EFFECTIVE NONLINEAR SUSCEPTIBILITIES OF METAL-INSULATOR AND
METAL-INSULATOR-METAL NANOLAYERED STRUCTURES
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 Electro-Optics
By
Mallik Mohd Raihan Hussain
Dayton, Ohio
May, 4242 EFFECTIVE NONLINEAR SUSCEPTIBILITIES OF METAL-INSULATOR AND
METAL-INSULATOR-METAL NANOLAYERED STRUCTURES
Name: Hussain, Mallik Mohd Raihan
APPROVED BY:
Imad Agha, Ph.D. Andrew Sarangan, Ph.D. Advisory Committee Chairman Committee Member Associate Professor, Department of Professor, Department of Physics, and, Department of Electro- Electro-Optics and Photonics Optics and Photonics
Partha Banerjee, Ph.D. Michael Scalora, Ph.D. Committee Member Committee Member Professor and Department Chair, Research Physicist, Charles M. Department of Electro-Optics and Bowden Research Facility, Photonics AMRDEC, US Army RDECOM
Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research Dean, School of Engineering and Innovation Professor, School of Engineering
ii © Copyright by
Mallik Mohd Raihan Hussain
All rights reserved
4242 ABSTRACT
EFFECTIVE NONLINEAR SUSCEPTIBILITIES OF METAL-INSULATOR AND
METAL-INSULATOR-METAL NANOLAYERED STRUCTURES
Name: Hussain, Mallik Mohd Raihan University of Dayton
Advisor: Dr. Imad Agha
Nonlinear electromagnetic radiation (second and third harmonic) from the metal-insulator and metal-insulator-metal structures were measured and compared against predictions from the hydrodynamic models of plasmonics. This model incorporated higher order terms stem- ming from electron tunneling and nonlocality. This study shows that, besides the linear optical parameter like permittivity, conductivity etc, changes in the nonlinear optical pa- rameters, namely, second and third order susceptibilities (χ(2) and χ(3), respectively) can also be used to probe and compare the higher-order terms of the hydrodynamic model of plasmonics. Two insulator materials (ZnO and Al2O3) were used in two separate sets of experiments, and atomic layer deposition was used to cover the gold substrate with variable thicknesses of these insulator films (nanometer to sub-nanometer range). Large reduction in second and third harmonic signals was measured after the insulator film was deposited over the gold substrate revealing the spilled-out electronic states in the insulator region at the vicinity of the metal-insulator interface, which are dubbed metal insulator gap states.
Then, the metal-insulator samples were spin-coated with Au-nanoparticle solution to pre- pare a metal-insulator-metal structures. For these structures, saturation and quenching of the third harmonic efficiencies were observed which were indicative of the capping of E-field enhancement due to the existence of higher order terms in the hydrodynamic model that accounts for nonlocality and quantum tunneling of electrons. A generalized 4 × 4 matrix
iii method was utilized to calculate the effective χ(2) and χ(3) parameters that confirm the changes of effective material properties for ultra-thin films. These nonlinear coefficients, besides the linear permittivity and conductivity σ, can be a useful material parameter to study the effects of higher-order terms of hydrodynamic model.
iv To
my mentor,
Dr. Joseph W. Haus,
my parents,
Nasima Khatun
&
Mohd Shakhawat Hussain Mallik.
v ACKNOWLEDGMENTS
A lot has changed since I started my journey as a Ph.D. student. My ex-advisor, Dr.
Joseph Haus suddenly passed away on January 33, 423;. I would not have come this far without him. He used to challenge me with hard problems and give me the space to work on novel ideas. I always admired his openness and humble, close-to-earth demeanor during technical/nontechnical discussions. After Dr. Haus’s death, I was kindly picked up by my current advisor Dr. Imad Agha. I would like to thank him from the deepest of my heart for supporting me till the end. I appreciate his patience and encouragements during every meeting. I have truly found a friend in him and I am looking forward to grow this friendship into a life-long research collaboration tackling the frontiers of optics. I would like to extend my sincere gratitude towards Dr. Andrew Sarangan. His support through the Laser-assisted Si wet etch project has helped me immensely. I have learned a lot from his teaching style and attitude towards solving a problem. In October 423;, Dr. Partha
Banerjee suggested me to work with Dr. Behzad Bordbar on the ONR project and he finally offered me a position as a ’Research Engineer’. That opportunity has been a blessing to me and I can never thank Dr. Banerjee enough for trusting in me. I will keep working hard. I would like to thank Drs. Michael Scalora, Domenico de Ceglia, Maria Vincenti,
Parag Banerjee and Zhengning Gao for making me a part of a wonderful collaboration. I look forward to contribute more to this collaboration in future. I would like to thank Dr.
Shekhar Guha for promptly replying to my emails and queries. I found his book titled
“Laser Beam Propagation in Nonlinear Optical Media” to be very rigorous and detailed.
The book helped me with many concepts. I would also like to thank Dr. Qiwen Zhan for his insightful suggestions on how to incorporate evanescent/inhomogeneous waves in TMM.
I have had many friends in the department over the years. I would like to thank all of them for being my well-wishers. I would like to thank the University of Dayton for creating such
vi a warm sense of community and being my home away from home. Lastly, I would like to thank my dear wife Syeda Prem Ara Bahar for supporting me through thick and thin. The world is going through a pandemic. In such situation, I can not defend my dissertation in person and thank and shake hands with everyone who congratulates me. Yet, I feel everyone’s blessings are with me.
vii TABLE OF CONTENTS
ABSTRACT ...... iii
DEDICATION ...... v
ACKNOWLEDGMENTS ...... vi
LIST OF FIGURES ...... x
LIST OF TABLES ...... xv
CHAPTER I. INTRODUCTION ...... 3
3.3 Problem statement and novel contributions ...... 5 3.4 Organization of ideas ...... 7
CHAPTER II. LIGHT PROPAGATION IN STRATIFIED MEDIA ...... 9
4.3 Notation convention ...... 32 4.3.3 Subscripts and superscripts ...... 32 4.3.4 Vectors and scalars ...... 33 4.3.5 Tensors ...... 34 4.3.6 Matrices ...... 34 4.4 Material birefringence and layer geometry ...... 35 4.5 Maxwell’s equations ...... 37 4.6 Free mode calculation ...... 43 4.6.3 Free K-mode calculation ...... 44 4.6.4 Free E-mode calculation ...... 46 4.6.5 Boundary conditions for free modes ...... 4; 4.6.6 Reflectance and transmittance ...... 56 4.7 Bound mode calculation ...... 64 4.7.3 Bound K-mode calculation ...... 64 4.7.4 Bound E-mode calculation ...... 66 4.7.5 Boundary conditions with free and bound modes ...... 67 4.8 A interim summary of the TMM algorithm ...... 74 4.9 What is effective susceptibility? ...... 76 (2),2ω 4.9.3 deff calculation ...... 77 (3),3ω 4.9.4 deff calculation ...... 78 4.: Theoretical models of susceptibility ...... 79 4.:.3 Herman-Hayden model ...... 79 4.:.4 Effective medium model ...... 79 4.:.5 Nonlocality and tunneling ...... 82
CHAPTER III. EXPERIMENTS AND RESULTS ...... 83
5.3 Experimental setup ...... 83 5.4 Results ...... 86 5.4.3 Effective d(2),2ω calculation ...... 87 5.4.4 Effective d(3),3ω calculation ...... 8: 5.5 Electron density in metal-induced-gap-states ...... 92
viii 5.6 Effects of nonlocality and quantum tunneling ...... 95 5.7 Summary of findings ...... 96
CHAPTER IV. CONCLUSION ...... 98
BIBLIOGRAPHY ...... 9:
APPENDICES
A. Surface Characterization ...... :8
A.3 Thickness and growth rate ...... :8 A.4 Surface roughness ...... :9 A.5 Surface coverage ...... :; A.6 Chemical composition ...... :; A.7 Au nanoparticle distribution ...... ;2
B. Singular Value Decomposition ...... ;4
C. MATLAB Codes ...... ;5
ix LIST OF FIGURES
3.3 Miniaturization of electronics has opened a new paradigm of nanoplasmonics that can serve as an interface between electronics and photonics [38]...... 4
4.3 A stratified layered structure with arbitrary numbers of layers and layer thick- nesses...... 9
4.4 (a)Resonant metamaterial structures (e.g. split-ring resonators show resonance in dispersion relation) are used to design left-handed-materials, whereas, (b) a non-resonant metamaterial structure is designed to engineer exotic anisotropic behaviour...... :
4.5 The transformation between the laboratory axes system and the principle axes system of the crystal. The symbols of Euler angles (Φ, Θ, Ψ) are uppercase and color coded to green, red and green respectively. Symbols for angle of incidence and azimuthal angle (θ, φ) are lowercase...... 36
4.6 Vector diagram of EM fields (not to scale and color coded for easy correspondence). 38
4.7 An example case of the normal surface of a positive uniaxial medium with an arbitrary direction of oˆ with respect to the laboratory axes system (xˆyˆzˆ). Four (1),Ω (1),Ω free K-modes (Kl ) along with their angles (θl ) with respect to zˆ is marked. 44
4.8 Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of an isotropic material for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 47
4.9 Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of a positive uniaxial material (arbitrary orientation of optic axis, oˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 47
4.: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of a negative uniaxial material (arbitrary orientation of optic axis, oˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 48
4.; Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of a type-3 hyperbolic uniaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 48
4.32 Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of a type-4 hyperbolic uniaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 49
x 4.33 Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of positive biaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 49
4.34 Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of negative biaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 4:
4.35 Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of type-3 hyperbolic biaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 4:
4.36 Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of type-4 hyperbolic biaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦...... 4;
4.37 Generalized Snell’s law for (linear) free K-modes. The components of K parallel to the interface remains the same for all free K-modes. An isotropic/uniaxial (1),Ω interface is chosen for an example case. Both normal surfaces and Klk are marked for reference...... 52
4.38 Boundary condition of free E-modes for linear cases. The D¯, P¯andE¯ are pictori- ally represented...... 55
4.39 The reflectance (R) and transmittance (T ) of glass(n = 1.5)/Air(n = 1) for (a) s-polarized and (b) p-polarized light...... 59
4.3: The reflectance (R) and transmittance (T ) of Air(n = 1)/Calcite(no = 1.655, ne = 1.485) for (a) s-polarized and (b) p-polarized light. Optic axis direction is 45◦ from zˆ in the zˆyˆ plane. The results match with transmittance/reflectance found in Figure 5.5 of Reference [59]...... 5:
4.3; The reflectance (R) and transmittance (T ) of Air(n = 1)/calcite (no = 1.655, ne = 1.485)/Air(n = 1) for (a) s-polarized and (b) p-polarized light. The thick- ness of calcite is 1500 nm. The Euler angles of rotation to transform oˆ into the the direction of wˆ is [30◦, 60◦, 0◦]...... 5;
4.42 The reflectance (R) and transmittance (T ) of glass(n = 1.5)/Chiolite(no = 1.349, ne = 1.342) for (a) s-polarized and (b) p-polarized light. Optic axis direction is 45◦ from zˆ in the zˆyˆ plane. The results match with transmittance/reflectance found in Figure 5.7 of Reference [59]...... 62
xi 4.43 The reflectance (R) and transmittance (T ) of glass(n = 1.5)/Chiolite(no = 1.349, ne = 1.342)/air(n = 1) for (a) s-polarized and (b) p-polarized light. Optic axis direction is 45◦ from zˆ in the zˆyˆ plane. The thickness of Chiolite crystal is 1500 nm...... 63
4.44 Generalized Snell’s law for bound (nonlinear) cases. A uniaxial layer is chosen (1),Ω (2),Ω for an example case. The normal surfaces and Klk , Klk are marked for reference...... 65
4.45 Maker fringe pattern from a Quartz layer for second harmonic. Quartz layer thickness is 4477.4µm. The blue line is the simulated SH signal, black line is the envelope of the simulated SH signal and the red circles are experimental SH signal collected from Reference [3:, 63]...... 6;
4.46 Maker fringe pattern from a ZnO layer for SH signal. ZnO layer thickness is 511µm. The blue line is the simulated SH signal, black line is the envelope of the simulated SH signal and the red circles are experimental SH signal collected from Reference [3;]...... 6;
5.3 Experimental setup: Mode-locked Ti:Sapphire laser, chopper; HWP: half wave plate L3: focusing lens; L4: collimating lens; P3 and P4: filtering prisms; SPF: short-pass filter; L5: detector lens; PD: (silicon) photodiode [94]...... 83
5.4 The SH and TH signals are collected, first, from an ultra-flat Au surface, then, thin film of insulator (on top of the Au surface), and lastly, from a thin film of insulator coated with AuNPs...... 85
5.5 SH efficiencies measured for samples with (a) Al2O3 and Al2O3 + AuNP , and (b) ZnO and ZnO + AuNP as insulator layer...... 86
5.6 TH efficiencies measured for samples with (a) Al2O3 and Al2O3 + AuNP , and (b) ZnO and ZnO + AuNP as insulator layer...... 86
(2) 5.7 |d15 | for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures. . . . 87
(2) 5.8 |d31 | for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures. . . . 88
(2) 5.9 |d33 | for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures. . . . 88
(2) 5.: Re{deff } for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures. 89
(2) 5.; Im{deff } for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures. 89
(3) 5.32 |d16 | for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures. . . . 8:
xii (3) 5.33 Re{deff } for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures. 8;
(3) 5.34 Im{deff } for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures. 8;
5.35 The insulator layer (Al2O3) is considered as effective medium. The permittivity of Al2O3 is modified by the MIGS spillage and surface scattering due to the Au surface and, then, by the surface coverage ratio [:;]...... 92
5.36 The reflected SH efficiency (left axis) as a function of ALD Al2O3 cycles. Corre-
spondingly, the coverage factor, fAl2O3 (right axis) varies as a function of ALD cycles from 2, i.e. no Al4O5 coverage to 3, i.e. full Al4O5 coverage at cycle 69. The MIGS factor α has been shown to vary from a = 0.0 to a = 0.7...... 94
5.37 For different values of MIGS factor α, the RMS error between simulated and measured harmonic power is calculated. The minimum RMS error is found for MIGS factor α = 0.13...... 95
5.38 (a) Comparison between four different simulations. The Classical results omit nonlocality and quantum effects; the Quantum effects incorporate electron tun- neling and the nonlocal calculation incorporates spatial dispersion effects. The nonlocal + quantum curve includes both effects. (b) Field enhancement distri- bution near the gap region for an MIM with a gap size of 0.2nm for nonlocal + quantum effects...... 96
A.3 The growth rate (change of thickness per unit cycle of ALD) was measured for each insulator...... :8
A.4 The AFM image showing the topography of bare Au surface and Au surface
coated with (a) Al2O3 and (b) ZnO. Each image is labeled with the number of ALD cycles. The inset shows a length scale of 100nm. The bar on the right shows the heigt range...... :9
A.5 The schematic of nucleation and growth model. (a) pill box, (b) nucleation inside pill-boxes, The yellow represents Au and the white represents the deposited insulator, (c) critical point 3: when the nucleation circles touch each other and the sides of the pill box, (d) critical point 4: when the nucleation circles touch the corners of the pillbox. At this point the growth envelop is marked by the black curve...... ::
A.6 (a) The measured surface roughness of Al2O3 in the context of (b) the expected surface roughness from nucleation and growth model...... ::
A.7 XPS valence band spectra showing the Au valence band transitioning to the Zn (left) or Al (right) as the number of ALD cycle increases...... ;2
xiii A.8 The SEM image of Au surface spin-coated with Au nanoparticle solution. Au nanoparticles are (marked in green) randomly distributed and sparse enough to not cause any internal coupling of E-fields. The density of Au nanoparticles. . . ;3
xiv LIST OF TABLES
4.3 List of subscripts/superscripts...... 32
4.4 List of directions (unit vectors), angles, distances and planes related to the ge- ometry of the layers and material anisotropy (see Figure 4.5 for reference). . . 35
4.5 List of constants...... 37
4.6 List of vectors, directions, magnitude and angles related to EM fields...... 37
4.7 List of material tensors...... 39
4.8 List of matrices in the TMM engine [c() = cos(), s() = sin()]...... 72
xv CHAPTER I
INTRODUCTION
After the invention of laser in 3;82, the field of nonlinear light-matter interaction blos- somed to its fullest potential. Since then, researchers have used lasers to investigate the higher-order susceptibilities of materials in the optical frequency region. The linear prop- erties of material e.g. linear permittivity and conductivity provide insight into emerging properties like dispersion, absorption, electronic band structure etc. Similarly, the non- linear material properties i.e. susceptibilities carries the signature of material resonances that can only be probed at high intensity. Comprehensive understanding of different non- linear optical processes has created a whole range of applications [3–5], from academic research oriented tools to widespread industrial devices. For example, the second-order nonlinear effects are utilized in frequency doubled lasers [6], optical parametric oscillators
(OPOs) [7], optical parametric amplifiers (OPAs) [8], and quantum light sources [9]. The third-order nonlinear effects has given rise to self-phase and cross-phase modulation and intensity induced changes in refractive indices [3]. The process of four-wave mixing [:] is proven to be hugely beneficial in optical signal processing, optical switching [;], electro- optic modulators-demodulators [32], and optical correlators [33]. Also, second-harmonic imaging microscopy [34, 35] and surface second-harmonic generation to probe material in- terfaces [36, 37] are emerging as advanced characterization tools. While these applications readily justify further research in the field of nonlinear optics, the conjugation of nanoplas- monics with nonlinear optics needs further discussion to instigate completely new ideas of applications. Due to the high absorption coefficient of plasmonic materials in the optical frequency range, they may seem unsuitable at first. In spite of this apparent demerit, plas- monics offer several advantages. The benefits of plasmonic materials will be sought in light of the latest nanofabrication technology.
3 Figure 3.3: Miniaturization of electronics has opened a new paradigm of nanoplasmonics that can serve as an interface between electronics and photonics [38].
Plasmonics can be a natural interface for electronics and photonics in terms of size and operational speed (Figure 3.3), resulting into smaller, less costly and more efficient devices.
The advantages of introducing plasmonics to achieve nonlinear optical effects are two-fold:
(3) light confinement (i.e. large wavevectors along the plane of interface and large electric
field normal to the plane of interface may exist) and (4) surface or bulk plasmon-polariton modes. Although the high intensity of laser light has enabled photons of different frequencies to interact with each other through the bulk material, the efficiency of power exchange is limited by the phase-matching conditions, dispersion relations and degree of anisotropy of the material. Usually, the nonlinear optical devices that utilizes bulk material properties are much thicker then the wavelength of the light. With the advent of nanotechnology, photonic
4 crystals and metamaterials can be easily fabricated with thicknesses in range of or much smaller than the quarter-wavelength. Such length scales are not enough for efficient phase- matching. So, electric field localization i.e. large wavevector and high field enhancements are required. The introduction of metal or metal-like layers enables propagation modes with large wavevectors to exist causing highly confined light and, thus, more efficient light-matter interaction. Also, in the field of electronics, miniaturization has reached the nano-scale range. But the operation frequency e.g. the clock frequency is much slower then the optical frequency. The worlds of optics and electronics can be bridged by utilizing the physics of plasmonics. Further research in this field, namely, integrated optics, can enable nonlinear optical devices to be put into photonic integrated circuits reducing the footprint of regular nonlinear optical devices.
3.3 Problem statement and novel contributions
The discussion in the above section asserts the significance of combining nonlinear optics with the field of nanoplasmonics. With that motivation in mind, this dissertation attempts to address the topic of measuring effective nonlinear optical susceptibilities of nanostructures that contains metallic/plasmonic materials as components using a “transfer matrix method” technique. These nonlinear susceptibilities, as measured from the experiment, will also be derived from a detailed physical model known as hydrodynamic model. The novel contribution of this work is as follows:
• The effective values of second and third order susceptibilities were calculated from
the measured second and third harmonic efficiencies, respectively. In the literature,
the second and third order susceptibilities are measured using a well-known technique
called “Maker fringes technique” [39–42]. But this technique has not been developed
5 for measurement in reflection direction. Also, the analytic results in Reference [39]
only accounts for the first reflection from any interface. This dissertation addresses
both reflection and transmission measurements and can handle multiple reflections
at once. A generalized transfer matrix method (TMM) technique that incorporates
nonlinearity was employed to calculate these susceptibility values. Inside the TMM
engine, the propagation modes are sorted based on the real and imaginary part of the
component of wavevector along surface normal. Singular value deposition (SVD) was
utilized to calculate the electric field vectors. This work will soon be published.
• Effective medium theory was used to model the second order susceptibility due to
metal-interfaces. In comparison to the TMM technique (briefly introduced above;
complete explanation in Chapter II) where the susceptibility parameters were directly
fitted, a phenomenological parameter was fitted which is derived from a physical model
of susceptibility. This parameter, dubbed as metal-induced-gap-state parameter, in
turn, provided the areal density of free electrons in the vicinity of metal-insulator
interface. The work was published in Reference [43].
• A more detailed model was utilized to predict the second and third harmonic effi-
ciencies of metal-insulator-metal interfaces. The model, as mentioned above, is called
the hydrodynamic model and it accounts for the resonance terms from bound and
free electrons, convection terms, pressure terms or nonlocal terms and tunneling cur-
rent terms. The experimental confirmation of the comparative effects of these terms
in harmonic generation is also one of the novel contribution of this project. These
results were published in Reference [44].
6 3.4 Organization of ideas
This research offers three novel contributions as discussed above. The chapters of this dissertation has been organized to explain the theoretical background of each contribution.
In chapter II, light propagation in stratified media will be discussed. A generalized 4 × 4
transfer matrix method (TMM) is utilized to calculate Fresnel coefficients for any combi-
nation of isotropic and birefringent layers with arbitrary orientation of optic axes. The
method is then modified to include optical nonlinearity and generate Maker-fringe patterns
for thin layers. The formulae to calculate the second and third order susceptibilities for
(2),2ω (3),3ω second and third harmonic generation, respectively, (i.e. deff and deff , the symbols
are formally introduced in Chapter II, see Table 4.7 for quick reference) using the TMM
engines are introduced. Then, the hydrodynamic model is introduced to predict the sec-
ond and third harmonic generation. The Lorentz-Drude model is discussed that extends
to the hydrodynamic model including several higher order terms stemming from different
physical phenomena at nanometer length scales. Emphasis is given on electron Fermi pres-
sure terms and other diffusion-like terms that can result into a shift of resonance frequency
peaks and line broadening. Chapter III discusses the experiment in details. Then, the
(2),2ω (3),3ω deff and deff are calculated from the measured second and third harmonic radiation
originated from a thin-film of bare insulator (Al2O3 and ZnO) deposited on Au (metal-
insulator sample) and from the same insulator with metal-nanoparticles dispersed on top
(metal-insulator-metal sample). Measurements are done for different thicknesses of the
insulators. Then we predict the areal density of delocalized electrons in metal-induced-
gap-states from the SH efficiencies while taking the surface roughness and nucleation and
growth model of atomic layer deposition into account. Then, the measured second and
third harmonic efficiencies emanating from the thin-insulator films are compared against
7 the prediction of hydrodynamic model. Finally, in Chapter IV, conclusions were drawn and the scope of future work is discussed. Details of sample thickness measurement using ellipsometry, surface roughness characterization using atomic force microscopy, chemical composition characterization using X-ray photoelectron spectroscopy and Au nanoparticle distribution characterization using scanning electron microscopy are given in Appendix A.
The singular value decomposition is briefly described in Appendix B. The MATLAB codes are also appended at the end (Appendix C) for easy reference.
8 CHAPTER II
LIGHT PROPAGATION IN STRATIFIED MEDIA
The analysis of the propagation of light through a stratified media is crucial for designing
filters, waveguides, cavities, mirrors and anti-reflection/high-reflection coating in a required wavelength range [45, 46]. Figure 4.3 shows an arbitrary layered structures where the ith
layer has a thickness of ai. if the layers repeat periodically and the period thickness, a, is comparable to the quarter of the wavelength, Λ, inside that layer medium i.e. a ≈ Λ/4, the periodic layered structures are known as photonic crystals. For much thinner layers, the photons traversing the layers can not register them as separate medium. In stead, the layers can be combined into an “effective medium” with an “effective refractive index”.
The “effective medium theory” takes the geometry and symmetry properties of the layered structures into account [47] and calculates the corresponding “effective permittivitty”.
Figure 4.3: A stratified layered structure with arbitrary numbers of layers and layer thick- nesses.
9 In some case, if the layered structure shows resonance peaks in the dispersion curve for a certain frequency band, the structure is considered as a “resonant” metamaterial for that frequency range. These resonance peaks may be caused by the presence of specific geometry of metallic structures with the critical dimension, a, such that a ≈ Λ/10. Such structures are used to design materials with less-then-one and negative refractive index (left-handed materials) [48].
(a) (b)
Figure 4.4: (a)Resonant metamaterial structures (e.g. split-ring resonators show resonance in dispersion relation) are used to design left-handed-materials, whereas, (b) a non-resonant metamaterial structure is designed to engineer exotic anisotropic behaviour.
If a Λ/10, the stratified structures can be thought of as “non-resonant metamaterial”.
They are designed to achieve materials with effective anisotropy not found in the nature.
Also, exotic features like hyperbolic [49, 4:], epsilon-near-zero [4;] and zero-index [52] dispersion regions can be engineered by properly designing non-resonant metamaterials.
Our goal in this chapter is to develop a simulation tool, namely a 4 × 4 transfer matrix method (TMM), to study the electromagnetic wave propagation through layered structures
: containing any sort of birefrengent layers (with arbitrary direction of the optic axes) or isotropic layers. Then, we will modify this tool to include nonlinearity and calculate the second and third harmonic radiation traveling in both reflection and transmission directions.
The benchmarks used to check the accuracy of the TMM are:
• benchmark 3, Free mode calculation: Conservation of momentum and energy
gives rise to the boundary conditions for EM propagation interfaces. In benchmark 3,
the conservation of momentum is used to derive the generalized Fresnel equation [4]
and generalized Snell’s law [53]. All propagation modes were calculated and the shape
of the the indicatrix were assessed for accuracy. Next, Fresnel reflection and trans-
mission coefficients were calculated and reflectance and transmittance were checked
for conservation of energy. The reflectance and transmittance spectrum were also
compared against the reported values in the literature.
• benchmark 4, Bound mode calculation: While employing nonlinearity in our
TMM engine a significant milestone is to benchmark the results against the well-
known Maker’s fringe patterns [39, 3:]. The fringe patterns show the interference of
SH/TH reflected/transmitted from multiple interfaces.
These benchmarks will confirm the dependability of our TMM engine. We will use our
TMM engine in Chapter III to calculate effective nonlinear susceptibility. The algorithm used in this TMM engine is discussed in great details in Reference [54–57]. In the following sections the algorithm is revisited.
; 4.3 Notation convention
Before beginning the discussion on TMM, the notation conventions will be introduced.
In this section, the focus is put on the features (fonts, bold/unbold, capital/non-capital, hats, bars etc) of the symbols and their subscripts or superscripts, but the symbols of the specific parameters are not introduced. They were introduced later in the appropriate contexts. Also, Table 4.4, 4.5, 4.6, 4.7 and 4.8 enlists all the symbols used in the TMM
engine and can be used for quick reference. The symbols are classified as constants, scalars,
vectors, unit vectors (directions), tensors, matrices, angles and planes.
4.3.3 Subscripts and superscripts
Table 4.3 discusses the symbols and meanings of sub/superscripts of vectors and scalars
related to those vectors. The same subscripts and superscripts are also used to describe
tensors and matrices later. If a superscript is omitted in any equation then that equation
holds for all values of that superscript. The subscripts follow Einstein summation convention
in which repeated summation implied over terms that carry the same subscripts index in a
formula.
Table 4.3: List of subscripts/superscripts.
Symbol Meaning Set of choice 2: Rectified or dc ω: Fundamental frequency (FF), Ω Harmonic number 4ω: Second harmonic (SH), 5ω: Third harmonic (TH). 2 or vacuum, Layer number l 3, 4, 5, ... or, material’s name, or interface number l : l + n : lth to (l + n)th interface 3: Linear, E-mode: free, 4: Three wave mixing, E-mode: bound, h Number of wave mixed 5: Four wave mixing, E-mode: bound, 5+: Not studied.
32 Table 4.3 Continued from previous page Symbol Meaning Set of choice 3 to 6: if h = 1, K-modes k 3 to 32: if h = 2, see Section 4.6.3 3 to 42: if h = 3, c: Crystal axes system, a Axes system l: Laboratory axes system. 1 to 3: components along x,ˆ y,ˆ zˆ Components or u,ˆ v,ˆ wˆ respectively, c or, degree of freedom k: component in the xˆyˆ plane or plane of interface.
4.3.4 Vectors and scalars
Symbols used for vectors are uppercase letters with bold fonts (e.g. E). Unit vectors
(or, directions) are symbolized by the same letter in lowercase and bold fonts with a hat (
ˆ) on top (e.g. eˆ). Symbols used for scalar quantity are not in bold font (e.g. E). If the
scalar represents the magnitude of a field vector, the same uppercase letter is used. The
convention used for subscripts and superscripts for an example field vector, E and its scalar components E, is as follows:
(h),Ω,l E(h),Ω,l Ek,ac ∈ k,a , (4.3) v u 3 (h),Ω,l uX (h),Ω,l2 |Ek | = t Ek,ac , for a = l or c, (4.4) c=1 (h),Ω,l E(h),Ω,l (h),Ω,l eˆk,a = k,a /|Ek |, (4.5)
(h),Ω,l (h),Ω,l eˆk,ac ∈ eˆk,a . (4.6)
The angles (e.g. θ) have the sub/superscript as mentioned in Equation 4.3 but without the
(h),Ω,l component number (i.e. θk,a ). The angular frequency, ω, has no sub/superscript.
33 4.3.5 Tensors
Symbols used for tensors are not in bold font. They have a bar (¯) on top. The elements of a C1 × C2 × ... × Cn-dimensional tensor (e.g. ¯) is as follows:
(h),Ω,l (h),Ω,l ac1c2...cn ∈ ¯a , (4.7) where h, Ω, l are discussed in Table 4.3. Also, cn = 1, 2, 3, ..., Cn and Cn is the total degree of freedom in the nth dimension. The tensor notations can be contracted based on symmetry relations. For this study, all tensors are contracted into a 4-D matrix format. Therefore,
(h),Ω,l (h),Ω,l ac1c2 ∈ ¯a , (4.8) where, 1 to 3, if h = 1, c1 = 1 to 3, and c2 = 1 to 6, if h = 2, (4.9) 1 to 10, if h = 3.
The values of c2 depends on the number of waves mixed (i.e. h) which in turn gives the number of K-modes available. Please find more details in Section 4.6.3. The tensors are listed in Table 4.7 in the context of Maxwell’s equation and the constitutive relations.
4.3.6 Matrices
The matrices are represented by uppercase letters in double-struck font with a bar above
(e.g. M¯ ). All matrices are 4-dimensional. Thus,
M(h),Ω,l M¯ (h),Ω,l c1c2 ∈ C1×C2 , (4.:) where, c1 = 1, 2, ..., C1, c2 = 1, 2, ..., C2, and C1,C2 are the total degree of freedom. Also, h,
Ω, l represents the sub/superscript as discussed in Table 4.3. Abstract vectors (e.g. an array of values which does not represent any spatial components) are considered as 3-D matrices
(C2 = 1). The matrices are listed in Table 4.8 in the context of boundary conditions.
34 4.4 Material birefringence and layer geometry
Figure 4.3 represents the geometry of the layered structures studied for this research.
The thicknesses of layers and the special directions, planes and angles introduced by the anisotropy of materials are enlisted in Table 4.4.
Table 4.4: List of directions (unit vectors), angles, distances and planes related to the geometry of the layers and material anisotropy (see Figure 4.5 for reference).
Symbol Meaning Laboratory axes: xˆ, yˆ, zˆ Stratified media has C∞v symmetry. xˆyˆ is the plane of symmetry and zˆ is the surface/interface normal. Principle axes: The principle axes used for this study are determined based on the electric permittivity tensor. The principle axes of the crystal uˆ, vˆ, wˆ are different from its crystallographic axis. Also, the principle axes based on magnetic permeability and conductivity can be different in a generalized scenario. Such generalization is out of scope for this study. uˆl, vˆl, wˆl are the principle axes of the lth layer. Biaxial: oˆ1 and oˆ2 are the two optic axes lying in the uˆwˆ plane. oˆ is the direction that bisects the smaller angle between oˆ1 and oˆ2. oˆ k wˆ. Uniaxial: oˆ is the optic axis and oˆ k wˆ. oˆ Isotropic: Degenerate k-modes (i.e. oˆ ) exist in all directions. oˆl represents the bisection axis (biaxial) or optic axis (uniaxial) lth of the layer. Euler angles: Φ, Θ, Ψ These angles transform laboratory axes into principle axes. Φl, Θl, Ψl are the Euler angles of the lth layer. Rotation axis (compound): rˆ The axis of rotation of second Euler angle, Θ (see Figure 4.5) is the rotation axis. rˆl is the rotation axis of the lth layer. Plane of incidence: Generally, if multiple input beams are mixed, plane of reflection/ zˆkˆ transmission may not be parallel to the multiple planes of incidence. For this study, when φ = 0 (see Figure 4.5 and Table 4.6), zˆkˆ k zˆxˆ. Propagation distance or layer thickness: a The layer thickness (unit: m) is measured along zˆ (see Figure 4.5). al is the thickness of lth layer.
35 Figure 4.5: The transformation between the laboratory axes system and the principle axes system of the crystal. The symbols of Euler angles (Φ, Θ, Ψ) are uppercase and color coded to green, red and green respectively. Symbols for angle of incidence and azimuthal angle (θ, φ) are lowercase.
xˆyˆ plane is the plane of interface and zˆ is the surface normal. θ is the angle of incidence between zˆ and kˆ. φ is the azimuthal angle between xˆzˆ and kˆzˆ planes. When φ = 0,
xˆzˆ k kˆzˆ. The transformation from laboratory axes system (xˆ − yˆ− zˆ) to the principle axes system of the crystal (uˆ − vˆ − wˆ) uses Euler angles (Ψ, Θ, Φ). Both axes systems are color coded to red-blue-green, respectively. The primed (0) and double-primed (00) xˆ − yˆ − zˆ are the intermediary versions of the axes system after each rotation operation. The order of rotation operation is: R¯ z(Φ) (around zˆ) → R¯ x(Θ) (around rˆ) → R¯ z(Ψ) (around wˆ). The xˆ −yˆ plane (plane of interface) is colored white and the uˆ −vˆ plane of the crystal is colored gray for comparison. The broken yellow line marking the intersection of xˆ − yˆ and uˆ − vˆ planes is the direction of the rotation axis, rˆ and xˆ0 k rˆ.
36 4.5 Maxwell’s equations
In this section, the preliminaries of Maxwell’s equations are concisely discussed. The constants, scalars, vectors and tensors that go into the Maxwell’s equation will be introduced in the following few tables. First, the constants used in the TMM engine are given in Table
4.5. Table 4.5: List of constants.
Symbol Value Meaning Unit (S.I.) c 2.997924 × 108 Speed of light ms−1 0 8.854 × 10−12 Vacuum permeability C2s2kg−1m−3 µ0 4π × 10−7 Vacuum permittivity kgmC−2 √ j −1 imaginary unit ∼
Next, the field vectors and the corresponding scalars and angles are listed in Table 4.6.
Table 4.6: List of vectors, directions, magnitude and angles related to EM fields.
Vectors, unit vectors and magnitudes related to EM fields Symbol Meaning Unit (S.I.) (Vector/Unit Vector/Magnitude) E, eˆ, |E| Electric field V m−1 D, dˆ, |D| Electric displacement Cm−2 H, hˆ, |H| Magnetic field Am−1 B, ˆb, |B| Magnetic displacement V s2m−2 P, pˆ, |P | Polarization Cm−2 J, ˆj, |J| Current density Am−2 K, kˆ, |K| Wavevector rad m−1 ˆ 2π Wavelength Λ, λ, |Λ| = |K| m S, sˆ, |S| Poynting vector W m−2 Angles related to EM fields Angle of K-modes: θ This angle is between interface degree normal, zˆ and wavevector, kˆ Azimuthal angle: This angle is between xˆzˆ plane φ degree and plane of incidence, zˆkˆ. φ = 0, unless otherwise stated. Walk-off angle: w This angle is between sˆ and kˆ, degree or, eˆ and dˆ.
37 Table 4.6 Continued from previous page Symbol Meaning unit (S.I.) Angular frequency: - Rectified/DC: 2, - Fundamental frequency: ω, Ω - Second Harmonic: 2ω, rad s−1 - Third harmonic: 3ω, . . - nth harmonic: nω. Charge density: ρ ρl is the charge density Cm−2 of the lth layer.
Figure 4.6: Vector diagram of EM fields (not to scale and color coded for easy correspon- dence).
The EM radiation is assumed to be a plane wave with wavevector K. The materials of all
the layers are nonmagnetic (¯µ = ¯I3×3) and have no free currents and charges (J = 0, ρ = 0).
For any angular frequency (Ω), the Maxwell’s equations can be written as follows:
K × E = −ΩB, (4.;)
K × H = ΩD, (4.32)
K · B = 0, (4.33)
K · D = 0. (4.34)
Also, the time averaged Poynting vector is,
1 hSi = Re{E × H∗}. (4.35) 2
38 Figure 4.6 describes the relative orientation of field vectors. E, D, K and S lie on the same
plane and B, H are perpendicular to this plane. Also, E and S are perpendicular to each
other and K and S are perpendicular to each other. The angle between E and D, or, S
and K is known as walk-off angle.
w = cos−1(eˆ.dˆ) = cos−1(sˆ.kˆ), and − π/2 ≤ w ≤ π/2. (4.36) K θ = cos−1 l3 , and 0 ≤ θ ≤ π. (4.37) |Kl|
The material tensors will be listed in Table 4.7. These tensors will be used in the constitutive relations next.
Table 4.7: List of material tensors.
Symbol Meaning Refractive index [3 × 3]: - In crystal axes (uˆ, vˆ, wˆ), (1) n¯ nu 0 0 (1) n¯c = 0 nv 0 , where nu, nv, nw ∈ {all complex number}. 0 0 nw Electric permittivity (relative) [3 × 3]: In any axes system, ¯(1) = 1 +χ ¯(1).
- In crystal axes (uˆ, vˆ, wˆ), (1) c11 0 0 (1) (1) (1) (1) 2 (1) ∆12 = Re{c11} − Re{c22}, ¯c = (¯nc ) = 0 c22 0 , and, (1) (1) . (1) ∆23 = Re{c22} − Re{c33}, 0 0 c33 Biaxial (1) (1) (1) : c11, c22, c33 are sorted such that (1) (1) (1) (1) (1) ¯ , χ¯ Re{c11} < Re{c22} < Re{c33}. Positive: ∆12 > ∆23, Negative: ∆12 < ∆23. Uniaxial (1) (1) (1) : c11, c22, c33 are sorted such that (1) (1) Re{c11} − Re{c22} = ∆12 = 0. (1) (1) Positive: ∆23 > 0 and Re{c22} ≥ 0 , or, ∆23 < 0 and Re{c33} < 0 , (1) (1) Negative: ∆23 > 0 and Re{c22} < 0 , or, ∆23 < 0 and Re{c33} ≥ 0 . Isotropic (1) (1) (1) : Re{c11} = Re{c22} = Re{c33}.
- In laboratory axes (xˆ, yˆ, zˆ), (1) A¯ (1) A¯ −1 A¯ ¯l = 3×3¯c ( 3×3) . (see Table 4.8 for )
39 Table 4.7 Continued from previous page Symbol Meaning Electric susceptibility (second order) [3, 4]: [unit: m/V ] [Expanded: 3 × 3 × 3, Contracted: 3 × 6] Contracted 3 × 6 version is used, unless otherwise specified. In any axes system, 1 optical rectification: d¯(2),0 = χ¯(2),0. 2 1 second harmonic generation: d¯(2),2ω = χ¯(2),2ω. 2 d¯(2), χ¯(2) - In crystal axes (uˆ, vˆ, wˆ), (2) (2) (2) (2) (2) (2) dc11 dc12 dc13 dc14 dc15 dc16 ¯(2) (2) (2) (2) (2) (2) (2) dc = dc21 dc22 dc23 dc24 dc25 dc26. (2) (2) (2) (2) (2) (2) dc31 dc32 dc33 dc34 dc35 dc36
- In laboratory axes (xˆ, yˆ, zˆ), ¯(2) A¯ ¯(2) A¯ −1 A¯ dl = 3×3dc ( 6×6) . (see Table 4.8 for ) Electric susceptibility (third order) [3, 58]: [unit: m2/V 2] [Expanded: 3 × 3 × 3 × 3, Contracted: 3 × 10] Contracted 3 × 10 version is used, unless otherwise specified. In any axes system , 1 optical Kerr effect: d¯(3),ω = χ¯(3),ω. 3 1 third harmonic generation: d¯(3),3ω = χ¯(3),3ω. 3 d¯(3), χ¯(3) - In crystal axes (uˆ, vˆ, wˆ), (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) dc11 dc12 dc13 dc14 dc15 dc16 dc17 dc18 dc19 dc1,10 ¯(3) (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) dc = dc21 dc22 dc23 dc24 dc25 dc26 dc27 dc28 dc29 dc2,10. (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) dc31 dc32 dc33 dc34 dc35 dc36 dc37 dc38 dc39 dc3,10
- In laboratory axes (xˆ, yˆ, zˆ), ¯(3) A¯ ¯(3) A¯ −1 A¯ dl = 3×3dc ( 10×10) . (see Table 4.8 for ) Magnetic permeability (relative) [3 × 3]: Non-magnetic media is assumed. µ¯ In both crystal axes (uˆ, vˆ, wˆ) and laboratory ¯ ¯ axes (xˆ, yˆ, zˆ), µ¯c =µ ¯l = I3×3. (see Table 4.8 for I)
The constitutive relationships in laboratory axes system (xˆyˆzˆ) can be written as follows:
0 Dl = El + Pl, (4.38)
0 Bl = µ µ¯lHl. (4.39)
3: Here, Linear Nonlinear z }| { z }| { n (1),Ω X (h),Ω Pl = Pl + Pl , (4.3:) h=2,3 The values of Ω and h, along with their interpretation, are given in Table 4.3. Section 4.6.3 and 4.7.3 constitute the detailed discussion on the number of free and bound P modes. We assume no depletion at fundamental frequency (Ω = ω). Thus, only free E-modes (h = 1) exist and optical Kerr effects and other higher order mixing (i.e. h > 1) at Ω = ω is ignored.
In laboratory axis, at fundamental frequency (Ω = ω),
4 4 (1),ω X (1),ω (1),ω 0 X (1),ω (1),ω (1),ω Pl = Pk,l exp jKk,l3 z = χ¯l Ek,l exp jKk,l3 z , (4.3;) k=1 k=1
(1),ω where, χ¯l is given in Table 4.7.
In laboratory axis, at second harmonic frequency: (Ω = 2ω), - For free wave (h = 1):
4 4 (1),2ω X (1),2ω (1),2ω 0 X (1),2ω (1),2ω (1),2ω Pl = Pk,l exp jKk,l3 z = χ¯l Ek,l exp jKk,l3 z , (4.42) k=1 k=1
(1),2ω where, χ¯l is given in Table 4.7. - For bound wave (h = 2):
10 (2),2ω X (2),2ω (2),2ω Pl = Pk,l exp jKk,l3 z k=1 (1),ω (1),ω m · Ep,l1 Eq,l1 (1),ω (1),ω m · E E p,l2 q,l2 4 4 (1),ω (1),ω 0 X X (2),2ω (1),ω (1),ω m · Ep,l Eq,l = 2 d¯ exp j K + K z 3 3 , l p,l3 q,l3 (1),ω (1),ω (1),ω (1),ω p=1 q=p m · (Ep,l2 Eq,l3 + Eq,l2 Ep,l3 ) m · (E(1),ωE(1),ω + E(1),ωE(1),ω) p,l1 q,l3 q,l1 p,l3 (1),ω (1),ω (1),ω (1),ω m · (Ep,l2 Eq,l1 + Eq,l2 Ep,l1 ) (4.43)
(1, if p = q, where, d¯(2),2ω is given in Table 4.7 and m = l 2, if p 6= q.
3; In laboratory axis, at third harmonic frequency: (Ω = 3ω), - For free wave (h = 1):
4 4 (1),3ω X (1),3ω (1),3ω 0 X (1),3ω (1),3ω (1),3ω Pl = Pk,l exp jKk,l3 z = χ¯l Ek,l exp jKk,l3 z , (4.44) k=1 k=1
(1),3ω where, χ¯l is given in Table 4.7. - For bound wave (h = 3):
20 (3),3ω X (3),3ω (3),3ω Pl = Pk,l exp jKk,l3 z k=1 4 4 4 0 X X X ¯(3),3ω (1),ω (1),ω (1),ω = 3 dl exp j Kp,l3 + Kq,l3 + Kr,l3 z × p=1 q=p r=q (1),ω (1),ω (1),ω m · Ep,l1 Eq,l1 Er,l1 (1),ω (1),ω (1),ω m · E E E p,l2 q,l2 r,l2 (1),ω (1),ω (1),ω m · Ep,l3 Eq,l3 Er,l3 (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω Ep,l2 Eq,l3 Er,l3 + Ep,l3 Eq,l2 Er,l3 + ... m · (1),ω (1),ω (1),ω ...Ep,l3 Eq,l3 Er,l2 (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω Ep,l3 Eq,l2 Er,l2 + Ep,l2 Eq,l3 Er,l2 + ... m · (1),ω (1),ω (1),ω ...Ep,l2 Eq,l2 Er,l3 (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω Ep,l3 Eq,l3 Er,l1 + Ep,l3 Eq,l1 Er,l3 + ... m · (1),ω (1),ω (1),ω ...Ep,l1 Eq,l3 Er,l3 , (4.45) (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω Ep,l1 Eq,l1 Er,l3 + Ep,l1 Eq,l3 Er,l1 + ... m · (1),ω (1),ω (1),ω ...Ep,l3 Eq,l1 Er,l1 (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω Ep,l1 Eq,l2 Er,l2 + Ep,l2 Eq,l1 Er,l2 + ... m · (1),ω (1),ω (1),ω ...Ep,l2 Eq,l2 Er,l1 (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω Ep,l1 Eq,l1 Er,l2 + Ep,l1 Eq,l2 Er,l1 + ... m · (1),ω (1),ω (1),ω ...Ep,l1 Eq,l1 Er,l2 (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω Ep,l1 Eq,l2 Er,l3 + Ep,l1 Eq,l3 Er,l2 + ... (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω m · ...E E E + E E E + ... p,l2 q,l1 r,l3 p,l2 q,l3 r,l1 (1),ω (1),ω (1),ω (1),ω (1),ω (1),ω ...Ep,l3 Eq,l2 Er,l1 + Ep,l3 Eq,l1 Er,l2 1, if p = q = r, ¯(3),3ω where, dl is given in Table 4.7 and m = 3, if p = q, or, q = r, or, r = p, 6, if p 6= q 6= r.
42 4.6 Free mode calculation
In this section, we introduce the wave equation in matrix format. The wave equation can be derived from Equation 4.;, 4.32, 4.38 and 4.39 as:
2 0 0 K × K × E − Ω µ µˆl E = P , (4.46)
As discussed in Table 4.3, the superscript h represents the number of waves mixed. h = 1 represents the linear modes of propagation for any frequency Ω. All the K-modes for h = 1 are known as free K-modes. Each free K-mode corresponds to a free E-modes. Since we are assuming no depletion of power of the fundamental frequency (ω) compared to the power transferred into the mixed modes, we are only interested in the mixing of waves with fundamental frequency. Mixing only fundamental frequencies give rise to higher harmonics and AC Kerr effect. Only second and third harmonic generation are of interest for this study.
For the free modes of any frequency (Ω = ω, 2ω, or, 3ω), Equation 4.46 can be written as,
(1),Ω (1),Ω (1),Ω 2 0 0 (1),Ω (1),Ω Kl × Kl × El − Ω µ El = Pl , (4.47)
(1),Ω (1),Ω (1),Ω 2 0 0 (1),Ω (1),Ω =⇒ Kl × Kl × El − Ω µ ¯l El = 0, (4.48) | {z } | {z } U(1),Ω V(1),Ω 3×3 3×3 where,
(1),Ω 2 (1),Ω 2 (1),Ω (1),Ω (1),Ω (1),Ω (Kl2 ) + (Kl3 ) −Kl1 Kl2 −Kl1 Kl3 (1),Ω (1),Ω (1),Ω (1),Ω (1),Ω (1),Ω (1),Ω U = −K K (K )2 + (K )2 −K K , (4.49) 3×3 l2 l1 l1 l3 l2 l3 (1),Ω (1),Ω (1),Ω (1),Ω (1),Ω 2 (1),Ω 2 −Kl3 Kl1 −Kl3 Kl2 (Kl1 ) + (Kl2 ) and, Ω2 V(1),Ω = Ω2µ00¯(1),Ω = ¯(1),Ω. (4.4:) 3×3 l c2 l
The wave matrix, W¯ = U¯ − V¯. Thus, from Equation 4.48, 4.49, 4.4:, the following can be
written,
43 W¯ (1),Ω (1),Ω 3×3 El = 0, (4.4;) W¯ (1),Ω (1),Ω or, 3×3 eˆl = 0. (4.52)
4.6.3 Free K-mode calculation
Equation 4.4; is the matrix format of wave propagation equation derived from Maxwell’s equations and constitutive relations. For Equation 4.52 to have a nontrivial solution, the determinant of W¯ should be zero. Thus,
W¯ (1),Ω det( 3×3 ) = 0. (4.53)
(a)
Figure 4.7: An example case of the normal surface of a positive uniaxial medium with an arbitrary direction of oˆ with respect to the laboratory axes system (xˆyˆzˆ). Four free (1),Ω (1),Ω K-modes (Kl ) along with their angles (θl ) with respect to zˆ is marked.
44 For any ith layer, Figure 4.7 shows the solution of Equation 4.53 graphically for a certain
(1),Ω (1),Ω value of Kl|| . The unknowns are the Kl3 . A positive uniaxial medium is chosen as an example case. The four K-modes of propagation is marked. To increase the readability of the solution process, following symbols are chosen for the time being.
(1),Ω α := Kl1 , (4.54)
(1),Ω β := Kl2 , (4.55)
(1),Ω γ := Kl3 , and (4.56)
V(1),Ω Vpq ∈ 3×3 , (4.57)
where p, q are the row and column positions of V¯. To solve Equation 4.53, the matrix W¯ is
partitioned in to an unknown part (named, B¯) containing γ and a known part (named, B¯0)
containing the rest. Thus, −γ2 0 −αγ B¯ 2 = 0 −γ −βγ , and, (4.58) −αγ −βγ 0
0 0 0 2 B11 B12 B13 β − V11 −αβ − V12 −V13 B¯0 0 0 0 2 = B21 B22 B23 = −αβ − V21 α − V22 −V23 . (4.59) 0 0 0 2 2 B31 B32 B33 −V31 −V23 α + β − V33 Therefore, from Equation 4.53, the following can be written,
det B¯ + B¯0 = 0, (4.5:)
4 3 2 =⇒ c4γ + c3γ +c2γ + c1γ + c0 = 0. (4.5;)
Here,
0 2 2 c4 =B33 + α + β , (4.62)
0 0 0 0 c3 =β B23 + B32 + α B13 + B31 , (4.63)
0 0 0 0 0 0 0 c2 = B23 + B32 + B13 + B31 − B33 B11 + B22 −
0 2 0 2 0 0 B11β − B22α + B12 + B21 αβ (4.64)
45 0 0 0 0 0 0 0 0 c1 =α(−B31B22 + B21B32 − B13B22 + B12B23)+
0 0 0 0 0 0 0 0 β B13B21 + B12B31 − B11B32 − B11B23 (4.65)
0 0 0 0 0 0 0 0 0 0 c0 = − B13 B31B22 − B21B32 + B12 B23B31 − B21B33
0 0 0 0 0 − B11(B23B32 − B22B33) (4.66)
Since Equation 4.5; is a fourth order polynomial, it has four solutions for γ. The solutions may be degenerate or distinct, real, imaginary or complex. These four γ solutions provide
(1),Ω the four free K-modes i.e. K1−4,l. The free K-modes are always sorted in ascending order of Re{γ} (when Im{γ} = 0) or Re{sˆl3} (when Im{γ} 6= 0). Thus, the sorted order of free transmitted slow(t-) transmitted fast(t+) K-modes looks like: for the case when all free K-modes are distinct reflected fast(r+) reflected slow(r-) transmitted p-polarized(tp) transmitted s -polarized(ts) and for the case when degenerate K-modes exist. It is very reflected s-polarized(rs) reflected p-polarized(rp) important to keep track of the sorting order to avoid ambiguities in K-modes.
4.6.4 Free E-mode calculation
In the above discussion, four free K-modes were calculated. For each K-mode (or each value of γ), Equation 4.53 can provide us the free E-modes. The free E-modes are numer- ically calculated by using singular value deposition (SVD) algorithms (see Appendix B for some details of the SVD method). The following figures (Figure 4.8 to 4.36) show the index ellipsoid of materials with different birefringence types. The surface of the index ellipsoid also contains free E-mode polarization directions. The E directions are parallel to the index ellipsoid surfaces and the length of the vector is not to scale.
46 Figure 4.8: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of an isotropic material for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦.
Figure 4.9: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of a positive uniaxial material (arbitrary orientation of optic axis, oˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦. 47 Figure 4.:: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of a negative uniaxial material (arbitrary orientation of optic axis, oˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦.
Figure 4.;: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of a type-3 hyperbolic uniaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦.
48 Figure 4.32: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of a type-4 hyperbolic uniaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦.
Figure 4.33: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of positive biaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦.
49 Figure 4.34: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of negative biaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦.
Figure 4.35: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of type-3 hyperbolic biaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦.
4: Figure 4.36: Index ellipsoid indicating the directions of free K-modes and free E-modes (scaled for easy visualization) of type-4 hyperbolic biaxial material (for optic axis, oˆ k zˆ) for 0◦ ≤ θ < 90◦ and 0◦ ≤ φ < 360◦.
4.6.5 Boundary conditions for free modes
Boundary conditions can also be discussed in terms of K-modes and E-modes. The
boundary condition of K-mode takes conservation of momentum into account. According
to the K-mode boundary condition, the component of wavevector along the xˆyˆ plane for
all the K-modes should be equal at the interfaces For example, the boundary condition for
the interface of the ith and (i + 1)th layer is,
(1),Ω,i (1),Ω,i+1 Kk,lk = Kk,lk , (4.67)
where, the subscript k = 1, 2, 3, or, 4. This is also known as “Snell’s law”. Figure 4.37
graphically explains the Snell’s law for a isotropic-uniaxial (positive) interface. Notice, this
(1),Ω,i Kk,lk component is constant across all the layers and is set by the incident angle in the
(1),Ω,1 first (i = 1) layer i.e. θ1/2,lk .
4; (a)
Figure 4.37: Generalized Snell’s law for (linear) free K-modes. The components of K parallel to the interface remains the same for all free K-modes. An isotropic/uniaxial interface is (1),Ω chosen for an example case. Both normal surfaces and Klk are marked for reference.
52 Meanwhile, the boundary condition of E-modes accounts for the conservation of energy.
In laboratory axes system, for any ith layer,
(h),Ω,i X (h),Ω,i El = Ek,l , (4.68) k (h),Ω,i X (h),Ω,i Hl = Hk,l , (4.69) k
0 where Ω, h, k are given in Table 4.3, c and µ is given in Table 4.5 and µl is given in Table
4.7. Using Equation 4.; and 4.39, the Equation 4.69 can be written in terms of E field vectors as,
(h),Ω,i c (h),Ω,i (h),Ω,i Hk,l = 0 Kk,l × Ek,l Ωµ µl c (h),Ω,i (h),Ω,i (h),Ω,i = 0 Kk,l × eˆk,l |Ek |, (4.6:) Ωµ µl | {z } (h),Ω,i hk,l
(h),Ω,i c (h),Ω,i (h),Ω,i where, h is defined as hk,l := 0 Kk × eˆk . Notice, h is a vector along H, Ωµ µ¯l but it is not the unit vector of H. Thus, the hat (ˆ) is missing. Equation 4.68 and 4.69 can be rewritten as,
(h),Ω,i X (h),Ω,i (h),Ω,i El = eˆk,l |Ek |, (4.6;) k (h),Ω,i X (h),Ω,i (h),Ω,i Hl = hk,l |Ek |, (4.72) k
Equation 4.6; and 4.72 is showing the resultant E and H fields generated by the vector addition of all the free E-modes. According to the free E-mode boundary condition, the components of the resultant E and H field along the xˆyˆ plane should be equal at the interfaces. Apparently,
53 (h),Ω,i X (h),Ω,i (h),Ω,i El1 = eˆk,l1 |Ek |, (4.73) k (h),Ω,i X (h),Ω,i (h),Ω,i Hl2 = hk,l2 |Ek |, (4.74) k (h),Ω,i X (h),Ω,i (h),Ω,i El2 = eˆk,l2 |Ek |, (4.75) k (h),Ω,i X (h),Ω,i (h),Ω,i Hl1 = hk,l1 |Ek |. (4.76) k
Three new matrices, E¯, D¯ and P¯ will be introduced next. E¯ is matix of size 4 × 1. Matrix notation is chosen for E¯ so that it can be differentiated from a vector in physical space. E¯ consists of the magnitude of the four free E-modes. Thus, in any ith layer,
(h),Ω,i |E1 | (h),Ω,i (1),Ω,i |E | E¯ = 2 . (4.77) 4×1 (h),Ω,i |E3 | (h),Ω,i |E4 |
D¯ (1),Ω,i Secondly, the 4×4 matrix is defined as,
(1),Ω,i (1),Ω,i (1),Ω,i (1),Ω,i eˆ1,l1 eˆ2,l1 eˆ3,l1 eˆ4,l1 (1),Ω,i (1),Ω,i (1),Ω,i (1),Ω,i h h h h D¯ (1),Ω,i 1,l2 2,l2 3,l2 4,l2 4×4 = (1),Ω,i (1),Ω,i (1),Ω,i (1),Ω,i . (4.78) eˆ1,l2 eˆ2,l2 eˆ3,l2 eˆ4,l2 (1),Ω,i (1),Ω,i (1),Ω,i (1),Ω,i h1,l1 h2,l1 h3,l1 h4,l1
Thirdly, the matrix P¯ accounts for the phase accumulation along zˆ due to propagation
through ith layer that has a thickness of ai.
(h),Ω,i i exp(−jK1,l3 a ) 0 0 0 (h),Ω,i 0 exp(−jK ai) 0 0 P¯(1),Ω,i 2,l3 4×4 = (h),Ω,i 0 0 exp(−jK ai) 0 3,l3 (h),Ω,i i 0 0 0 exp(−jK4,l3 a ) (4.79)
Figure 4.38 is a visualization of the boundary condition at the interface of the (i−1)th layer and the ith layer.
54 (a)
Figure 4.38: Boundary condition of free E-modes for linear cases. The D¯, P¯andE¯ are picto- rially represented.
55 According to the free E-mode boundary condition,
D¯ (1),Ω,i−1E¯(1),Ω,i−1 D¯ (1),Ω,iP¯(1),Ω,iE¯(1),Ω,i 4×4 4×1 = 4×4 4×4 4×1 , (4.7:) −1 E¯(1),Ω,i−1 D¯ (1),Ω,i−1 D¯ (1),Ω,iP¯(1),Ω,iE¯(1),Ω,i 4×1 = 4×4 4×4 4×4 4×1 , (4.7;) E¯(1),Ω,i−1 M¯ (1),Ω,i−1:iE¯(1),Ω,i 4×1 = 4×4 4×1 . (4.82)
M¯ (1),Ω,i−1:i Here, the matrix 4×4 is defined as,
−1 M¯ (1),Ω,i−1:i D¯ (1),Ω,i−1 D¯ (1),Ω,iP¯(1),Ω,i 4×4 = 4×4 4×4 4×4 . (4.83)
And, for a total of N layers,
M¯ (1),Ω,1:N M¯ (1),Ω,1:2M¯ (1),Ω,2:3 M¯ (1),Ω,N−1:N 4×4 = 4×4 4×4 ... 4×4 , (4.84) E¯(1),Ω,1 M¯ (1),Ω,1:N E¯(1),Ω,N and, 4×1 = 4×4 4×1 (4.85)
M¯ (1),Ω,1 This 4×4 matrix is the 4 × 4 anisotropic transfer matrix of the TMM engine. To confirm that the engine works, the reflectance and transmittance for a range of incident angle and for several combinations of stratified layers are calculated using the engine and checked for energy conservation. Few reflectance and transmittance are also compared against the values found in the literature [59].
4.6.6 Reflectance and transmittance
transmitted slow(t-) transmitted fast(t+) The choice of sorting order for the TMM engine developed above is reflected fast(r+) reflected slow(r-) transmitted p-polarized(tp) transmitted s -polarized(ts) for the case when all free K-modes are distinct and when two reflected s-polarized(rs) reflected p-polarized(rp) or more K-modes are degenerate. But, in the TMM developed by Yeh [5:], the choice of
56 reflected s-polarized(rs) transmitted s -polarized(ts) sort order is To use the reflection and transmission formula reflected p-polarized(rp) transmitted p-polarized(tp) derived by Yeh [5:], the following transformations are done to re-sort,
0 − Y¯ (1),Ω,1:N S¯ 1 M¯ (1),Ω,1:N S¯ 4×4 = 4×4 , (4.86) |K1,l| c −1 S¯ |K2,l| Refractive index, nY = , (4.87) Ω |K3,l| |K4,l| |θ1,l| −1 S¯ |θ2,l| K-mode angles, θY = , (4.88) |θ3,l| |θ4,l| |w1,l| −1 S¯ |w2,l| Walk-off angles, wY = , (4.89) |w3,l| |w4,l| 0 0 1 0 S¯ 1 0 0 0 where, the Yeh’s matrix, = . The reflection and transmission coefficient for 0 1 0 0 0 0 0 1 a layered structure is as follows:
Y21Y33 − Y23Y31 r12 = , (4.8:) Y11Y33 − Y13Y31 Y41Y33 − Y43Y31 r14 = , (4.8;) Y11Y33 − Y13Y31 Y11Y23 − Y21Y13 r32 = , (4.92) Y11Y33 − Y13Y31 Y11Y43 − Y41Y13 r34 = , (4.93) Y11Y33 − Y13Y31 Y33 t11 = , (4.94) Y11Y33 − Y13Y31 −Y31 t13 = , (4.95) Y11Y33 − Y13Y31 −Y13 t31 = , (4.96) Y11Y33 − Y13Y31 Y11 t33 = . (4.97) Y11Y33 − Y13Y31
In this context, the interpretation of subscripts 1, 2, and 3 of reflection and transmission
coefficients depend on the birefrigence type and orientation of the optic axis of the first
57 and last layer of the structure. The subscripts of the Yeh’s matrix indicate the row and column number of the element in Yeh’s matrix. The reflectance and transmittance can be calculated using the following formulae:
1 1 1 1 2 Re{nY 2cos θY 2 + wY 2 cos wY 2 } R12 = |r12| 1 1 1 1 , (4.98) Re{nY 1cos θY 1 − wY 1 cos wY 1 } 1 1 1 1 2 Re{nY 4cos θY 4 + wY 4 cos wY 4 } R14 = |r14| 1 1 1 1 , (4.99) Re{nY 1cos θY 1 − wY 1 cos wY 1 } 1 1 1 1 2 Re{nY 2cos θY 2 + wY 2 cos wY 2 } R32 = |r32| 1 1 1 1 , (4.9:) Re{nY 3cos θY 3 − wY 3 cos wY 3 } 1 1 1 1 2 Re{nY 4cos θY 4 + wY 4 cos wY 4 } R34 = |r34| 1 1 1 1 , (4.9;) Re{nY 3cos θY 3 − wY 3 cos wY 3 } 2 2 2 2 2 Re{nY 1cos θY 1 − wY 1 cos wY 1 } T11 = |t11| 1 1 1 1 , (4.:2) Re{nY 1cos θY 1 − wY 1 cos wY 1 } 2 2 2 2 2 Re{nY 3cos θY 3 − wY 3 cos wY 3 } T13 = |t13| 1 1 1 1 , (4.:3) Re{nY 1cos θY 1 − wY 1 cos wY 1 } 2 2 2 2 2 Re{nY 1cos θY 1 − wY 1 cos wY 1 } T31 = |t31| 1 1 1 1 , (4.:4) Re{nY 3cos θY 3 − wY 3 cos wY 3 } 2 2 2 2 2 Re{nY 3cos θY 3 − wY 3 cos wY 3 } T33 = |t33| 1 1 1 1 . (4.:5) Re{nY 3cos θY 3 − wY 3 cos wY 3 }
In the absence of absorption in the media, the total power flow should be conserved. Thus,
P1 = −R12 − R14 + T11 + T13 = 1, (4.:6)
P3 = −R32 − R34 + T31 + T33 = 1. (4.:7)
In the next few figures (Figure 4.39a to 4.42b) the reflectance and transmittance for several
layered structures are calculated. The subscripts are changed to s/p or +/- values to
conform with the notations used in the literature.
58 (a)
(b)
Figure 4.39: The reflectance (R) and transmittance (T ) of glass(n = 1.5)/Air(n = 1) for (a) s-polarized and (b) p-polarized light.
59 (a)
(b)
Figure 4.3:: The reflectance (R) and transmittance (T ) of Air(n = 1)/Calcite(no = 1.655, ◦ ne = 1.485) for (a) s-polarized and (b) p-polarized light. Optic axis direction is 45 from zˆ in the zˆyˆ plane. The results match with transmittance/reflectance found in Figure 5.5 of Reference [59]. 5: (a)
(b)
Figure 4.3;: The reflectance (R) and transmittance (T ) of Air(n = 1)/calcite (no = 1.655, ne = 1.485)/Air(n = 1) for (a) s-polarized and (b) p-polarized light. The thickness of calcite is 1500 nm. The Euler angles of rotation to transform oˆ into the the direction of wˆ is [30◦, 60◦, 0◦].
5; (a)
(b)
Figure 4.42: The reflectance (R) and transmittance (T ) of glass(n = 1.5)/Chiolite(no = ◦ 1.349, ne = 1.342) for (a) s-polarized and (b) p-polarized light. Optic axis direction is 45 from zˆ in the zˆyˆ plane. The results match with transmittance/reflectance found in Figure 5.7 of Reference [59]. 62 (a)
(b)
Figure 4.43: The reflectance (R) and transmittance (T ) of glass(n = 1.5)/Chiolite(no = 1.349, ne = 1.342)/air(n = 1) for (a) s-polarized and (b) p-polarized light. Optic axis direction is 45◦ from zˆ in the zˆyˆ plane. The thickness of Chiolite crystal is 1500 nm.
63 4.7 Bound mode calculation
To calculate the bound modes, the sequence of discussion will be as same as before.
The discussion bound K-modes is followed by the discussion on bound E-mode calculation which leads to boundary conditions.
4.7.3 Bound K-mode calculation
When multiple waves are mixed e.g. second/third harmonic (SH/TH) generation or sum/difference frequency generation, the newly produced frequency propagates in the non- linear media. An important question follows the above statement. Since there is no incident wave at the newly generated frequency, how does this wave follow Snell’s law and main- tain boundary conditions? The answer to this question is the concept of bound waves.
Bound waves play the same “role” in satisfying the boundary condition as the incident wave at fundamental frequency plays for the case free waves at fundamental frequency. In a sense, bound waves are the source of the transverse component of K-modes along the interface. Similar to the mixing of angular frequencies, for a certain frequency the free
K-modes also mix in a number of ways. The total number of bound modes depend on the total number of free-K-modes of the source frequencies. For example, for the case of SHG, the fundamental frequency has 4 free K-modes. Thus, these 4 free fundamental 4(4 + 1) K-modes can be mixed into P4 k = = 10 bound K-modes for SH generation. k=1 2 1 For the case of THG, the free FF K-modes can be mixed into P4 k + P4 k2 = 2 k=1 k=1 1 4(4 + 1) 4(4 + 1)(2 · 4 + 1) + = 20 bound K-modes for TH generation. Such mixing 2 4 6 happens through the P polarization vector. Figure 4.44 graphically presents the meaning of
Equation 4.:8 and the generation of bound and free K-modes for SH. Similar visualization
for TH can also be done. The bound K-modes are derived from the free K-modes.
64 Figure 4.44: Generalized Snell’s law for bound (nonlinear) cases. A uniaxial layer is chosen (1),Ω (2),Ω for an example case. The normal surfaces and Klk , Klk are marked for reference.
65 For any ith layer, the generalized Snell’s law that accounts free or bounds modes are:
K(1),2ω,i K(2),2ω,i For SH: K(1),ω,i = p,lk = k,lk , (4.:8) p,lk 2 2 (2),2ω (1),ω (1),ω and, Kk = Kp + Kq , (4.:9) where, k = 1, 2, 3, ..., 10, (4.::) p = 1, 2, 3, 4, (4.:;) q = p, p + 1, ...4. (4.;2)
K(1),3ω,i K(2),3ω,i For TH: K(1),ω,i = p,lk = k,lk , (4.;3) p,lk 3 3 (3),3ω (1),ω (1),ω (1),ω and, Kk = Kp + Kq + Kr , (4.;4) where, k = 1, 2, 3, ..., 20, (4.;5) p = 1, 2, 3, 4, (4.;6) q = p, p + 1, ...4, (4.;7) r = q, q + 1, ...4. (4.;8)
4.7.4 Bound E-mode calculation
For bound modes of nth harmonic generated from h wave mixing, the wave equation from Equation 4.4; is modified into a inhomogeneous wave equation as follows:
W¯ (h),nω (h),nω (h),nω 3×3 Ek,l = Pk,l , (4.;9) −1 (h),nω W¯ (h),nω (h),nω Ek,l = 3×3 Pk,l , (4.;:) E(h),nω eˆ(h),nω = k,l (4.;;) k,l (h),nω |Ek,l | (h),nω c (h),nω (h),nω hk,l := 0 Kk,l × eˆk,l (4.322) nωµ µ¯l
66 Here,
W¯ (h),nω U¯ (h),nω V¯ (h),nω 3×3 = 3×3 − 3×3 . (4.323)
(h),nω 2 (h),nω 2 (h),nω (h),nω (h),nω (h),nω (Kl2 ) + (Kl3 ) −Kl1 Kl2 −Kl1 Kl3 (h),nω (h),nω (h),nω (h),nω (h),nω (h),nω (h),nω U¯ = −K K (K )2 + (K )2 −K K , 3×3 l2 l1 l1 l3 l2 l3 (h),nω (h),nω (h),nω (h),nω (h),nω 2 (h),nω 2 −Kl3 Kl1 −Kl3 Kl2 (Kl1 ) + (Kl2 ) (4.324)
(nω)2 and, V(h),nω = (nω)2µ00¯(h),nω = ¯(h),nω. (4.325) 3×3 l c2 l
(2),2ω (2),2ω For SH: n = 2, and h = 2. El derived from Equation 4.;:. Equation 4.43 takes El
(2),2ω as input and provides the Pl .
(3),3ω (3),3ω For TH: n = 3, and h = 3. El derived from Equation 4.;:. Equation 4.45 takes El
(3),3ω as input provides the Pl .
4.7.5 Boundary conditions with free and bound modes
To include nonlinearity to the TMM engine, the boundary conditions need to be mod- ified. But once the bound E-modes are included into the boundary condition, the TMM does not remain as a linear relation between input and output for the higher harmonics.
The total (free and bound) E-mode boundary conditions are developed below.
For the sake of brevity, only the boundary conditions for the SH is derived and explained.
The boundary condition for TH follows the same derivation as SH.
For second harmonic, h = 2 and n = 2. Following the discussion of 4.6.5, the matrix
E¯(2),2ω,i 10×1 containing the magnitude of all the bound E-modes are defined as, (2),2ω,i |E1 | (2),2ω,i |E2 | E¯(2),2ω,i (2),2ω,i 10×1 := |E3 | . (4.326) . . . (2),2ω,i |E10 |
67 D¯ (2),2ω,i The dynamic matrix 4×10 for bound modes of SH is defined as,
(2),2ω,i (2),2ω,i (2),2ω,i (2),2ω,i eˆ1,l1 eˆ2,l1 eˆ3,l1 ... eˆ10,l1 (2),2ω,i (2) (2),2ω,i (2),2ω,i h h h . . . h D¯ (2),2ω,i 1,l2 2,l2 3,l2 10,l2 4×10 := (2),2ω,i (2),2ω,i (2),2ω,i (2),2ω,i . (4.327) eˆ1,l2 eˆ2,l2 eˆ3,l2 ... eˆ10,l2 (2),2ω,i (2) (2),2ω,i (2),2ω,i h1,l1 h2,l1 h3,l1 . . . h10,l1
P¯(2),2ω,i And, the propagation matrix 10×10 for bound modes of SH is defined as,
(2),2ω,i i exp(−jK1,l3 a ) 0 ... 0 (2),2ω,i i . (2),2ω,i 0 exp(−jK a ) . P¯ := 2,l3 . (4.328) 10×10 . . . .. 0 (h),2ω,i i 0 0 ... exp(−jK10,l3 a )
D¯ (1),2ω,i P¯(1),2ω,i At SH frequency, the free dynamic matrix, 4×4 and propagation matrix 4×4 is defined in Equation 4.77 and 4.79, respectively. Now, at the boundary of (i−1)th layer and ith layer, the component of the total (free and bound) E and H field along the interface remains the same. Thus, the boundary condition for SH can be written in matrix format as:
D¯ (1),2ω,i−1E¯(1),2ω,i−1 D¯ (2),2ω,i−1E¯(2),2ω,i−1 4×4 4×1 + 4×10 10×1 =
D¯ (1),2ω,iP¯(1),2ω,iE¯(1),2ω,i D¯ (2),2ω,iP¯(2),2ω,iE¯(2),2ω,i 4×4 4×4 4×1 + 4×10 10×10 10×1 , (4.329)
−1 E¯(1),2ω,i−1 D¯ (1),2ω,i−1 D¯ (1),2ω,iP¯(1),2ω,iE¯(1),2ω,i =⇒ 4×1 = 4×4 4×4 4×4 4×1
−1 D¯ (1),2ω,i−1 D¯ (2),2ω,iP¯(2),2ω,iE¯(2),2ω,i + 4×4 4×10 10×10 10×1
−1 D¯ (1),2ω,i−1 D¯ (2),2ω,i−1E¯(2),2ω,i−1 − 4×4 4×10 10×1 ,
E¯(1),2ω,i−1 M¯ (1),2ω,i−1:iE¯(1),2ω,i N¯ (2),2ω,i−1:iE¯(2),2ω,i =⇒ 4×1 = 4×4 4×1 + 4×4 10×1
O¯ (2),2ω,i−1:iE¯(2),2ω,i − 4×4 10×1 . (4.32:)
68 Here the matrices M¯ , N¯ and O¯ are introduce, so that,
−1 M¯ (1),2ω,i−1:i D¯ (1),2ω,i−1 D¯ (1),2ω,iP¯(1),2ω,i 4×4 = 4×4 4×4 4×4 , (4.32;)
−1 N¯ (2),2ω,i−1:i D¯ (1),2ω,i−1 D¯ (2),2ω,iP¯(2),2ω,i 4×10 = 4×4 4×10 10×10 , (4.332)
−1 O¯ (2),2ω,i−1 D¯ (1),2ω,i−1 D¯ (2),2ω,i−1 4×10 = 4×4 4×10 . (4.333)
For a total of N layers in which n layers have nonlinearity, the total system equation can be written in matrix format as,
E¯(1),2ω,1 M¯ (1),2ω,1:N E¯(1),2ω,N C¯ (2),2ω 4×1 = 4×4 4×1 + 4×1 . (4.334)
where,
C¯ (2),2ω X C¯ (2),2ω,i 4×1 = 4×1 , and, n = {all nonlinear layer numbers}. (4.335) i∈n if the 1st layer (i = 1) is nonlinear, then
C¯ (2),2ω,1 O¯ (2),2ω,1 E¯(2),2ω,1 4×1 = 4×10 · 10×1 . (4.336) if the last layer i.e. N th layer (i = N) is nonlinear, then
C¯ (2),2ω,N M¯ (2),2ω,1:N−1 N¯ (2),2ω,N−1:N E¯(2),2ω,N 4×1 = 4×4 · 4×10 · 10×1 . (4.337)
Otherwise, for any ith nonlinear layer,
C¯ (2),2ω,i M¯ (1),2ω,1:i−1 N¯ (2),2ω,i−1:i M¯ (1),2ω,1:i O¯ (2),2ω,i E¯(2),2ω,i 4×1 = 4×4 · 4×10 − 4×4 · 4×10 · 10×1 . (4.338)
. (1),2ω,1 (1),2ω,N |E1 | |E1 | (1),2ω,1 (1),2ω,N |E | (1),nω,1:N |E | (2),2ω 2 = M¯ 2 + C¯ . (4.339) (1),2ω,1 4×4 (1),2ω,N 4×1 |E3 | |E3 | (1),2ω,1 (1),2ω,N |E4 | |E4 | The last layer i.e. N th layer does not have any power in the reflection K-modes (mode 5
and 6). Also, the first layer does not have any power in the transmission K-modes (mode 3
69 and mode 4), since the harmonics are generated inside the layers and there is no incident light at the second harmonic frequency. Equation 4.339 becomes,
0 (1),2ω,N |E1 | (1),2ω,N 0 (1),2ω,1:N (2),2ω M¯ |E2 | C¯ (1),2ω,1 = 4×4 + 4×1 . (4.33:) |E3 | 0 (1),2ω,1 |E4 | 0
Thus, the transmitted E-modes of SH are,
" # " #!−1 " # |E(1),2ω,N | M(1),2ω,1:N M(1),2ω,1:N C(2),2ω 1 = − 11 12 1 . (4.33;) (1),2ω,N M(1),2ω,1:N M(1),2ω,1:N C(2),2ω |E2 | 21 22 2
Thus, the reflected E-modes of SH are, " # " # |E(1),2ω,1| C(2),2ω 3 = 3 (1),2ω,1 C(2),2ω |E4 | 4 −1 "M(1),2ω,1:N M(1),2ω,1:N # "M(1),2ω,1:N M(1),2ω,1:N #! "C(2),2ω# − 31 32 11 12 1 . M(1),2ω,1:N M(1),2ω,1:N M(1),2ω,1:N M(1),2ω,1:N C(2),2ω 41 42 21 22 2 (4.342)
Equation 4.33; and 4.342 can be written for TH E-modes following the same derivation procedure discussed above. Although the derivation is not shown here, the TMM codes were written for the TH too and both SH and TH were calculated.
If the magnitude of the E-field is known (from Equations 4.342 and 4.33;) the average power, P can be written as, 1 P = c0Atf (|E|)2 . (4.343) 2
Here, A = spot size of the laser beam, t = pulse width of the laser beam, f = repetition rate of laser pulse. Since the different modes of the SH/TH field interfere with each other, the reflected or transmitted power shows a fringe patter (peaks and troughs) for a range of incident angle of the FF beam. Next, two example cases were picked up from the literature [3:, 3;, 5;, 62] to confirm that the TMM is correctly predicting the reflected and
6: transmitted power. The match between actual experimental values and the prediction from
TMM is very close. Finally, all the different matrices are summarized for a quick reference in Table 4.8.
Figure 4.45: Maker fringe pattern from a Quartz layer for second harmonic. Quartz layer thickness is 4477.4µm. The blue line is the simulated SH signal, black line is the envelope of the simulated SH signal and the red circles are experimental SH signal collected from Reference [3:, 63].
Figure 4.46: Maker fringe pattern from a ZnO layer for SH signal. ZnO layer thickness is 511µm. The blue line is the simulated SH signal, black line is the envelope of the simulated SH signal and the red circles are experimental SH signal collected from Reference [3;].
6; Table 4.8: List of matrices in the TMM engine [c() = cos(), s() = sin()].
Symbol Meaning 1 0 0 ¯I Identity matrix: ¯I e.g. 3×3 = 0 1 0 0 0 1 Euler rotation matrix:
R¯ n×n is an Euler rotation matrix in n-degrees of freedom (see Figure 4.5). - In 5 degrees of freedom [64]: A¯ 3×3 = R¯ z,3×3(Ψ)R¯ x,3×3(Θ)R¯ z,3×3(Φ), and, c(Ψ) s(Ψ) 0 1 0 0 R¯ R¯ z,3×3(Ψ) = −s(Ψ) c(Ψ) 0, x,3×3(Θ) = 0 c(Θ) s(Θ), and, 0 0 1 0 −s(Θ) c(Θ) c(Φ) s(Φ) 0 R¯ z,3×3(Φ) = −s(Φ) c(Φ) 0. 0 0 1 - In 8 degrees of freedom [65]: A¯ 6×6 = R¯ z,6×6(Ψ)R¯ x,6×6(Θ)R¯ z,6×6(Φ), and, c2(Ψ) s2(Ψ) 0 0 0 2c(Ψ)s(Ψ) s2(Ψ) c2(Ψ) 0 0 0 −2c(Ψ)s(Ψ) R¯ 0 0 1 0 0 0 z,6×6(Ψ) = , 0 0 0 c(Ψ) s(Ψ) 0 A¯ 0 0 0 −s(Ψ) c(Ψ) 0 −c(Ψ)s(Ψ) c(Ψ)s(Ψ) 0 0 0 c2(Ψ) − s2(Ψ) c2(Θ) 0 s2(Θ) 0 2c(Θ)s(Θ) 0 0 1 0 0 0 0 2 2 R¯ s (Θ) 0 c (Θ) 0 −2c(Θ)s(Θ) 0 x,6×6(Θ) = , 0 0 0 c(Θ) 0 −s(Θ) 2 2 −c(Θ)s(Θ) 0 c(Θ)s(Θ) 0 c (Θ) − s (Θ) 0 0 0 0 s(Θ) 0 c(Θ) c2(Φ) s2(Φ) 0 0 0 2c(Φ)s(Φ) s2(Φ) c2(Φ) 0 0 0 −2c(Φ)s(Φ) R¯ 0 0 1 0 0 0 z,6×6(Φ) = . 0 0 0 c(Φ) s(Φ) 0 0 0 0 −s(Φ) c(Φ) 0 −c(Φ)s(Φ) c(Φ)s(Φ) 0 0 0 c2(Φ) − s2(Φ) - In 32 degrees of freedom:
A¯ 10×10 = R¯ z,10×10(Ψ)R¯ x,10×10(Θ)R¯ z,10×10(Φ). The R¯ 10×10 matrices were not studied. Wave matrix: W¯ The wave equation (homogeneous or inhomogeneous) can be termed in matrix format. See Equation 4.4;, 4.;9
72 Table 4.8 Continued from previous page Symbol Meaning E-modes: E¯ The solution of homogeneous wave equation are the free E-modes. The solution of inhomogeneous wave equation are the bound E-modes. See Equation 4.4;, 4.;9 Dynamic matrix: Dynamic matrix transforms the singular E and H modes into components along xˆ and yˆ in the plane of interface. - For h = 1, (1) (1) (1) (1) eˆ1,l1 eˆ2,l1 eˆ3,l1 eˆ4,l1 (1) (1) (1) (1) h h h h D¯ (1) 1,l2 2,l2 3,l2 4,l2 4×4 = (1) (1) (1) (1) . eˆ1,l2 eˆ2,l2 eˆ3,l2 eˆ4,l2 (1) (1) (1) (1) h1,l1 h2,l1 h3,l1 h4,l1 - For h = 2, D¯ (2) (2) (2) (2) eˆ1,l1 eˆ2,l1 eˆ3,l1 ... eˆ10,l1 (2) (2) (2) (2) h h h . . . h D¯ (2) 1,l2 2,l2 3,l2 10,l2 4×10 = (2) (2) (2) (2) . eˆ1,l2 eˆ2,l2 eˆ3,l2 ... eˆ10,l2 (2) (2) (2) (2) h1,l1 h2,l1 h3,l1 . . . h10,l1 - For h = 3, (3) (3) (3) (3) eˆ1,l1 eˆ2,l1 eˆ3,l1 ... eˆ20,l1 (3) (3) (3) (3) h h h . . . h D¯ (2) 1,l2 2,l2 3,l2 20,l2 4×20 = (3) (3) (3) (3) . eˆ1,l2 eˆ2,l2 eˆ3,l2 ... eˆ20,l2 (3) (3) (3) (3) h1,l1 h2,l1 h3,l1 . . . h20,l1 Propagation matrix: Propagation matrix introduces phase to the singular E and H modes due to propagation in the bulk. - For h = 1, (1) (1) (1) (1) P¯ = ¯I4×4 · [exp −jK a exp −jK a exp −jK a ... = 4×4 1,l3 2,l3 3,l3 (1) T ... exp −jK4,l3a ] P¯ - For h = 2, P¯(2) ¯I (1) (1) (1) 10×10 = 10×10 · [exp −jK1,l3a exp −jK2,l3a exp −jK3,l3a ... (1) T ... exp −jK10,l3a ] - For h = 3, P¯(2) ¯I (1) (1) (1) 20×20 = 20×10 · [exp −jK1,l3a exp −jK2,l3a exp −jK3,l3a ... (1) T ... exp −jK20,l3a ]
73 Table 4.8 Continued from previous page Symbol Meaning Transfer matrix or M matrix: The transfer matrix that transforms E-modes of ith layer into E-modes of (i − 1)th layer (i.e. E¯(1),i−1 = M¯ (1),i−1:iE¯(1),i) is defined as: M¯ M¯ (1),i−1:i = (D¯ (1),i−1)−1 · D¯ (1),i · P¯(1),i (If i = 1, then, M¯ (1),i−1:i = P¯(1),i). Again, the transfer matrix that transforms E-modes of nth layer into E-modes of the 1st layer is: M¯ (1),1:n Qn M¯ (1),i−1:i M¯ (1),1:2 M¯ (1),2:3 M¯ (1),n−1:n = i=1 = · · ... · . N matrix: th N¯ If the i layer is a nonlinear layer, then, N¯ (h),i−1:i = (D¯ (1),i−1)−1 · D¯ (h),i · P¯(h),i. For second harmonic, h = 2, and for third harmonic, h = 3. O matrix: th O¯ If the i layer is a nonlinear layer, then, O¯ (h),i = (D¯ (1),i)−1 · D¯ (h),i For second harmonic, h = 2, and for third harmonic, h = 3. Constant matrix: if the 1st layer is nonlinear, then C¯ (h),1 = O¯ (h),1 · E¯(h),1. if the last layer i.e. N th layer is nonlinear, then C¯ (h),N M¯ (h),1:N−1 N¯ (h),N−1:N E¯(h),N C¯ = · · . Otherwise, for any ith nonlinear layer, C¯ (h),i = (M¯ (h),1:i−1 · N¯ (h),i−1:i − M¯ (h),1:i · O¯ (h),i) · E¯(h),i. For second harmonic, h = 2, and for third harmonic, h = 3. If n = {l : lth layer exhibit nonlinearity}, C¯ (h) P C¯ (h),i = i∈n . Yeh’s Matrix: The algorithm developed in this dissertation uses elaborate sorting of the E-modes to avoid ambiguities resulting in discontinuities in transmittance or reflectance vs incident angle plot. But the sort order of chosen by Yeh was different. Y¯ Thus the Yeh’s matrix is the following: 0 0 1 0−1 0 0 1 0 Y¯ 1 0 0 0 M¯ 1 0 0 0 = . 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1
4.8 A interim summary of the TMM algorithm
Before moving forward, a quick overview of the algorithm will be discussed below to summarize the key points of the algorithm. The MATLAB codes are given in Appendix C.
74 3. The objective is to calculate the E-field of the nth harmonic frequency i.e. nω gen-
erated by h = n wave mixing at the transmission and reflection side of the stratified
media. The E-field can be used to calculate the power which is needed to compare
against the Maker’s fringe pattern.
4. Find free-K-modes of fundamental wavelength by solving det W¯ (1),ω = 0,
5. Find free-E-modes of fundamental wavelength by solving W¯ (1),ωE(1),ω = 0. The
magnitudes of the solutions form E¯(1),ω.
6. Find D¯ (1),ω, P¯(1),ω and M¯ (1),ω. The input/output relation of the linear system is:
E¯(1),ω,1 = M¯ (1),ω,1:N E¯(1),ω,N
7. Find E¯(1),ω,i at all the interfaces for a given input power, angle of incidence and
polarization.
8. Find P¯(h),nω,i using the constitutive relations.
−1 9. Find E(n),ω,i by using E(n),ω,i = W¯ (h),nω,i P (h),nω,i. The magnitudes of the solu-
tions form E¯(h),nω.
:. Find D¯ (h),nω, P¯(h),nω and M¯ (1),nω. The input/output relation of the linear system is:
E¯(1),ω,1 = M¯ (1),ω,1:N E¯(1),ω,N + C¯ (h)
The E-fields of the harmonic can be calculated from this equation which in turn gives
the power radiated in the reflection and transmission direction.
75 4.9 What is effective susceptibility?
The nonlinear susceptibility d introduced in Chapter II (see Table 4.7) is a third rank
(3 × 3 × 3) tensor with 49 components. The objective is to shrink down the d¯ matrix as much as possible. One trick is to utilize intrinsic permutation symmetries. This reduces the number of independent tensor components to 3: and allows the use of a contracted (3 × 6) matrix. Other material symmetry conditions (e.g. Kleinmann symmetry due to operating in frequency ranges far from any resonance frequency, crystal symmetries due to molecular structures, C∞v symmetry of layered structures) reduce the number of independent nonzero elements even further. A detailed description of symmetry relations can be found in [66,
(h),nω 67]. The simplifications introduced by deff further reduces the matrix into a scalar value. These are discussed in great detail in Chapter 5 of [68]. Essentially, the objective of introducing effective susceptibility is to rewrite Equation 4.43 and 4.45 in terms of unit vectors eˆ and take all the directional components of the formula into the definition of
(h),nω ¯(h),nω deff . So, it is a sum over the nonzero values of the d matrix, weighted by the applicable angular projections of the components of the unit vectors of free E-modes, i.e.
(1),ω (1),nω th eˆk,l of fundamental frequency and eˆk,l of the n harmonic. Graphically, the meaning of deff can be explained using Figure 4.38. For given K-mode and E-mode at fundamental frequency, there exists several bound K-modes (a total of 10 in figure 4.38, since it is showing second harmonic boundary conditions, rays marked in green). For a certain bound K-mode,
(h),nω (1),nω the components of Pk,l along one of the free modes eˆk,l can be termed as the effective
(h),nω polarization Peff along that free mode direction. Therefore, deff can be such that,
76 (1),2ω (h),nω 0 (h),nω Y (1),ω Peff := eˆk,l · Pk,l = 2 deff |Ep,l |, (4.344) n (h),nω (1),2ω ¯(h),nω (1),ω (1),ω (1),ω =⇒ deff := eˆk,l ·d : eˆk,l eˆk,l ... eˆk,l , (4.345) | {z } | {z } Free E-modes generated Free E-modes mixed
where the subscript k of each variable is independent of each other and can assume any
(2),2ω (3),3ω value listed in Table 4.3. In the following subsections, the values of deff and deff will
be discussed for isotropic layers.
4.9.3 (2),2ω calculation deff
(2),2ω This study uses isotropic layers with C∞v. For such case, the deff is defined in
Equation (A4:) of Reference [69,6:]. Those relations will be derived here for clarification.
(2),2ω Considering C∞v, the direction xˆ and yˆ bears no difference and d¯ becomes, 0 0 0 0 d15 0 ¯(2),2ω d = 0 0 0 d15 0 0 (4.346) d31 d31 d33 0 0 0
(2),2ω Using Equation 4.344 and 4.43, the deff can be defined as,
(1),ω (1),ω m · eˆk,l1 eˆk,l1 (1),ω (1),ω T m · eˆ eˆ eˆ(1),2ω k,l2 k,l2 k,l1 (1),ω (1),ω (2),2ω (1),2ω (2),2ω m · eˆk,l eˆk,l d = eˆ d¯ 3 3 . (4.347) eff k,l2 (1),ω (1),ω (1),ω (1),ω (1),2ω m · (ˆek,l2 eˆk,l3 +e ˆk,l2 eˆk,l3 ) eˆ k,l3 m · (ˆe(1),ωeˆ(1),ω +e ˆ(1),ωeˆ(1),ω) k,l1 k,l3 k,l1 k,l3 (1),ω (1),ω (1),ω (1),ω m · (ˆek,l2 eˆk,l1 +e ˆk,l2 eˆk,l1 )
(1),2ω (1),2ω (1),ω For compactness, let’s assume, c2 = cos(θ4,l ), s2 = sin(θ4,l ), c1 = cos(θ1,l ) and
(1),ω s1 = sin(θ1,l ). For the case of the p-polarized second harmonic generation at the reflection direction by mixing two p-polarized fundamental frequency, m = 1 and the unit vectors are,
(2),2ω h iT (1),2ω h iT eˆ4,l = c2 0 s2 , eˆ1,l = c1 0 s1 . Thus, Equation 4.347 can be rewitten as:
77 2 (c1) 0 h i 2 (2),2ω ¯(2),2ω (s1) deff = c2 0 s2 d , (4.348) 0 2c1s1 0 (2),2ω 2 2 deff = 2c1s1c2d15 + s2 (c1) d31 − (s1) d33 . (4.349)
4.9.4 (3),3ω calculation deff
Using the symmetry arguments made in Reference [67] and assuming Kleinmann sym- metry, the d¯(3),3ω can be written as follows: 3d16 0 0 0 0 d16 0 d16 0 0 ¯(3),3ω d = 0 3d16 0 d16 0 0 0 0 d16 0 (4.34:) 0 0 3d16 0 d16 0 d16 0 0 0
(1),3ω Following symbol convention similar to Equation 4.348, let’s assume, c3 = cos(θ4,l ),
(1),3ω (1),ω (1),ω (3),3ω s3 = sin(θ4,l ), c1 = cos(θ1,l ) and s1 = sin(θ1,l ). Thus, deff can be defined as,
3 (c1) 0 (s )3 1 0 (3),3ω h i (3),3ω 0 d = c 0 s d¯ , (4.34;) eff 3 3 3c · (s )2 1 1 2 3s1 · (c1) 0 0 0
for p-polarized third harmonic generation from three p-polarized waves mixed.
78 4.: Theoretical models of susceptibility
The goal of this section is to introduce few models found in the literature that deals
(h),nω with the technique of calculating the SH/TH power from an effective susceptibility, deff .
4.:.3 Herman-Hayden model
The model that is the closest to the TMM engine in terms of assumptions and imple- mentation of boundary conditions is the “Herman-Hayden” model. This model is widely used to predict/fit the effective nonlinear susceptibility values [42,69,6;–76]. A well-known approach is the fitting of Maker’s fringe pattern [39]. The analytic equations that are used for the fitting are,
2πa sin (∆Ka)2 I(2),2ω ∝ (d(2),2ω)2 (I(1),ω)2, (4.352) eff Λ ∆Ka 2πa sin (∆Ka)2 I(3),3ω ∝ (d(3),3ω)2 (I(1),ω)3, (4.353) eff Λ ∆Ka
(1),ω (2),2ω where, ∆K = Kl3 −Kl3 and I is the intensity of radiation. As can be seen, for a → 0,
the dependence of I(2),2ω becomes quadratic with respect to thickness a.
4.:.4 Effective medium model
The effective susceptibility defined above provides a simplified constitutive relation
among the E-fields of generated frequency and the frequencies that are mixed. But this
sense of “effective”-ness does not account for composite materials/layers when the mate-
rials’ feature size is much smaller then the wavelength and the assumption of material
properties being equal to their bulk counterpart does not hold anymore. In this section,
theoretical models of susceptibility is discussed in light of “effective medium theory” so that
individual material properties of a composite system are combined to create one material
79 property [77–79]. In a sense these materials can be thought of as resonant/non-resonant metamaterials. Various considerations goes into the effective combination of material prop- erties. Some of these parameters are:
3. Geometry of constituent materials,
4. Fill fraction or doping concentration of the materials,
5. Modification of electronic properties due physical contact and/or chemical changes,
such as change in electron energy levels, presence of electron tunneling and spilling
current etc. A clever manipulation of such behaviour requires robust models developed
from first principles. But in an essence, they are simply careful manipulation of the
system Hamiltonian and thus electronic properties.
Many models (quantum, semiclassical or classical) exist that attempt to solve for an emer- gent “effective” property. The theory of left-handed materials, metamaterials with exotic anisotropy, ENZ materials and, even, superconductors, topological insulator, Weyl semimet- als etc fall into the same category of problem in a sense that the attempt is to modify the effective Hamiltonian or energy levels of the system.
For this study, the scope was limited to the dynamics of electrons that results in a change in the effective linear/nonlinear susceptibilities. Since the dynamics of a large collection of electrons is similar to the dynamics of a fluid, the classical version of the model is known as the “hydrodynamic” model of electrons. It is derived from the linearized Navier–Stokes equation [7:] that is used to describe the motion of the interacting electron gas. The complete quantum approach to solve the problem is known as “driven quantum transport” problem and deals with equations known as “Master Equations” [7;]. These are the quan-
7: tum counterpart of hydrodynamic model that constitutes analogous terms for different types of dynamics. To understand different terms of the hydrodynamic model, let’s first examine the Lorentz-Drude model [82].
In classical electromagnetism, the Polarization, P is defined as,
P = Nehri (4.354)
where, N is the average number of dipoles per unit volume, e is the electron charge and
hri is the average displacement of electron cloud from equilibrium position caused by an
external E field. The rate equation or equation of motion, can be written as,
2 Ne P¨ + γP˙ + (ω0) P + E = 0, (4.355) |{z} | {z } m decay term resonance term | {z } driving term
where, γ is decay constant, ω0 is the resonance frequency of the system, m is the electron
mass. Thus, the linear permittivity for bound electrons can be written as,
2 (1),Ω (ωp,bound) bound = 1 + 2 2 , (4.356) Ω − ω0 + jΩγbound r Ne where, ω = is defined as Plasma frequency. Ω is the frequency of the driving p,bound m0 field (for electromagnetic wave as driving field, values are given in Table 4.6). Equation
4.355 is true for any dielectric. This can also be true for the contribution of the bound electrons in a plasmonic or metallic material. Thus the subscript bound was used. But for the free electrons of plasmonics, the resonance frequency, ω0, does not exist. Thus the
modified version of Equation 4.356 is known as Drude model of plasmonics [83, 84]. The
linear permittivity for free electrons becomes,
2 (1),Ω (ωp,free) free = 1 + 2 . (4.357) Ω + jΩγfree
. Equation 4.356 and 4.357 can be considered as bare-bone hydrodynamic model with contribution from both free and bound electrons. The scope of this study is the two terms:
7; nonlocalilty and quantum tunneling terms. The following subsection briefly describes them.
In the next chapter in Section 5.6, the effects of these parameters during nonlinear harmonic generation will be discussed.
4.:.5 Nonlocality and tunneling
At the interface between different materials, a charge gradient exists and a pressure-like term proportional to (∇ · P ) [85–88], can be introduced in Equation 4.355. This term is known as convection term. Equation 4.357 can be written as,
(ω )2 (1),Ω = 1 + p,free , (4.358) free 2 2 2 Ω − β |K| + jΩγtunneling +jΩγfree | {z } | {z } Nonlocal term tunneling term p where, β is proportional to Fermi velocity i.e. β ∝ EF /m and EF is the Fermi energy level.
On the other hand, tunneling current can be introduced as an effective conductivity term.
In quantum correction model [89–93], the modified conductivity changes the γ. Instead of considering γ as a phenomenological parameter, the new conductivity and in turn γ can be calculated more accurately by using quantum conductivity theory [94–96]. Equation 4.358 can be rewritten for 2ω and 3ω. These coupled equations are solved for E field and thus, the SH/TH power is calculated. Although, Landau damping [97] was not used in the final calculation, the concept is very relevant and will be given attention in future calculation.
The idea of Landau damping is that the convection term can have imaginary parts (i.e.
β2 → β2 − jωD, where the newly-introduced D is diffusion constant).
82 CHAPTER III
EXPERIMENTS AND RESULTS
The chapter explains the experimental setup (Section 5.3) followed by a discussion on results (Section 5.4). Using the SH/TH power extracted from the metal-insulator (MI)
(2),2ω (3),3ω and metal-insulator-metal (MIM) samples their respective deff and deff values are
calculated. Following this discussion on nonlinear susceptibilities, some results achieved by
employing hydrodynamic model of plasmonics will be discussed.
5.3 Experimental setup
Figure 5.3: Experimental setup: Mode-locked Ti:Sapphire laser, chopper; HWP: half wave plate L3: focusing lens; L4: collimating lens; P3 and P4: filtering prisms; SPF: short-pass filter; L5: detector lens; PD: (silicon) photodiode [94].
The experimental setup is illustrated in Figure 5.3. Our source is a mode-locked
Ti:Sapphire laser (Spectra-Physics Tsunami) with a 100fs pulse width at a 80MHz pulse
83 repetition rate. The wavelength is fixed at 810nm, and the average output power is
≈ 0.45W . The laser output intensity and wavelength are fixed. The pulse travels through a half-wave plate for polarization control and an aspheric focusing lens (L3:A482TM-B). The focused spot size on top of the sample is ≈ 25µm, and the peak power is calculated to be
6.15GW/cm2. We use an ND filter to drop the intensity and confirm the squared and cubic dependency of SH and TH, respectively, on the input intensity incident on the samples.
The repetition rate of the laser is high enough for the utilization of a lock-in amplifier for detection. The laser beam is incident on the sample surface at an oblique angle of 68◦. After the sample, two lenses with focal length of 150cm (L4: ACA476-372-UV) and 100cm (L5:
ACA476-322-UV) are used to collimate and refocus the beam on a detector. The harmonic light scattered from the sample is separated from the collimated fundamental beam using two fused silica prisms and a short-pass filter placed between the second and the third lens.
The stray light is reduced below measurable limits by placing the detector inside a dark tube. The optics forms a demagnified image of the sample surface on the photodetector sur- face. The demagnification helps to improve the tolerance of the two-lens imaging system. A
TE-cooled UV-enhanced Si detector (EO Systems: UVS − 025 − TE2 − H) is placed at the focal length of L5 to collect the signal. The detector’s (photodiode, PD) output was then measured through a lock-in amplifier (LA), SR:52. The detector was cooled to −40◦C and a low gain configuration was used. A lock-in amplifier reduced the noise-equivalent-power below 10fW . The input and output arms were so placed that the specular reflection (the incident angle is kept at 68◦) from the sample can be collected. The details of the setup i.e. alignment procedure, loss calibration was discussed in great details in Reference [94]. Fig- ure 5.4 shows the steps of sample preparation. At each step of the preparation SH and TH signals were collected from multiple positions of every samples. MI samples were fabricated with Al2O3 or ZnO films of controlled thickness. The thickness change of the insulator
84 Figure 5.4: The SH and TH signals are collected, first, from an ultra-flat Au surface, then, thin film of insulator (on top of the Au surface), and lastly, from a thin film of insulator coated with AuNPs.
film over our gold surfaces required sub-nanometer control. This was achieved using atomic layer deposition (ALD) [98–9;]. Surface characterization and chemical composition charac- terization of these layers can be found in the Appendix A. By varying the number of ALD cycles, MI samples were fabricated with different insulator thicknesses (the total number of the ALD cycles of Al2O3 : 8, 16, 24, 32, 80, and 160; the total number of the ALD cycles of the ZnO : 5, 10, 15, 20, 40, 80, and 160). The SH and TH signals from the MI samples are
attributed to the surface and bulk nonlinearities. A solution containing Au nanoparticles
(AuNPs) with a 10.5nm diameter was then placed on the MI sample and distributed evenly
over the surface by spin coating to produce MIM samples. The distribution of nanoparticles
were discussed in Appendix A. The samples were dried using N2 gas. The experimental procedure to measure SH and TH from MIM samples follows the same step described above for the MI samples.
85 5.4 Results
The SH and TH efficiencies measured using the experimental setup (Figure 5.3) are given in Figure 5.5 and 5.6, respectively. The results were published and analyzed in
Reference [43] and [44].
(a) (b)
Figure 5.5: SH efficiencies measured for samples with (a) Al2O3 and Al2O3 + AuNP , and (b) ZnO and ZnO + AuNP as insulator layer.
(a) (b)
Figure 5.6: TH efficiencies measured for samples with (a) Al2O3 and Al2O3 + AuNP , and (b) ZnO and ZnO + AuNP as insulator layer.
86 5.4.3 Effective d(2),2ω calculation
Since the TMM engine discussed in Chapter II, have successfully reproduced all the benchmark results and simulations compared reasonably well with the results found in the literature, the TMM engine was deemed applicable for further implementation as a tool to calculate the higher order susceptibilities. The necessary codes and functions are given in
Appendix C for reference. Next, the code C is applied to calculate the d(2),2ω for given SH efficiencies measured experimentally. The code uses fminsearch() function to reduce the error in power calculation by changing the matrix elements of d¯(2),2ω. First it minimizes
the error in power calculation that is collected from the bare Au surface. The initial guess
of d¯(2),2ω,Au is taken from Reference [:2]. The newly calculated d¯(2),2ω,Au is then used afterward to calculate d(2),2ω of the insulator.
(a) (b)
(2) Figure 5.7: |d15 | for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures.
87 (a) (b)
(2) Figure 5.8: |d31 | for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures.
(a) (b)
(2) Figure 5.9: |d33 | for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures.
Figure 5.7,5.8 and 5.9 shows the non-zero components of the d¯(2),2ω. Using Equation
4.348 the effective d(2),2ω is calculated. Figure 5.: and 5.; shows the calculated real and
88 (2),2ω ¯(2),2ω imaginary parts of deff values. The bulk values of the components of d for ZnO found in the literature [3;,5;,72,:3] are d15 ≈ 2 pm/V , d31 ≈ 2 pm/V and d33 ≈ 10 pm/V .
(a) (b)
(2) Figure 5.:: Re{deff } for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures.
(a) (b)
(2) Figure 5.;: Im{deff } for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures.
89 5.4.4 Effective d(3),3ω calculation
The MATLAB codes to calculate d(3),3ω is given in Appendix C. Figure 5.32 shows the non-zero components of the d¯(3),3ω (Kleinmann symmetry is assumed). Using Equation
4.34; the effective d(3),3ω is calculated. Similar to the d(2),2ω calculations discussed above,
first the error in TH power calculation from the bare Au surface is minimized. The initial guess of d¯(3),3ω,Au is taken from Reference [:4, :5]. The newly calculated d¯(3),3ω,Au is then used afterward to calculate d(3),3ω of the insulator. Figure 5.33 and 5.34 shows the calculated
(3),3ω real and imaginary parts of deff values. The bulk value found in the literature are 23
2 2 4 2 2 pm /V (for Al2O3) and 1.85 × 10 pm /V (for ZnO)[:6–:8].
(a) (b)
(3) Figure 5.32: |d16 | for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures.
8: (a) (b)
(3) Figure 5.33: Re{deff } for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures.
(a) (b)
(3) Figure 5.34: Im{deff } for (a) Al2O3 and (b) ZnO as insulator for MI and MIM structures.
8; 5.5 Electron density in metal-induced-gap-states
At the beginning of Section 5.4, the efficiencies of second and third harmonic generation is presented in Figure 5.5a, 5.5b. In this section the SH efficiencies from MI samples will be used to calculate the delocalized electron density in ultra-thin layers deposited on top of Au. To describe the gradual drop of SH power in Figure 5.6a, two steps were taken to calculate the effective medium (see Equation 5.3).
l h Al2O3 Au,modi air = fAl2O3 − α + (1 − fAl2O3 ) (5.3)
First, the surface coverage fAl2O3 was determined (see Appendix A.5). Second, the per- mittivity of Au, Au,mod was introduced (with modification in its γ term to account for surface scattering) in the insulator region that is stemming from the spilled electrons from
Au surface [:9, ::]. The parameter α is known as the metal-induced-gap-states (MIGS) parameter that acts like a weight to Au permittivity when calculating the effective medium.
These were combined to calculate the effective permittivity of the insulator layer.
Figure 5.35: The insulator layer (Al2O3) is considered as effective medium. The permittivity of Al2O3 is modified by the MIGS spillage and surface scattering due to the Au surface and, then, by the surface coverage ratio [:;].
92 Now, the generation of second harmonic light from the Au interface is due to Lorentz, convective, and gas pressure forces acting on (3) free and (4) bound electrons in Au and,
(5) bound electrons. The contribution from bound electrons is usually negligible and is ignored. Therefore, nonlinearity is primarily induced by free electrons which produce a quadruple-like second-harmonic polarization in bulk Au with Eω as pump (fundamental frequency) electric field. Thus,
P (2),2ω,V ∝ (E(1),ω,V · ∇)E(1),ω,V + ∇(E(1),ω,V · E(1),ω,V ). (5.4)
On the other hand, the dipole like surface polarization can be written as,
(1),ω (1),ω El1 El1 0 0 0 0 0 d15 0 (1),ω (1),ω (2),2ω,S 0 E E P = 0 0 0 d 0 0 l3 l3 , (5.5) 15 0 d d d 0 0 0 31 31 33 (1),ω (1),ω 2El1 El3 0
where, the d matrix is the second order susceptibility due to surface. The nonzero values
of d matrix due to C∞v symmetry are d15, d33 and d31. For this case, the assumption was
d31 → 0. Also,
(1),ω,Au d15 = Aηχ (a + 1)b, (5.6) ! 1 Au d = Bηχ(1),ω,Au 2a + 1 + b, (5.7) 33 2 l where, A and B are Rudnik-Stern parameters [;2]. These are determined by fitting the ex- perimentally measured SH/TH power and they capture the non-ideality of a poly-crystalline
−1 −1 2 (1),ω,Au and rough Au surface. a = ω(ω+jγω) , b = 2ω(2ω+jγ2ω) and η = −e/(2mω ). χ
is the linear susceptibility determined from Drude model parameters. Au is total dielectric
constant of Au, due to both free and bound electrons (values for Au was collected from [;3]).
93 Figure 5.36: The reflected SH efficiency (left axis) as a function of ALD Al2O3 cycles.
Correspondingly, the coverage factor, fAl2O3 (right axis) varies as a function of ALD cycles from 2, i.e. no Al4O5 coverage to 3, i.e. full Al4O5 coverage at cycle 69. The MIGS factor α has been shown to vary from a = 0.0 to a = 0.7.
For the second order susceptibility values given in Equation 5.6, the SH power was
calculated for different values of α. Figure 5.36 shows the SH efficiencies marked in different
colors. Figure 5.37 shows that for α = 0.13 the match with the experimental values were
the best. Since the penetration depth of electrons in the Au/Al2O3 interface is ∆ ≈ 0.4nm,
and the electron density in Au is N = 5.9 × 1022cm−3 [43, ;4], the electron density (areal
density, ρ) in the metal induced gap states [::] can calculated as:
ρ = αN∆ = 3.53 × 1014cm−2. (5.8)
94 Figure 5.37: For different values of MIGS factor α, the RMS error between simulated and measured harmonic power is calculated. The minimum RMS error is found for MIGS factor α = 0.13.
For the error margin in the measured values please refer to the detailed calculation in
Reference [43]. Equation 5.8 is a significant finding in terms of changes in second order effective susceptibilities of ultra-thin insulator films in the presence of plasmonic materials.
5.6 Effects of nonlocality and quantum tunneling
In this section, the SH/TH efficiencies presented in Figure 5.5a, 5.5b, 5.6a and 5.6b will be used to draw a conclusion on the comparative effects of nonlocality and tunneling in the hydrodynamic model. Using the model discussed in Section 4.:.5, a coupled system was solved and the field enhancement at the pump wavelength has been calculated for the MIM structure with the finite-element-method (COMSOL) [44]. The results shown in Figure 5.38. It is apparent from Figure 5.38 that the effect of nonlocality of nonlinear
susceptibility is much stronger than that of quantum tunneling, when both effects are
present simultaneously. This was also third significant finding of the study.
95 Figure 5.38: (a) Comparison between four different simulations. The Classical results omit nonlocality and quantum effects; the Quantum effects incorporate electron tunneling and the nonlocal calculation incorporates spatial dispersion effects. The nonlocal + quantum curve includes both effects. (b) Field enhancement distribution near the gap region for an MIM with a gap size of 0.2nm for nonlocal + quantum effects.
5.7 Summary of findings
The significant and novel contributions that this study offered are as follows,
3. A robust TMM engine was developed that can handle isotropic or anisotropic material
with any orientation of the principle axes. Implementation of SVD in the context of
TMM engine has not been found by the author.
4. Second and third order nonlinearity was added to the TMM engine to generate Maker’s
fringe pattern and calculate the effective nonlinear susceptibilities. The TMM engine
was able to calculate them in the reflection direction. The technique does not invoke
Kleinmann symmetries for the second harmonic case. In the absence of d(3),3ω matrix
for C∞v symmetry, the isotropic matrix was used assuming Kleinmann symmetry.
96 But the technique is independent of these assumptions and it can function for any
given susceptibility matrix.
5. The nonlinear susceptibility measurement technique was applied to measure the areal
electron density in the metal-induced-gap-states at the metal-insulator interface.
6. The nonlocal and tunneling terms of the hydrodynamic model was studied to compare
their effects on nonlinear susceptibilities and electric-field enhancement.
97 CHAPTER IV
CONCLUSION
(2) (3) The effective nonlinear susceptibilities, χeff and χeff , of ultra-thin films in the presence of plasmonics may play a crucial role in regards to the miniaturization of nonlinear optical devices and making them more efficient. With that motivation in mind, the goal of this
(2) (3) dissertation was to experimentally measure the real and imaginary parts of χeff and χeff
of the ultra-thin films of insulators deposited on metal and compare them against the
contribution of different higher-order terms of the hydrodynamic model of plasmonics. For
(2) (3) larger thicknesses of insulator layers (> 10nm), the χeff and χeff values were measured
to be very close to their bulk values found in the literature. To explain the susceptibility
at the relatively smaller thicknesses (≤ 10nm), physical models were introduced. First,
the metal-insulator interface was modeled. Due to the electron spilling into the insulator
region (these electronic states are known as metal-induced-gap-states), the conductivity in
the transverse direction near the interface increases causing the gradual drop of second and
third harmonic efficiency. The surface coverage ratio (calculated from the nucleation and
growth model of surface roughness) was taken into account by assuming an effective medium
approach. The amount of spill was determined as the areal density of free or delocalized
electrons in the metal-induced-gap-states. Second, the metal-insulator-metal interface was
modeled. Several competing theories exist that describe the higher order terms of the
hydrodynamic model of free and bound electrons in plasmonics-based structures. Some
of these higher order terms are due to the nonlocality of electronic wavefunction, quantum
tunneling, Landau damping etc. and different theories claim their manifestation of existence
through changes in real and imaginary part of the effective permittivity and susceptibility.
The conclusion drawn from the second and third harmonic signals from metal-insulator-
metal interfaces is that the nonlocal effects of electronic wavefunction drowns out the effects
98 of tunneling for the studied thicknesses of insulators. Since a generalized TMM engine was
(2) (3) developed during this study, it can be used in future as a tool to determine the χeff , χeff of more exotic composite materials e.g. metallo-dielectric layers, hyperbolic metamaterial with nano-layers or nano-pillar-like array structures [49,4:], epsilon-near-zero material [4;], zero- index [52] and negative index (left-handed) material [48]. This tool can be further developed to study different linear and nonlinear surface effects e.g. D’yakonov waves on the surface of an anisotropic media [;5–;7], surface plasmon-polariton [;8], bulk plasmon-polariton [4:] etc. Although this study serves towards the purpose of understanding second and third harmonic generation, the TMM tool can be modified to include other three-wave and four- wave mixing phenomena e.g. optical rectification, electro-optic effects (Pockel effect, DC
Kerr effect), self-phase and cross-phase modulation, electric-field induced second harmonic generation, AC Kerr effect etc. Moreover, the second and third harmonic generation from the bulk and surface of topological insulators [;9], Z4 insulators [;:] and Weyl semimetals
[;;] can also be of great experimental significance since these signals can probe the quantum hydrodynamics of their zero-loss metal-like surface states [322].
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:7 APPENDIX A
Surface Characterization
The physical parameters that needed to be measured after each atomic layer deposition step are: 3. layer’s thickness, 4. surface height profile and surface roughness, 5. surface coverage ratio, 6. chemical composition. Each characterization parameter will be described briefly below:
A.3 Thickness and growth rate
Ellipsometry (Woollam®, α-SE) was used to measure the thickness and growth rate of every sample after atomic layer deposition (ALD). Figure A.3 shows the measured growth rate for both Al2O3 and ZnO. The growth rate (GR) of Al2O3 on Au was measured to be
0.15nm/cycle. Meanwhile, the growth of ZnO on Au was measured to be 0.18nm/cycle.
Figure A.3: The growth rate (change of thickness per unit cycle of ALD) was measured for each insulator.
:8 A.4 Surface roughness
Atomic force microscopy (AFM) was used to characterize surface roughness. Two key parameters are retrieved from AFM measurement: global roughness and local roughness.
(a)
(b)
Figure A.4: The AFM image showing the topography of bare Au surface and Au surface coated with (a) Al2O3 and (b) ZnO. Each image is labeled with the number of ALD cycles. The inset shows a length scale of 100nm. The bar on the right shows the heigt range.
AFM was performed using a Bruker®Dimension Icon system with a Bruker PFQNE-
AL tip of radius 7 nm. AFM data analysis was conducted using Gwyddion®Version 4.63.
Figure A.4 shows the AFM images. Grain boundaries are visible and average grain size is
:9 ≈ 30nm. For the first few cycles the layers nucleate and each nucleation centers gradually grow in size. The growth model is known as “nucleation and growth” model [43, 99] (see
Figure A.5).
Figure A.5: The schematic of nucleation and growth model. (a) pill box, (b) nucleation inside pill-boxes, The yellow represents Au and the white represents the deposited insulator, (c) critical point 3: when the nucleation circles touch each other and the sides of the pill box, (d) critical point 4: when the nucleation circles touch the corners of the pillbox. At this point the growth envelop is marked by the black curve.
Figure A.6: (a) The measured surface roughness of Al2O3 in the context of (b) the expected surface roughness from nucleation and growth model.
:: √ The nucleation density is, ND ≈ 9.5 × 1011cm−2. The pill box size, L = 1/ ND ≈
10.2nm. Global roughness is 15nm. If n is the number of ALD cycles and n1 and n2
represents the number of ALD cycles needed to reach the critical point 3 and 4, respectively,
(see Figure A.5), the formula to calculate local roughness (LR) is as follows: n × GR, when, n < n , 2 s 2 LR = 2 L (A.3) n × GR, − (n × GR) − √ when, n ≥ n2. 2
The measured surface roughness is given in Figure A.6 in the context of nucleation and growth model.
A.5 Surface coverage
The surface coverage fraction, f, is defined as follows:
πr2 , when, n < n 2 1 L f = [πr2 − 4r2(ζ − cos ζ sin ζ)] (A.4) , when, n1 < n < n2 L2 1, when, n > n2.
Here, r = radius of the nucleation = n × GR and ζ = cos−1 (L/2r). n, GR, L are defined in
Section A.3 and A.4
A.6 Chemical composition
X-ray photoelectron spectroscopy (XPS) was conducted using the PHI 7222 VersaProbe
II from Physical Electronics, Inc., using Al K-a (36:8.8 eV) and a beam diameter of 322
µm. Fine spectra of C 3s were used for calibration. Figure A.7 shows the valence band
spectra for both insulators.
:; Figure A.7: XPS valence band spectra showing the Au valence band transitioning to the Zn (left) or Al (right) as the number of ALD cycle increases.
A.7 Au nanoparticle distribution
The Au nanoparticles (AuNP) (diameter: 20nm) solution was spin-coated to prepare a metal-insulator-metal sample with evenly and randomly distributed Au nanoparticle. The particle density is ≈ 20 particle/µm2. Figure A.8 shows an example case of AuNP distribu- tion. The particles are identified using an image processing tool called houghTransformation().
;2 Figure A.8: The SEM image of Au surface spin-coated with Au nanoparticle solution. Au nanoparticles are (marked in green) randomly distributed and sparse enough to not cause any internal coupling of E-fields. The density of Au nanoparticles.
;3 APPENDIX B
Singular Value Decomposition
There are many forms of singular value decomposition (SVD). For the sake of brevity, only “compact SVD” [323] of square matrices will be discussed, since this form of factoriza- tion is relevant to the scope of this study. If A¯b = 0, for any matrix A (size: N × N) with det(A¯) = 0, then those vectors b that satisfy A¯b = 0 are known as the singular vectors of
A¯. A¯ can be decomposed into matrix U¯, S¯ and V¯, such that, A1 0 ... 0 0 A ... 0 † ¯ ¯ ¯ ¯ † h i 2 h i A = USV = U1U2 ... UN . . . V1V2 ... VN . (B.3) . . .. . . . . . 0 0 ... AN
Here, the subscript † indicates transpose and complex conjugate operation. S¯ is a diagonal
matrix containing the singular values of A¯ on its diagonal. The singular values may be
sorted in a descending order. The columns of U¯ and V¯ represent the singular vectors of A¯
also sorted in descending order of the corresponding singular values. U¯ and V¯ are unitary
matrices such that U¯ †U¯ = V¯ †V¯ = ¯I, where ¯I is an identity matrix of size N × N. The space
spanned by the singular vectors are known as the null space of the matrix A¯. The rank is
determined by setting a lower limit for the singular value. The number of singular vectors
that have singular value larger than the lower threshold of singular values is defined as the
rank, R, of the matrix. The rank of matrix A¯ indicates the degeneracy of singular vectors
and the dimension, D of the null space such that. D = N − R. For N = 3, R = 1 means
a 4D null space. In terms of electromagnetic wave propagation in a media, R = 1 means
either the medium is isotropic or the propagation is along an optic axis. If R = 2, the null
space becomes 3D which represents any arbitrary direction of propagation in a generalized
birefringent media except along the optic axis.
;4 APPENDIX C
MATLAB Codes
(2),2ω Script for deff Listing C.3: d4effAnisoTMM.m g l o: b a l c mu2 eps2 lambda3 lambda4 lambda5 ... lambda_array k_array k2_array omega_array ... layer_thickness n d layer_materials layer_birefrengence ... layer_OA_eulerAngles layer_birefrengence_sign scale scale 4 no_of_layers ... 7 no_of_wavelengths degen_thres_3 degen_thres_4 ... allow_imaginary_index allow_evanascent_field_both allow_evanascent_field ... layer_nonlinearity var_insulator_pos var_thickness_current ... E_ff_input isPPolarized P_sh_exp A t_pulse f_pulse aoi th_step isMIM isZnOThere ... d53_Au_real d37_Au_real d55_Au_real d53_Au_imag d37_Au_imag d55_Au_imag . . . 32 kk_wave th_ff w_ff th_sh w_sh % ’n’ is the refractive index % ’d’ is the susceptibility for nonlinear waves % ’aoi’ is the angle of incidence
37 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% Electro - magnetic Input %%%%% c = 4;;.9;467:; %nm/fs, speed of light mu2 = 6* pi *3 e4 ; % kg*nm/C^4 FYI : mu2 = 6* pi *3 e -9 SI u ni t eps2 = 3/(mu2* c ^4); % C^4* f s ^4/( kg*nm^5) 42 lambda3 = :32; % nm, input fundamental lambda4 = lambda3 /4; % nm, SH lambda5 = lambda3 /5; % nm, TH lambda_array = [lambda3 , lambda4 ]; 47 no_of_wavelengths = l e n g t h (lambda_array) ;
I _ l a s e r = 46e - 5 4 ; % kg . f s ^ -5; this is peak intesity or radiance at ... % fundamental wavelength: 46GW/cm^4 M = 3 / 3 . 7 ; % System Magnification 52 r a d i u s = 34e -8*M; % m A = pi * r a d i u s ^4; % m^4 t_pulse = :2e - 3 7 ; % s, pulse width f_pulse = :2e8 ; % Hz, Repetation rate
57 E_ff_input = s q r t (4*I_laser/(c*eps2 )); % kg.nm.fs^-4/C % p-polarized % peak E field % E_ff_input = 3 ; %%% 3 e43 V/m = 3 kg .nm. f s ^ -4/C isPPolarized = 3 ;
;5 62 k2_array = 4.* pi ./lambda_array; % 3/nm k_array = z e r o s ( 3 , l e n g t h ( k2_array ) ) ; omega_array = c*k2_array ; % f s ^ -3 a o i = 8:; % degree, angle of incidence th_step = 2 . 3 ; 67 allow_imaginary_index = 3 ; % allow_evanascent_field = 3 ;% allow_evanascent_field_both = 3 ;% s c a l e = 3 ; % to scale up/down the refractive index 72 s c a l e 4 = 3 ; i f s c a l e ~= 3 || s c a l e 4 ~= 3 dis p ( ’epsilon_yy and/or epsilon_zz has been scaled for better visibility !!! ’ ) end
77 s e t ( groot , ’defaulttextinterpreter ’ , ’ l a t e x ’ ); s e t ( groot , ’defaultLegendInterpreter ’ , ’ l a t e x ’ ); s e t ( groot , ’defaultAxesTickLabelInterpreter ’ , ’ l a t e x ’ ); f i g 3 = f i g u r e ( ) ; ax3 = axes ; f i g 4 = f i g u r e ( ) ; ax4 = axes ; 82 f i g 5 = f i g u r e ( ) ; ax5 = axes ; f i g 6 = f i g u r e ( ) ; ax6 = axes ; f i g 7 = f i g u r e ( ) ; ax7 = axes ; f i g 8 = f i g u r e ( ) ; ax8 = axes ; f i g 9 = f i g u r e ( ) ; ax9 = axes ; 87 f i g : = f i g u r e ( ) ; ax: = axes ; f i g ; = f i g u r e ( ) ; ax; = axes ; f i g 3 2 = f i g u r e ( ) ; ax32 = axes ; f i g 3 3 = f i g u r e ( ) ; ax33 = axes ; f i g 3 4 = f i g u r e ( ) ; ax34 = axes ; 92 f o r sample_no = 3:6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%Material & Geometry Input %%%%%
97 findMaterialandGeometryProperties(sample_no) ;
i f ( l e n g t h (layer_materials)~= l e n g t h (layer_birefrengence)) ... || ( l e n g t h (layer_materials)~= s i z e (layer_OA_eulerAngles , 3 ))... || ( l e n g t h (layer_materials)~= l e n g t h (layer_thickness)) :2 e r r o r ( ’Not all input provided!’ ); end no_of_layers = l e n g t h (layer_thickness);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :7 %%%% Load Experimental Data %%%% i f sample_no == 3 isZnOThere = 2; isMIM = 2; sample_type = ’Al4O5_MI . t xt ’ ; ;2 e l s e i f sample_no == 4
;6 isZnOThere = 2; isMIM = 3 ; sample_type = ’Al4O5_MIM . txt ’ ; e l s e i f sample_no == 5 ;7 isZnOThere = 3 ; isMIM = 2; sample_type = ’ZnO_MI. t xt ’ ; e l s e i f sample_no == 6 isZnOThere = 3 ; 322 isMIM = 3 ; sample_type = ’ZnO_MIM. txt ’ ; end
327 i f isZnOThere == 2 thickness_per_ALD_cycle = 2 . 3 7 ; %nm/ALD Cycle ZnO: 2 . 3 : , Al4O5 : 2 . 3 7 e l s e thickness_per_ALD_cycle = 2 . 3 : ; end 332 eff_filename = strcat( ’SH_efficiency_ ’ , sample_type); sh_efficiency_data = importdata(eff_filename); var_insulator_thickness = thickness_per_ALD_cycle .*... (sh_efficiency_data.data( 3 ,:)); 337 sh_efficiency = sh_efficiency_data.data( 4 ,:); sh_efficiency_sigma = sh_efficiency_data.data( 5 ,:); d _ f i t = z e r o s ( l e n g t h (sh_efficiency) ,8);
var_insulator_pos = 4; 342 no_of_thicknesses = l e n g t h (var_insulator_thickness); d_eff = z e r o s ( 3 ,no_of_thicknesses);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Initiate Material properties %%%%%%%%%% 347 n = ones(no_of_layers ,no_of_wavelengths , 5 ); % refractive index... % n(what layer , what wavelength, what birefrengence axis) layer_birefrengence_sign = z e r o s ( 3 ,no_of_layers); % assuming isotropic by default 352 % I do not assign it right away to keep flexibility for % in-situ changes of index due to dispersion nonlinear_layer_pos = f i n d (layer_nonlinearity~=2); no_of_NL_layers = l e n g t h (nonlinear_layer_pos); d = z e r o s (no_of_NL_layers , 5 , 8); 357 make_n_matrix() ;
%%%%%%%%%%%%%%%%%%%%%%%%% %%%%Simulation Input %%%%
362 degen_thres_3 = 3e - 3 4 ; % handles the rotation axis degeneracy degen_thres_4 = 3e - ; ; % handle the rank of system matrix
;7 marker_type = [ ’ r - - o ’ ; ’ b : ^ ’ ]; i f sample_no == 3 367 s t a r t = 3 ; e l s e s t a r t = 4; end f o r ll = start:no_of_thicknesses 372 var_thickness_current = var_insulator_thickness( ll ); i f var_thickness_current == 2 d37_guess_real = r e a l (9* exp (3 i * deg4rad (372))); d37_guess_imag = imag (9* exp (3 i * deg4rad (372))); d53_guess_real = r e a l (3* exp (3 i * deg4rad (:3))); 377 d53_guess_imag = imag (3* exp (3 i * deg4rad (:3))); d55_guess_real = r e a l (39.:* exp (3 i * deg4rad ( -65))); d55_guess_imag = imag (39.:* exp (3 i * deg4rad ( -65))); e l s e d37_guess_real = 2; % pm/V 382 d37_guess_imag = 2; % pm/V d53_guess_real = 2; % pm/V d53_guess_imag = 2; % pm/V d55_guess_real = 2; % pm/V d55_guess_imag = 2; % pm/V 387 end P_sh_exp = sh_efficiency(ll) * (2.7* c * eps2 *A*t_pulse*f_pulse )... .* abs (E_ff_input) .^ 4 ; kk_wave = 3 ; errorInSimulatedHarmonicPower ([]) ; 392 kk_wave = 4; options = optimset( ’ Display ’ , ’ i t e r ’ , ’ PlotFcns ’ , @optimplotfval); [d_fit(ll ,:) , f_val] = fminsearch( @errorInSimulatedHarmonicPower ,... [ d37_guess_real , d53_guess_real , d55_guess_real ,... 397 d37_guess_imag, d53_guess_imag, d55_guess_imag] , options ); i f var_thickness_current == 2 d37_Au_real = d_fit(ll , 3 ); d53_Au_real = d_fit(ll , 4 ); d55_Au_real = d_fit(ll , 5 ); 3:2 d37_Au_imag = d_fit(ll , 6 ); d53_Au_imag = d_fit(ll , 7 ); d55_Au_imag = d_fit(ll , 8 ); end t3 = th_ff(var_insulator_pos , 3 , 3 ); 3:7 w3 = w_ff(var_insulator_pos , 3 , 3 ); t4 = th_sh(var_insulator_pos , 3 , 3 ); w4 = w_sh(var_insulator_pos , 3 , 3 ); c3_sq_ff = ( cos ( t3+w3) ) . ^ 4 ; s3_sq_ff = ( s i n ( t3+w3) ) . ^ 4 ; 3;2 s3_4_ff = s i n ( 4 . * ( t3+w3)); c4_sh = cos ( t4+w4);
;8 s4_sh = s i n ( t4+w4);
d_eff(ll) = ((d_fit(ll , 3 ) +3 i*d_fit(ll , 6 ) ) . * c4_sh . * s3_4_ff ) + . . . 3;7 ( s4_sh.*((d_fit(ll , 4 ) +3 i*d_fit(ll , 7 ) ) . * c3_sq_ff + . . . ( d _ f i t ( l l , 5 ) +3 i*d_fit(ll , 8 ) ) . * s3_sq_ff ) ) ; end
i f sample_no == 3 || sample_no == 4 422 a x i s 3 = ax3 ; a x i s 4 = ax4 ; a x i s 5 = ax5 ; axis6 = ax9 ; a x i s 7 = ax: ; a x i s 8 = ax33 ; e l s e a x i s 3 = ax6 ; a x i s 4 = ax7 ; 427 a x i s 5 = ax8 ; axis6 = ax; ; a x i s 7 = ax32 ; a x i s 8 = ax34 ; end 432 i f sample_no == 3 || sample_no == 5 marker = marker_type( 3 ,:); e l s e marker = marker_type( 4 ,:); end 437 end
hold ( ax3 , ’ o f f ’ ); hold ( ax4 , ’ o f f ’ ); hold ( ax5 , ’ o f f ’ ); 442 hold ( ax6 , ’ o f f ’ ); hold ( ax7 , ’ o f f ’ ); hold ( ax8 , ’ o f f ’ ); hold ( ax9 , ’ o f f ’ ); hold ( ax: , ’ o f f ’ ); 447 hold ( ax; , ’ o f f ’ ); hold ( ax32 , ’ o f f ’ ); hold ( ax33 , ’ o f f ’ ); hold ( ax34 , ’ o f f ’ );
452 x l a b e l ( ax3 , ’$Al_4O_5$ Thickness, $[nm]$’ ); y l a b e l ( ax3 , ’ $|d ^{(4) }_{37}|, [pm/V]$’ ); legend ( ax3 , ’$Al_4O_5$ ’ , ’$AuNP+Al_4O_5$ ’ );
x l a b e l ( ax4 , ’$Al_4O_5$ Thickness, $[nm]$’ ); 457 y l a b e l ( ax4 , ’ $|d ^{(4) }_{53}|, [pm/V]$’ ); legend ( ax4 , ’$Al_4O_5$ ’ , ’$AuNP+Al_4O_5$ ’ );
x l a b e l ( ax5 , ’$Al_4O_5$ Thickness, $[nm]$’ ); y l a b e l ( ax5 , ’ $|d ^{(4) }_{55}|, [pm/V]$’ ); 462 legend ( ax5 , ’$Al_4O_5$ ’ , ’$AuNP+Al_4O_5$ ’ );
x l a b e l ( ax6 , ’$ZnO$ Thickness , $[nm]$’ );
;9 y l a b e l ( ax6 , ’ $|d ^{(4) }_{37}|, [pm/V]$’ ); legend ( ax6 , ’$ZnO$ ’ , ’$AuNP+ZnO$ ’ ); 467 x l a b e l ( ax7 , ’$ZnO$ Thickness , $[nm]$’ ); y l a b e l ( ax7 , ’ $|d ^{(4) }_{53}|, [pm/V]$’ ); legend ( ax7 , ’$ZnO$ ’ , ’$AuNP+ZnO$ ’ );
472 x l a b e l ( ax8 , ’$ZnO$ Thickness , $[nm]$’ ); y l a b e l ( ax8 , ’ $|d ^{(4) }_{55}|, [pm/V]$’ ); legend ( ax8 , ’$ZnO$ ’ , ’$AuNP+ZnO$ ’ );
x l a b e l ( ax9 , ’$Al_4O_5$ Thickness, $[nm]$’ ); 477 y l a b e l ( ax9 , ’Effective $Re\{d^{(4)}\}, [pm/V]$’ ); legend ( ax9 , ’$Al_4O_5$ ’ , ’$AuNP+Al_4O_5$ ’ );
x l a b e l ( ax: , ’$Al_4O_5$ Thickness, $[nm]$’ ); y l a b e l ( ax: , ’Effective $Im\{d^{(4)}\}, [pm/V]$’ ); 482 legend ( ax: , ’$Al_4O_5$ ’ , ’$AuNP+Al_4O_5$ ’ );
x l a b e l ( ax; , ’$ZnO$ Thickness , $[nm]$’ ); y l a b e l ( ax; , ’Effective $Re\{d^{(4)}\}, [pm/V]$’ ); legend ( ax; , ’$ZnO$ ’ , ’$AuNP+ZnO$ ’ ); 487 x l a b e l ( ax32 , ’$ZnO$ Thickness , $[nm]$’ ); y l a b e l ( ax32 , ’Effective $Im\{d^{(4)}\}, [pm/V]$’ ); legend ( ax32 , ’$ZnO$ ’ , ’$AuNP+ZnO$ ’ );
492 x l a b e l ( ax33 , ’$Al_4O_5$ Thickness, $[nm]$’ ); y l a b e l ( ax33 , ’SH Efficiency $(\times 32^{ -32}) $ ’ ); legend ( ax33 , ’$Al_4O_5/Au/Glass$ ’ , ’$AuNP+Al_4O_5/Au/Glass$ ’ );
x l a b e l ( ax34 , ’$ZnO$ Thickness , $[nm]$’ ); 497 y l a b e l ( ax34 , ’SH Efficiency $(\times 32^{ -32}) $ ’ ); legend ( ax34 , ’$ZnO/Au/Glass$ ’ , ’$AuNP+ZnO/Au/ Glass$ ’ );
(3),3ω Script for deff Listing C.4: d5effAnisoTMM.m g l o b a l c mu2 eps2 lambda3 lambda4 lambda5 ... : lambda_array k_array k2_array omega_array ... layer_thickness n d d5 d_Au layer_materials layer_birefrengence ... layer_OA_eulerAngles layer_birefrengence_sign scale scale 4 no_of_layers ... 7 no_of_wavelengths degen_thres_3 degen_thres_4 ... allow_imaginary_index allow_evanascent_field_both allow_evanascent_field ... layer_nonlinearity var_insulator_pos var_thickness_current ... E_ff_input isPPolarized P_th_exp A t_pulse f_pulse aoi th_step isMIM isZnOThere ... d38_Au_real d38_Au_imag . . .
;: 32 kk_wave th_ff w_ff th_sh w_sh % ’n’ is the refractive index % ’d’ is the susceptibility for nonlinear waves % ’aoi’ is the angle of incidence
37 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% Electro - magnetic Input %%%%% c = 4;;.9;467:; %nm/fs, speed of light mu2 = 6* pi *3 e4 ; % kg*nm/C^4 FYI : mu2 = 6* pi *3 e -9 SI u ni t eps2 = 3/(mu2* c ^4); % C^4* f s ^4/( kg*nm^5) 42 lambda3 = :32; % nm, input fundamental lambda4 = lambda3 /4; % nm, SH lambda5 = lambda3 /5; % nm, TH lambda_array = [lambda3 , lambda4 , lambda5 ]; 47 no_of_wavelengths = l e n g t h (lambda_array) ;
I _ l a s e r = 46e - 5 4 ; % kg . f s ^ -5; M = 3 / 3 . 7 ; % System Magnification r a d i u s = 34e -8*M; % m 52 A = pi * r a d i u s ^4; % m^4 t_pulse = :2e - 3 7 ; % s, pulse width f_pulse = :2e8 ; % Hz, Repetation rate
E_ff_input = s q r t (4*I_laser/(c*eps2 )); % kg.nm.fs^-4/C 57 isPPolarized = 3 ; k2_array = 4.* pi ./lambda_array; % 3/nm k_array = z e r o s ( 3 , l e n g t h ( k2_array ) ) ; omega_array = c*k2_array ; % f s ^ -3 a o i = 8:; % degree, angle of incidence 62 th_step = 2 . 3 ;
allow_imaginary_index = 3 ; % allow_evanascent_field = 3 ;% allow_evanascent_field_both = 3 ;% 67 s c a l e = 3 ; s c a l e 4 = 3 ; i f s c a l e ~= 3 || s c a l e 4 ~= 3 dis p ( ’epsilon_yy and/or epsilon_zz has been scaled for better visibility !!! ’ ) end 72 s e t ( groot , ’defaulttextinterpreter ’ , ’ l a t e x ’ ); s e t ( groot , ’defaultLegendInterpreter ’ , ’ l a t e x ’ ); s e t ( groot , ’defaultAxesTickLabelInterpreter ’ , ’ l a t e x ’ );
77 f i g 3 = f i g u r e ( ) ; ax3 = axes ; f i g 4 = f i g u r e ( ) ; ax4 = axes ; f i g 5 = f i g u r e ( ) ; ax5 = axes ; f i g 6 = f i g u r e ( ) ; ax6 = axes ; f i g 7 = f i g u r e ( ) ; ax7 = axes ; 82 f i g 8 = f i g u r e ( ) ; ax8 = axes ; f i g 9 = f i g u r e ( ) ; ax9 = axes ;
;; f i g : = f i g u r e ( ) ; ax: = axes ;
f o r sample_no = 3:6 87 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%Material & Geometry Input %%%%%
findMaterialandGeometryProperties(sample_no) ;
92 i f ( l e n g t h (layer_materials)~= l e n g t h (layer_birefrengence)) ... || ( l e n g t h (layer_materials)~= s i z e (layer_OA_eulerAngles , 3 ))... || ( l e n g t h (layer_materials)~= l e n g t h (layer_thickness)) e r r o r ( ’Not all input provided!’ ); end 97 no_of_layers = l e n g t h (layer_thickness);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Initiate Material properties %%%%%%%%%% i f sample_no == 3 :2 isZnOThere = 2; isMIM = 2; sample_type = ’Al4O5_MI . t xt ’ ; e l s e i f sample_no == 4 isZnOThere = 2; :7 isMIM = 3 ; sample_type = ’Al4O5_MIM . txt ’ ; e l s e i f sample_no == 5 isZnOThere = 3 ; isMIM = 2; ;2 sample_type = ’ZnO_MI. t xt ’ ; e l s e i f sample_no == 6 isZnOThere = 3 ; isMIM = 3 ; sample_type = ’ZnO_MIM. txt ’ ; ;7 end
i f isZnOThere == 2 thickness_per_ALD_cycle = 2 . 3 7 ; %nm/ALD Cycle ZnO: 2 . 3 : , Al4O5 : 2 . 3 7 e l s e 322 thickness_per_ALD_cycle = 2 . 3 : ; end
eff_filename = strcat( ’TH_efficiency_ ’ , sample_type); th_efficiency_data = importdata(eff_filename); 327 var_insulator_thickness = thickness_per_ALD_cycle .*... (th_efficiency_data.data( 3 ,:)); th_efficiency = th_efficiency_data.data( 4 ,:); th_efficiency_sigma = th_efficiency_data.data( 5 ,:); d _ f i t = z e r o s ( l e n g t h (th_efficiency) ,4); 332 var_insulator_pos = 4; no_of_thicknesses = l e n g t h (var_insulator_thickness);
322 d_eff = z e r o s ( 3 ,no_of_thicknesses);
337 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Initiate Material properties %%%%%%%%%% n = ones(no_of_layers ,no_of_wavelengths , 5 ); layer_birefrengence_sign = z e r o s ( 3 ,no_of_layers); no_of_NL_layers = l e n g t h (nonlinear_layer_pos); 342 d = z e r o s (no_of_NL_layers , 5 , 42); make_n_matrix() ;
%%%%%%%%%%%%%%%%%%%%%%%%% %%%%Simulation Input %%%% 347 degen_thres_3 = 3e - 3 4 ; % handles the rotation axis degeneracy degen_thres_4 = 3e - ; ; % handle the rank of system matrix
marker_type = [ ’ r - - o ’ ; ’ b : ^ ’ ]; 352 i f sample_no == 3 s t a r t = 3 ; e l s e s t a r t = 4; end 357 f o r ll = start:no_of_thicknesses var_thickness_current = var_insulator_thickness( ll ); i f var_thickness_current == 2 d38_guess_real = 4.79 e: ; % pm^4/V^4 d38_guess_imag = 32.7 e8 ; % pm^4/V^4 362 % ref: http://dx.doi.org/32.3586/OL.63.222539 e l s e i f isZnOThere == 3 d38_guess_real = 2;%3.:7 e6 ; % pm^4/V^4 d38_guess_imag = 2; % pm^4/V^4 367 e l s e d38_guess_real = 2;%45; % pm^4/V^4 d38_guess_imag = 2; % pm^4/V^4 end end 372 P_th_exp = th_efficiency(ll) * (2.7* c * eps2 *A*t_pulse*f_pulse )... .* abs (E_ff_input) .^ 4 ; kk_wave = 3 ; errorInSimulatedHarmonicPower ([]) ;
377 kk_wave = 5; options = optimset( ’ Display ’ , ’ i t e r ’ , ’ PlotFcns ’ , @optimplotfval); [d_fit(ll ,:) , f_val] = fminsearch( @errorInSimulatedHarmonicPower ,... [ d38_guess_real , d38_guess_imag] , options); i f var_thickness_current == 2 382 d38_Au_real = d_fit(ll , 3 ); d38_Au_imag = d_fit(ll , 4 ); end
323 t3 = th_ff(var_insulator_pos , 3 , 3 ); w3 = w_ff(var_insulator_pos , 3 , 3 ); 387 t4 = th_sh(var_insulator_pos , 3 , 3 ); w4 = w_sh(var_insulator_pos , 3 , 3 ); c3_cb_ff = ( cos ( t3+w3) ) . ^ 5 ; s3_cb_ff = ( s i n ( t3+w3) ) . ^ 5 ; c3_sq_ff = ( cos ( t3+w3) ) . ^ 4 ; 392 s3_sq_ff = ( s i n ( t3+w3) ) . ^ 4 ; c3 = cos ( t3+w3); s3 = s i n ( t3+w3); c4_th = cos ( t4+w4); s4_th = s i n ( t4+w4); 397 d_c = d_fit(ll , 3 ) +3 i*d_fit(ll , 4 ); d_eff(ll) = [c4_th 2 s4_th ] * . . . [5* d_c 2 2 2 2 d_c 2 d_c 2 2 ;... 2 5*d_c 2 d_c 2 2 2 2 d_c 2 ;... 2 2 5*d_c 2 d_c 2 d_c 2 2 2 ] * [ c3_cb_ff 2 s3_cb_ff 2 2 ... 3:2 5* s3_sq_ff * c3 5* c3_sq_ff * s3 2 2 2 ]’; end
i f sample_no == 3 || sample_no == 4 a x i s 3 = ax3 ; 3:7 a x i s 4 = ax4 ; a x i s 5 = ax5 ; axis6 = ax9 ; e l s e a x i s 3 = ax6 ; 3;2 a x i s 4 = ax7 ; a x i s 5 = ax8 ; axis6 = ax: ; end
3;7 i f sample_no == 3 || sample_no == 5 marker = marker_type( 3 ,:); e l s e marker = marker_type( 4 ,:); end 422 p l o t ( axis3 , var_insulator_thickness( 4 : end ), abs ( d _ f i t ( 4 : end , 3 ) + . . . 3 i . * d _ f i t ( 4 : end , 4 )).’,... marker , ’linewidth ’ , 3 . 7 ); hold ( axis3 , ’ on ’ ); p l o t ( axis4 , var_insulator_thickness( 4 : end ), r e a l ( d_eff ( 4 : end )) .’,... 427 marker , ’linewidth ’ , 3 . 7 ); hold ( axis4 , ’ on ’ ); p l o t ( axis5 , var_insulator_thickness( 4 : end ), imag ( d_eff ( 4 : end )) .’,... marker , ’linewidth ’ , 3 . 7 ); hold ( axis5 , ’ on ’ ); 432 e r r o r b a r ( axis6 , var_insulator_thickness , th_efficiency*3 e32 ,... th_efficiency_sigma*3 e32 , marker , ’linewidth ’ , 3 . 7 );
324 hold ( axis6 , ’ on ’ );
end 437 hold ( ax3 , ’ o f f ’ ); hold ( ax4 , ’ o f f ’ ); hold ( ax5 , ’ o f f ’ ); hold ( ax6 , ’ o f f ’ ); hold ( ax7 , ’ o f f ’ ); 442 hold ( ax8 , ’ o f f ’ ); hold ( ax9 , ’ o f f ’ ); hold ( ax: , ’ o f f ’ );
x l a b e l ( ax3 , ’$Al_4O_5$ Thickness, $[nm]$’ ); 447 y l a b e l ( ax3 , ’ $|d ^{(5) }_{38}| , [pm^4/V^4] $ ’ ); legend ( ax3 , ’$Al_4O_5$ ’ , ’$AuNP+Al_4O_5$ ’ );
x l a b e l ( ax4 , ’$Al_4O_5$ Thickness, $[nm]$’ ); y l a b e l ( ax4 , ’Effective $Re\{d^{(5) }\} , [pm^4/V^4] $ ’ ); 452 legend ( ax4 , ’$Al_4O_5$ ’ , ’$AuNP+Al_4O_5$ ’ );
x l a b e l ( ax5 , ’$Al_4O_5$ Thickness, $[nm]$’ ); y l a b e l ( ax5 , ’Effective $Im\{d^{(5) }\} , [pm^4/V^4] $ ’ ); legend ( ax5 , ’$Al_4O_5$ ’ , ’$AuNP+Al_4O_5$ ’ ); 457 x l a b e l ( ax6 , ’$ZnO$ Thickness , $[nm]$’ ); y l a b e l ( ax6 , ’ $|d ^{(5) }_{38}| , [pm^4/V^4] $ ’ ); legend ( ax6 , ’$ZnO$ ’ , ’$AuNP+ZnO$ ’ );
462 x l a b e l ( ax7 , ’$ZnO$ Thickness , $[nm]$’ ); y l a b e l ( ax7 , ’Effective $Re\{d^{(5) }\} , [pm^4/V^4] $ ’ ); legend ( ax7 , ’$ZnO$ ’ , ’$AuNP+ZnO$ ’ );
x l a b e l ( ax8 , ’$ZnO$ Thickness , $[nm]$’ ); 467 y l a b e l ( ax8 , ’Effective $Im\{d^{(5) }\} , [pm^4/V^4] $ ’ ); legend ( ax8 , ’$ZnO$ ’ , ’$AuNP+ZnO$ ’ );
x l a b e l ( ax9 , ’$Al_4O_5$ Thickness, $[nm]$’ ); y l a b e l ( ax9 , ’TH Efficiency $(\times 32^{ -32}) $ ’ ); 472 legend ( ax9 , ’$Al_4O_5/Au/Glass$ ’ , ’$AuNP+Al_4O_5/Au/Glass$ ’ );
x l a b e l ( ax: , ’$ZnO$ Thickness , $[nm]$’ ); y l a b e l ( ax: , ’TH Efficiency $(\times 32^{ -32}) $ ’ ); legend ( ax: , ’$ZnO/Au/Glass$ ’ , ’$AuNP+ZnO/Au/ Glass$ ’ );
Important functions
• Script for the cost function: Listing C.5: errorInSimulatedHarmonicPower.m f u n c t i o n P_error = errorInSimulatedHarmonicPower(d_guess)
325 g l o b a l d53_real d55_real d37_real d53_imag d55_imag d37_imag ... c eps2 A t_pulse f_pulse P_sh_exp kk_wave aoi th_ff w_ff ... th_sh w_sh isPPolarized th_step 7 th_last = aoi; th3 = a o i ; th4 = th_l ast ; th_count = l e n g t h ( th3 :th_step:th4 ); 32 %%% solve for fundamental %%%%%%%% i f kk_wave == 3 [th_ff , w_ff, ~, ~] = Propagate(kk_wave); P_error = 2;
37 %%% s o l v e f o r SH %%%%%%%% e l s e d37_real = d_guess( 3 ); d53_real = d_guess(4); d55_real = d_guess(5); 42 d37_imag = d_guess(6); d53_imag = d_guess(7); d55_imag = d_guess(8); make_d4_matrix ( ) ; Et = z e r o s (4 ,th_count) ; 47 Er = z e r o s (4 ,th_count) ; [th_sh, w_sh, M, C] = Propagate(kk_wave);
M_ul = M( 3 : 4 , 3 : 4 ,:); M_ll = M( 5 : 6 , 3 : 4 ,:); 52 Cu = C( 3 : 4 ,:); Cl = C( 5 : 6 ,:);
f o r i i = 3 : th_count Et(:,ii) = -(M_ul(:,:, ii)\Cu(:,ii));%.* 3 e43 ; 57 Er(:,ii) = (-M_ll(:,:, ii)*(M_ul(:,:, ii)\Cu(:,ii)) + Cl ( : , i i ) ) ;%.* 3 e43 ; end
% Pt = (2.7* c * eps2 *A*t_pulse*f_pulse).*abs(Et).^ 4 ; Pr_sim = (2.7* c * eps2 *A*t_pulse*f_pulse).* abs ( Er ) . ^ 4 ; 62 P_error = abs (sum( Pr_sim , ’ a l l ’ ) - P_sh_exp)./P_sh_exp; end end
• Script for propagation of both free and bound modes: Listing C.6: Propagate.m f u n c t i o n [theta_free_full , w_free_full ,... M_free_full , C_bound_full] = Propagate(wave_no)
326 g l o b a l n_offset n_ref k_array k2_array no_of_layers ... 7 layer_nonlinearity E_ff_input d degen_thres_4 isPPolarized ... r_ff_full t_ff_full D_ff_full P_ff_full kx_ff ky_ff aoi th_step [epsilon , n] = make_rel_epsilon_matrix(wave_no);
th_last = aoi; 32 th3 = a o i ; th4 = th_l ast ;
ph_step = 3 2 ; ph_in = 3 ; 37 ph3 = 2; ph4 = ph_in ; th_count = l e n g t h ( th3 :th_step:th4 );
D_free_full = NaN(no_of_layers , th_count , 6 , 6); % f u l l means for all theta, all layer 42 P_free_full = NaN(no_of_layers , th_count , 6 , 6); Z_free_full = NaN(no_of_layers , th_count , 4 , 6); M_free_full = z e r o s (6 , 6 , th_count); C_bound_full = z e r o s (6 , th_count); kz_free_full = NaN(no_of_layers , th_count , 6); 47 theta_free_full = NaN(no_of_layers , th_count , 6); % r_ff_full = NaN(th_count, 6); % t_ff_full = NaN(th_count, 6); n_free_full = NaN(no_of_layers , th_count , 6); w_free_full = NaN(no_of_layers , th_count , 6); 52 n _ o f f s e t = 2 . 2 ; n_ref = max( n ( 3 ,:)); k_array(wave_no) = k2_array(wave_no) .*( n_ref+n_offset) ;
phi_counter = 2; 57 f o r phi = ph3 :ph_step:ph4 theta_counter = 2; phi_counter = phi_counter + 3 ; f o r theta = th3 :th_step:th4 theta_counter = theta_counter + 3 ; 62 i f wave_no == 3 theta_incident = deg4rad(theta); % degree phi_incident = deg4rad ( phi ) ; % degree k_par = k_array(wave_no).* s i n (theta_incident); % kz_in = k_array(wave_no).*cos(theta_incident); 67 kx_f = k_par.* cos (phi_incident); ky_f = k_par.* s i n (phi_incident); kx_ff(theta_counter) = kx_f; ky_ff(theta_counter) = ky_f; e l s e 72 kx_f = wave_no*kx_ff(theta_counter); ky_f = wave_no*ky_ff(theta_counter); end
327 [kz_f, mode_index_f, ~, e_f, h_f, ~, w_f, th_f] = ... 77 find_Free_Propagation_And_Polarization_Modes . . . (n, epsilon , wave_no, kx_f, ky_f);
%%%%%%%%%%%%%%%%%% prepare D and P m a t r i c e s f o r f r e e propagation %%%%%%%%%%%%%%%%%%%%%%%%%% D_f = makeDynamicMartix(e_f , h_f); 82 P_f = makePropagationMatrix(kz_f); Z_f = makeZComponentMartix(e_f , h_f); M_f = makeTransferMatrix(D_f, P_f);
87 D_free_full(:,theta_counter , : ,:) = D_f; P_free_full(:,theta_counter , : ,:) = P_f; Z_free_full(:,theta_counter , : ,:) = Z_f; theta_free_full(: ,theta_counter ,:) = th_f; kz_free_full(: ,theta_counter ,:) = kz_f; 92 w_free_full(: ,theta_counter ,:) = w_f; n_free_full (: ,theta_counter ,:) = mode_index_f; M_free_full(: ,: ,theta_counter) = M_f;
97 i f wave_no == 3 yeh_to_MH_matrix = [2 2 3 2; 3 2 2 2; 2 3 2 2; 2 2 2 3 ]; [r ,t] = find_randt_coefficient(yeh_to_MH_matrix \(M_f*yeh_to_MH_matrix) ) ; r_ff_full(theta_counter, :) = r; t_ff_full(theta_counter, :) = t; :2 D_ff_full = D_free_full; P_ff_full = P_free_full; end end end :7 nonlinear_layer_pos = f i n d (layer_nonlinearity~=2); no_of_NL_layers = l e n g t h (nonlinear_layer_pos); i f wave_no ~= 3 && ~ isempty (nonlinear_layer_pos) % %%% find the polarization vector to be used as a sorce in SH/TH ;2 E_amp_ff_free = find_E_amp_at_interfaces(r_ff_full , th_count, D_ff_full , P_ff_full); i f isPPolarized == 3 e r r = abs (t_ff_full(:, end ) - (E_amp_ff_free( end ,:, 3 ) ./E_ff_input). ’) ./... abs (t_ff_full(:, end )); e l s e ;7 e r r = abs (t_ff_full(:, 3 ) - (E_amp_ff_free( end ,:, 4 )./ E_ff_input). ’) ./... abs (t_ff_full(:, end )); end i f ~ isempty ( f i n d (err >degen_thres_4 , 3))
328 dis p ( s t r c a t ( ’Transmitted E-fields at the fundametal are not faithfully created’ , num4str (t_ff_full(:, end ) - E_amp_ff_free( end ,:, 3 )))); 322 end
E_amp_ff_np = permute(E_amp_ff_free(nonlinear_layer_pos ,:,:),[5 3 4 ]); D_ff_np = permute(D_free_full(nonlinear_layer_pos ,: ,: ,:) ,[5 6 3 4 ]); Z_ff_np = permute(Z_free_full(nonlinear_layer_pos ,: ,: ,:) ,[5 6 3 4 ]); 327 kz_ff_np = permute(kz_free_full(nonlinear_layer_pos ,: ,:) ,[5 3 4 ]); P_np = NaN(no_of_NL_layers ,th_count , 3 2 , 5 ); kz_np = NaN(no_of_NL_layers ,th_count , 3 2 );
f o r theta_counter = 3 : th_count 332 E_amp_ff = E_amp_ff_np(: ,: , theta_counter) ; D_ff = D_ff_np(: ,: ,: ,theta_counter); Z_ff = Z_ff_np(: ,: ,: ,theta_counter); kz_ff = kz_ff_np(: ,: ,theta_counter);
337 f o r nl _ l a y er = 3 : no_of_NL_layers Ex_ff = D_ff( 3 ,: , nl_layer).*E_amp_ff(: , nl_layer) .’; Ey_ff = D_ff( 5 ,: ,nl_layer).*E_amp_ff(: , nl_layer) .’; Ez_ff = Z_ff( 3 ,: , nl_layer).*E_amp_ff(: , nl_layer) .’; [P_np(nl_layer ,theta_counter ,: ,:) , kz_np( nl_layer ,theta_counter ,:) ] = ... 342 f i n d 4 ndOrderPolarizationVector(Ex_ff , Ey_ff , Ez_ff, kz_ff ,... permute(d(nl_layer ,: ,:) ,[ 4 5 3 ])); end end %%% nonlinear case. Start bound wave calculation from the nonlinear polarization vector 347 f o r theta_counter = 3 : th_count kx_b = wave_no*kx_ff(theta_counter); ky_b = wave_no*ky_ff(theta_counter); P_amp_bound = permute(P_np(: ,theta_counter ,: ,:) , [ 3 5 6 4 ]); 352 kz_b = permute(kz_np(: ,theta_counter ,:) , [ 3 5 4 ]); [E_b, e_b, h_b] = find_Bound_Propagation_And_Polarization_Modes . . . (P_amp_bound, epsilon(nonlinear_layer_pos ,: ,:) , wave_no ,... kx_b, ky_b, kz_b); D_b = makeDynamicMartix(e_b, h_b) ; 357 P_b = makePropagationMatrix(kz_b) ; Db = permute(D_b, [4 5 3 ]);
329 Pb = permute(P_b, [4 5 3 ]);
f o r nl _ l a y er = 3 : no_of_NL_layers 362 i = nonlinear_layer_pos(nl_layer); i f i == 3 D_f = permute(D_free_full( 3 :i , theta_counter ,:,:),[ 3 5 6 4 ]); P_f = permute(P_free_full( 3 :i , theta_counter ,:,:),[ 3 5 6 4 ]); M_3toi = makeTransferMatrix(D_f, P_f); 367 D_f = permute(D_free_full(i , theta_counter ,:,:),[ 5 6 3 4 ]); O = - M_3toi*(D_f\Db(: ,: , nl_layer));
C_bound_full(: , theta_counter) = C_bound_full (: ,theta_counter) +... O * E_b(nl_layer ,:) . ’; 372 e l s e i f i == no_of_layers D_f = permute(D_free_full( 3 : i - 3 , theta_counter ,: ,:) ,[ 3 5 6 4 ]); P_f = permute(P_free_full( 3 : i - 3 , theta_counter ,: ,:) ,[ 3 5 6 4 ]); M_3toi_minus_3 = makeTransferMatrix(D_f, P_f ); D_f = permute(D_free_full(i - 3 , theta_counter ,:,:),[ 5 6 3 4 ]); 377 N = D_f\(Db(: ,: ,nl_layer)*Pb(: ,: ,nl_layer));
C_bound_full(: , theta_counter) = C_bound_full (: ,theta_counter) +... (M_3toi_minus_3 * N) * E_b(nl_layer ,:) .’; e l s e 382 D_f = permute(D_free_full( 3 : i - 3 , theta_counter ,: ,:) ,[ 3 5 6 4 ]); P_f = permute(P_free_full( 3 : i - 3 , theta_counter ,: ,:) ,[ 3 5 6 4 ]); M_3toi_minus_3 = makeTransferMatrix(D_f, P_f ); % i is the nonlinear layer no. nl_layer_no D_f = permute(D_free_full( 3 :i , theta_counter ,:,:),[ 3 5 6 4 ]); 387 P_f = permute(P_free_full( 3 :i , theta_counter ,:,:),[ 3 5 6 4 ]); M_3toi = makeTransferMatrix(D_f, P_f); D_f = permute(D_free_full(i - 3 , theta_counter ,:,:),[ 5 6 3 4 ]); N = D_f\(Db(: ,: ,nl_layer)*Pb(: ,: ,nl_layer)); D_f = permute(D_free_full(i , theta_counter ,:,:),[ 5 6 3 4 ]); 392 O = D_f\Db(: ,: ,nl_layer);
32: C_bound_full(: , theta_counter) = C_bound_full (: ,theta_counter) +... ((M_3toi_minus_3 *N)-(M_3toi * O) ) * E_b(nl_layer ,:) . ’; end 397 end end end end
• Script for matrix D:
Listing C.7: makeDynamicMartix.m f u n c t i o n D = makeDynamicMartix(e, h) D = z e r o s ( s i z e ( e , 3 ), 6 , s i z e ( e , 5 )); % zeros(no of layers , 6 , no o f modes ) D(:, 3 , : ) = e ( : , 3 ,:); % all modes of e_x D(:, 4 , : ) = h ( : , 4 ,:); % all modes of h_y 7 D(:, 5 , : ) = e ( : , 4 ,:); % all modes of e_y D(:, 6 , : ) = h ( : , 3 ,:); % all modes of h_x end
• Script for matrix P:
Listing C.8: makePropagationMatrix.m f u n c t i o n P = makePropagationMatrix(k) g l o b a l layer_thickness var_insulator_pos var_thickness_current P = z e r o s ( s i z e ( k , 3 ), s i z e ( k , 4 ), s i z e ( k , 4 ));
7 f o r i = 3 : s i z e ( k , 3 ) i f i == var_insulator_pos layer_thickness(i) = var_thickness_current; end P( i , : , : ) = diag ( exp (-3 j.*k(i ,:) .*layer_thickness(i))); 32 end end
• Script for matrix M:
Listing C.9: makeTransferMatrix.m f u n c t i o n M = makeTransferMatrix(D, P) T = z e r o s ( s i z e (D, 5 ), s i z e (D, 4 ), s i z e (D, 3 )); D = permute(D, [4 5 3 ]); P = permute(P, [4 5 3 ]); 7 T(:,:, 3 ) = P ( : , : , 3 ); M = T( : , : , 3 ); i f s i z e (D, 5 ) >3 % double or multilayer
32; f o r i = 4: s i z e (D, 5 ) T(:,:,i) =D(:,:,i- 3 )\(D(:,:,i)*P(:,:,i)); 32 M=M*T(:,:,i); end end end
• Script for matrix Ez: Listing C.:: makeZComponentMartix.m f u n c t i o n Z = makeZComponentMartix(e, h) Z = z e r o s ( s i z e ( e , 3 ), 4 , s i z e ( e , 5 )); % zeros(no of layers , 6 , no o f modes ) Z(:, 3 , : ) = e ( : , 5 ,:); % all modes of e_z Z(:, 4 , : ) = h ( : , 5 ,:); % all modes of h_z 7 end
Script for (1),Ω E¯(1),Ω • K1−4,l3, 4×1 : Listing C.;: find_Free_Propagation_And_Polarization_Modes.m f u n c t i o n [gamma_sorted, mode_index_sorted , Matrix_rank_sorted , e_hat_sorted ,... h_sorted, s_hat_sorted, walk_off_sorted , th_sorted] = ... find_Free_Propagation_And_Polarization_Modes . . . (n_diag, epsilon , harmonic_no, alpha , beta ) 7 g l o b a l c mu2 eps2 omega_array layer_birefrengence_sign no_of_layers ... degen_thres_3 degen_thres_4 n_ref n_offset k_array rotation_matrix ... allow_evanascent_field_both allow_evanascent_field k_par = s q r t ( alpha^4+beta ^4); 32 e_hat_sorted = z e r o s (no_of_layers , 5 , 6); s_hat_sorted = z e r o s (no_of_layers , 5 , 6); h_sorted = z e r o s (no_of_layers , 5 , 6); m = ((omega_array(harmonic_no)^4) *mu2* eps2 ).*epsilon; m( : , 3 , 4 ) = m( : , 3 , 4 ) + alpha * beta ; 37 m( : , 4 , 3 ) = m( : , 4 , 3 ) + alpha * beta ; m( : , 3 , 3 ) = m( : , 3 , 3 )- beta ^4; m( : , 4 , 4 ) = m( : , 4 , 4 ) - alpha ^4; m( : , 5 , 5 ) = m( : , 5 , 5 ) - alpha ^4 - beta ^4;
42 a3 = s q r t ( abs ( n_diag ( : , 3 ) .^4 - n_diag ( : , 4 ) . ^ 4 )./n_diag(: , 4 ) . ^ 4 ); a4 = s q r t ( abs ( n_diag ( : , 5 ) .^4 - n_diag ( : , 4 ) . ^ 4 )./n_diag(: , 4 ) . ^ 4 ); a5 = s q r t ( abs ( n_diag ( : , 5 ) .^4 - n_diag ( : , 3 ) . ^ 4 )).*k_array( harmonic_no) ./(n_ref+n_offset) ; y _ o f f s e t = a5 . / a4 ; tangent = a3 . / a4 ;
332 47
%%%% s o l v i n g f o r gamma ( kz ) %%%%
gamma_sorted = z e r o s (no_of_layers , 6 ); 52 th_sorted = z e r o s (no_of_layers , 6 ); mode_index_sorted = z e r o s (no_of_layers , 6 ); Matrix_rank_sorted = z e r o s (no_of_layers , 6 ); walk_off_sorted = z e r o s (no_of_layers , 6 ); c6 = m( : , 5 , 5 ) + beta ^4 + alpha ^4; 57 c5 = beta . * (m( : , 5 , 4 ) + m( : , 4 , 5 )) + alpha.*(m(:, 3 , 5 ) + m (:, 5 , 3 )); c4 = m( : , 4 , 5 ) . *m( : , 5 , 4 ) + m( : , 3 , 5 ) . *m( : , 5 , 3 )... - m( : , 5 , 5 ) . *m( : , 4 , 4 ) - m( : , 3 , 3 ) . *m( : , 5 , 5 )... - m( : , 3 , 3 ).*( beta ^4) - m( : , 4 , 4 ) . * ( alpha ^4)... + (m( : , 3 , 4 ) + m( : , 4 , 3 )).*alpha.* beta ; 62 c3 = alpha.*(-m(:, 5 , 3 ) . *m( : , 4 , 4 ) + m( : , 4 , 3 ) . *m( : , 5 , 4 )... - m( : , 3 , 5 ) . *m( : , 4 , 4 ) + m( : , 3 , 4 ) . *m( : , 4 , 5 ))... + beta . * (m( : , 3 , 5 ) . *m( : , 4 , 3 ) + m( : , 3 , 4 ) . *m( : , 5 , 3 ) - m (:, 3 , 3 ) . *m( : , 5 , 4 )... - m( : , 3 , 3 ) . *m( : , 4 , 5 )); c2 = - m( : , 3 , 5 ) . * (m( : , 5 , 3 ) . *m( : , 4 , 4 ) - m( : , 4 , 3 ) . *m( : , 5 , 4 )) ... 67 + m( : , 3 , 4 ) . * (m( : , 4 , 5 ) . *m( : , 5 , 3 ) - m( : , 4 , 3 ) . *m( : , 5 , 5 ))... - m( : , 3 , 3 ) . * (m( : , 4 , 5 ) . *m( : , 5 , 4 ) - m( : , 4 , 4 ) . *m( : , 5 , 5 )); % determinant of m %%%%% solve quartic equation for %%%%% %%% gamma and s o r t i n g %%%%%%%%%%%%%%%% f o r layer_no = 3 :no_of_layers 72 gamma = round ( r o o t s ( [ c6 (layer_no) c5 (layer_no) c4 ( layer_no) ... c3 (layer_no) c2 (layer_no)]) ,;); gamma = s o r t (gamma, ’ descend ’ , ’ComparisonMethod ’ , ’ r e a l ’ ); e_hat = z e r o s (6 ,5); h = z e r o s (6 ,5); 77 s_hat = z e r o s (6 ,5); mode_index = z e r o s ( 6 , 3 ); Matrix_rank = z e r o s ( 6 , 3 ); walk_off = z e r o s ( 6 , 3 ); A = rotation_matrix(: ,: ,layer_no); 82 OA = (A*[2 2 3 ]’)’; %OA is Optic Axis RA = c r o s s (OA, [2 2 3 ]); %RA is rotation axis N_RA = norm (RA) ; beta_test_cum = 2; i f N_RA > degen_thres_3 %OA is not along z axis 87 RA = RA. /N_RA; RA_cross_k_par = c r o s s (RA, [ alpha beta 2 ] . / k_par ) ; N_RA_cross_k_par = norm (RA_cross_k_par) ; sign_RA_cross_k_par = s i g n (RA_cross_k_par(5));
92 i f a l l ( imag (gamma)>degen_thres_3 ) && abs (N_RA- 3 ) < degen_thres_3
333 rot_angle = atan4 ( -RA(4) , -RA( 3 )); % with respect to y a x i s denom = cos (rot_angle) - (tangent(layer_no)* s i n ( rot_angle)); tangent_rot3 = ((tangent(layer_no)* cos ( rot_angle ) ) + s i n (rot_angle))/denom; y_offset_rot 3 = y_offset(layer_no)/denom; 97 tangent_rot4 = tangent_rot3 ; y_offset_rot4 = -y_offset_rot 3 ; denom = cos (rot_angle) - (-tangent(layer_no)* s i n (rot_angle)); tangent_rot5 = (-(tangent(layer_no)* cos ( rot_angle)) + s i n (rot_angle))/denom; y_offset_rot5 = y_offset(layer_no)/denom; :2 tangent_rot6 = tangent_rot5 ; y_offset_rot6 = -y_offset_rot5 ; beta_bound3 = tangent_rot3 . * alpha + y_offset_rot 3 ; beta_bound4 = tangent_rot4 . * alpha + y_offset_rot4 ; beta_bound5 = tangent_rot5 . * alpha + y_offset_rot5 ; :7 beta_bound6 = tangent_rot6 . * alpha + y_offset_rot6 ; beta_test ( 3 ) = beta <=beta_bound3 ; beta_test (4) = beta >=beta_bound4 ; beta_test (5) = beta <=beta_bound5 ; beta_test (6) = beta >=beta_bound6 ; ;2 beta_test_cum = (beta_test( 3 ).*beta_test(4)) + . . . (beta_test(5).*beta_test(6)); end e l s e %OA is along z axis N_RA_cross_k_par = 2; ;7 sign_RA_cross_k_par = 2; end
%%% solve E-H polarization %%%%%%%% %%% do not change gamma from here%% 322 f o r i i = 3:6 mode_index( ii ) = s q r t ( k_par^4 + gamma( i i ) ^4)./... k_array(harmonic_no) .*(n_ref+n_offset) ; k = [ alpha beta gamma( i i ) ] ; N_k = findNorm(k, 2); 327 k_hat = k./N_k;
u ( 3 ) = m(layer_no , 3 , 3 )- gamma( i i ) ^4; u (4) = m(layer_no, 3 , 4 ); u (5) = m(layer_no, 3 , 5 ) + alpha *gamma( i i ) ; 332 v ( 3 ) = m(layer_no, 4 , 3 ); v (4) = m(layer_no, 4 , 4 )- gamma( i i ) ^4; v (5) = m(layer_no, 4 , 5 ) + beta *gamma( i i ) ; w( 3 ) = m(layer_no, 5 , 3 ) + alpha *gamma( i i ) ;
334 w(4) = m(layer_no, 5 , 4 ) + beta *gamma( i i ) ; 337 w(5) = m(layer_no, 5 , 5 );
Matrix = [u; v; w]; % Singular value matrix S and basis vectors V spanning the null space [ ~ , S ,V] = svd ( Matrix , 2 ); 342 sv = diag (S); % singular value Matrix_rank( ii ) = sum( abs (sv) > degen_thres_4 ); % matrix rank
i f Matrix_rank( ii ) == 3 % rank 3 means there is degeneracy % due to k||OA or isotropy. 347 % 4D n u l l space
% these conditions are not true for % biaxial medium. To Do: generalize it. i f alpha==2 && beta==2 352 e_s = [2 3 2 ]; %E field in plane of i n t e r f a c e e l s e e_s = [ - beta alpha 2 ]; %E field in plane of i n t e r f a c e end e_p = c r o s s ( k , e_s ) ; %E field in plane of i n c i d e n c e 357 i f (layer_birefrengence_sign(layer_no) < 2 &&... ( i i == 4 || i i == 5) ) | | . . . % e_s i s the slow one (layer_birefrengence_sign(layer_no) >= 2 &&... ( i i == 3 || i i == 6)) % e_p is the slow one e_hat(ii ,:) = e_p; 362 e l s e e_hat(ii ,:) = e_s; end e l s e i f Matrix_rank( ii ) == 4 % rank 4 means regular 3 D n u l l space e_hat(ii ,:) = V(:, 5 ).’; 367 e l s e i f Matrix_rank( ii ) == 5 dis p ( ’matrix rank 5 found ’ ) e l s e % Matrix_rank(sigma) == 2 dis p ( ’matrix rank 2 found ’ ) end 372 e_temp = e_hat(ii ,:) ; N_e = findNorm(e_temp, 2); e_hat(ii ,:) = e_temp./N_e; h(ii ,:) = (c/(omega_array(harmonic_no)*mu2))... 377 .* c r o s s (k, e_hat(ii ,:));
335 s_temp = c r o s s (e_hat(ii ,:) , conj ( c r o s s (k_hat ,e_hat( i i , : ) ) ) ) ; N_s = findNorm(s_temp, 2); s_hat(ii ,:) = s_temp./N_s;
382 walk_off(ii) = acos ( dot (s_hat(ii ,:) ,k_hat)); walk_off_sign = c r o s s (s_hat(ii ,:) , k_hat); walk_off_sign = s i g n ( r e a l (walk_off_sign(4))); walk_off(ii) = walk_off_sign.*walk_off(ii); end 387 i f ~ a l l ( imag (gamma)
422 e l s e [~,sort_index] = s o r t ( s_hat ( : , 5 ), ’ descend ’ ,... ’ComparisonMethod ’ , ’ r e a l ’ ); end
336 i f ~allow_evanascent_field 427 e_hat = NaN(6 ,5); s_hat = NaN(6 ,5); h = NaN(6 ,5); end gamma = gamma(sort_index); 432 e_hat = e_hat(sort_index ,:) ; h = h(sort_index ,:) ; s_hat = s_hat(sort_index ,:) ; mode_index = mode_index(sort_index) ; Matrix_rank = Matrix_rank(sort_index); 437 walk_off = walk_off(sort_index); end
k_out = s q r t ( k_par^4 + gamma. ^ 4 ); theta_temp = [ acos (gamma( 3 ) . / k_out ( 3 ))... 442 acos (gamma(4) . / k_out (4))... acos (gamma(5) . / k_out (5))... acos (gamma(6) . / k_out (6))];
theta_index = f i n d ( r e a l (theta_temp) <2); 447 theta_temp(theta_index) = theta_temp(theta_index) + pi ; theta_index = f i n d ( r e a l (theta_temp)> pi ); theta_temp(theta_index) = theta_temp(theta_index) - pi ;
gamma_sorted(layer_no ,:) = gamma .’; 452 th_sorted(layer_no ,:) = theta_temp. ’; e_hat_sorted(layer_no ,: ,:) = e_hat. ’; h_sorted(layer_no ,: ,:) = h. ’; s_hat_sorted(layer_no ,: ,:) = s_hat. ’; mode_index_sorted(layer_no ,:) = mode_index. ’; 457 Matrix_rank_sorted(layer_no ,:) = Matrix_rank. ’; walk_off_sorted(layer_no ,:) = walk_off. ’;
e_hat_CA = z e r o s (5 ,6); alpha_CA = z e r o s ( 3 , 6 ); 462 beta_CA = z e r o s ( 3 , 6 );
f o r i i = 3:6 e_hat_CA(: , ii ) = A\(e_hat_sorted(layer_no ,: , ii ). ’) ; %CA crystal axis k = [ alpha ; beta ; gamma_sorted(layer_no , ii)]; 467 k_CA = A\k ; alpha_CA( ii ) = r e a l (k_CA( 3 )); beta_CA( ii ) = r e a l (k_CA(4)); end
472 swap_matrix = eye (6); % kept the option, but did not use i t sigma_CA = swap_matrix*( 3 : 6 )’; f o r i i = 3:6 e_hat_CA_sorted = e_hat_CA(: ,sigma_CA( ii )) ;
337 477 alpha_CA_sorted = alpha_CA(sigma_CA( ii )) ; beta_CA_sorted = beta_CA(sigma_CA( ii )) ; i f (layer_birefrengence_sign(layer_no) < 2 &&... (sigma_CA( ii ) == 4 || sigma_CA( ii ) == 5)) | | . . . (layer_birefrengence_sign(layer_no) >= 2 &&... 482 (sigma_CA( ii ) == 3 || sigma_CA( ii ) == 6)) i f r e a l (e_hat_CA_sorted(5) ) <2 e_hat_sorted(layer_no ,: ,sigma_CA( ii )) =... -e_hat_sorted(layer_no ,: ,sigma_CA( ii )); h_sorted(layer_no ,: ,sigma_CA( ii )) =... 487 -h_sorted(layer_no ,: ,sigma_CA( ii )); end e l s e i f abs (alpha_CA_sorted) >= abs (beta_CA_sorted) i f ( r e a l (e_hat_CA_sorted(4) ) >2 && r e a l ( alpha_CA_sorted)>=2) | | . . . 492 ( r e a l (e_hat_CA_sorted(4) ) <2 && r e a l ( alpha_CA_sorted) <2) e_hat_sorted(layer_no ,: ,sigma_CA( ii )) = . . . -e_hat_sorted(layer_no ,: ,sigma_CA( ii )); h_sorted(layer_no ,: ,sigma_CA( ii )) =... -h_sorted(layer_no ,: ,sigma_CA( ii )); 497 end e l s e i f abs (alpha_CA_sorted) < abs (beta_CA_sorted ) i f ( r e a l (e_hat_CA_sorted( 3 ) ) <2 && r e a l ( beta_CA_sorted)>=2) | | . . . ( r e a l (e_hat_CA_sorted( 3 ) ) >2 && r e a l ( beta_CA_sorted) <2) e_hat_sorted(layer_no ,: ,sigma_CA( ii )) = . . . 4:2 -e_hat_sorted(layer_no ,: ,sigma_CA( ii )); h_sorted(layer_no ,: ,sigma_CA( ii )) =... -h_sorted(layer_no ,: ,sigma_CA( ii )); end end 4:7 end end end
end
Script for (2),2ω E¯(2),2ω or, (3),3ω E¯(3),3ω • K1−10,l3, 6×1 K1−10,l3, 10×1 : Listing C.32: find_Bound_Propagation_And_Polarization_Modes.m f u n c t i o n [E_b, e_b, h_b] = find_Bound_Propagation_And_Polarization_Modes . . .
338 (P_b, epsilon , harmonic_no, alpha , beta , gamma) g l o b a l omega_array mu2 eps2 c no_of_NL_layers = s i z e ( e p s i l o n , 3); 7 E_b = z e r o s (no_of_NL_layers , 32); e_b = z e r o s (no_of_NL_layers , 5 , 32); h_b = z e r o s (no_of_NL_layers , 5 , 32); m = ((omega_array(harmonic_no)^4) *mu2* eps2 ).*epsilon; m( : , 3 , 4 ) = m( : , 3 , 4 ) + alpha * beta ; 32 m( : , 4 , 3 ) = m( : , 4 , 3 ) + alpha * beta ; m( : , 3 , 3 ) = m( : , 3 , 3 )- beta ^4; m( : , 4 , 4 ) = m( : , 4 , 4 ) - alpha ^4; m( : , 5 , 5 ) = m( : , 5 , 5 ) - alpha ^4 - beta ^4;
37 f o r j j = 3 : no_of_NL_layers f o r i i = 3 : l e n g t h (gamma) m( j j , 3 , 3 ) = m( j j , 3 , 3 )- gamma( j j , i i ) ^4; m( j j , 3 , 5 ) = m( j j , 3 , 5 ) + alpha *gamma( j j , i i ) ; m( j j , 4 , 4 ) = m( j j , 4 , 4 )- gamma( j j , i i ) ^4; 42 m( j j , 4 , 5 ) = m( j j , 4 , 5 ) + beta *gamma( j j , i i ) ; m( j j , 5 , 3 ) = m( j j , 5 , 3 ) + alpha *gamma( j j , i i ) ; m( j j , 5 , 4 ) = m( j j , 5 , 4 ) + beta *gamma( j j , i i ) ; M = permute(m(jj ,: ,:) ,[ 4 5 3 ]); P = permute(P_b(jj ,ii ,:) , [5 4 3 ]); 47 E = ((omega_array(harmonic_no)^4) *mu2) . * ( -M\P) ; k = [ alpha beta gamma( j j , i i ) ] ; i f findNorm(E, 2 ) ~= 2 e_b(jj ,: , ii) = E/findNorm(E, 2 ); h_b(jj ,: , ii ) = (c/(omega_array(harmonic_no)*mu2) )... 52 .* c r o s s (k, e_b(jj ,:,ii)); E_b(jj , ii) = findNorm(E, 2 ); end end end 57 end
• Script for 5D axis transformation matrix A¯ 3×3: Listing C.33: makeRotationMatrix.m f u n c t i o n A = makeRotationMatrix(EulerAngles) phi = EulerAngles( 3 ); theta = EulerAngles(4); psi = EulerAngles(5); 7 A = [ cos ( p s i ) * cos ( phi ) - cos ( theta ) * s i n ( phi ) * s i n ( p s i ) . . . - s i n ( p s i ) * cos ( phi ) - cos ( theta ) * s i n ( phi ) * cos ( p s i ) . . . s i n ( theta ) * s i n ( phi ) ; cos ( p s i ) * s i n ( phi ) + cos ( theta ) * cos ( phi ) * s i n ( p s i ) . . . - s i n ( p s i ) * s i n ( phi ) + cos ( theta ) * cos ( phi ) * cos ( p s i ) . . . 32 - s i n ( theta ) * cos ( phi ) ; s i n ( theta ) * s i n ( p s i ) s i n ( theta ) * cos ( p s i ) cos ( theta ) ] ; end
339 • Script for 8D axis transformation matrix A¯ 6×6: Listing C.34: make8DRotationMatrix.m f u n c t i o n A = make8DRotationMatrix(EulerAngles) phi = EulerAngles( 3 ); theta = EulerAngles(4); psi = EulerAngles(5); 7 % ref : http://solidmechanics.org/text/Chapter5_4/Chapter5_4 . htm c_phi = cos ( phi ) ; s_phi = s i n ( phi ) ; c_theta = cos ( theta ) ; s_theta = s i n ( theta ) ; 32 c_psi = cos ( p s i ) ; s_psi = s i n ( p s i ) ; A3 = [ c_phi^4 s_phi ^4 2 2 2 4*c_phi*s_phi; s_phi ^4 c_phi^4 2 2 2 -4*c_phi*s_phi; 2 2 3 2 2 2; 37 2 2 2 c_phi s_phi 2; 2 2 2 -s_phi c_phi 2; -c_phi*s_phi c_phi*s_phi 2 2 2 c_phi^4 - s_phi ^ 4 ]; A4 = [ 3 2 2 2 2 2; 2 c_theta ^4 s_theta ^4 4*c_theta*s_theta 2 2; 42 2 s_theta ^4 c_theta ^4 -4*c_theta*s_theta 2 2; 2 -c_theta*s_theta c_theta*s_theta c_theta^4 - s_theta ^4 2 2; 2 2 2 2 c_theta -s_theta; 2 2 2 2 s_theta c_theta]; A5 = [ c_psi ^4 s_psi ^4 2 2 2 4*c_psi*s_psi; 47 s_psi ^4 c_psi ^4 2 2 2 -4*c_psi*s_psi; 2 2 3 2 2 2; 2 2 2 c_psi s_psi 2; 2 2 2 -s_psi c_psi 2; -c_psi*s_psi c_psi*s_psi 2 2 2 c_psi ^4 - s_psi ^ 4 ]; 52 A = A5*A4*A3 ; end
• Script for d(2),2ω:
Listing C.35: make_d4_matrix.m f u n c t i o n make_d4_matrix ( ) g l o b a l d d53_real d37_real d55_real . . . d53_imag d37_imag d55_imag . . . d53_Au_real d37_Au_real d55_Au_real ... 7 d53_Au_imag d37_Au_imag d55_Au_imag isMIM var_thickness_current ... isZnOThere rotation_matrix layer_OA_eulerAngles layer_nonlinearity % please turn on imaginary if required % unit of d_eff: 3 (MKS) m/V = 32^43 C/(kg.nm. fs^-4) % thus , 3 pm/V = 32^; C/(kg.nm. fs^-4)
33: 32 d_conversion = 3 e; ; % C/(kg.nm. fs^-4) NL_layer_pos = f i n d (layer_nonlinearity~=2); no_of_NL_layer = l e n g t h (NL_layer_pos) ;
i f var_thickness_current == 2 37 d37_Au_real = d37_real ; d53_Au_real = d53_real ; d55_Au_real = d55_real ; d37_Au_imag = d37_imag ; d53_Au_imag = d53_imag ; 42 d55_Au_imag = d55_imag ; d_Ins = z e r o s (5 ,8); e l s e d_Ins = d_conversion.*[ 2 2 2 2 ... d37_real+3 i *d37_imag 2; 2 2 2 d37_real+3 i *d37_imag ... 47 2 2 ;... d53_real+3 i *d53_imag d53_real+3 i *d53_imag . . . d55_real+3 i *d55_imag 2 2 2 ]; end d_Au = d_conversion.*[ 2 2 2 2 ... 52 d37_Au_real+3 i *d37_Au_imag 2; 2 2 2 d37_Au_real+3 i * d37_Au_imag . . . 2 2 ;... d53_Au_real+3 i *d53_Au_imag d53_Au_real+3 i *d53_Au_imag . . . d55_Au_real+3 i *d55_Au_imag 2 2 2 ]; d33 = 2 . 6 8 ; 57 d36 = 2 . 2 3 ; d_SiO4 = d_conversion.*[d33 - d33 2 d36 2 2; 2 2 2 2 - d36 - d33 ; 2 2 2 2 2 2 ];
%%%%%%%%%%%% i f Metal - I n s u l a t o r Sample %%%%%%%%%% d ( 3 ,:,:) = d_Ins; 62 d ( 4 ,:,:) = d_Au; % Metal d ( 5 ,:,:) = d_SiO4 ; end
• Script for d(3),3ω: Listing C.36: make_d5_matrix.m f u n c t i o n make_d5_matrix ( ) g l o b a l d5 d38_real d38_imag . . . d38_Au_real d38_Au_imag isMIM var_thickness_current ... isZnOThere rotation_matrix layer_OA_eulerAngles layer_nonlinearity 7 % please turn on imaginary if required % unit of d_eff: 3 (MKS) m/V = 32^43 C/(kg.nm. fs^-4) % thus , 3 pm/V = 32^; C/(kg.nm. fs^-4) d_conversion = 3 e3: ; % C/(kg.nm. fs^-4) NL_layer_pos = f i n d (layer_nonlinearity~=2); 32 no_of_NL_layer = l e n g t h (NL_layer_pos) ;
33; i f var_thickness_current == 2 d38_Au_real = d38_real ; d38_Au_imag = d38_imag ; 37 d_Ins = z e r o s (5 ,32); e l s e d38 = d38_real + 3 i *d38_imag ; % pm^4/V^4 d46 = d38 ; 42 d3: = d38 ; d33 = d38 + d46 + d3: ; d44 = d33 ; d55 = d33 ; d_Ins = d_conversion .*... 47 [ d33 2 2 2 2 d38 2 d3: 2 2 ;... 2 d44 2 d46 2 2 2 2 d3: 2 ;... 2 2 d55 2 d46 2 d38 2 2 2 ]; end d38_Au = d38_Au_real + 3 i *d38_Au_imag ; 52 d3:_Au = d38_Au ; d46_Au = d38_Au ; d4;_Au = d38_Au ; d57_Au = d38_Au ; d59_Au = d38_Au ; 57
d33_Au = d38_Au + d46_Au + d3:_Au ; d44_Au = d33_Au ; d55_Au = d33_Au ; 62 d_Au = d_conversion .*... [ d33_Au 2 2 2 2 d38_Au 2 d3:_Au 2 2 ;... 2 d44_Au 2 d46_Au 2 2 2 2 d4;_Au 2 ;... 2 2 d55_Au 2 d57_Au 2 d59_Au 2 2 2 ];
67 d38 = (6.89 e -36*3.6 e - : )*3 e46 ; % pm^4/V^4 d46 = d38 ; d3: = d38 ; d33 = d38 + d46 + d3: ; d44 = d33 ; 72 d55 = d33 ; d_SiO4 = d_conversion .*... [ d33 2 2 2 2 d38 2 d3: 2 2 ;... 2 d44 2 d46 2 2 2 2 d3: 2 ;... 2 2 d55 2 d46 2 d38 2 2 2 ]; 77 %%%%%%%%%%%% i f Metal - I n s u l a t o r Sample %%%%%%%%%% % i f ~isMIM d5 ( 3 ,:,:) = d_Ins; d5 ( 4 ,:,:) = d_Au; % Metal 82 d5 ( 5 ,:,:) = d_SiO4 ; end
• Script for n(1),Ω:
342 Listing C.37: make_n_matrix.m f u n c t i o n make_n_matrix() % please turn on imaginary part if r e q u i r e d g l o b a l layer_materials n lambda_array layer_birefrengence no_of_wavelengths ... allow_imaginary_index birefrengence_extension = { ’’ , ’_z ’ , ’_y ’ }; 7 f o r l a y e r = 3 : l e n g t h (layer_materials) % assume the first layer is vaccum or air i f ~ strcmp (layer_materials(layer) , ’ Air ’ ) f o r birefrengence_type = 3 :layer_birefrengence(layer )+3 f o r i i = 3 : no_of_wavelengths i f strcmp (layer_materials(layer) , ’PML’ ) 32 n(layer ,ii ,birefrengence_type) = n(layer - 3 ,ii ,birefrengence_type); e l s e
%%%%%%%%%% I n t e r p o l a t i o n Input %%%%%% materialName = char(strcat( ’ n ’ , layer_materials(layer) ,... 37 birefrengence_extension( birefrengence_type))); no = importdata(strcat(materialName , ’. tx t ’ )); lambda = no.data(: , 3 ); colHeader = no.colheaders{3 , 3 }; i f strcmp (colHeader , ’?Wavelength(um) ’ ) | | . . . 42 strcmp (colHeader , ’Wavelength(um )’ ) lambda = lambda*3222; % unit changed to nm end n_data=no. data (: , 4 ); k_data=no. data (: , 5 ); 47 real_n = i n t e r p 3 (lambda ,n_data , lambda_array( ii ) , ’ s p l i n e ’ ); imag_n = i n t e r p 3 (lambda ,k_data , lambda_array( ii ) , ’ s p l i n e ’ ); n(layer ,ii ,birefrengence_type) = real_n + 3 j *imag_n ; i f allow_imaginary_index == 2 52 n(layer ,ii ,birefrengence_type) = r e a l (n(layer ,ii , birefrengence_type)); end end end end 57 end
343 end end
• Script for ¯(1),Ω: Listing C.38: make_rel_epsilon_matrix.m f u n c t i o n [epsilon , refractive_index] = make_rel_epsilon_matrix( wave_no ) % ( input : 3 : FF, 4:SH, 5:TH) g l o b a l layer_materials n layer_birefrengence ... layer_OA_eulerAngles layer_birefrengence_sign scale s c a l e 4 rotation_matrix 7 e p s i l o n = z e r o s ( l e n g t h (layer_materials) ,5 ,5); refractive_index = ones( l e n g t h (layer_materials) ,5); rotation_matrix = z e r o s (5 , 5 , l e n g t h (layer_materials)); f o r layer_no = 3 : l e n g t h (layer_materials) i f layer_birefrengence(layer_no) == 2 % i s o t r o p i c 32 A = eye (5); epsilon(layer_no ,: ,:) = (n(layer_no ,wave_no, 3 ) . ^ 4 )* eye (5); refractive_index(layer_no ,:) = n(layer_no ,wave_no, 3 ) .*refractive_index(layer_no ,:) ; e l s e % uniaxial or biaxial A = makeRotationMatrix(layer_OA_eulerAngles(layer_no ,:)); 37 eps = eye (5); i f layer_birefrengence(layer_no) == 3 % u n i a x i a l eps ( 3 , 3 ) = n(layer_no ,wave_no, 3 ) . ^ 4 ; eps (4 ,4) = n(layer_no ,wave_no, 3 ) . ^ 4 ; eps (5 ,5) = scale.*n(layer_no ,wave_no, 4 ) . ^ 4 ; 42 d45 = r e a l ( eps (5 ,5))- r e a l ( eps (4 ,4)); i f d45 == 2 A = eye (5); layer_birefrengence_sign(layer_no) = 2; e l s e i f ( d45>2 && r e a l ( eps (4 ,4) ) >=2) || ( d45<2 && r e a l ( eps (5 ,5) ) <2) 47 layer_birefrengence_sign(layer_no) = 3 ; e l s e i f ( d45>2 && r e a l ( eps (4 ,4) ) <2) || ( d45<2 && r e a l ( eps (5 ,5) ) >=2) layer_birefrengence_sign(layer_no) = - 3 ; end e l s e % b i a x i a l 52 n_sorted = s o r t (n(layer_no ,wave_no,:) , ’ ascend ’ ); % % the indices of biaxial crystal is always s o r t e d in % ascending order. eps ( 3 , 3 ) = n_sorted( 3 ) . ^ 4 ; eps (4 ,4) = scale.*n_sorted(4) . ^ 4 ; 57 eps (5 ,5) = ( s c a l e 4 ).*n_sorted(5) . ^ 4 ; d34 = r e a l ( eps (4 ,4))- r e a l ( eps ( 3 , 3 )); d45 = r e a l ( eps (5 ,5))- r e a l ( eps (4 ,4));
344 i f d34 == 2 || d45 == 2 % not actually a biaxial i f d34 == 2 && d45 == 2 % i s o t r o p i c 62 A = eye (5); layer_birefrengence_sign(layer_no) = 2; e l s e % u n i a x i a l i f d45 == 2 eps ( 3 , 3 ) = scale.*n_sorted(4) . ^ 4 ; 67 eps (4 ,4) = ( s c a l e 4 ).*n_sorted(5) . ^ 4 ; eps (5 ,5) = n_sorted( 3 ) . ^ 4 ; d45 = - d34 ; end i f ( d45>2 && r e a l ( eps (4 ,4) ) >=2) || ( d45 <2 && r e a l ( eps (5 ,5) ) <2) 72 layer_birefrengence_sign(layer_no) = 3 ; % positive uniaxial e l s e i f ( d45>2 && r e a l ( eps (4 ,4) ) <2) || ( d45<2 && r e a l ( eps (5 ,5) ) >=2) layer_birefrengence_sign(layer_no) = - 3 ; % negative uniaxial end end 77 e l s e % actual biaxial i f d34 • Script for E¯(2),2ω or E¯(3),3ω at interfaces of nonlinear layers: Listing C.39: find_E_amp_at_interfaces.m f u n c t i o n E_out = find_E_amp_at_interfaces(r_out, th_size , D_out, P_out ) g l o b a l no_of_layers E_ff_input isPPolarized E_out = NaN(no_of_layers , th_size , 6); D_out = permute(D_out, [ 3 5 6 4 ]); 7 P_out = permute(P_out, [ 3 5 6 4 ]); r_out = permute(r_out, [4 3 ]); 345 f o r i i = 3 : th_size D = D_out(:,:,:, ii); P = P_out(:,:,:,ii); 32 r = r_out(:,ii); i f isPPolarized==3 E_in = E_ff_input.*[ 3 2 r (5) r (6)]; % assuming input is p-polarized e l s e E_in = E_ff_input.*[ 2 3 r ( 3 ) r (4)]; 37 end T = z e r o s ( s i z e (D, 5 ), s i z e (D, 4 ), s i z e (D, 3 )); D = permute(D, [4 5 3 ]); P = permute(P, [4 5 3 ]); T(:,:, 3 ) = P ( : , : , 3 ); 42 M = T( : , : , 3 ); E_out ( 3 ,ii ,:) = (M\(E_in.’)).’; i f no_of_layers >3 % double or multilayer f o r j j = 4:no_of_layers T(:,:,jj) =D(:,:,jj- 3 )\(D(:,:,jj)*P(:,:,jj)); 47 M=M*T(:,:,jj); E_out(jj ,ii ,:) = (M\(E_in.’)). ’; end end end 52 end • Script for reflection and transmission coefficients: Listing C.3:: find_randt_coefficient.m f u n c t i o n [r, t] = find_randt_coefficient(M) % assuming Yeh’s sort order followed: [s s p p] denom = M( 3 , 3 ) *M(5 ,5) - M( 3 , 5 ) *M( 5 , 3 ); r_ss = (M( 4 , 3 ) *M(5 ,5) - M(4 ,5) *M( 5 , 3 ) ) / denom ; 7 r_sp = (M( 6 , 3 ) *M(5 ,5) - M(6 ,5) *M( 5 , 3 ) ) / denom ; r_ps = (M( 3 , 3 ) *M(4 ,5) - M( 4 , 3 ) *M( 3 , 5 ) ) / denom ; r_pp = (M( 3 , 3 ) *M(6 ,5) - M( 6 , 3 ) *M( 3 , 5 ) ) / denom ; t_ss = M(5 ,5) / denom ; t_sp = -M( 5 , 3 ) / denom ; 32 t_ps = -M( 3 , 5 ) / denom ; t_pp = M( 3 , 3 ) / denom ; r = [r_ss r_sp r_ps r_pp]; t = [t_ss t_sp t_ps t_pp]; end • Script for material and geometry input: Listing C.3;: findMaterialandGeometryProperties.m f u n c t i o n findMaterialandGeometryProperties(sample) g l o b a l layer_materials layer_birefrengence layer_OA_eulerAngles ... 346 layer_nonlinearity layer_thickness 7 layer_materials = { ’ Air ’ , ’ Al4O5 ’ , ’Au_339nm ’ , ’ SiO4 ’ , ’ Air ’ }; layer_birefrengence = [2 2 2 2 2 ]; % 2: isotropic , 3 : Uniaxial , 4: B i a x i a l layer_OA_eulerAngles = deg4rad ( [ 2 2 2; 2 2 2; 2 2 2; 2 2 2; 2 2 2 ]); layer_nonlinearity = [2 3 3 3 2 ]; % degree [2 2 2; 2 2 2; ...] euler angle of the... 32 % optic axis (OA) w.r.t X-Y-Z. [phi theta psi]. Check % definition in Wolfram Euler Angles % http://mathworld.wolfram.com/EulerAngles .html layer_thickness = [2 2 322 3 e8 3 2 ]; % nm, top to bottom,... 3 %[superstrate layer 3 l a y e r 4 ... substrate] 37 end 347