2-a.^4);

%% check causality figure subplot(2,2,1) plot(w,kr);title('k_r'); grid on; xlabel('\omega') subplot(2,2,2) plot(w,imag(hilbert(ki))); title('H_\omega(k_i(\omega))'); grid on; xlabel('\omega') subplot(2,2,3) plot(w,ki) ;title('k_i'); grid on; xlabel('\omega') subplot(2,2,4) plot(w,-imag(hilbert(kr))); title('H_\omega(k_r(\omega))') grid on ; xlabel('\omega')

%% find Xr and Xi and check causality xr=(kr.^2-ki.^2).*(c.^2./w.^2)-1; xi=2.*kr.*ki.*(c^2./w.^2); % xr=(c/v)^2.*((1-a.^2)./(1+a.^2).^2) )%-1 % xi=2*(c/v)^2.*(a./(1+a.^2).^2) figure subplot(2,2,1) plot(w,xr);title('\chi_r'); grid on; xlabel('\omega') subplot(2,2,2) plot(w,imag(hilbert(xi))); title('H_\omega(\chi_i(\omega))'); grid on; xlabel('\omega') subplot(2,2,3) plot(w,xi) ;title('\chi_i'); grid on; xlabel('\omega') subplot(2,2,4) plot(w,-imag(hilbert(xr))); title('H_\omega(\chi_r(\omega))') grid on ; xlabel('\omega')

160

APPENDIX B

MATLAB Code for Baseband Propagation

%% Baseband propagation clc clear all close all N=2^8; % z=10; Z=0:1:20; umax(1)=1; L=400; NI=35; tau=5; wn=100; for z=1:1:21; d_z=Z(z)/NI; h = L/N; n = [-N/2:1:N/2-1]'; t = n*h; tau=5; ub0=umax(1)*(1-(tanh(0.5*t)).^2); ub0=umax(1)*exp(-t.^2/tau^2); %.*cos(w0*t); plot(t,abs(ub0)); ub=ub0; Eb0(z)=h*sum(abs(ub0).^2) w =2*n*pi/L; Numb=-(w.^2+i*w); %without simplification Denb=(1+w.^2); c1=1 ; c2=-1; % Numb=-c1*w.^2+c2*w.^4; % diffusion term alone % Denb=1; Pb=Numb./Denb; % aviobj = avifile('mynegativeindex2.avi','fps',5); for m = 1:1:NI ubp = fftshift(fft(ub)); ubp = exp(d_z*Pb).*ubp; ub = ifft(fftshift(ubp)); end % movie(M) % aviobj = close(aviobj); Efb(z)=h*sum(abs(ub).^2) figure plot(t,abs(ub),'r') ylabel('u'); title(strcat(' Base band Pulse in Time Domain after propagation of a distance Z=',num2str(z))) hold grid on end plot(Z,Efb) grid on;

161

APPENDIX C

MATLAB Code for Envelope Propagation

%Pulse propagation in a negative index material clc N=2^8; z=10; umax(1)=1; L=400; NI=35; d_z=z/NI; h = L/N; n = [-N/2:1:N/2-1]'; t = n*h; tau=20; wn=4; kn=-wn.^2/(1+wn^2); % u0=umax(1)*(1-(tanh(10*t)).^2); u0=umax(1)*exp(-t.^2/tau^2); u=u0; E0=h*sum(abs(u0).^2); a=(1+wn^2); b=(-i*kn-i*wn-wn^2-i*wn^2-i*wn^2*kn); c=(-1+2*i*wn-2*kn*wn); d=(1+i*kn); w =2*n*pi/L; Num=(-b-i*w*c+w.^2*d); Den=(a+w.^2+2*w*wn); P=Num./Den;

for m = 1:1:NI up = fftshift(fft(u)); up = exp(d_z*P).*up; u = ifft(fftshift(up)); end Ef=h*sum(abs(u).^2); figure subplot(2,1,1) plot(t,abs(u0)) ylabel('|u|'); title(strcat('Initial pulse in Time Domain: w_n=',num2str(wn),',\tau=',num2str(tau))) xlabel( 'Time T') grid on subplot(2,1,2) plot(t,abs(u),'k') ylabel('|u|'); xlabel( 'Time T') title(strcat(' Envelope Pulse in Time Domain after propagation of a distance Z=',num2str(z))) grid on

162

APPENDIX D

MATLAB Code for Beam Propagation in LC

%%% Propagation of Gaussian Beam in the LC. clear all; clc; load nref; N=100; % point in X L=75; %Length of X w0=2; delz=0.1; M=1000; %0.01*1000 = length of z =10 umax(1)=1; lambda=0.633; h = L/N ; %dx a=pi/h; k0=2*pi/lambda n = [-N/2:1:N/2-1]'; % Indices x1 = n*h; % Grid points x2=x1'; e1=[-a:2*a/N:a-2*a/N]'; e2=e1'; [ee1,ee2] = meshgrid(e1); load dn no= 1.5038; ne=1.6954; u0=umax(1)*exp(-((x1.^2)./w0^2))*exp((-x2.^2)./w0^2); energy=h^2*sum(sum(abs(u0.^2))) figure,mesh(x1,x2,abs(u0)),view(90,0) dn=meshgrid(dn); Num=-(ee1.^2+ee2.^2); Dem=i*2*k0*nref; P=Num./Dem; zprop(1)=0; u=u0; uold=u0; for k = 1:1:M % Start time evolution propagation in z zprop(k+1)=zprop(k)+delz; du=-(1./no).*(dn).*((u-uold)./delz); u=uold+du; uold=u; up = fftshift(fft2(u)); up = exp(delz.*P).*up; u = ifft2(fftshift(up)); umax(k+1)=max(max(abs(u))); end figure,mesh(x1,x2,abs(u)) ,view(90,0) zr=zprop(length(zprop)); figure plot(zprop,umax) title(strcat('\psimax:Plot vs. Propagation dist z' ,',z_r_* =',num2str(zr))); ylabel('Beam Intensity') xlabel('z') energyp=h^2*sum(sum(abs(u.^2)))

163

APPENDIX E

MATLAB Code for TMM function [R,A,T,M,t,f] = LambdaMultidielMod(n,L,lambda,mum,thi,ni,ns,pol)

Y=2.6544*10^-3;

R=0; A=0 ;T=0; for r=1:1:length(lambda)

M=[1,0;0,1]; for m=1:1:length(n), k=(2*pi/lambda(r)).*n(m); thr=asin(ni*sin(thi)./n(m)); gamma=k.*L(m).*cos(thr); if pol==1 eta=Y*n(m).*cos(thr); %% TE elseif pol==0 eta=Y*n(m)./cos(thr); %% TM end

a=cos(gamma); d=cos(gamma); b=((1i.*mum(m))./eta).*sin(gamma); c=((1i.*eta)./mum(m)).*sin(gamma); M= M*[a,b;c,d];

end

eta0=Y*ni*cos(thi); thm=asin(ni*sin(thi)/ns); if pol==1 etam=Y*ns*cos(thm); else etam=Y*ns/cos(thm); end BC=M*[1 ;etam]; B=BC(1); C=BC(2); R(r)=((eta0*B-C)./(eta0*B+C)).*conj((eta0*B-C)./(eta0*B+C) ); A(r)=( 4*eta0*real(B.*conj(C)-etam))./((eta0*B+C).*conj(eta0*B+C) ); T(r)=(4*eta0*real(etam))./((eta0*B+C).*conj(eta0*B+C)); t(r)=(2*eta0)/(eta0*B+C); f(r)=atan(-imag(eta0*B+C)/real(eta0*B+C)); end

164

APPENDIX F

MATLAB Code for Anisotropic TMM

% Birefringence in 1D finite photonic bandgap structure clear all; clc; syms gamm;

NL= 15 ; %# of layer-pairs N= 2; t=[41.7*10^-9 125*10^-9]; % vector thickness of each layers mu0=4*pi*10^-7; mu= 1*mu0 ; c=3*10^8; eps0=8.8*10^-12; epsxx=eps0*[6 3]; epsxy=eps0*[0 0]; epsxz=[0 0]; epsyx=[0 0]; epsyy=eps0*[2 7]; epsyz=[0 0]; epszx=[0 0]; epszy=[0 0]; epszz=eps0*[4 4]; w= 2*pi*10^14*linspace(3,15,200); lambda=2*pi*c./w;

% air epsxxa=eps0*3; epsxya=0; epsxza=0; epsyxa=0; epsyya=eps0*7; epsyza=0; epszxa=0; epszya=0; epszza=eps0*4; phi=0.1*pi/180; %% loop on wavelength

for m=1:1:length(lambda) %length(freq) beta =sin(phi)*2*pi/lambda(m); alpha =cos(phi)*2*pi/lambda(m); D=eye(4,4); T=eye(4,4); %% Treat the first layer (air)

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B=[(2*pi*c/lambda(m))^2*mu*epsxxa-beta^2-gamm^2, (2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta, (2*pi*c/lambda(m))^2*mu*epsxza+alpha*gamm; (2*pi*c/lambda(m))^2*mu*epsyxa+alpha*beta, (2*pi*c/lambda(m))^2*mu*epsyya-alpha^2-gamm^2, (2*pi*c/lambda(m))^2*mu*epsyza+beta*gamm; (2*pi*c/lambda(m))^2*mu*epszxa+alpha*gamm, (2*pi*c/lambda(m))^2*mu*epszya+beta*gamm, (2*pi*c/lambda(m))^2*mu*epszza-alpha^2-beta^2]; g=eval(solve(det(B))); g=sort(real(g),'descend'); g1=g(1); g2=g(3); g3=g(2); g4=g(4);

p1=[((2*pi*c/lambda(m))^2*mu*epsyya-alpha^2- g1^2)*((2*pi*c/lambda(m))^2*mu*epszza-alpha^2-beta^2)- ((2*pi*c/lambda(m))^2*mu*epsyza+beta*g1)^2;

((2*pi*c/lambda(m))^2*mu*epsyza+beta*g1)*((2*pi*c/lambda(m))^2*mu*epszx a+alpha*g1)- ((2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta)*((2*pi*c/lambda(m))^2*mu*ep szza-alpha^2-beta^2);

((2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta)*((2*pi*c/lambda(m))^2*mu*ep syza+beta*g1)- ((2*pi*c/lambda(m))^2*mu*epsxza+alpha*g1)*((2*pi*c/lambda(m))^2*mu*epsy ya-alpha^2-g1^2)]; p1=p1./norm(p1);

p2=[((2*pi*c/lambda(m))^2*mu*epsyya-alpha^2- g2^2)*((2*pi*c/lambda(m))^2*mu*epszza-alpha^2-beta^2)- ((2*pi*c/lambda(m))^2*mu*epsyza+beta*g1)^2;

((2*pi*c/lambda(m))^2*mu*epsyza+beta*g2)*((2*pi*c/lambda(m))^2*mu*epszx a+alpha*g2)- ((2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta)*((2*pi*c/lambda(m))^2*mu*ep szza-alpha^2-beta^2);

((2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta)*((2*pi*c/lambda(m))^2*mu*ep syza+beta*g2)- ((2*pi*c/lambda(m))^2*mu*epsxza+alpha*g2)*((2*pi*c/lambda(m))^2*mu*epsy ya-alpha^2-g2^2)]; p2=p2./norm(p2);

p3=[((2*pi*c/lambda(m))^2*mu*epsyya-alpha^2- g3^2)*((2*pi*c/lambda(m))^2*mu*epszza-alpha^2-beta^2)- ((2*pi*c/lambda(m))^2*mu*epsyza+beta*g3)^2;

((2*pi*c/lambda(m))^2*mu*epsyza+beta*g3)*((2*pi*c/lambda(m))^2*mu*epszx a+alpha*g3)- ((2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta)*((2*pi*c/lambda(m))^2*mu*ep szza-alpha^2-beta^2);

((2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta)*((2*pi*c/lambda(m))^2*mu*ep

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syza+beta*g3)- ((2*pi*c/lambda(m))^2*mu*epsxza+alpha*g3)*((2*pi*c/lambda(m))^2*mu*epsy ya-alpha^2-g3^2)]; p3=p3./norm(p3);

p4=[((2*pi*c/lambda(m))^2*mu*epsyya-alpha^2- g4^2)*((2*pi*c/lambda(m))^2*mu*epszza-alpha^2-beta^2)- ((2*pi*c/lambda(m))^2*mu*epsyza+beta*g4)^2;

((2*pi*c/lambda(m))^2*mu*epsyza+beta*g4)*((2*pi*c/lambda(m))^2*mu*epszx a+alpha*g4)- ((2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta)*((2*pi*c/lambda(m))^2*mu*ep szza-alpha^2-beta^2);

((2*pi*c/lambda(m))^2*mu*epsxya+alpha*beta)*((2*pi*c/lambda(m))^2*mu*ep syza+beta*g4)- ((2*pi*c/lambda(m))^2*mu*epsxza+alpha*g4)*((2*pi*c/lambda(m))^2*mu*epsy ya-alpha^2-g4^2)]; p4=p4./norm(p4);

q1=(lambda(m)/(2*pi*mu))*[ beta*p1(3)-g1*p1(2);- alpha*p1(3)+g1*p1(1);alpha*p1(2)-beta*p1(1)];

q2=(lambda(m)/(2*pi*mu))*[ beta*p2(3)-g2*p2(2);- alpha*p2(3)+g2*p2(1);alpha*p2(2)-beta*p2(1)];

q3=(lambda(m)/(2*pi*mu))*[ beta*p3(3)-g3*p3(2);- alpha*p3(3)+g3*p3(1);alpha*p3(2)-beta*p3(1)];

q4=(lambda(m)/(2*pi*mu))*[ beta*p4(3)-g4*p4(2);- alpha*p4(3)+g4*p4(1);alpha*p4(2)-beta*p4(1)];

Dtemp=[ p1(1) p2(1) p3(1) p4(1); q1(2) q2(2) q3(2) q4(2); p1(2) p2(2) p3(2) p4(2); q1(1) q2(1) q3(1) q4(1)];

%------loop on the other layers------for n=1:1:N B=[(2*pi*c/lambda(m))^2*mu*epsxx(n)-beta^2-gamm^2, (2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta, (2*pi*c/lambda(m))^2*mu*epsxz(n)+alpha*gamm; (2*pi*c/lambda(m))^2*mu*epsyx(n)+alpha*beta, (2*pi*c/lambda(m))^2*mu*epsyy(n)-alpha^2-gamm^2, (2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*gamm; (2*pi*c/lambda(m))^2*mu*epszx(n)+alpha*gamm, (2*pi*c/lambda(m))^2*mu*epszy(n)+beta*gamm, (2*pi*c/lambda(m))^2*mu*epszz(n)-alpha^2-beta^2];

g=eval(solve(det(B))); g=sort(real(g),'descend'); g1=g(1); g2=g(3); g3=g(2);

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g4=g(4);

p1=[((2*pi*c/lambda(m))^2*mu*epsyy(n)-alpha^2- g1^2)*((2*pi*c/lambda(m))^2*mu*epszz(n)-alpha^2-beta^2)- ((2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*g1)^2;

((2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*g1)*((2*pi*c/lambda(m))^2*mu*eps zx(n)+alpha*g1)- ((2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta)*((2*pi*c/lambda(m))^2*mu* epszz(n)-alpha^2-beta^2);

((2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta)*((2*pi*c/lambda(m))^2*mu* epsyz(n)+beta*g1)- ((2*pi*c/lambda(m))^2*mu*epsxz(n)+alpha*g1)*((2*pi*c/lambda(m))^2*mu*ep syy(n)-alpha^2-g1^2)]; p1=p1./norm(p1);

p2=[((2*pi*c/lambda(m))^2*mu*epsyy(n)-alpha^2- g2^2)*((2*pi*c/lambda(m))^2*mu*epszz(n)-alpha^2-beta^2)- ((2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*g2)^2;

((2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*g2)*((2*pi*c/lambda(m))^2*mu*eps zx(n)+alpha*g2)- ((2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta)*((2*pi*c/lambda(m))^2*mu* epszz(n)-alpha^2-beta^2);

((2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta)*((2*pi*c/lambda(m))^2*mu* epsyz(n)+beta*g2)- ((2*pi*c/lambda(m))^2*mu*epsxz(n)+alpha*g2)*((2*pi*c/lambda(m))^2*mu*ep syy(n)-alpha^2-g2^2)]; p2=p2./norm(p2);

p3=[((2*pi*c/lambda(m))^2*mu*epsyy(n)-alpha^2- g3^2)*((2*pi*c/lambda(m))^2*mu*epszz(n)-alpha^2-beta^2)- ((2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*g3)^2;

((2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*g3)*((2*pi*c/lambda(m))^2*mu*eps zx(n)+alpha*g3)- ((2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta)*((2*pi*c/lambda(m))^2*mu* epszz(n)-alpha^2-beta^2);

((2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta)*((2*pi*c/lambda(m))^2*mu* epsyz(n)+beta*g3)- ((2*pi*c/lambda(m))^2*mu*epsxz(n)+alpha*g3)*((2*pi*c/lambda(m))^2*mu*ep syy(n)-alpha^2-g3^2)]; p3=p3./norm(p3);

p4=[((2*pi*c/lambda(m))^2*mu*epsyy(n)-alpha^2- g4^2)*((2*pi*c/lambda(m))^2*mu*epszz(n)-alpha^2-beta^2)- ((2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*g4)^2;

((2*pi*c/lambda(m))^2*mu*epsyz(n)+beta*g4)*((2*pi*c/lambda(m))^2*mu*eps zx(n)+alpha*g4)- ((2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta)*((2*pi*c/lambda(m))^2*mu* epszz(n)-alpha^2-beta^2);

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((2*pi*c/lambda(m))^2*mu*epsxy(n)+alpha*beta)*((2*pi*c/lambda(m))^2*mu* epsyz(n)+beta*g4)- ((2*pi*c/lambda(m))^2*mu*epsxz(n)+alpha*g4)*((2*pi*c/lambda(m))^2*mu*ep syy(n)-alpha^2-g4^2)]; p4=p4./norm(p4); q1=(lambda(m)/(2*pi*mu))*[ beta*p1(3)-g1*p1(2);- alpha*p1(3)+g1*p1(1);alpha*p1(2)-beta*p1(1)]; q2=(lambda(m)/(2*pi*mu))*[ beta*p2(3)-g2*p2(2);- alpha*p2(3)+g2*p2(1);alpha*p2(2)-beta*p2(1)]; q3=(lambda(m)/(2*pi*mu))*[ beta*p3(3)-g3*p3(2);- alpha*p3(3)+g3*p3(1);alpha*p3(2)-beta*p3(1)]; q4=(lambda(m)/(2*pi*mu))*[ beta*p4(3)-g4*p4(2);- alpha*p4(3)+g4*p4(1);alpha*p4(2)-beta*p4(1)]; Dold=Dtemp; Dtemp=[p1(1) p2(1) p3(1) p4(1); q1(2) q2(2) q3(2) q4(2); p1(2) p2(2) p3(2) p4(2); q1(1) q2(1) q3(1) q4(1)];

D=inv(Dold)*Dtemp; P=[ exp(i*g1*t(n)) 0 0 0; 0 exp(i*g2*t(n)) 0 0 ; 0 0 exp(i*g3*t(n)) 0; 0 0 0 exp(i*g4*t(n))]; T=T*D*P; end %------calculate the Final T matrix------T=T^NL; tss(m)=T(3,3)./(T(1,1)*T(3,3)-T(1,3)*T(3,1));Tss=abs(tss).^2; tpp(m)=T(1,1)./(T(1,1)*T(3,3)-T(1,3)*T(3,1));Tpp=abs(tpp).^2; tsp(m)=-T(3,1)./(T(1,1)*T(3,3)-T(1,3)*T(3,1));Tsp=abs(tsp).^2; tps(m)=-T(1,3)./(T(1,1)*T(3,3)-T(1,3)*T(3,1));Tps=abs(tps).^2; end subplot(2,2,1) plot(lambda,Tss); title('T_s_s') xlabel('lambda') grid on subplot(2,2,2) plot(lambda,Tpp); title('T_p_p') xlabel('lambda') grid on subplot(2,2,3) plot(lambda,Tsp); title('T_s_p') xlabel('lambda') grid on subplot(2,2,4) plot(lambda,Tps); grid on title('T_p_p') xlabel('lambda')

169

APPENDIX G

MATLAB Code for Mie Scattering Coefficients

% This function finds The electric and magnetic dipole components % or the Mie coefficients a1 and b1 and then the effective % permeability and permittivity %% binary alloy clc clear all close all c=3*10^8; theta=pi/6;% angle in radians

%------polaritonic material CuCl------f1=0.74; r1=35*10^-9; A=632; w0=2*pi*0.7751*10^15; lambda0=2*pi*c/w0; epsinf=5.59; gamma1=7.58*10^10; mu1=1;

%------plasmonic material Ag------f2=0.74; r2=10*10^-9; wp=2*pi*2.18*10^15; eps0=1; gamma2=2*pi*4.35*10^12; mu2=mu1;

%------host------eps_perp=1.96; eps_para=4; mu3=mu1; %------w=linspace(0.99,1.005,500)*w0; w_norm=w/w0; lambda=(3*10^8*2*pi)./w; eps1=epsinf+A*gamma1./(w0-w-i*gamma1); eps2=eps0*(1-wp^2./(w.^2+i*w*gamma2)); eps3=eps_para*eps_perp/(eps_para*cos(theta)^2+eps_perp*sin(theta)^2);

% ------x31=(eps3*mu3)^0.5*w*r1/c; x32=(eps3*mu3)^0.5*w*r2/c; x1=(eps1*mu1).^0.5.*w*r1/c;

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x2=(eps2*mu2).^0.5.*w*r2/c; sb1x1=myspbessel1(x1); sb1x2=myspbessel1(x2); srb1primex31=myriccatibessel1prime(x31); srb1primex32=myriccatibessel1prime(x32); sb1x31=myspbessel1(x31); sb1x32=myspbessel1(x32); srb1primex1=myriccatibessel1prime(x1); srb1primex2=myriccatibessel1prime(x2); sh1x31=mysphankel1(x31); sh1x32=mysphankel1(x32); srh1primex31=myriccatihankel1prime(x31); srh1primex32=myriccatihankel1prime(x32);

T11H=(sb1x1.*srb1primex31*mu1- sb1x31.*srb1primex1*mu3)./(sh1x31.*srb1primex1*mu3- sb1x1.*srh1primex31*mu1); T12H=(sb1x2.*srb1primex32*mu2- sb1x32.*srb1primex2*mu3)./(sh1x32.*srb1primex2*mu3- sb1x2.*srh1primex32*mu2);

T12E=(sb1x2.*srb1primex32.*eps2- sb1x32.*srb1primex2*eps3)./(sh1x32.*srb1primex2*eps3- sb1x2.*srh1primex32.*eps2); T11E=(sb1x1.*srb1primex31.*eps1- sb1x31.*srb1primex1*eps3)./(sh1x31.*srb1primex1*eps3- sb1x1.*srh1primex31.*eps1); mu1eff=mu3*((x31.^3-i*3*f1*T11H)./((x31.^3+i*1.5*f1*T11H))); eps1eff=eps3*((x31.^3-i*3*f1*T11E)./((x31.^3+i*1.5*f1*T11E))); eps2eff=eps3*((x32.^3-i*3*f2*T12E)./((x32.^3+i*1.5*f2*T12E))); mu2eff=mu3*((x32.^3-i*3*f2*T12H)./((x32.^3+i*1.5*f2*T12H)));

%% ------clc CA=0.68; CB=1-CA; fAB=(f1+f2)/2;%0.67; a1=3*i./(2*x31.^3).*fAB.*T11E; b1=3*i./(2*x32.^3).*fAB.*T12E; a2=3*i./(2*x31.^3).*fAB.*T11H; b2=3*i./(2*x32.^3).*fAB.*T12H; for m=1:length(a1) d1=-2*(CA+CB)*(1+a1(m))*(1+b1(m)); d2=CA*eps3*(2*(1-2*a1(m))*(1+b1(m))-(1+a1(m))*(1- 2*b1(m)))+CB*eps3*(2*(1-2*b1(m))*(1+a1(m))-(1+b1(m))*(1-2*a1(m))); d3=(CA+CB)*eps3^2*(1-2*a1(m))*(1-2*b1(m)); s=roots([d1 d2 d3]); epseff(m,1)=s(1,1); epseff(m,2)=s(2,1);

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e1=-2*(CA+CB)*(1+a2(m))*(1+b2(m)); e2=CA*mu3*(2*(1-2*a2(m))*(1+b2(m))-(1+a2(m))*(1- 2*b2(m)))+CB*mu3*(2*(1-2*b2(m))*(1+a2(m))-(1+b2(m))*(1-2*a2(m))); e3=(CA+CB)*mu3^2*(1-2*a2(m))*(1-2*b2(m)); s2=roots([e1 e2 e3]); mueff(m,1)=s2(1,1); mueff(m,2)=s2(2,1); end

% for m=1:length(a1) % s = eval(solve('CA*(eps3-x-a1(m)*(2*eps3+x))/(eps3+2*x- 2*a1(m)*(eps3-x))+ CB*(eps3-x-b1(m)*(2*eps3+x))/(eps3+2*x- 2*b1(m)*(eps3-x))=0','x')); % epseff(m,1)=s(1,1); % epseff(m,2)=s(2,1); % s= eval(solve('CA*(mu3-x-a2(m)*(2*mu3+x))/(mu3+2*x-2*a2(m)*(mu3- x))+ CB*(mu3-x-b2(m)*(2*mu3+x))/(mu3+2*x-2*b2(m)*(mu3-x))=0','x')); % mueff(m,1)=s(1,1); % mueff(m,2)=s(2,1); % end %% Method 1 (not sure which one is right ) neff1=epseff(:,1).^0.5.*mueff(:,1).^0.5; neff2=epseff(:,2).^0.5.*mueff(:,2).^0.5; neff3=epseff(:,1).^0.5.*mueff(:,2).^0.5; neff4=epseff(:,2).^0.5.*mueff(:,1).^0.5; figure subplot(2,2,1) plot(lambda,real(neff1),lambda,imag(neff1),'r--'); legend('Real(n_e_f_f)','Imag(n_e_f_f)') xlabel('\lambda') grid on; subplot(2,2,2) plot(lambda, real(neff2),lambda,imag(neff2),'r--'); legend('Real(n_e_f_f)','Imag(n_e_f_f)') xlabel('\lambda') grid on; subplot(2,2,3) plot(lambda,real(neff3),lambda,imag(neff3),'r--'); legend('Real(n_e_f_f)','Imag(n_e_f_f)') xlabel('\lambda') grid on; subplot(2,2,4) plot(lambda, real(neff4),lambda,imag(neff4),'r--'); legend('Real(n_e_f_f)','Imag(n_e_f_f)') xlabel('\lambda') grid on;

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