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

Fig. 1.2.6: Combination of thin wires and SRR

It is well known that metal at optical frequencies is characterized by an electric permittivity and that varies with frequency according to (Drude) relation: – plasma frequency, – amplitude of decreasing the plasma oscillations. Thus when = 0 and the medium has negative permittivity. Typical values of plasma frequency are in the ultraviolet range while for 1013rad/s. It should be noticed that for , fulfilled too, so the dominant part of epsilon is an imaginary part that is associated with losses (light absorption).

Pendry was the first who proposed the method based on the significance of dependences of plasma frequency on the Fig. 1.2.7 density and mass of electrons motion. The system consists of thin infinite in z-direction metallic wires with radius r that are periodically arranged on a horizontal plane (xy). The unit cell is square and has length an of its side (Fig. 1.2.7). Free electrons move in the direction of the incident field and move only in the direction of wires. The effective electron density will be . Thus plasma frequency can be significantly reduced if r becomes smaller. For instance for r=10-6m, a=1mm the effective concentration could be reduced down to seven orders. For effective mass the situation is vice versa, => effective mass is bigger than a mass of free electron up to 4 orders. That leads us to the fact that the plasma frequency lays in range of microwave regime with wavelength . The last result shows that the main condition is fulfilled and we can construct the material with negative electric permittivity in the microwave regime.

Only some gyrotropic media has negative magnetic permeability, but we can construct such media in the lab even using non-magnetic metallic elements. Let us consider the three-dimensional repetition of stricter that contains ‘split ring resonator’ (RSS, Fig. 1.2.8) Fig. 1.2.8 with radius r. The rings are concentric and set at one plane (yz) that has a-length sides. As for E, we can assume that incident and the current in each ring defines by induced source . When the distance between rings -> 0 then the magnetic losses will be negligible and magnetic flux , where l ‘distance between RRSs. Inductance . The magnetic field from all rings are uniformly distributed and the mutual inductance between two RRSs could be found by . Thus the voltage across RRS circuit – ohmic resistance of each ring. The induced magnetic dipole moment per unit volume where . Finally, the relative effective magnetic permeability will be . From the last equation, magnetic permeability becomes negative when the resonance frequency of Fig. 1.2.9

Lorentzian variation of the medium’s magnetic permeability – – corresponding plasma frequency (Fig. 1.2.9). The resonant wavelength that depends on L and C of the structure can be made big enough to satisfy the condition of effective medium. Using such method we can extend the technology of creating the three-dimentional isotropic metamaterial by adding two RSSs to other two planes (xz, xy) and equations will be similar to already obtained.

Nowadays we can consider the possibility of using metamaterials in different ways [19-20]. For instance, to create the light interrupter (absorber), ‘perfect’ lens or even black hole. It was thought that optical signal could not be stored statically, and a motion was the essential criterion to use storing feature of the metamaterial. Recently it has been discovered the method of slowing down or even completely stopping the light.

Light has zigzag propagation along waveguide with LH core. The ray experiences negative Goos-Hanchen lateral displacements. Each time it strikes the interface between positive and negative index mediums. The effective thickness is smaller than natural one. By decreasing

Fig. 1.2.10 the physical thickness effective one could vanish. Thus at degeneracy point (power flow and cladding are almost equal), we obtain the situation when the group velocity of energy flux becomes zero and the light will not propagate further (Fig. 1.2.10).

Metamaterials provide tools to enhance the sensitivity and resolution of sensors significantly. In agriculture, the sensors are based on the resonant material. They employ SRR to gain better sensitivity. Wireless strain sensors are widely used in biomedicine. In particular, nested SRR based on strain sensors have been developed to enhance the sensitivity.

The first metamaterial based on absorber utilizes three layers: two metallic layers and a dielectric one. The simulated absorptivity is about 99% at Fig. 1.2.11: (a) Multiple SRR (b) Sierpinski SRR 11.48 GHz. (c) Spiral Resonator

Landy could achieve an absorptivity of 88% experimentally [21]. The difference between simulated and measured results occurred due to fabrication errors.

Superlens can have properties that go beyond the diffraction limit using metamaterials. The structure was shown to have resolution capabilities that go beyond ordinary microscopes [22].

Cloaking could be created by eliminating the electromagnetic field generated by an object or by guiding the electromagnet wave around the object [23]. Unfortunately it is far from practical implementation.

Photonic quasicrystals are modifications of photonic crystals. In the quasicrystalline structures, there is no periodicity but a long-range order is present. Due to this, along with the crystals, they have a discrete diffraction pattern, but unlike them, they can have types of symmetry that are forbidden. Another modification of photonic crystals is photonic-crystalline layered systems. These nanostructures consist of several quasi-homogeneous layers where the permittivity varies periodically along two directions and does not change along the third direction.

Metamaterial acts as a phase compensator when the wave passes through a (double positive) DPS slab having positive phase shift while DNG slab has opposite phase shift so when wave exits from a DNG slab the total phase difference is equal to zero.

The dissipative losses of energy are relatively high to inhibit the further exploitation of metamaterials. Theoretically some works [24] show that it is possible to eliminate losses in photonic metamaterials using active gain inclusions and even overcome [25].

Metamaterials provide a flexible platform for technological advancement and open up new possibilities for the development of various microwave and optical devices that include focusing systems, absorbers, resonators, antennas. New technology offers a unique perspective for development of new electromagnetic materials. These materials are constructed using unit cells with predetermined unique properties. Recently the study of mechanical (e.g. acoustic) metamaterials has been developed. The future of metamaterials lies in the field of optics and medicine. Despite the progress made in experimental and theoretical studies commercially successful metamaterials have not been created. The only exception is a radar in which the losses are a necessary condition for work. The main problem of fabrication metamaterials is based on creation structures smaller than the wavelength at the limit of current fabrication approaches.

1.3 Numerical calculations

Numerical methods for simulating physical processes are very effective instruments to investigate photonic crystals and metamaterials. Therefore, the development of methods for the theoretical description of the properties of photonic crystals and metamaterials is a very important task.

One of the most popular numerical methods for deriving the optical properties of stacked grating structures is Fourier modal method (FMM) [26] based on the spectral concept. Its development began more than fifty years ago with volume gratings consisting of simple sinusoidal permittivity variations [27]. Photonic crystals [28], metamaterials [29], and plasmonics [30] are the fields where FMM is highly exploited.

The method is based on analysis of eigenvalue problem. The problem is derived from Maxwell’s equations for ‘eld distribution. The field ‘eld distribution can propagate or decay forward or backward in the direction of invariance with only a change in the global phase and amplitude. The direction of invariance is defined by invariance under translation in one spatial direction. It is possible to solve the obtained eigenvalue problem electively by an expansion in truncated Fourier series. Fourier-Bloch modes create the solution of the problem. These modes span the subspace of general solutions within such a layer of translation invariance. Correct boundary conditions are necessary to combine solutions of neighboring layers. Thus the optical properties of several stacked layers with identical directions of invariance can be easily defined by the obtained solution.

The periodicity in photonic crystals has its advantage and disadvantage. Disadvantage relates to the restriction of the Fourier modal method with respect to specific geometries, and it requires translation invariance. The advantage includes a correct formulation of periodic boundary conditions. As a result, the Fourier modal method can be applied only to systems that can be separated into several planar layers. These structures can be fabricated by modern techniques such as focused ion beam etching or electron beam lithography. That is the reason why FMM is applicable for simulation physical processes with grating systems.

The FMM contains the scattering matrix approach. This approach is closely related to the transfer matrix method. It is stable for thick periodic layers [31]. A numerical method was developed for the scattering matrix [28]. It allows ones to simulate the properties of one-dimensional or two-dimensional photonic-crystal layers. This method works well for dielectric or semiconductor photonic-crystalline layered structures. The main idea of this method consists of two steps. The first one is splitting the photonic-crystal layered system into several planar periodic layers that are translationally invariant in the normal direction. The second one is solving the Maxwell equations for each layer by expanding the fields in the Floquet-Bloch modes. The exact solution is described by infinitely many terms of series. However, it is necessary to consider truncated series for numerical calculations. Then the solutions for each layer are combined using boundary conditions for the electromagnetic field and the scattering matrix formalism.

The use of big number of harmonics allows ones to obtain accurate solution but the computational time grows as well. The scattering matrix for modern computers is calculated in a reasonable time for photonic-crystalline layered structures consisting of substances with small variations of the dielectric permittivity and magnetic permeability. The example of such structures is systems consisting only of dielectrics and semiconductors. However, the convergence of the approximate solution to the exact one becomes too slow with increasing number of harmonics for two-dimensional photonic-crystalline metal’dielectric layered structure. As a result, a large number of modes is required and the calculation of the optical characteristics becomes impossible even on a supercomputer for specific tasks. Therefore optimization methods like factorization rules and adaptive spatial resolution [32-34] are developed in order to accelerate the calculation of the scattering matrix of metallic photonic-crystalline layered structures.

Lifeng Li provided a detailed analysis of the convergence behavior of the products in Fourier space and formulated the so-called factorization rules [35]. This work led to a systemically optimized eigenvalue problem of the Fourier modal method [36]. As it was shown Fourier-convolution of two discontinuous functions experiences a jump. This leads to the wrong result due to low convergence rate. The example of it is the convolution of dielectric permittivity and normal electric field component that should be continuous. As a result, the factorization rules were formulated. They consist of the Fourier expansion of function directly and expansion of inverse function value. These rules make possible to improve the convergence of the scattering matrix method depending on the proper component of electromagnetic in a specific location.

The second issue occurs from the oscillating Floquet-Fourier behavior of a discontinuous function expansion. This problem is also known as the Gibbs phenomenon and it prevents rapid convergence. The adaptive spatial resolution (ASR) was suggested [37] to increase the resolution near the jump of the discontinuous function as the partial solution of this issue.

The full FMM consists of two parts. The first is the analysis of eigenmodes defining the layer with z homogeneity. The seconds one consists of three optimization approaches that increase the convergence: in thickness sense (scattering matrix approach), for Flouqet-Fourier series by consideration convolution of jump discontinuity functions (factorization rules) and by consideration Gibbs phenomenon (ASR). Full FMM method can be used to calculate single layered waveguides, photonic crystal systems with superlattice or more complex systems. The only difference in the numerical implementation of the optical characteristic evaluations consists of two points. The first one is a choice of the scalar product rule. The second is a choice of the finite-dimensional subspace in which all vectors are projected onto.

However, the application of full FMM is only straight-forward when the lateral interfaces coincide with the directions of periodicity. Therefore, it is possible to calculate the optical properties of skewed geometries, whereas even simple shapes such as cylinders have to be approximated by zig-zag lines. The concept is extended for factorization rules [38], aperiodic structures [39], nonlinear material response [40].

Ordinary frequency solvers use the ‘nite element method which has the advantage of adaptive mesh usage for an appropriate description of arbitrary shapes. The disadvantages of frequency domain solvers are a restriction to linear material response as well as increased calculation time compared to time domain solvers.

1.4 Improving the FMM convergence

The first approach is given by the analysis of the boundary conditions based on the discontinuities of physical properties of the photonic crystal. The analysis of the convergence of infinite series multiplication extends it. The continuous product of two functions with jump discontinuities must take this into account studying the multiplication in a truncated Fourier space. Factorization rules, as one of the best procedures, improve the convergence of the Fourier modal method. The product between permittivity and electric ‘eld component with complementary jump discontinuities defines the normal component of the electric displacement. Factorization rules make the normal component of the electric displacement continuous considering its calculation in Fourier space.

Another way to increase the convergence is to reduce the in’uence of the oscillations around the jump discontinuities that are due to the truncation. These oscillations occurred by Gibbs phenomenon become smaller with an increase of the spatial resolution at the transitions between metal and dielectric.

Wavelet analysis and Lanczos sigma factor can be considered as a solution of problems based on Gibbs phenomenon.

In current work FMM with Flouqet-Fourier basis is used eliminating the possibility to use wavelet analysis.

1.5 Optimization methods for photonic problems

Progress in the production of effective micro/nanomaterials for photonics goals is formulated as an optimization problem. There is a large class of methods that make it possible to find the global optimum for linear or convex optimization problems effectively. Depending on the type, the tasks can be formulated as LP, ILP, NLP, unconstrained, convex, discrete optimization problems.

For a linear problem simplex methods [41] are used. Relaxation LP with branching and cutting hyperplanes are used to solve integer linear programming (ILP) problem.

However, an explicit analytical form (black box) for an optimization task is not always available. Therefore conventional approaches are not effective for black box objective nonlinear function. Nowadays a lot of approaches are being developed to find global minimum. The following methods are examples of such approaches.

Global search [42] algorithm starts a local solver. It finds minimum of constrained nonlinear multivariable function from multiple start points. The algorithms use multiple start points to sample multiple basins of attraction. Firstly trial points are generated. Then basins, counters and threshold are initialized. Finally, the trial points are examined to understand if it is possible to run local solver.

Pattern search [43] algorithm searches a set of points around the current point. During search the point with the lowest value of the objective function is taken from a set of points without gradient approach. The algorithm iteratively searches a set of points around the current point. This set of points is called mesh The mesh is formed by adding the current point to a scalar multiple set of vectors called a pattern. If the pattern search algorithm finds a point on the mesh that improves the objective function at the current point. The new point becomes the current one at the next step of the algorithm.

Genetic algorithm [44] is based on creation of initial random population and generation of a sequence of new populations. A new population is defined by the individuals in the current generation. The task in this method is to find the way of calculation the new population efficiently.

Simulated annealing [45] starts with an initial solution at higher temperature. The changes of an initial solution are accepted with higher probability. So the exploration capability of the algorithm is high and the search space can be explored widely. As the algorithm continues to run, the temperature decreases gradually like in the annealing process. As a result the acceptance probability of non-successful moves is decreasing.

The particle swarm algorithm [46] begins with creating the initial particles and assigning them initial velocities. Firstly, it evaluates the objective function at each particle location and determines the best (lowest) function value and location. Secondly, new velocities are chosen by the information of the current velocity, the particles’ individual best locations and the best locations of particles’ neighbors. Finally, it iteratively updates the particles’ locations, velocities, and neighbors. A new location is obtained by adding a modified velocity to old one. A modified velocity keeps particles within bounds. The cycle of three steps is proceeding until the algorithm reaches a stopping criterion.

The range of machine learning applications is increased significantly. Machine learning principles of building the model to predict unknown variables can be used to solve an optimization problem. Bayesian optimization methods [47, 48] have a great potential to be the most effective. The conventional idea of them is to create surrogate model (e.g. Gaussian process model (GPM)). The model can predict the location of minimum target function value efficiently. The algorithm is based on next loop:

‘ Creating training and test sets of parameters that define the objective function

‘ Updating procedure of GPM in order to obtain posterior distribution over functions

‘ Finding a new point that maximizes an acquisition function a(x).

Bayesian optimization methods will be tested in one- and two-dimensional metallic structures for normal and inclined incidence.

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2. Fourier analysis for micro/nano-structures

This chapter reviews the mathematical formulation of Fourier series with respect to the convergence for piecewise periodic functions occurred in micro/nano-structures as well as numerical calculation approach.

2.1 Spectral concept

Spectral methods take on global approach and use basis functions that are non-zero over the whole domain.

Let – Fourier series of function on the interval , where :

, (2.1.1)

where – scalar product and – Fourier coefficients:

(2.1.2)

A truncated sum of Fourier series (2.1.1) is partial sum:

(2.1.3)

The Fourier series converges if residual

Riesz’Fischer theorem states that a measurable function on [”’, ”]n is square integrable if the corresponding Fourier series converges in the space L2. If ” is square-integrable

(2.1.4)

It means that converges to in L2 norm.

If f is continuously di’erentiable in R, the Fourier series fs of its derivative f’ can be obtained by di’erentiating fs term by term:

(2.1.5)

For any piecewise function f, (2.1.1) gives:

(2.1.6)

If f is square-integrable on [”’, ”]n and piecewise continuous, it has a generalized Fourier series:

Or

(2.1.7)

in L2 n norm.

Consider , where – arbitrary coefficients.

Then

Applying the Pythagorean Theorem to the perpendicular components :

or

(2.1.8)

It shows the Fourier approximation of a function to be the best one in the form(2.1.1).

Nanostructure materials have a special type of discontinuities ‘ the piecewise constant function. Let’s consider a typical example of such one-dimensional function defined on [0,2L]:

, (2.1.9)

where – Heaviside step function.

is odd function, because , where . [-L, L] interval is considered instead of .

Thus Fourier series for(2.1.9):

(2.1.10)

The Fourier series cannot describe the function properly due to the discontinuities (fig.2.1.1).

Fig. 2.1.1: thick black solid line ‘ original function for L = 4 (2.9). Truncated Fourier series for 42 and 82 harmonics (21 and 42 terms ‘ non-zero according to (2.10)) are presented by blue and red solid lines correspondently. Over- and undershoot deviations arise at the jump discontinuities. Green and purple dashed lines correspond to g and h in (2.11) equations respectively.

2.2 Gibbs phenomenon

Let us review Gibbs phenomenon. For the description of piecewise constant functions, it is enough to represent it using a linear combination of

, (2.2.1)

where and

It is possible because piecewise continuously di’erentiable functions can be defined as a linear combination of a continuously monotonic di’erentiable function in R with discontinuity for each one.

Fourier coefficients of :

Anagogic procedure for h gives us:

Thus

Final results fully coincide with (2.1.10) expression.

For simplicity consider one saw function – . Residual of it:

where

The goal is to minimize residual by minimizing integration interval. So consider the closest point to the first value of [0,2L] interval ‘

Last inequality shows that for every large N and x ‘ [0,2L] we have an overshoot of at least 18 % near each jump discontinuity. is the position of the overshoot and due to this formulation shifts towards zero while a number of harmonics tends to infinity. This is called Gibbs phenomenon, and it manifests the non-uniform convergence at a jump discontinuity. The undershoot deviation appears symmetrically at point. This is a crucial point for the convergence. Thus, it is impossible to describe a function correctly using a truncated Fourier series at jump discontinuities.

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