# Electromagnetic Numerical Tools for the diffraction of light by periodic structures

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## Overview of the popular electromagnetic methods

Maxwell’s equations are the set of equations that describe the propagation of electromagnetic waves in different medium. Therefore, the study of the interaction between electromagnetic waves and complex structures necessitates the use of numerical modeling methods. From many decades, theorists and researchers spared no effort in the development of efficient and powerful methods that tackle the problem of nano-scaled diffraction of optical wave especially when the studied structure is in the order of the incident wavelength. Two well-known approaches exist to deal with the electromagnetic diffraction problem. In the first one, Maxwell’s equations are treated in time space and the other approach deals with them as a matter of spatial frequency so that in Fourier space.
For time domain, the most exploited method is the finite difference time domain method briefly known as FDTD [123]. It has been introduced by Yee in 1966 and developed over the years. Its algorithm is based on the discretization of the time and space simultaneously. Notwithstanding, its implementation is considered straightforward and simple. In contrast, due to the discretization meshes a large number of variables appears turning the algorithm into a time consuming and memory exhausting technique. Moreover, the Finite Element Method (FEM) is also one of the leading spatial domain methods [112]. Its solution rests on the description of a global function of a global domain in terms of nodes of sub-functions in sub-domains recognized as finite element. As the FEM discretizes the domain appropriately, the advantage of this method compared to the FDTD appears with the minimization of the geometrical discretization error. Although, the meshing of Maxwell’s equations leads to an algebraic system where the number of variables is proportional to the number of discretized meshes. Apparently, the FEM is considered as a powerful method. But, when complex structures are investigated, giant matrices full of zeros appear with the small meshes. However, storing this type of system is not an easy job. Consequently, in the temporal regime of Maxwell’s equations, the matrices must be reorganized appropriately leading to a lack of execution time and computer memory.
Another drawback of the FEM occurs when handling with diffractive structures. At this level, the calculated electromagnetic fields suffer from singularities at the boundaries of the structures and generating the slow convergence of the method.
The other branch of methods dedicated for the study of diffraction of light by periodic structures is known as Fourier space methods. This family will be the center of the concern of this manuscript. Their algorithms are based on the projection of the electromagnetic fields on the modes of the structure. Three basic methods are the backbone of this family, the Differential Method (DM) [81], the Fourier Modal Method (FMM) also known as Rigorous Coupled Wave Analysis (RCWA) [48, 73, 75], and the Chandezon Method so-called (C-Method) [16, 56]. Indeed, the DM and the FMM share multiple common concepts.
Apparently, the study done by Tamir et al. of the interaction of electromagnetic wave with variable dielectric sinusoidal profiles in transverse electric polarization (TE) was the first step on the appearance of the FMM at the mid of 1960’s [107]. Simultaneously, the same problem was addressed by Yeh and al. for the transverse magnetic polarization (TM) [124]. It appears that Burkhardt was the first theorist that reformulated the FMM in the form of truncated Eigenvalue matrix problem to study the diffraction of light for both TE and TM polarization [14]. In 1973, Kaspar has extended the work of Burkhardt to deal with complex and non-sinusoidal profiles [45]. One of the most published theory of the FMM is the paper of Peng [91]. He deduced that the FMM might be unworkable on all types of surface relief gratings. In his study, he mentioned that the regularity of a linear system doesn’t assure the convergence of an infinite determinant. Therefore, he concluded that other mathematical solution must be applied for surface relief gratings to determine the characteristic solutions of the grating region. Without being up-to-date to the result of Peng, Knop was investigated, in 1978, the use of FMM on surface relief lamellar gratings [48]. He extracted an eigenvalue problem for the TM polarization case by coupling two first-order differential equations. The qualitative leap of this method was in 1981 where Moharam and Gaylord figured out the volume gratings with inclined surface [73]. Moreover, they elaborated the use of the representation of an arbitrary structures as a succession of rectangular step index distribution known as the staircase approximation. Indeed, they also derived the eigenvalue differential equation for TE and TM polarizations [74]. Consequently, their work and all the previous other works have turned the Fourier Modal Method into the most popular tool for the modeling of diffraction gratings. In 2005, Hugonin et al. have demonstrated that applying a non-linear and complex coordinate transformation to the propagation equations of the FMM can turn the method into an aperiodic method used for the modeling of guided optical structures instead of optical periodic gratings [36]. Basically, this coordinate transformation play the role of Perfectly Matched Layers (PML) that suppress the incoming waves from the neighboring cells allowing to model artificially periodized guided structures as an open boundary semi-infinite structure.
At the same time of the appearance of FMM, the development of the Differential Method was on in full swing. We’re are talking about a more rigorous method aiming to reduce the number of variables during the integration of the differential form of Maxwell’s equations by dealing with the problem in the harmonic space. In case of 1D diffraction grating, the harmonic time dependence and the periodicity along the periodization axis added to the invariance with respect to a given axis help to treat the system of differential equations smoothly [15, 92]. Dating to the 70’s, the first application of the DM was realized through Numerov numerical integration algorithm [80]. This integration allows the modeling of finite and infinite conductive diffraction gratings in TE and TM polarization. Although, numerical instabilities have been observed during this process especially when modeling deep grooves in TM polarization. This weakness has attracted the attention of researchers, involving in the algorithm of the DM and FMM, along 20 years without any valid interpretation. In 1996, series of papers ,introduced by Granet, Lalanne and Li, was the clue to unblock the limitations of the differential method [27, 51, 55]. The researchers referred this numerical instabilities to two reasons. The first error is due to the numerical integration process where the exponential components of the evanescent Fourier modes grow exponentially imposing numerical fluctuations. For that, the efforts was devoted to find the best mathematical solution that suit well with the integration step. Firstly the Schmidtt orthogonality has been used to tackle this problem, but this method doesn’t show a big impact on the stability of the method. Another solution rests on the discretized integration has not also shown its well functionality. Finally, the use of the S-Matrix algorithm has demonstrated that it is the most effective algorithm to handle this problem [54]. Despite this solution, the divergence of the method weren’t completely treated where another source of instabilities has appeared. This error is originated from the slow convergence of the Fourier series product describing the electromagnetic field at the interface of the grating especially in TM polarization. In other words, the multiplication of two discontinuous periodic functions which must give a continuous displacement field is not respected in the Fourier domain. For that, Li proposed to use what is called the ’inverse rule’ [55]. Basically, these errors take place when two discontinuous periodic functions are multiplied. Therefore, the inverse rule will be applied for functions with complementary jump discontinuities along the periodization axis. This case appears, for example, in TM polarization between the incident electric field and the permittivity distribution at the surface of the modulated zone. Indeed, the same inverse rule has been applied to FMM as same instabilities was appeared during tests. At the beginning of the 21th century, a dramatic reformulation of the differential method has been applied by Popov et al. They introduced the Fast Fourier Factorization (FFF) to the propagation matrix of the Differential Method [94]. Since then, the efficiency of the method has incredibly been enhanced especially for non-lamellar gratings illuminated by TM polarized light. This reformulation allows to correctly describe the evolution of the incident field with respect to the grating’s profile enabling an accurate and rigorous modeling technique for a wide class of diffraction problems.
Similarly, the C-Method is a frequency method applied on arbitrary shaped gratings. Its algorithm is based on the description of the grating’s surface as continuous function [58]. After applying a coordinate transformation along the propagation axis and the periodic axis, the coefficients of Maxwell’s equations become spatially dependent. In that case, the corrugation layer is replaced by a simple layer with a constant thickness and an equivalent complex permittivity.
Last but not least, the integral theory is another approach to deal with the diffraction problem. This theory is known as the integral method [70]. In that case, the fields at any point of the cartesian space is expressed as a set of integral functions. Therefore, the determination of the fields at any point of the space is reduced to the determination of the unknown functions following the periodic axis. The first use of the method featured with the perfectly conducting grating in the 1970’s. At that time, the other electromagnetic methods was not adapted for such problem. But, this method is considered hard to be implemented and memory and time exhausting techniques while dealing with multi-layered diffraction gratings constituted by a stack of different refraction index.

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### Electromagnetism applied to 1D diffraction gratings

Basically, electromagnetism is the phenomenon of interaction of electric fields with magnetic fields. This branch of science is the fruit of works of several scientists. In 1861, James Maxwell assembled these theorems into a concise set of equations and completed their laws by a consistent and coherent model so-called Maxwell’s equations. This set describes the distribution of the electric and magnetic fields and their change with respect to time. This chapter is intended to review the basics of diffraction of light, ruled by Maxwell’s equations, of ideally 1D infinite periodic structure arrangements known as diffraction gratings.

The basics of diffraction gratings

’No single tool has contributed more to the progress of modern physics than the diffraction grating, especially in its reflecting form’ are the opening words of a research article published in 1949 by the spectroscopist George R. Harrison. A diffraction grating is a set of closely spaced grooves or periodic arrangement. When a plane wave excites this arrangement, the diffraction of light by one of this groove interferes constructively or destructively with the light diffracted from the other grooves. Consequently, the light is split and diffracted into several plane waves (either transmitted or reflected) traveling in different directions and known as diffracted orders. (θR/T ) of each reflected (R) or transmitted (T ) order can accurately be predicted depending on the angle of incidence θinc and the period of the grating Λ as follows, k′ = kx + m 2π
sin(θr/t ) = ninc sin(θinc) + m
However, this equation can be expressed in a simplest form as, nsup/sub sin(θr/t ) = ninc sin(θinc) + m Λ (2.2)
With, nsup/sub and ninc represent the refractive index of the superstrate layer or the substrate layer and the refractive index of the incident region respectively, and m ∈ Z is the number of the diffracted order.
On the other hand, the drop-off of a given order m occurs when the angle of incidence is chosen in a way that the refracted or transmitted plane wave is excited with grazing angle, which mathematically expressed as sin(θr/t ) = ±1. Under this condition, the cut-off wavelength of the mth diffracted order λm can be calculated as, λm = Λ ±nsup/sub −nincsin(θinc) (2.3)

#### MAXWELL’S EQUATIONS FORMALISM: TRANSIENT REGIME

Aside the calculated angles and cut-off wavelengths, there is no information brought by these laws concerning the distribution of energy on each diffracted order. Although, the transmission and reflection coefficients could be expressed as Rayleigh expansions. But, calculating the efficiency of each order is not straightforward. These values depend on the incident angle, the polarization of the plane wave, the opto-geometrical dimensions of the grating and the different refractive indices of the structure. Accordingly, electromagnetic numerical methods based on Maxwell’s Equations have to be developed to tackle this problem.

1 General Introduction
1.1 Objectives and Motivation
1.2 Overview of the popular electromagnetic methods Nomenclature
I Electromagnetic Numerical Tools for the diffraction of light by periodic structures
2 Electromagnetism applied to 1D diffraction gratings
2.1 The basics of diffraction gratings
2.2 Maxwell’s equations formalism: Transient regime
2.3 Maxwell’s equations in the harmonics regime
2.4 The decomposition into TE and TM polarization
2.4.1 Introduction
2.4.2 Polarization of the field exciting the diffraction grating
2.5 Fourier expansion of the field
2.5.1 The general form of the field outside the modulated zone in TE polarization
2.5.2 TM polarization
3 Principles of the Differential Method associated with Fast Fourier Factorization
3.1 The differential Theory in TM polarization
3.1.1 The classical Differential method: the transient equations of the electric and the magnetic fields inside the modulated zone
3.1.2 The harmonic Fields equations inside the modulated zone
3.1.3 Harmonic propagation equations
3.1.4 FFF associated to the differential theory in TM polarization
3.1.5 Fields outside the modulated zone
3.2 Formulation of the differential theory in TE polarization
3.2.1 Fields inside the modulated zone
3.2.2 Field outside the modulated zone
3.3 Numerical integration of the matrix differential system
3.3.1 The Runge-Kutta integration algorithm
3.3.2 Runge-Kutta algorithm applied to the Differential theory in Fourier space
3.4 The linear relation that links the modulated region with the homogeneous zones
3.5 Use of the Shooting Method
3.5.1 Defining the initial conditions
3.5.2 The transition from the stationary fields into the forward and backward representation of the fields
3.5.3 T-matrix of a given section
3.6 Scattering Matrix algorithm (S-Matrix)
3.6.1 Introduction
3.6.2 Definition of the S-Matrix
3.6.3 The total S-matrix of the entire structure
3.7 The intensities of the diffracted order
3.8 RCWA extracted from the differential theory
3.9 Conclusion
4 DM-FFF: Validation and comparison with other electromagnetic computational methods
4.1 Validation of the methods
4.1.1 Introduction
4.1.2 Numerical validation
4.2 The DM-FFF compared to other electromagnetic computational methods
4.2.1 Introduction
4.2.2 Sinusoidal Metallic Grating
4.2.3 Discontinuous structure: A Trapezoidal metallic grating as an example 68
4.3 Conclusion
5 The differential theory and the lossless permitivitty metals: Problem and solution
5.1 Definition of the problem
5.2 Implementation of Graded Index Layer (GIL) at the metal-dielectric interface with pure negative and real permitivitty metallic gratings
5.3 Application of the GIL on a triangular metallic grating with quasi-real negative permitivitty
5.4 Conclusion
6 DM-FFF applied to visual security structures: A full study
6.1 Introduction
6.2 Spectrum to color transformation
6.3 All dielectric structures as a reflection visual security device
6.3.1 Geometry of the structure
6.3.2 The impact of the buffer layer on the chromatic response of the structure
6.3.3 Impact of the period on the chromatic response of the structure
6.3.4 Impact of varying the amplitude of the grating A
6.3.5 Impact of the incident angle θinc
6.4 Use of the DM-FFF for the inverse tailoring of structural color
6.5 Conclusion
II Electromagnetic Numerical Tools for the propagation of light in guided structures
7 DM-FFF applied to optical guided structures: Theoretical interpretation
7.1 Introduction
7.2 Modelisation of integrated optics structures using the DM-FFF
7.2.1 Illustration of the problem
7.2.2 Perfectly Matched Layer as non-linear complex coordinate transformation
7.3 Formulation of the aperidic DM-FFF (a-DM-FFF)
7.3.1 Propagation equation in TM polarization
7.3.2 Propagation equation in TE polarization
7.3.3 Input/Output Ψ as an equivalent guided zone
7.3.4 The use of a transition matrix that directly links the amplitude of the eigen modes with the stationary harmonic vectors
7.4 Validation of the method
7.4.1 TE polarization
7.4.2 TM polarization
7.5 Conclusion
8 A-DM-FFF compared to the a-FMM: Application on complex shaped photonic guided structures
8.1 Application of the a-DM-FFF on curvilinear guided reflector
8.1.1 Geometry of the structure
8.1.2 TM polarization: Dielectric high contrast index structure
8.1.3 TM polarization: Metallic 2D pillar
8.1.4 TE polarization
8.1.5 Conclusion
8.2 Application of the a-DM-FFF on resonant cavities: Microdisk resonators as examples
8.2.1 Introduction
8.2.2 Problem of doublet resonances
8.2.3 Microdisk cavities excited by plane waves
8.2.4 Microdisk-guide coupling
8.3 Conclusion
III Experimental results of Bragg grating filters
9 Fabrication and Characterization of Bragg Reflection filters associated with ion-exchanged waveguides
9.1 Priciples of Bragg grating waveguide used as wavelength filter
9.2 Ion Exchanged waveguides principle
9.2.1 Principle of ion exchange
9.2.2 Description and Fabrication of optical waveguides
9.2.3 Characterization of the guides
9.3 Roll to roll Grating with ion exchanged waveguide: direct interaction
9.4 Roll to roll Grating with ion exchanged waveguide: Hybridization by MEMO
9.4.1 Hybridization process
9.4.2 Simulated results
9.4.3 Spectral analysis
9.5 Conclusion
10 Conclusion and Perspectives
10.1 Conclusion
10.2 Perspectives
References
Appendix A The multiplication of two Fourier series: Toeplitz matrix formulation
Appendix B The normal of the surface for the different used geometries
B.1 Rectangular Profile
B.2 Sinusoidal Profile
B.3 Trapezoidal or triangular Profile
B.4 Curvilinear profile
Appendix C Test ECH mask datasheet
Appendix D List of Publications
D.1 Journal Publications
D.2 Oral contributions in international conferences
D.3 Oral Contributions in French National conferences
Résumé

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