Campus Monterrey
School of Engineering and Sciences
Modelling of the intensity-dependent refraction of conductive oxide thin-films with near-zero-permittivity through a nonlinear transfer
matrix approach
A thesis presented by
Adriana P´erez Casanova
Submitted to the
School of Engineering and Sciences
in partial fulfillment of the requirements for the degree of Master of Science
in
Nanotechnology
Monterrey, Nuevo Le´on, November, 2020
A mi mam´a, a mi pap´a.
I would like to express my deepest and most sincere gratitude to my thesis advisor, Dr.
Israel De Le´on, from whose exceptional career I’ve learned so much, for welcoming me in his research group, for his guidance throughout my masters studies, for his time and patience, and specially for sharing his knowledge and passion for research. I could not have wished for a better advisor for my masters studies. I would also like to thank the rest of the members of the Nanophotonics & Metamaterials Research Group, for kindly allowing me to learn from them all.
Special thanks to Dr. Servando L´opez and Dr. Raul Hern´andez for sharing their time and knowledge for the evaluation of this work.
I would like to acknowledge the support from Tecnol´ogico de Monterrey for giving me a full tuition scholarship and to CONACyT for the scholarship that helped me with my living expenses.
I am very grateful for the unconditional support of Dominik, Diana, Baldx, Luis and Miriam. Thanks to all those others friends and colleagues, who have been by my side in during my masters studies. Thanks to all those who have helped me in one way or another, from a technical discussion to a good laugh.
Finally, I thank my family, for their unconditional love and support. Thanks dad for your love and interest on the things I do. Thanks Pichi for your patience, support, and for helping me with this document’s figure edition. Thanks Marisol, for caring so deeply for me. Thank you mom, for your love, for being my eternal inspiration, for reminding me to enjoying life and do things with passion.
iv
Modelling of the intensity-dependent refraction of conductive oxide thin-films with near-zero-permittivity
through a nonlinear transfer matrix approach by
Adriana P´erez Casanova Abstract
This thesis is intended to contribute to the analytical understanding of intensity-dependent re- fraction in homogeneous thin layers of transparent conductive oxides in the frequency region where the real part of the relative permittivity vanishes. The motivation for this investigation is the extraordinarily large and ultra-fast optical nonlinearity displayed by transparent conduc- tive oxides in the near-zero permittivity region which turn them into a promising base material of new photonic devices. In order to achieve this goal, a simplified numerical model has been developed for studying the behaviour of the intensity-dependent refractive index, under steady state conditions, in homogeneous, one-dimensional layered systems of TCOs. The numerical model is based on an adaptation of the Nonlinear Transfer Matrix Method that enables the method to obtain the refractive index values as a function of local intensity values inside the material. The most important capability of this model is the ability to accurately relate exper- imentally acquired measurements with the material’s microscopic properties as long as local saturation effects remain negligible.
v
2.1 Visualization of the change in refractive index ∆n, as a function of the applied optical field intensity I when the refractive index is limited by the saturation behaviour of the material. The dashed arrow indicates the effects of field en- hancement mechanisms that might increase the values of n2 which is only reflected as an increase in the slope but not an increase in the maximum at- tainable value of change in refraction ∆nmax .The shadowed zone represent the zone where saturation effects are no longer negligible . . . 5 2.2 Measured intensity dependent transmittance (T) and reflectance (R) by using
a z-scan technique . . . 6 2.3 Linear relative permittivity, Re(ε) (purple), and Im(ε) (pink) imaginary of an
ITO film measured via spectroscopic ellipsometry (symbols) and estimated by the Drude model (lines) reported in [1] . . . 9 2.4 a. Schematic showing the conduction band electron configurations during the
hot-electron mechanism. In the conduction band, solid lines and dashed lines are constructed with non-parabolic and parabolic dispersions, respectively. b.
Schematic showing the Fermi-Dirac distribution for different electron temper- atures. . . 11 4.1 Visualization of a multilayered media where refraction can be studied by us-
ing the TMM. At each layer, the multiple reflections and transmissions are simplified as a superposition of the form of a forward (E+) and a backward (E−) field. The layers are characterized by the refractive index ni, thickness li, while the interfaces are characterized by their corresponding Fresnel coef- ficients ti, ri. . . 16 4.2 Comparison of the linear Reflectance, R, and Transmittance, T, as a function
of the incident angle, θ0, obtained with the FDTD software and the TMM routine for a ITO slab deposited on a glass substrate in the linear regime. . . 22 4.3 Scheme of the sub-division of the TCO slab. When the sublayer thickness is
sufficiently small, local fields, thus intensity profile I(z) and local refractive index n(z) can be obtained from the NLTMM. Substrate and incident media are considered as semi-infinite media . . . 23
vi
5.1 Comparison between average reported refractive index values, ¯n, for different incident intensities and averaged values of the obtained local-intensity depen- dent refractive index values, n(z), for the corresponding incident intensities using the NLTMM and the iterative approach. Figure on the left correspond to data and calculations of Experimental setup 1 while figure on the right corre- spond to Experimental setup 2. Dotted vertical line divides the region where no saturation effects of the measured refractive index values are present (left) from the region where saturation effects are present (right) . . . 34 5.2 Comparison between the measured reflectance, R, and transmittance, T , val-
ues as a function of the input intensity, I0, obtained from Experimental setup 1 and the results obtained when using the NLTMM combined with the simple or iterative approach to compute N0 for the same experimental setup. . . 35 5.3 Comparison between the measured reflectance, R, and transmittance, T , val-
ues as a function of the input intensity, I0, obtained from Experimental setup 2 and the results obtained when using the NLTMM combined with the simple or iterative approach to compute N0 for the same experimental setup. . . 35 5.4 Scheme describing how the local-intensity-dependent change in refraction
function. ∆n(Ilocal), is extracted from local intensity profiles. Each input in- tensity, I0,n, has a corresponding local intensity profile, Ilocal(z), for the same values of local intensity (red x marker), a shared point of the local-intensity- dependent refraction function is expected Pn ← ∆n(Ilocal(z)) . . . 37 5.5 a. Intensity profile inside the material’s slab for different iterations of the
NLTMM. b. Refractive index profile inside the material’s slab for different iterations of the NLTMM. c. Change in refraction as a function of the corre- sponding local intensity obtained for different iterations of the NLTMM. . . . 38 5.6 Local-intensity-dependent change in the refractive index for the whole range
of input intensities obtained from plotting the Local-intensity-dependent change in the refractive index for each of the iterations of the NLTMM. Figure on the left correspond to data and calculations of Experimental setup 1 while figure on the right correspond to Experimental setup 2. . . 39 5.7 Local-intensity-dependent change in the refractive index for the range of in-
put intensities where saturation effects are still negligible. The n2,eff and βeff values are extracted from dotted red line and dotted green line respectively.
Figure on the left correspond to data and calculations of Experimental setup 1 while figure on the right correspond to Experimental setup 2. . . 39 5.8 Optimized values of the complex proportionality factor N0 for the material
described in Experimental setup 2. . . 40 5.9 Comparison between the measured reflectance, R, and transmittance, T , val-
ues as a function of the input intensity, I0, obtained from Experimental setup 2 and the results obtained when using the NLTMM combined with the opti- mization routine to compute N0 for the same experimental setup. . . 40 5.10 Local-intensity-dependent change in the refractive index for the whole range
of input intensities obtained from plotting the Local-intensity-dependent change in the refractive index for each of the iterations of the NLTMM combined with the optimization routine for the material described in Experimental setup 2. . 41
vii
Drude parameters values for the whole range of input intensities involved in Experimental setup 2. . . 42 5.12 Comparison between the reported values of ωp(I0) and γ(I0) for the material
described in Experimental setup 2 and averaged values of the local-intensity- dependent Drude parameters ωp(z) and γ(z) computed using the proposed minimization algorithm. . . 42
viii
List of Tables
1.1 Typical values of nonlinear refractive index . . . 2 4.1 Iterative approach used to include the intensity-dependent refraction phenomenon
in the Transfer Matrix Method. This algorithm is referred in this document as the Nonlinear Transfer Matrix method. . . 24 5.1 Nelder-Mead algorithm. . . 31 5.2 Description of the numerical routine that combines the Nelder-mead optimiza-
tion algorithm with the Nonlinear Transfer Matrix Method to obtain the values of the change in refractive index that reproduce the experimentally acquired measurements of R and T. . . 32 5.3 Mean Squared error between experimentally measured values and reproduced
values of Transmittance and Reflectance. . . 35 5.4 Obtained values of n2 and β for the two different Experimental setups. Ex-
perimental setup 1 is based on Ref. [1] while Experimental setup 2 is based on Ref. [2]. . . 37
ix
Abstract v
List of Figures viii
List of Tables ix
1 Introduction 1
1.1 Motivation . . . 1
1.2 Problem Statement . . . 2
1.3 Solution Overview . . . 2
1.4 Main contributions . . . 3
1.5 Thesis organization . . . 3
2 Theoretical background 4 2.1 Intensity-dependent refraction . . . 4
2.1.1 Experimental observations of the intensity-dependent refraction in the ENZ region . . . 5
2.1.2 Description of the intensity-dependent refraction using the standard perturbative model . . . 7
2.2 Linear optical properties of Transparent conductive oxides . . . 8
2.3 ENZ enhancement mechanisms of nonlinear response in TCOs . . . 9
2.3.1 Microscopic description of intensity-dependent refraction nonlineari- ties . . . 9
2.3.2 Fermi smearing and change in the electron effective mass . . . 9
2.3.3 Electric field enhancement . . . 10
2.4 Summary . . . 11
3 Current techniques for modelling nonlinear refraction in Transparent Conduc- tive Oxides 12 3.1 Two temperature Model . . . 12
3.2 The hydrodynamic model . . . 12
3.3 A model based on nonlinear susceptibilities . . . 13
3.4 Summary . . . 14
x
4 Method of Analysis 15
4.1 Transfer Matrix Method . . . 15
4.1.1 Light propagation in multilayered media . . . 16
4.1.2 Light propagation in multi-layer optical systems at normal incidence . 16 4.1.3 Light propagation in multi-layer optical systems at oblique incidence 19 4.1.4 Numerical implementation . . . 20
4.2 Nonlinear Transfer Matrix Method . . . 21
4.2.1 Modifications to the linear Transfer Matrix Method . . . 22
4.2.2 Iterative approach . . . 23
4.3 Summary . . . 24
5 Local-intensity-dependent change in refraction 25 5.1 The proportionality factor N0 . . . 25
5.1.1 Some important remarks . . . 27
5.2 Analysis of the average change in the refractive index . . . 27
5.2.1 Simple approach: Perturbative regime . . . 27
5.2.2 Iterative approach: Non-perturbative regime . . . 28
5.3 Analysis of the experimentally measured transmittance and reflectance . . . . 28
5.3.1 The multi-objective minimization problem . . . 29
5.3.2 The multi-objective optimization Algorithm . . . 30
5.3.3 Numerical implementation . . . 31
5.4 Analysis of the average change in refractive index using a minimization algo- rithm . . . 32
5.5 Comparison with experimental results . . . 33
5.5.1 Validation of results obtained from the experimentally measured change in refractive index. . . 34
5.5.2 Validation of results obtained from the analysis of the experimentally measured transmittance and reflectance . . . 38
5.5.3 Validation of results obtained from the analysis of the average change in refractive index using a minimization algorithm . . . 41
5.6 Local saturation . . . 42
5.7 Summary . . . 43
6 Conclusions and future work 44 6.1 Conclusions . . . 44
6.2 Future work . . . 45
6.2.1 Local saturation effects . . . 45
6.2.2 Drude parameters obtained from experimentally measured change in refractive index. . . 45
6.2.3 Results obtained from the analysis of the experimentally measured transmittance and reflectance. . . 45
Bibliography 49
xi
Introduction
1.1 Motivation
Nonlinear optical phenomena are considered to be the platform that will enable next gen- eration technologies [3, 4] in the fields of all-optically data processing [5] and storage [6], microscopy, spectroscopy, among others. A long-standing goal in the field of nonlinear optics has been the study and development of materials with strong nonlinear responses that doesn’t require long interaction lengths for displaying a large nonlinear response while remaining versatile enough to be integrated in nanophotonics systems. Materials with vanishingly small permittivity, also known as epsilon-near-zero materials (ENZ), have recently be proven to be a promising platform to enhance nonlinear optical effects [4, 7, 8].
There are many materials that posses ENZ properties, each one of them with their own advantages and drawbacks. ENZ materials can be classified in two main branches: Structured materials and non-structured (homogeneous) materials. Structured materials such as meta- materials, waveguides or materials with some kind of inclusion can be engineered in such a way that the ENZ spectral region and bandwidth can be widely modified. However, structured materials can achieve ENZ behaviour just on distances larger than the structural unit [8]. ENZ homogeneous materials such as metals, semi-metals and doped semiconductors can achieve ENZ behaviour, but tunability of the ENZ region might be more limited. For example, metals are natural ENZ materials but their spectral range of vanishing permittivity is typically lim- ited to UV wavelengths [9]. For noble metals such as gold and silver, ENZ condition can be achieved in the visible wavelength range [10]. Nevertheless, noble metals have key limitations related to spectral tunability of the ENZ region and high losses.
In view of this limitations, degenerately doped semiconductors, which are semicon- ductors with such a high level of doping that they start showing metallic behaviour, such as Transparent Conductive Oxides (TCOs) like aluminium-doped zinc oxide (AZO) and tin- doped indium oxide (ITO), have been widely studied lately as alternative, non-structured ENZ materials [1, 4, 8, 11–13]. Studies have positioned TCOs as a promising base material of new photonic devices for all-optical modulation due to their extraordinarily large and ultra-fast optical nonlinearity in its epsilon-near-zero- frequency region. For a comparison of typical values of nonlinear response see Table 1.1.
In addition, TCOs have shown many favorable characteristics like broad tunability of
1
CHAPTER 1. INTRODUCTION 2
Material n2(cm2/GW) Reference ITO 1.1 × 10−10 [1]
AZO 5.2 × 10−12 [14]
GaAs 3.3 × 10−13 [15]
Si 2.7 × 10−14 [15]
As2S3 1.7 × 10−14 [16]
Table 1.1: Typical values of nonlinear refractive index
optical properties, ENZ region in the NIR spectral range and the proximity of their low- permittivity wavelengths to the telecommunications wavelengths (∼ 1,550 nm), well estab- lished fabrication methods and their compatibility with standard, silicon-compatible fabrica- tion processes [12, 13] .
Knowing that TCOs in the ENZ frequency regime are highly promising nonlinear ma- terials, with great potential for applications for nonlinear photonic devices, it is of interest to design photonic devices based on them. For doing so, a general formalism to relate theory and experiment and model the nonlinear response over a spectral region is needed. Several ways of modelling the nonlinear response of such materials have been proposed based on different approaches. However, there is still work to be done in order to be able to fully understand this materials and develop a way of predicting optical nonlinearities enabled by them. One of the simplest ways of studying their behaviour is by developing a way of relating experimentally observed quantities with microscopic parameters described in physical models that describe the nonlinearities while working with low dimensional systems (1D), to avoid excess of com- plexity while providing a valuable insight of the behaviour. This thesis aims to contribute to the current understanding of the nonlinear optical response, under steady state conditions, of TCOs by the study of low dimensional multilayered systems.
1.2 Problem Statement
Nonlinear response in homogeneous TCOs layers have been both studied form a microscopic perspective and characterized from experimental measurements before. Nevertheless a sim- plified model that allows to relate experimental measurements with local-intensity-dependent properties of these materials hasn’t been proposed yet. This model can both, help in the understanding of the phenomena occurring within the material and serve as a powerful char- acterization tool for doped oxides.
1.3 Solution Overview
In this thesis, a simplified numerical model to describe the Intensity-dependent refraction, under steady state conditions, in homogeneous, one-dimensional layered systems of TCOs is developed. This model, allows to relate experimentally acquired measurements with the material’s microscopic properties. The numerical model is based on an adaptation of the Nonlinear Transfer Matrix Method (NLTMM) that enables the method to account for the dependence of the microscopic parameters on the local intensity values within the material.
1.4 Main contributions
The usual approach for extracting the intensity-dependent refraction from experimental mea- surements is by using a numerical method (Inverse Transfer Matrix method) to extract the average value of refractive index of the material that is being studied. Other ways for describ- ing this phenomena are trough microscopic models to describe how the dielectric function and other microscopic parameters change when light interacts with it. This models include the Two temperature model [1], hydrodynamic models [17, 18] and other extended Drude models [2, 19]. Main contribution of the present work is a numerical model that extracts from experimentally measured quantities the value of change in refractive index as a function of local values of intensity in each point inside the material instead of as an average is proposed.
A second contribution of the present work is a numerical method that extracts the values of the Drude parameters as a function of the local intensity.
1.5 Thesis organization
In chapter 2 the theoretical framework for describing the enhancement of the nonlinear op- tical response in TCOs at ENZ region is introduced, special emphasis is made in Intensity- dependent refraction processes that arise from a free-carrier absorption (intraband) processes.
In chapter 3, current approaches to model processes mentioned in chapter 2 are described in- cluding the Two temperature model, the hydrodynamic model which accounts for non-local effects and a model based on the nonlinear susceptibilities that accounts for local saturation effects. In chapter 4, the theoretical framework corresponding to the core of the method of analysis of the proposed numerical model is addressed. At the end of this chapter, the main modifications to the NLTMM that will enable the numerical routine to account for the local- intensity-dependent parameters are described and justified. In chapter 5, different approaches for extracting the local-intensity-dependent parameters of interest like change in refraction and Drude parameters from experimental measurements are discussed. Results of the pro- posed numerical model are validated using published results from experimental studies per- formed on thin films of ITO. The obtained results are reviewed and possible ways to account for the local saturation effects of the material are discussed. In chapter 6, conclusions are drawn and possible future additions for the presented work are listed.
Chapter 2
Theoretical background
TCOs have proven to be a suitable platform to enhance different nonlinear optical processes such as intensity-dependent refraction, frequency mixing and harmonic generation. In this chapter and throughout the whole work we focus only on intensity-dependent refraction en- hanced by TCOs in the ENZ region. In Section 2.1 the Intensity-dependent refraction is described. Then, in Sections 2.2 and 2.3, the linear optical properties of degenerately doped oxides followed by the description of the physical processes on the microscopic scale that enhance nonlinear response will be reviewed.
2.1 Intensity-dependent refraction
The refractive index of many optical materials depends on the intensity of the light interacting with them [15]. This phenomena is generally described by the relation
n = n0+ n2I, (2.1)
where n0 represent the linear (low-intensities) refractive index, n2 is the nonlinear refractive index and I denotes the time averaged intensity of the optical field given, in terms of the real part of the linear refractive index and the vacuum permittivity,ε0, by
I = Re(n0)ε0c|E|2. (2.2)
The nonlinear refractive index, n2, represents the rate at which the refractive index increases with increasing optical intensity and is defined as:
n2 = 3χ(3)
4cε0n0Re(n0), (2.3)
where χ(3) is the third-order nonlinear susceptibility, n0 the linear refractive index, ε0 the vacuum permittivity and c the speed of light. From the definition of n2, and the relation between the material’s relative electric permittivity ( which will be referred simply as “per- mittivity”, ε, throughout the whole document), and relative permeability µ and the refrac- tive index, n ≡ √
εµ it is easy to see that, for a non-magnetic material such as conductive oxides (µ = 1), when the real part of the permittivity vanishes, a high change in refrac- tion is expected. However, it is important to note that both n and ε are complex quantities
4
thus, the real and imaginary parts of the refractive index will be non-vanishing and given by Re(n) =pIm(ε)/2.
The maximum change in the refractive index, ∆nmax, that can be obtained in a material, even when in the ENZ region, is often limited by the damage threshold or the saturation be- haviour of the material at high intensities [4]. Thus, the nonlinear refractive index n2 which might be increased for a particular material using suitable field-enhancement mechanisms must be understood as the slope of this change with respect to the applied optical intensity within the intensity range where saturation effects are negligible and not as the possible at- tainable change in refractive index ∆n as described in Fig. 2.1.
Figure 2.1: Visualization of the change in refractive index ∆n, as a function of the applied optical field intensity I when the refractive index is limited by the saturation behaviour of the material. The dashed arrow indicates the effects of field enhancement mechanisms that might increase the values of n2 which is only reflected as an increase in the slope but not an increase in the maximum attainable value of change in refraction ∆nmax.The shadowed zone represent the zone where saturation effects are no longer negligible
2.1.1 Experimental observations of the intensity-dependent refraction in the ENZ region
Refractive index and the relative permittivity (which will be referred simply as “permittiv- ity”, ε from now on) of a non-magnetic, non-polar material (µ = 1), are related through n = √
ε, thus, high sensitivity of refractive index values are expected for changes in relative permittivity. This can be visualized by using simple differentiation: dn/dε = 1/(2√
ε), thus
∆n/∆ε ≈ dn/dε = 1/(2√
ε) so the change in refraction for a lossless material is expected to be of the form:
∆n = ∆ε 2√
ε. (2.4)
CHAPTER 2. THEORETICAL BACKGROUND 6
Therefore, in the frequency region where the real part of the permittivity vanishes (i.e ENZ region), a high change in refraction is expected. This combined with the observation that the nonlinear refractive index coefficient n2also attains high values in the ENZ region, turned out to be the motivation for performing experimental studies of the intensity dependent refraction in transparent conductive oxides at ENZ wavelengths [1, 2, 20]. Experimental studies on the optical nonlinearity of ITO [1] and AZO [20] were performed in the ENZ regime. For ITO, an enhancement of 170% of the linear refractive index was reported (Re(∆n) = 0.72 ± 0.025).
In the case of the study of AZO, the reported change was of ∆n = 0.4. Both results exhibit a huge enhancement of the nonlinear optical response associated with the ENZ spectral region.
Measurements of intensity dependent refraction can be done for degenerate as well as a non-degenerate nonlinear effects. For the non-degenerate case, the optical excitation involves multiple and distinguishable optical waves with different frequencies while the degenerate case involves only a single optical wave or more but all of them with the same frequency.
Within the theoretical framework described in the past subsection, when dealing with a non- degenerate nonlinearity, the degeneracy factor must be modified. There are two widely used techniques to study the intensity-dependent refractive index in films of TCOs, the z-scan tech- nique for characterizing the intensity-dependent refractive index [21, 22] and the pump-probe transmission measurement for measuring transient change in the optical properties. For the z- scan technique, the sample is exposed to different optical intensities by translating it along the optical axis of the laser beam, and changes in reflectivity (∆R), transmittivity (∆T ) and op- tical phase (∆ϕ) are measured as functions of the optical intensity (I) and reported as shown in Fig. 2.2.
Figure 2.2: Measured intensity dependent transmittance (T) and reflectance (R) by using a z-scan technique
For pump-probe measurements, two optical beams interact with the material. A strong beam (pump), leads to a modification of the refractive index while a weak beam (probe), expe- rience this modification. In some experimental setups, both pump and probe beams come from a single source of laser pulse that splits into two parts; here, the pump modifies the material’s optical properties while the probe monitors this change (i.e. through R and T measurements).
Time dependency of these changes can be measured by delaying the probe beam before it
reaches the sample. In some experimental configurations, pump and probe beams might come from different sources at different angles and frequencies. The way in which experimental measurements are performed is important when designing numerical models whose inputs are experimentally measured parameters, as the model proposed in the present work.
When performing experimental measurements of the intensity dependent refraction, there are two main figures of interest used to quantify the effective nonlinear refractive in- dex coefficient n2,effand the effective nonlinear attenuation constant βeff [23]. Both quantities are frequency-dependent and must be inferred from laboratory measurements such as, ∆R,
∆T , ∆n and ∆ϕ. Values of n2,eff and βeff are computed from changes in refractive index which in turn are extracted from ∆R and ∆T . The quantities are defined as:
n2,eff = Re(∆n)
Iavg , βeff = ∆α
Iavg, (2.5)
where Iavgis the averaged value of the intensity profile inside the material, ∆n is the change in refraction with respect to the linear value and ∆α is the change in the absorption coefficient, α, with respect to the linear value. Since imaginary part of the refractive index is related with absorption, the change in absorption can also be expressed in terms of the change in the refractive index and incident wavelength λ as ∆α = 2πλ Im(∆n) .
2.1.2 Description of the intensity-dependent refraction using the stan- dard perturbative model
Although experimental studies were performed based on the observation that n2 attains very large values for ENZ and large values of ∆n were obtained, the observation is not completely correct. Shortly after experimental studies showed very large values of intensity-dependent refractive index, a theoretical and experimental study was performed to explore the conse- quences of vanishingly small permittivity on the nonlinear optical response [24]. Results from this study pointed out that the divergent behaviour of n2 caused by the factor Re(n0) in Eq. (2.3) denominator, its a numerical artifact. In this study, it is mentioned that expression in Eq.(2.1) describing the intensity-dependent refraction is attained by performing a power series expansion of the expression for the complex intensity-dependent index of refraction n as a function of the third order susceptibility contributions:
n = √ ε =
q
ε(1)+ 3χ(3)|E|2, (2.6)
where χ(3) is the third order nonlinear susceptibility. The mentioned expansion, n = n0
r
1 + 2n2I n0 ≈ n0
1 + 1
2
2n2I
n0
+ · · ·
, (2.7)
is done under the assumption that |2n2I/n0| 1, which is no longer valid for low values of the linear refractive index n0. They conclude that Eq. (2.1) is not a valid approximation of the intensity-dependent index of refraction and the intensity-dependent index of refraction should be obtained directly from the electric field and susceptibility values as in Eq.(2.6) instead of from nonlinear refractive index values as in Eq. (2.1). Another important conclusion of this
CHAPTER 2. THEORETICAL BACKGROUND 8
study was to quantitatively demonstrate that, although intensity-dependent nonlinearities of TCOs in the ENZ region is observed to be a non-perturbative response (i.e. higher nonlinear contributions are significantly larger than the linear contribution.); no evidence was found that, by modelling the nonlinear polarization of this materials using a power series expansion, the expression will diverge. More about modelling nonlinear refraction using this perturbative approach will be discussed in Chapter 3. By knowing that a diverging value of n2 is not the origin of the nonlinearity, and that there must be some enhancement either in the nonlinear susceptibility χ(3), the electric field |E| or in both, enhancement mechanisms enabled in ENZ materials are discussed in Section 2.3.
2.2 Linear optical properties of Transparent conductive ox- ides
Optical properties of materials are described by its dielectric function, ε(ω). Since highly- doped semiconductors like TCOs behave as metals, their optical response in the ENZ regime is well described by the Drude model [25, 26]:
ε(ω) = ε∞− ωp2
ω2+ iωγ = ε∞− ω2p
ω2+ γ2 + i ω2pγ
ω(ω2+ γ2), (2.8) where ε∞is the high frequency permittivity which works as a corrective parameter by includ- ing the effect of the Lorentz permittivity (bound electrons), ω the optical angular frequency, γ the electron damping term related to the absorption loss and ωp the plasma frequency given by:
ωp = s
N e2
ε0m∗e, γ = e
m∗eµe, (2.9)
where N is the free-electron volume density, e is the electron charge, ε0 is the vacuum permit- tivity, m∗eis the electron effective mass and µeis the electron mobility. For a Drude material, ENZ condition is attained at the bulk plasmon frequency, meaning that when ω = ωp/√
ε∞, the real part of the permittivity becomes zero. Drude model, has proven to reproduce in an accurate way the linear permittivity values of ENZ films measured via ellipsometry as shown in Fig. 2.3. Here, an ITO slab is analyzed and obtained values of Drude parameters are ωp = 2π × 473THz and γ = 0.0468ωp. The zero permittivity condition was reported to occur at the bulk plasmon wavelength λ = 1240nm and is displayed in Fig. 2.3 as the dotted vertical line [1].
Figure 2.3: Linear relative permittivity, Re(ε) (purple), and Im(ε) (pink) imaginary of an ITO film measured via spectroscopic ellipsometry (symbols) and estimated by the Drude model (lines) reported in [1]
2.3 ENZ enhancement mechanisms of nonlinear response in TCOs
2.3.1 Microscopic description of intensity-dependent refraction nonlin- earities
Intensity-dependent refractive index nonlinearities in TCOs can be differentiated into two classes depending on the relative energies of the bandgap and the optical pump. The intraband type is a free-carrier absorption process while the interband type is a free carrier generation process. When the energy of the optical pump used to illuminate the material is larger than the bandgap (interband), the carrier density inside the conduction band may increase and according to the definition of plasma frequency in Eq. (2.9) such increase in carrier density N will result in the reduction of the real part of the permittivity [4, 12, 27]. On the other hand, if the the energy of the optical pump used to illuminate the material is smaller than the bandgap (intraband), electron heating may cause a redistribution of electrons within the conduction band; this, combined with the non-parabolicity of the conduction band, a characteristic that degenerately doped semiconductor have, leads to an increase in the electron effective mass m∗e that will result, according to the definition of ωpin Eq. 2.9, in a redshift (reduction) of the plasma frequency [4, 19] and finally, a change in the complex refractive index of the material.
In the present work, we are focusing only on processes with intraband-transition-induced nonlinearities. Itraband-type nonlinearities occur when the energy of the optical beam (pump) is smaller than the bandgap of the material. Thus, the conduction-band electrons undergo intraband transitions via free-carrier absorption [12].
2.3.2 Fermi smearing and change in the electron effective mass
Nonlinear properties of metal-like materials such as TCOs are dominated by two mechanisms;
the Fermi-smearing also known as hot-electron mechanism and the mechanism associated
CHAPTER 2. THEORETICAL BACKGROUND 10
with the change in effective mass of the material. These mechanisms occur as follows:
For an intraband process, before the pump pulse interact with the material, the con- duction band electrons are in equilibrium and described by a room-temperature Fermi-Dirac distribution function, in which the Fermi energy resides inside the conduction band. Then, when illuminated, the excitation creates a highly-energized electron distribution. Next, dur- ing a period of a few picoseconds, the electrons relax into a different thermal distribution where temperature of the electrons in the conduction band, Te, increase up to values signifi- cantly higher than the lattice (phonon) temperature, Tp as shown in the scheme in Fig. 2.4a.
Such an increase in temperature in the conduction band electrons leads to a modification of the Fermi-Dirac distribution function. This results in an increased population for energies above the Fermi level and a decrease in population for energies below the Fermi level as shown in the scheme in Fig. 2.4b. This process, known as Fermi-smearing or hot-electron is then re- flected in the modification of the dielectric function of the material [15]. Finally, the excess of energy of the hot-electron distribution is absorbed by the lattice through electron-phonon interactions until thermal equilibrium between electrons and phonons is achieved (Te = Tp).
The experimental studies on this mechanism performed mainly in gold [28] revealed that this response is not instantaneous but in the order of hundred of femtoseconds. This time is deter- mined by the energy carried by the beam to energize and heat the conduction electrons plus the time that it takes to the excited electron plasma to relax until the electron temperature, Te, is in equilibrium with the lattice temperature Tp. This non-instantaneous nature, implies that the nonlinear optical response is then dependent of the incident optical pulse duration.
The mechanism related to the change in electron mass was described by Guo et al.
[19]. When dealing with an intraband process, the redshift of the plasma frequency cannot be related with the increase in carrier density, N, associated with the interband processes.
Thus, the redshift in plasma frequency must be a result of the electronic structure of the material. This reduction is then attributed to the conduction band non-parabolicity, which has been previously observed in different highly doped semiconductors [29, 30]. In the case of a parabolic band, the effective mass is constant. However, for non-parabolic conduction bands the electron effective mass, m∗e, becomes dependent of the electron wavevector thus, when hot-electron mechanism takes place, the average effective mass, me∗ of the electrons increases. A scheme showing the non-parabolicity of the conduction band is shown in Fig.
2.4a. Change in electron effective mass, produces then the plasma frequency redshift as a result, reflectivity is reduced as the material appears more dielectric [8].
2.3.3 Electric field enhancement
Small values of the electric permittivity for a material in the ENZ region result in an electric field enhancement mechanism that might greatly increase the the magnitude of the electric field within the material for a given incident pump field intensity. This field enhancement occurs in the material due to the electromagnetic boundary conditions [31]. In the absence of surface charge, boundary conditions ensure the continuity of the perpendicular component of the electric field displacement across an interface, D2⊥ = D⊥2. Thus in terms of the magni- tude of the electric field, ε1E1⊥ = ε2E2⊥, the normal component of the electric field inside the material, E2⊥will be enhanced by the factor ε1/ε2, where ε2 ≈ 0 when in the ENZ condition
Figure 2.4: a. Schematic showing the conduction band electron configurations during the hot- electron mechanism. In the conduction band, solid lines and dashed lines are constructed with non-parabolic and parabolic dispersions, respectively. b. Schematic showing the Fermi-Dirac distribution for different electron temperatures.
is achieved. In addition, this enhanced mechanism has a pronounced angular dependence ob- served for many nonlinear effects; for a p-polarized beam incident from air, with an incidence angle θ, the electric field within the ENZ medium of relative permittivity, ε, is:
|E| = |E0| r
cos2θ + sin2θ
ε (2.10)
Therefore, at an oblique incidence, electric field within the ENZ medium much larger than the incident electric field [4].
2.4 Summary
In this chapter we have introduced the formalism for describing intensity-dependent refrac- tion. The linear optical properties of transparent conductive oxides were described and the microscopic description of the mechanisms that enhance the intensity-dependent nonlinear- ities were described. In the next chapter, current techniques used for modelling nonlinear refraction in TCOs at ENZ region will be briefly reviewed.
Chapter 3
Current techniques for modelling nonlinear refraction in Transparent Conductive Oxides
No existing model has completely explained the physics of the optical nonlinearity of ENZ materials. As discussed on chapter 2, the free- electron dynamics of heavily doped semi- conductors like TCOs are metal-like and can be described by the Drude model. However, different ways of accounting for the nonlinear mechanisms through different types of mod- els have been reported. In this chapter, a small survey on different techniques for modelling nonlinear effects in TCOs is presented.
3.1 Two temperature Model
The hot-electron dynamics related to the enhanced nonlinear refraction in TCOs in the ENZ region (see Section 2.3), is to be modelled as a complex many-body problem that require detailed knowledge of the electronic band structure. Since that kind of knowledge is not available, the electronic dynamics are usually modelled via a phenomenological approach based on the Two-Temperature Model (TTM) and the Drude model [1]. The Two-temperature model [28] , is a microscopic model that has proven effective to explain electron dynamic of ultra fast nonlinear response of metals [28,32,33]. In this model, the temperature of electrons, TE, and phonons Tp, are coupled through a system of partial differential equations which are described elsewhere [1, 28, 33, 34]. Main result of this model is the transient response of the Teand Tp values and their relation with the optical pulse duration.
3.2 The hydrodynamic model
Recently, optical nonlocal phenomena (such as spatial dispersion) induced by optical excited carriers (conduction electrons) was demonstrated to have a contribution to the optical response of in a 10 nm-thick indium-doped cadmium oxide (CdO) thin films in the ENZ region [18].
The detectability of the nonlocality was attributed to the low damping rate of conduction
12
electrons and to the absence of interband transitions at infrared wavelengths, this last charac- teristic can also be observed in other TCOs thin films in the ENZ range. In order to account for nonlocal phenomena, the motion of electrons in the conduction band is described by a Hydrodynamic model [35] which treat conduction electrons as a viscoelastic fluid. By using this approach, the regular form of the Drude model is no longer valid and a modified form that involves Fermi velocities and a longitudinal wavenumber should be used. This form can be found elsewhere [18]. From this study is concluded that effects due ton nonlocality are, in general, considered to be quite small, nevertheless they can become important for very thin ENZ layers (∼ 10 nm). In that case, the hydrodynamic model should be adopted to study the optical properties and numerical tools as the nonlocal transfer matrix method [36] should be implemented to quantitatively study the impact of nonlocality arising free carriers.
3.3 A model based on nonlinear susceptibilities
In Section 2.1.2, it was mentioned that usual approximation for nonlinear optics to describe intensity dependent refraction is violated for low index materials in the ENZ region. Thus, the apparent divergence of the nonlinear refractive index, n2, is considered as a numerical artifact [24]. To overcome this, Boyd et al. concluded that the nonlinear optical response can be understood purely in terms of the third order susceptibility χ(3)and the electric field |E2|.
It was also pointed out, that although intensity-dependent refraction in TCOs in the ENZ frequency region is a non-perturbative response, it can be modeled following the approach used in a perturbative case (series expansion of the nonlinear polarization). In this Section, an approach based on nonlinear susceptibilities, proven to reproduce the experimentally observed values of the intensity-dependent refractive index without the need to a detailed microscopic model [24] is reviewed.
The contributions of nonlinear effects to refraction can be understood in terms of the nonlinear polarization, PN L(E), expressed as a power series expansion. By doing so, the polarization of a material is then expressed as [15]:
PT OT(E) = ε0E
∞
X
jodd
cjχ(j)|E|j−1, (3.1)
where ε0 is the permittivity of free space, cj is the degenearacy factor, χ(j) is the j−th order nonlinear susceptibility and E is the complex value of the amplitude of the applied electric field. Only odd orders of χ(j) are included since only those contribute to nonlinear refraction.
For a purely χ(3) process, assuming a centrosymmetric material (in order to be able to neglect both, the tensor nature of the susceptibility, and the magnetic material response), the polarization of a material illuminated with a monochromatic field is given by
PT OT = ε0Eχ(1)+ 3χ(3)|E|2 . (3.2) Here χ(1) ≡ ε(1) − 1, describes the linear response of the material. Thus, for a purely χ(3) process, the relative permittivity and the complex refractive index can both be expressed in terms of the susceptibility and complex electric field amplitude as
CHAPTER 3. CURRENT TECHNIQUES FOR MODELLING NONLINEAR REFRACTION IN TRANSPARENT CONDUCTIVE OXIDES14
ε = ε(1)+ 3χ(3)|E|2, n = q
ε(1)+ 3χ(3)|E|2. (3.3) These equations can be used instead of the common form in terms of n2 described by Eq. 2.1 to describe the complex intensity-dependent refractive index even for low-index materials.
This treatment achieve to both preserve the original definition of n2 and its relation with χ(3), and describe the behaviour of the refractive index for input intensities where saturation effects are still negligible as shown in [24].
However, for higher input intensities, higher-order nonlinear effects should be consid- ered. thus, the nonlinear refractive index values must be calculated as
n = q
ε(1)+ c3χ(3)|E|2+ c5χ5|E|4+ c7χ(7)|E|6+ ... (3.4) where ci’s are the corresponding degeneracy factors. This approach has been proven to re- produce with excellent agreement the intensity-dependent refraction for a ITO slab up to in- tensities of 275 GW/cm2 [24], where local saturation effect are present. For this case, the nonlinear refraction of ITO for the studied intensities is attributed to χ(3), χ(5) and χ(7). In this case, it was observed that contributions of and χ(5) and χ(7) become significant at higher intensities suggesting that by incorporating these contributions, local saturation effects can be accounted in the intensity-dependent refractive index description.
3.4 Summary
In this chapter, some of the current techniques for modelling the nonlinear refraction in TCOs at ENZ region were briefly reviewed. From phenomenological models that account for the transient response of the nonlinearities to simpler models based on the nonlinear susceptibil- ity. These approaches contribute to the understanding of the nonlinear effects enhanced by TCOs in the ENZ region and can be useful to analyse the results obtained with the proposed model. In the next chapters, the proposed method of analysis to model the intensity-dependent refraction in homogeneous slabs of TCOs in the ENZ region is described.
Method of Analysis
This chapter describes the method of analysis used to model the intensity-dependent refractive index in TCOs. As discussed in Section 2.1.1, the intensity-dependent refractive index of op- tical media is usually characterized experimentally through intensity-dependent transmission and reflection measurements on a thin film sample deposited on a substrate. Due to the low dimensionality of this system, it is possible to use a one-dimensional analysis technique to describe numerically the nonlinear optical phenomenon.
Here, we use the Transfer Matrix Method, a well known one-dimensional analysis tech- nique for linear optical systems, as a basis to design a numerical routine that extract from experimental measurements (∆T , ∆R, ∆n), the local-intensity-dependent refractive index values, ∆n(z), of a layer of the material of interest deposited on a substrate. The proposed approach, is designed in such a way that it is possible to obtain both the change in refrac- tive index, ∆n(z), and the nonlinear refractive index, n2, (i.e. change in the refractive index value for the intensity region where saturation effects are negligible) as a function of the lo- cal intensity values within the material. This is something other reported methods fail to do;
current methods for extracting the intensity-dependent parameters from experimental mea- surements, yield as a result parameters as a function of the incident optical beam intensity.
These techniques are useful for knowing the material’s response under a specific experimental configuration. However, without knowing the local-intensity-dependent parameters, it is not possible to predict the response of the material under different experimental configurations.
The proposed method, helps to overcome this limitation.
4.1 Transfer Matrix Method
The Transfer Matrix Method (TMM) [37] is a mathematical tool used when analyzing the propagation of plane electromagnetic waves in multilayer material systems with one-dimensional inhomogeneity. This method assumes a multilayer structure composed of optically isotropic and homogeneous layers with plane and parallel faces [38]. The method aims to relate an incident field (EA) with a transmitted field (EB). The relation is found in the same fashion as incident and transmitted fields are related in the derivation o Fresnel coefficients [39], by considering separately the s-polarized and p-polarized components of the incident planewave which allows to work with just the scalar field amplitudes.
15
CHAPTER 4. METHOD OF ANALYSIS 16
4.1.1 Light propagation in multilayered media
Inside a multilayer optical system as the one in the Fig. 4.1 diagram, light is reflected, re- fracted and absorbed within each layer in a way that can be described by the Fresnel equa- tions. Since multiple reflections take place at each layer, they can be treated mathematically as a superposition of two waves, one propagating forward and another propagating backwards.
In the TMM context, each part of the layered structure, interfaces and propagation media are represented by a matrix, this allows to operate this matrices onto a field of interest and obtain the optical response within the whole structure.
Figure 4.1: Visualization of a multilayered media where refraction can be studied by using the TMM. At each layer, the multiple reflections and transmissions are simplified as a superposi- tion of the form of a forward (E+) and a backward (E−) field. The layers are characterized by the refractive index ni, thickness li, while the interfaces are characterized by their corre- sponding Fresnel coefficients ti, ri.
4.1.2 Light propagation in multi-layer optical systems at normal inci- dence
For normal incidence, assuming a monochromatic plane wave propagating in ˆz direction, the forward (E+) and backward (E−) propagating fields are treated as the superposition of s and p-polarizations
E±= [ˆxA±+ ˆyB±]e±ikz. (4.1) where ˆx and ˆy are unit vectors pointing in the x and y directions respectively. Here, the s- and p-polarizations contributions to the transverse field are
E(s) = ˆyB±e±ik·z E(p) = ˆxA±e±ik·z (4.2)
since there are not in-plane polarization stated, fields can be treated as scalars while finding the optical response.
At an interface, the amplitudes of the incident and transmitted fields are related by
E+= 1
t(E0+− r0E0+) E−= 1
t[rE0++ (tt0 − rr0)E0−], (4.3) where (E0, r0, t0) correspond to the transmission medium and (E, r, t) correspond to the inci- dent medium while r and t are the Fresnel coefficients defined for normal incidence as
t = 2n
n + n0 r = n − n0
n + n0. (4.4)
The relationship between them is given by
t = 1 + r r = −r0 (4.5)
arranging the equations in a matrix form, the followong mathematical representation of the interface is obtained:
E+ E−
= 1 t
1 r r 1
E0+ E0−
. (4.6)
The boundary between two semi-infinite media is represented with the so-called Interface Matrix
I = 1 t
1 r r 1
, (4.7)
which should not be confused with an identity matrix. However, when light travels through a multilayered system, there is also a phase change acquired upon propagation in the different mediums. The transfer matrix that relates the fields after propagating a distance trough a homogeneous medium, known as the propagation matrix Φ is given by
Φ =e−ik0nl 0 0 eik0nl
. (4.8)
The interface I and propagation Φ matrices are the building blocks of the transfer matrix M that enables the relation between fields at different points of the system. A transfer matrix is defined as the combination of the effect of an interface and the propagation that follow in the transmission medium
M = IΦ = 1 t
e−ik0nl reik0nl re−ik0nl eik0nl
. (4.9)
For a system with N layers, as the one shown in Fig. 4.1, the fields in the N-th slab are related to the incident field by
EA+ EA−
= M1M2· · · MNEN0+ EN0−
, (4.10)
with
Mq = IΦ = 1 tq
e−iδq rqeiδq rqe−iδq eiδq
and δq = k0nqlq. (4.11)
CHAPTER 4. METHOD OF ANALYSIS 18
Thus, when considering the whole multilayered system, the input (EA) and output (EB) fields are related by
EA+ EA−
= M1M2· · · MNIBEB+ 0
, (4.12)
where
IB = 1 tB
1 rB rB 1
and Mq = 1 tq
e−iδq rqeiδq rqe−iδq eiδq
and δq = k0nqlq. (4.13)
the transfer matrix of the complete system is then
MS = M1M2· · · MNIB. (4.14)
Knowing the transfer matrix of the system, the system’s reflection and transmission coefficients can be obtained as
t = 1
M11 r = M21
M11. (4.15)
These coefficients relate the amplitudes of the incident (Ei), transmitted (Et) and reflected (Er) fields as
Et= tEi Er = rEi. (4.16)
The fraction of power transmitted and reflected in the system, also known as transmit- tance and reflectance, can be obtained from the system’s reflection and transmission coeffi- cients [40] as
R = |r|2, (4.17)
s-polarization: T = Re[nBcos θB]
Re[nAcos θA]|t|2, (4.18) p-polarization: T = Re[nBcos θ∗B]
Re[nAcos θ∗A]|t|2, (4.19) where nA is the refractive index of the incident media and nB is the refractive index of the output media as described in the scheme in Fig. 4.1. From the transference matrix, averaged local intensity for each point of the system can be determined while computing EN0+and EN0− as in Eq. (4.10)
Ilocal = 1
2cε0n|E|2. (4.20)
This tool, allows us to run simulations for any 1D system of N layers for the case of normally incident light.
4.1.3 Light propagation in multi-layer optical systems at oblique inci- dence
For oblique illumination,assuming a monochromatic plane wave propagating in the x-z plane, the forward (E+) and backward (E−) propagating fields are treated as the superposition of s and p-polarizations.
E±= [(ˆx cos θ ∓ ˆz sin θ)A±+ ˆyB±]e±ik±·r. (4.21) However, for the oblique incidence case, there are in-plane polarization states, thus it is not possible to treat the fields as scalar quantities. This can be overcome by working with the fields transverse to the z-axis, meaning the fields with polarization components parallel to the interfaces (in thex and y direction). The so called transverse field is given by
E±T = [(ˆxA±cos θ + ˆyB±]e±ik±·r, (4.22) where the transverse amplitudes can be defines as
A±T = A±cos θ BT± = B±. (4.23)
This, combined with the fact that the component of the wavevector parallel to the interface must be conserved (i.e. eikxx = eik0xx), allows to rewrite the transverse fields in a very similar way to those for the normal incidence case (see Eqs.(4.2) ):
E(s)T = ˆyBT±e±ikz·z E(p)T = ˆxA±Te±ikz·z, (4.24) this, since the exponential term in parenthesis will be common for all expressions thus, the factor can be removed from the calculations and added until the end.
By using this framework, the transverse Fresnel coefficients and the relation between them [37] must be defined as well
tT = 2nT
nT + n0T rT = nT − n0T
nT + n0T, (4.25)
tT = 1 + rT rT = −rT, (4.26)
where nT depends on the electric field polarization
n(s)T = n cos θ n(p)T = n
cos θ. (4.27)
Thus, within the context of the TMM, when dealing with an oblique incidence, it is possible to use the same form of the equations in Section 4.1.2, while using the following substitutions
k −→ kz = k0n cos θ (4.28)
n −→ nT, (4.29)
CHAPTER 4. METHOD OF ANALYSIS 20
where nT should be substituted with Eq. (4.27) according to the type of polarization. By Using the equivalence of transverse fields, the interface matrix I for the oblique incidence case is
ET+ ET−
= IET0+ ET0−
with I = 1 tT
1 rT rT 1
, (4.30)
and the propagation matrix for the oblique incidence case is
ET+ ET−
= ΦET0+ ET0−
with Φ =e−ik0nl cos θ 0 0 eik0nl cos θ
, (4.31)
where
cos θ = r
1 −n2Asin2θA
n2 . (4.32)
For a system with N layers as the one shown in Fig. 4.1, the fields in the N-th slab are related to the incident field in the same fashion as in Eq. 4.10 and the input (EA) and output (EB) fields in the same fashion as Eq.(4.12) while using the adaptation for the Transfer Mq and Interface I matrices as
IB = 1 tT,B
1 rT,B rT ,B 1
, Mq = 1 tT ,q
e−iδq rT ,qeiδq rT ,qe−iδq eiδq
, (4.33)
where
δq = k0nqlqcos θq and cos θ = s
1 − n2Asin2θA
n2q . (4.34)
As for the normal incidence case, when knowing the transfer matrix of the system, the system’s reflection and transmission amplitude coefficients, the amplitudes of the transmitted (Et) and reflected (Er) fields, the fraction of power transmitted T and reflected R, and the local averaged intensity for each point of the system can be determined in the same way as in Eqs. (4.15 - 4.20).
4.1.4 Numerical implementation
The Transfer Matrix Method formalism described in previous sections was implemented in MATLAB for the case of a single slab of a Transparent Conductive Oxide deposited on a glass substrate for both cases normal and oblique incidence. Here, incident media and substrate are assumed to be semi-infinite media and the refractive index of the substrate is assumed to be independent of the incident field intensity. The implementation assumes that the incident field is a completely p-polarized field for now, this due to the fact that most of the experimental studies a use a p-polarized source.
In the simulation environment, the following setup simulation must be introduced by the user:
• Frequency or wavelength λ0
