Essay: Parameter optimization of periodic dielectric structure for applications in photonics and optoelectronics

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ABSTRACT 3
ABBREVIATIONS 5
1. INTRODUCTION 7
1.1 PHOTONIC CRYSTALS 7
1.2 METAMATERIALS 12
1.3 NUMERICAL CALCULATIONS 18
1.4 IMPROVING THE FMM CONVERGENCE 21
1.5 OPTIMIZATION METHODS FOR PHOTONIC PROBLEMS 22
2. FOURIER ANALYSIS FOR MICRO/NANO-STRUCTURES 24
2.1 SPECTRAL CONCEPT 24
2.2 GIBBS PHENOMENON 26
2.3 NUMERICAL APPROACH 29
2.4 MAXWELL EQUATIONS IN RECIPROCAL SPACE 30
3. LIGHT PROPAGATION IN LAYERED SYSTEMS 34
3.1 ELECTROMAGNETIC FIELD EQUATION IN LINEAR LOCAL MEDIUM 34
3.2 TRANSLATION OPERATOR IN Z-DIRECTION 39
3.3 LIGHT PROPAGATION IN Z-DIRECTION 40
3.4 BOUNDARY CONDITIONS 42
3.5 SCATTERING MATRIX APPROACH 43
4. FACTORIZATION RULES 48
4.1 CONVOLUTION OF FOURIER SERIES 48
4.2 GENERAL FACTORIZATION RULES FOR MAXWELL’S EQUATIONS 50
5. SAMPLING PLAN 53
5.3 LATIN HYPERCUBE SAMPLING 54
5.4 SAMPLING BASED ON SOBOL’S SEQUENCES 55
6. BAYESIAN OPTIMIZATION 57
6.1 MAIN CONCEPT OF BAYESIAN ANALYSIS 57
6.2 GAUSSIAN PROCESS 59
6.3 VARYING THE HYPERPARAMETERS 61
6.4 EXPECTED IMPROVEMENT 62
7. NUMERICAL IMPLEMENTATION 64
7.1 HOMOGENEOUS LAYER 65
7.2 POWER FLUX AND SPECTRA 67
7.3 FIELD DISTRIBUTION 69
8. GRATING COUPLER PROBLEM FOR ONE DIMENSIONAL PERIODICITY 71
8.1 PROBLEM STATEMENT 71
8.2 RESULTS 73
9. TWO DIMENSIONAL PROBLEM 80
9.1 PROBLEM STATEMENT 80
9.2 RESULTS 83
10. CONCLUSION 90
10.1 SUMMARY 90
10.2 OUTLOOK 90
BIBLIOGRAPHY 92

Abstract

Parameter optimization of periodic dielectric structure for applications in photonics and optoelectronics
Ignatov Alexey
‘Submitted to the Skolkovo Institute of Science and Technology
on 01 June 2018

For the last decades the methods of creating nano/micro structures have been developed significantly. This fact initialized new investigations of periodic structures such as photonic crystals and metamaterials with periodicity. As a result a lot of design optimization problems have been arisen in photonics due to the increasing demand for fabrication of nano- and microstructures with desired optical properties.
The optical properties of two-dimensional metallic photonic crystals and metamaterials provide features such as negative refractive indices, chirality, band gaps and polarization effects. These effects can be used in different applications: antennas, sensors, cloaking devices, displays and superlenses.
Grating coupler is example of application of multilayered periodic structures. It plays an important role in integrated photonic systems. It consists of waveguide with etched surface generally. Its design causes light to propagate in desired direction of integrated optical system from outside medium or to scatter out of the waveguide. As a result, grating couplers require special design parameters that allow to couple light with optical system.
The development of effective grating couplers is a complex task, since the conditions of mode matching and directionality must be fulfilled. Moreover an ideal grating system has to eliminate high order re’ections occurred in a periodic structure. These points should be taken into account in order to build gratings with high-e’ciency.

The coupling problem or so-called mode matching problem arises with a demand to couple light into a desired output/input mode. The exponential attenuation of the light intensity with increasing length of the grating coupler is one of the main obstacles to solve the problem. This decay arises due to permanent period of the structure and can be solved partially using a structure with variable properties.
The grating coupler problem is one of the main problems in photonics. Its design of metal-dielectric or all-dielectric systems allows to conjugate the incident light with the integrated optical system. The effectiveness of conjugation is defined by ratio between energy flux of incident light and energy flux of light inside the waveguide.
The study of artificial materials has high potential for different applications. Among them thermal emitting structures can be used for important applications in near-‘eld thermal management. The approach of making surface based on photonic crystals is an effective tool to control the spectral, angular and coherence characteristics of thermal radiation. Emitter symmetry defines the polarization of thermal radiation and angular emission diagram. The mirror symmetry of emitting surface can be broken down by applying an external magnetic ‘eld due to spin-orbit interaction of electrons or by anisotropic permittivity tensor. The use of an external magnetic field to create a circularly polarized field limits radiation. Getting a high degree of circular polarization with the help of anisotropic materials is a technologically difficult task.
Alternatively the systems with lack of mirror symmetry having chiral morphology allow to obtain thermal emission with circular polarization. Thus to control the parameters of thermal emission by the geometry of emitting surface is another up-to-date problem.
Both of these problems require special geometric and optic parameters to conserve energy and to maximize light polarization degree correspondently.
These optimization problems define black box target functions which take a lot of time to obtain the function value. Thus the effective optimizer is required to be built.
This work is devoted to optimization the geometrical parameters of a grating coupler as well as the parameters of the thermal emitting surface. The solver and optimizer are built in order to solve the problems. The solver simulates optical properties of one- and two-dimensional structures using Fourier modal method in combination with a scattering matrix (S-matrix) approach and factorization rules. Scattering matrix approach allows to obtain field distribution with high accuracy for thick multilayered structures instead of transfer matrix formalism. Factorization rules minimize the number of harmonics necessary for FMM. They solve the problems of inaccuracy occurred for convolution of two functions with complementary high jump discontinuities. FMM method is the most powerful for description electromagnetic waves in periodic nanostructures.
The optimizer seeks the best optimal parameters for particular optical problem using Bayesian optimization approach based on Gaussian processes in a reasonable time. This approach uses the retraining of the Gaussian process model each time when a new optimal candidate, determined by the acquisition function, changes the samples from the test to the train one. The expected improvement function is used as an acquisition function.
.
Research Advisor:
Name.: Shapeev Alexander Vasilyevich
Degree: Ph.D.
Title: Assistant Professor

Co-advisor:
Name: Gippius Nikolay Alexeevich
Degree: Doctor of Physical and Mathematical Sciences
Title: Professor


Abbreviations
” – polar incidence angle
” – azimuth incidence angle
c – vacuum light velocity
” – permittivity
” – permeability
” – angular frequency
k – wave vector
k0 vacuum wave number
T – transmission
R – re’ection
A – absorption
D – di’raction
P – power
S – Poynting vector
E – electric ‘eld
H – magnetic ‘eld
D – electric displacement
B – magnetic induction
FMM – Fourier modal method
FFE – Floquet-Fourier expansion
FFT – fast Fourier transform
FFW – Floquet-Fourier waves
GP ‘ Gauss process
S-matrix scattering matrix
T-matrix transfer matrix
TE-polarization transverse electric polarization (S- polarization)
TM-polarization transverse magnetic polarization (P- polarization)
S-polarization polarization with electric ‘eld perpendicular to the incidence plane
P-polarization polarization with electric ‘eld parallel to the incidence plane
x, y, z – cartesian coordinates
– scalar functions
– variable integers
M, N – ‘xed positive integers
r – three-dimensional vector with spatial components
– mean of
G – reciprocal vector in xy-space
Li – period in i-direction
V volume of the unit cell
– periodic function in Bloch’s theorem
A – amplitude vector to describe the electric ‘eld as a linear combination of the eigensolutions
– material matrix to obtain the transverse electric and magnetic ‘eld from the amplitudes

1. Introduction

Nowadays plasmonics has great potential in many different ‘elds, for instance, in nano-optics, spectroscopy, sensing, and biology. The reason of this is an intensive development of approaches that allow ones to create structures with the sizes of several tens or even units of nanometers. As a result, a lot of new structures have been developed in two and three dimensions using both dielectric and metallic structures [1, 2].
Periodic nanostructures have interesting optical features. Examples of such structures are photonic crystals. It possible to slow down the radiation of atoms from a given structure [3], light redirection [4] or creation of circularly polarized waves [5] using such crystals.
Diffraction gratings have been evolved into the development of plasmonics, hybrid metallic-dielectric photonic crystals, metamaterials, and nanoantennas. Plasmon resonance analysis of such nanostructures allows ones to create effective metamaterials that are used in sensor manufacturing process.
This chapter is dedicated to the overview of photonic crystals, metamaterials as well as an explanation of problems that are met during seeking the optimal geometry for a particular photonic problem. This leads us directly to the main part of this work, the analysis and implementation of a numerical approach to simulate the interaction between light and photonic structure, its improvement, and optimization problems.

1.1 Photonic crystals

A photonic crystal is periodic optical nanostructure that has a band gap that forbids propagation of a certain frequency range of light. This feature of propagation of electromagnetic waves allows us to control light more efficiently than conventional optics could propose. Photonic crystal has a feature of a periodic change in the permittivity. The lattice constant of a photonic crystal is in the same order of magnitude as the wavelength of the propagating light. There are no big difficulties to fabricate photonic crystal in the microwave or infrared regime. The task of such crystal creation in visible or lower range could be competitive. However modern techniques allow ones to fabricate structures down to a few hundred/tens of nanometers.
Structures are divided into three types depending on their dimension: one-dimensional, two-dimensional and three-dimensional (Fig. 1.1.1). One-dimensional photonic crystals are materials in which the dielectric constant varies periodically in one direction. Such photonic crystals consist of layers of different materials parallel each other with different dielectric permittivities and exhibit their photonic crystal properties in a direction perpendicular to the layers. The simplicity of one-dimensional periodic structures makes it possible to use vacuum layer-by-layer deposition of films as a method of their creation [6, 7]. Nanolithography, anisotropic etching can also be used to create them.

Fig. 1.1.1: From left to right examples of a one-dimensional, two-dimensional, three-dimensional photonic crystal and a photonic-crystalline layered system are shown (photonic crystal slab). Different colors correspond to materials with different values of the permittivity

Fig. 1.1.2: SEM images of 1D photonic crystal formed by 50 grooves in silicon (cross-sectional view). Grooved silicon has been obtained by preferential etching of (110) Si in potassium hydroxide solution. [6]
Two-dimensional photonic crystals are materials in which the dielectric constant varies periodically in two directions (fig. 1.1.2). These alterations form a two-dimensional crystal lattice. Figure 1.1.3 shows an example of a two-dimensional photonic crystal ‘ DLC crystal (diamond-like carbon-based) [8]. The DLC-based photonic crystal structure was fabricated by using electron-beam lithography and inductively coupled plasma (ICP) etching techniques.

.Fig. 1.1.3: SEM image of the ‘nal diamond-like carbon-based 2-D photonic crystal structure (a =700 nm and r=0.29a = 203 nm)
Three-dimensional photonic crystals are materials in which the dielectric constant varies periodically in three directions. They can be represented as an array of volume regions ordered in a three-dimensional crystal lattice. Three-dimensional periodic structures create the greatest technological difficulties for experimental realization.
Figure 1.1.4 shows a three-dimensional photonic crystal of the “stack of firewood” type formed by rectangular parallelepipeds crossed at right angles [9]. The method of creating such crystal is based on the construction of multilayer structures by a photolithography method. These structures have a periodic change of the refractive index in each layer.

Fig. 1.1.4: Micrographs of the photonic crystal (photonic crystal slab). a, SEM top view of a completed four-layer structure. It shows good periodicity. The underlying layer structures are also evident. Scale bar, 20mm. b, SEM cross-sectional view of the same 3D photonic crystal. The rods are made of polycrystalline silicon. The spacing between adjacent rods is d (4:2mm), the rod width is w (1:2mm), and the layer thickness is 1.6mm. Scale bar, 5mm.
The optical properties of photonic crystals are very different from the optical properties of homogeneous continuous media. The propagation of radiation inside a photonic crystal due to the periodicity of the medium becomes similar to the motion of an electron inside an ordinary crystal under periodic potential [10].
As a result, electromagnetic waves have a band spectrum and a coordinate dependence in photonic crystal. This behavior is similar to existing Bloch waves of electrons in ordinary crystals. Band gaps could be formed under certain conditions in the band structure of photonic crystals, analogously to forbidden electronic bands in natural crystals. The spectrum of photonic crystals can have completely forbidden bands and partially forbidden ones in the frequency range depending on the specific elements’ material, size and lattice period. The propagation of radiation for completely forbidden bands is irrespectively impossible for its polarization and direction, and for partially forbidden ones, the propagation is possible only in the allocated directions. These unique properties of photonic crystals make it possible to create new types of waveguides [11], optical fibers [12], structures with a significant enhancement of the local electromagnetic field [13] and other revolutionary devices in the technology of optical communication, the physics of lasers and optical computer technology [14’16].
As a simple example for band gaps consider light in a homogeneous medium with a refractive index n, ” – the angular frequency, c – vacuum light velocity, – group velocity, k – the wave vector. Light has a linear dispersion relation for this case. Many concepts of solid-state physics can be applied to calculate the properties of photonic crystals and for this example particularly. Thus photonic crystals exhibit a dispersion relation with band gaps.

Fig. 1.1.5: Solid thin line indicates dispersion relation for 1D structure (fig. 1.1.1) with boundaries of the ‘rst Brillouin and the zone band gaps in the middle. Dispersion relation – dotted line in a homogeneous medium with a group velocity. Thick black vertical lines – the border of the ‘rst Brillouin zone
For a one-dimensional structure with a lattice constant the dispersion relation has the relation like on fig. 1.1.5. So the investigation area is reduced to the ‘rst Brillouin zone . The higher k faster dispersion relation stuck in the the ‘rst Brillouin zone. Band gaps occur close to the center and the boundary of the ‘rst Brillouin zone.
In case of metallo-dielectric photonic crystals, the plasmon-polaritons appear due to the interaction of collective electron excitations (so-called plasmons) and photons. They can exist strongly located at the transition between a dielectric and a metallic uniform layers for frequencies below the plasma frequency.
A localized plasmon occurs in metallo-dielectric gratings. As a result, plasmon-polariton can interact with different anomalies (Rayleigh or waveguide) in case of a metallic photonic crystal slab. The latter anomalies appear due to a uniform layer below or above the grating [17].
Photonic crystals find their application in technology. The development and study of various devices with photonic crystals as well as theoretical methods for their research, the intensive study of the properties of photonic crystals and practical implementation of theoretically predicted effects in photonic crystals is proceeding for last decades. The future of modern optoelectronics is associated with photonic crystals. A well-known example of a one-dimensional photonic crystal is distributed Bragg reflector. The usage of photonic crystals allows ones to create low-threshold and non-threshold lasers. Waveguides based on photonic crystals can be very compact and have small losses. Possibility to create superprisms occurs due to significant dispersion properties (fig. 1.1.5) of photonic crystals. Displays based on photonic crystals using a new image manipulation system can replace conventional displays. Metal-dielectric photonic crystals have an influence on spectra as well as on the excited-state lifetime of the atom/molecule/quantum dot placed inside a photonic crystal. Thus such a structure can change features of emission. For instance, the period of a photonic crystal, the spatial distribution of permittivity and permeability, and properties of incident wave change an excited-state lifetime. Thus it is possible to manipulate the lifetime of a quantum emitter by altering the parameters of the structure.

1.2 Metamaterials

Metamaterials are artificial structures with properties not available in nature also as negative index materials (NIM). They have metallic resonators in periodically distributed unit cells instead of atoms/molecules. Their precise shape, geometry, size, orientation, and arrangement provide them with smart properties capable of manipulating electromagnetic waves: by blocking, absorbing, enhancing or bending waves to achieve benefits that are not possible with natural materials. The interaction between the incident wave ” and meta-atoms defines the electromagnetic properties of medium. Thus magnetic permeability and electric permittivity could reach negative value. We can describe this behavior considering effective electric permittivity. It is necessary to fulfill the condition a << ” in order to avoid the dependence of these parameters from wavevector or parameters of neighbors cells.
The range of metamaterials can be obtained by consideration of photonic crystal structures that have a period much smaller than the wavelength of light. Due to the small size of the elements, light interacts with the metamaterial as with some effective homogeneous material which properties can be very different from the optical properties of materials occurring in nature. Therefore, metamaterials can be used in optical instruments for various applications.
Some materials with permittivity and permeability near zero (Zero-index material) can be used to achieve high directivity antennas. By using metamaterial in antenna we can increase bandwidth, reduce antenna size and increase radiation efficiency. The most favorite application is metamaterial absorber which is the fastest growing field in the electronics. The metamaterial absorber is thin, lightweight. It doesn’t require the use of expensive materials and can be used over a wide frequency range. The same concept can be applied to construct an absorber functioning at a different frequency.
Significant explorations to manipulate the electromagnetic field began at the end of 19th century. It was obtained that it is possible to do using artificial materials. The first study was done by Jagadish Chandra Bose investigating chiral properties in 1898. Karl Ferdinand Lindman studied wave interaction with metallic helices as artificial chiral media in the early twentieth century.
J.C. Bose showed the possibility of the existence of artificial material by conducting microwave experiment on the twisted structure. Later the physicist Victor Veselgo presented the theoretical investigation. He proved that such materials could transmit light. He showed that the phase velocity can be directed anti-parallel to the direction of Poynting vector (convey energy that has a group velocity) that is not possible for natural materials. In 1996 Pendry used an artificial wired medium which permittivity is negative to obtain artificial electric plasma. In 1999 magnetic plasma was studied with negative permeability using split-ring (C shape) resonators (SRR). He showed that a periodic array of wires and rings could give rise to a negative refractive index and also proposed a related negative-permeability design, the Swiss roll.
Smith et al(2004) realized gradient refractive index medium to bend electromagnetic waves and carried on the experimental demonstration of functioning electromagnetic metamaterials by horizontally stacking, periodically, split-ring resonators and thin wire structures. Metamaterial opened up a new exciting world for the scholars. Now the concept of the negative refractive index is widely accepted and focus of the research has moved toward applications. The word was first coined by Rodger M. Walser (2001) who gave the following definition ‘Metamaterials are defined as macroscopic composites having a man-made, three dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to a specific excitation.’ ‘Metamaterials are artificial periodic structures with lattice constants that are much smaller than the wavelength of the incident radiation. Therefore providing negative refractive index characteristics’
This word is a combination of ‘meta’ and ‘material’, Meta is a Greek word which means something beyond, altered, changed or something advance as presented in Sihovola [18]. In a precise way, metamaterials can have their electromagnetic properties altered to something beyond what can be found in nature. They are typically man-made material.
In order to describe the basic properties of metamaterial lets write Maxwell’s equations: or , where ”r ”r are relative permeability and permittivity respectively and .
To understand why such materials are called left-handed (LHM) let us take a look on the variation for fields in Maxwell’s equation: and same for H, where k ‘ wavevector. Then we can derive the form of Maxwell’s equations:
In the result, we see that for positive defined ”r ”r for E, H, k form right-hand triplet of
vectors (RHO) and for negative defined – (Fig. 1.2.1). RHO and LHO respectively
‘ left ‘handed system (LHO) (Fig. 1.2.1).

Materials can be classified in terms of ” and ”. (Fig. 1.2.2). The first quadrant (” >0, ”>0) represents right- handed material (RHM). The forward propagation of wave takes place in the first quadrant. It is commonly used in the material. Fig. 1.2.2
The right-hand thumb rule for the direction of propagation of wave S is fulfilled. The second quadrant (”< 0 and ” > 0) describes electric plasmas which support evanescent waves. It is also called ENG (epsilon negative) material. The fourth quadrant (”> 0 and ” < 0) also supports evanescent, corresponding to MNG (mu negative material) ”.
The third quadrant (”<0, ”<0) represents metamaterial (LHM or DNG – double negative material). It follows the left-handed rule because Fig. 1.2.4: First LHS (San Diego group) of propagation of wave takes place in the backward direction in this medium. Due to negative ” and negative ” the refractive index of the medium is calculated to be negative so-called NIM (negative index material). Electric vector E, electromagnetic vector H and wave vector k form the left-hand triplet as shown in figure 1.2.4.
The structure was built with a combination of the material with negative permittivity (thin wire) and negative permeability (SRR) (Fig. 1.2.4).
The correct value of the effective refractive index is for double negative relative effective coefficients where real parts of refractive are negative.
A metamaterial is usually implemented as a periodic structure. There are a lot of options to create the most effective and functional structure. The most common structure of the unit cell is a combination of SRR and wire structure (Fig. 1.2.5, 1.2.6). An array of unit cells may be used to get this structure.

Fig. 1.2.5: (a) Combination of wire and SRR as a unit cell (b) Pendry’s (1999) circular SRR (c) Equivalent circuit of circular SRR

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