Study of a Ferrite Circulator for Pbg Waveguides in the Microwave Band

JOSÉ LUIS ARRIETA CONCHA

STUDY OF A FERRITE CIRCULATOR FOR PBG WAVEGUIDES IN THE MICROWAVE BAND

ESTUDO DE UM CIRCULADOR DE FERRITA PARA GUIAS PBG NA FAIXA DE MICRO-ONDAS

CAMPINAS 2014 i ii

UNIVERSIDADE ESTADUAL DE CAMPINAS Faculdade de Engenharia Elétrica e de Computação

JOSÉ LUIS ARRIETA CONCHA

STUDY OF A FERRITE CIRCULATOR FOR PBG WAVEGUIDES IN THE MICROWAVE BAND

ESTUDO DE UM CIRCULADOR DE FERRITA PARA GUIAS PBG NA FAIXA DE MICRO-ONDAS

Thesis presented to the School of Electrical and Computer Engineering of the University of Campinas, in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering, in the field of Telecommunications and Telematics.

Tese apresentada à Faculdade de Engenharia Elétrica e de Computação da Universidade Estadual de Campinas para obtenção do título de Mestre em Engenharia Elétrica, na área de Telecomunicações e Telemática.

Orientador: Prof. Dr. HUGO ENRIQUE HERNÁNDEZ FIGUEROA

ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA TESE DEFENDIDA PELO ALUNO JOSE LUIS ARRIETA CONCHA E ORIENTADO PELO PROF. DR. HUGO ENRIQUE HERNÁNDEZ FIGUEROA

Assinatura do Orientador

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ABSTRACT

Optical communication networks are part of today’s information based society, in which optical fibers play a very important role, since they are widely used in such communication networks and other state of the art communication systems. However, the use of electronic devices in various electronic-optical networks represent a major bottleneck for the optimal utilization of the large bandwidth and high speed data rates allowed by optical fibers. In order to avoid such congestion of network traffic, devices based on photonic technologies are one of the proposed solutions, because they provide the capability to maintain transmission speeds with both high data rates and low losses. That is why such photonic devices are continuously being researched in order to improve their effectiveness, obtain greater confinement and guiding of the optical signal, as well as to reduce the insertion loss when it is integrated with other devices.

To achieve a greater understanding of photonic devices, the work presented in this paper allow for a pedagogical and yet thorough understanding of photonic crystals and magnetic effects. This pedagogical approach permits a hands-on experience for students and researchers on a complex subject, a kind of experimental implementation that usually demands sophisticated photonic fabrication resources. The microwave approach allows one to construct conceptually equivalent but hand-manipulable prototypes.

Here, we present the theoretical study, numerical analysis, computer modeling and electromagnetic simulations for a 3-port ferrite circulator in a photonic grid structure built for the 2.3 GHz to 2.9 GHz microwave range. For a photonic device operating in the optical communication range, we may expect a qualitative analogous behavior.

Keywords: Waveguide circulators, Photonic crystals, Ferrite (Magnetic Materials), Microwave devices.

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RESUMO

Redes de comunicação ópticas são parte da sociedade de informação de hoje, em que as fibras ópticas têm um papel muito importante, uma vez que são amplamente utilizados em redes de comunicação e outros sistemas de comunicação atuais. No entanto, o uso de dispositivos eletrônicos em várias redes eletro-ópticas representa um grande engarrafamento para a utilização ideal da grande largura de banda e as taxas de dados de alta velocidade permitida por fibras ópticas. A fim de evitar tal congestionamento de tráfego da rede, dispositivos baseados em tecnologias fotônicas são uma das soluções propostas, porque eles fornecem a capacidade de manter altas velocidades de transmissão com taxas elevadas de dados e baixas perdas. Por isso, tais dispositivos fotônicos são continuamente pesquisados, a fim de melhorar a sua eficácia, a obtenção do maior confinamento e encaminhamento do sinal óptico, e melhorar a perda de inserção quando ele é integrado com outros dispositivos.

Para alcançar uma maior compreensão dos dispositivos fotônicos, o trabalho apresentado aqui permite uma compreensão pedagógica e ainda completa de cristais fotônicos e efeitos magnéticos. Esta abordagem pedagógica permite uma experiência em mãos para os estudantes e pesquisadores sobre um tema complexo, um tipo de implementação experimental que normalmente exige sofisticados recursos de fabricação fotônicos. A abordagem em micro-ondas permite construir protótipos conceitualmente equivalentes e manipuláveis.

Aqui, apresentamos um estudo teórico, análise numérica, modelagem de computador e simulações eletromagnéticas para um circulador de ferrita de 3 portas em uma estrutura de grade fotônico construído para a faixa de micro-ondas de 2,3GHz - 2,9GHz.

Palavras chave: Circuladores de guia de ondas, Cristais fotônicos, Ferrita (Materiais magnéticos), Dispositivos de micro-ondas.

TABLE OF CONTENTS

1. INTRODUCTION ...... 1

2. PHOTONIC CRYSTALS ...... 3 2.1 BASIC DEFINITIONS ...... 3 2.2 MAXWELL’S EQUATIONS ...... 9 2.3 CRYSTALLINE STRUCTURE ...... 12 2.4 PHOTONIC DEVICES BASED ON TWO-DIMENSIONAL PHOTONIC CRYSTALS...... 17 2.4.1 Waveguide ...... 17 2.4.2 Resonant Cavity ...... 20 2.4.3 Waveguide Splitter ...... 21

3. FERRITES AND THE CIRCULATOR ...... 22 3.1 THEORY OF FERRIMAGNETIC COMPONENTS ...... 22 3.1.1 The Permeability Tensor ...... 23 3.2 CIRCULATORS ...... 24 3.3 FERRITE Y-JUNCTION CIRCULATORS...... 26 3.3.1 Description of the Ferrite Circulator Operation ...... 26 3.3.2 Design of a Ferrite Junction Circulator ...... 28

4. NUMERICAL ANALYSIS AND PROJECT DESIGN ...... 30 4.1 DESIGN AND MODELING OF THE PHOTONIC GRID ...... 30 4.2 PBG WAVEGUIDE DESIGN ...... 34 4.3 DESIGN OF THE FERRITE CYLINDER AND CIRCULATOR ...... 37

5. RESULTS ...... 38 5.1 COMPUTER SIMULATIONS ...... 38 5.1.1 Straight Photonic Waveguide ...... 38 5.1.2 Photonic Waveguide Y-Splitter ...... 41 5.1.3 3-Port Photonic Junction Circulator ...... 43

6. FINAL CONLUSIONS AND CONSIDERATIONS ...... 55

7. REFERENCES ...... 57

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DEDICATÓRIA

Dedico este trabalho a minha esposa Nella, que tem sido o meu apoio constante ao longo deste desafio e como a musa que me inspira a cada dia. Aos meus pais, Delia Rosa e Francisco; e minha irmã Cynthia, que têm me guiado e acompanhado durante toda a minha vida, sempre me dando o seu apoio constante para alcançar meus sonhos. Aos professores que tive durante toda a minha vida acadêmica e me orientaram para alcançar meus objetivos. Enfim, a todos os meus amigos e colegas, muitos dos quais também andam este caminho e de algum modo são parte desta conquista.

...... Dedico este trabajo a mi esposa Nella, quien ha sido mi apoyo constante a lo largo de este reto y por ser la musa que me inspira día a día. A mis padres, Delia Rosa y Francisco; y mi hermana Cynthia, quienes me han guiado y acompañado a lo largo de mi vida, brindándome siempre su apoyo constante para alcanzar mis sueños. A todos los profesores que tuve a lo largo de mi vida académica y que supieron orientarme para alcanzar mis metas. Finalmente, a todos mis amigos y compañeros de estudio, muchos de los cuales también recorren este camino y de alguna manera participan de este logro.

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AGRADECIMENTOS

Agradeço, primeiramente, a Deus, por nunca me desamparar, à minha esposa e a minha família, pelo apoio incondicional, e aos meus amigos, que sempre torceram por mim.

Agradeço ao Prof. Dr. Hugo Enrique Hernández Figueroa pela orientação deste trabalho, ao CNPq pelo suporte financeiro e aos colegas do Laboratório de Eletromagnetismo Aplicado e Computacional (LEMAC) pelas dúvidas solucionadas e pelas conversas.

Agradeço, também, ao Prof. Sombra, que confeccionou as cerâmicas utilizadas nesse projeto e ao Sr. Antunes do Talher de Mecânica do CCS pela ajuda com a fabricação do protótipo utilizado neste projeto.

Enfim, agradeço a todos que, de alguma forma, colaboraram para que esse trabalho fosse possível.

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“Everything must be made as simple as possible. But not simpler.” Albert Einstein

“The only way of discovering the limits of the possible is to venture a little way past them into the impossible” Arthur C. Clarke

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LIST OF FIGURES

Figure 1: Scattering regimes - (a) Homogeneous Medium (b) Incoherent scattering, as in geometrical optics (c) Coherent scattering as in a photonic crystal.3 ...... 5 Figure 2: Examples of photonic crystals a) one-dimensional (1D), b) two-dimensional (2D) and three-dimensional (3D). The different shades represent different materials with different dielectric constants 4...... 6 Figure 3: A two-dimensional photonic crystal with square lattice arrangement of dielectric columns. The material is homogeneous along the z-axis and periodic along the x and y-axis...... 7

Figure 4: a) square and b) triangular lattice photonic structures cross sections. (εa and εb are the relative permittivity of the background material, respectively, a: lattice constant, L: cavity length). 5 ...... 12 Figure 5: A schematic view of a) a square lattice photonic crystal with primitive vectors; and b) its representation in reciprocal space with reciprocal primitive vectors. 5 ...... 13 Figure 6: A schematic view of a) a triangular lattice photonic crystal with primitive vectors; and b) its representation in reciprocal space with reciprocal primitive vectors. 5 ...... 13 Figure 7: The square lattice. On the left is the network of lattice points in real space. In the middle is the corresponding reciprocal lattice. On the right is the construction of the first Brillouin zone 2...... 16 Figure 8: The triangular lattice. On the left is the network of lattice points in real space. In the middle is the corresponding reciprocal lattice, which in this case is a rotated version of the original. On the right is the Brillouin zone construction. In this case, the first Brillouin zone is a hexagon centered on the origin 2...... 16 Figure 9: Line-defect waveguide formed by a missing row in a square lattice of dielectric rods in air. 2 ...... 18 Figure 10: 퐸푧 field in a right-angle bend waveguide, showing 100% transmission at the PBG range. 2 ...... 19 Figure 11: Point-defect cavity formed by a single missing rod in a square lattice. The cavity supports a single mode inside the TM band gap, whose electric field 퐸푧 is shown. 2 ...... 20

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Figure 12: Left: 퐸푧 field in a “T” splitter for a waveguide, showing essentially 100% transmission (50% in each branch). Right: Abstract model, treating junction as weak resonance, predicting 100% transmission when 1휏1 = 1휏2 + 1휏3. 2 ...... 21 Figure 13: Ferrite circulator, principle of operation...... 25 Figure 14: Stripline Y-junction circulator ...... 27 Figure 15: a) Dipolar mode of a dielectric disk (퐻푖푛푡 = 0 표푟 ∞). b) Analogous pattern for a magnetized disk (퐻푖푛푡 푓표푟 푐푖푟푐푢푙푎푡푖표푛). The pattern for the magnetized disk has been rotated to isolate port 3...... 28 Figure 16: Unit-cell array configurations for different 푟푎 ratios, as viewed by the MPB tool ... 32 Figure 17: (a) Unit cell (highlighted by the parallelogram) to a hexagonal arrangement and (b) respective Brillouin zone. The irreducible zone is a triangular wedge of vertices Γ, K and M. ... 32 Figure 18: Band Gap diagram calculated for a hexagonal array of dielectric columns surrounded by air, with geometric ratio 푟푎 = 0.2167, for the TM polarization parameters. .... 33 Figure 19: From left to right: Super-cell configurations for five, four, three, two and one dielectric elements (on each side) respectively. Each super-cell has a line-defect along the middle to allow for the guided mode propagation...... 35 Figure 20: TM guided mode for the super-cell configuration of three elements per side...... 36 Figure 21: Straight Photonic Waveguide with two coaxial ports ...... 38 Figure 22: S-parameters for the straight photonic waveguide. Input signal sweeps from 1.5GHz to 3.5GHz...... 39

Figure 23: Zoom in on the S21 parameter (transmission coefficient). In this case it shows good transmission (under -1dB in losses) from 2.35 퐺퐻푧 < 푓 < 2.90 퐺퐻푧...... 40 Figure 24: Three-Port Photonic Waveguide Y-Splitter ...... 41

Figure 25: S-parameters for the photonic waveguide splitter. S21 and S31 are overlapping each other. They both show equal signal output with about -0.78dB less than ideally expected, caused by losses in the waveguide as well as some impedance mismatch (reflections)...... 42 Figure 26: 3-Port Photonic Junction Circulator ...... 43 Figure 27: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 1500 Gauss. Signal input is done only at Port-1 since all ports are symmetric ...... 44

Figure 28: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 1700 Gauss. Signal input is done only at Port-1 since all ports are symmetric ...... 45 Figure 29: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 2300 Gauss. Signal input is done only at Port-1 since all ports are symmetric ...... 46 Figure 30: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 2125 Gauss. Signal input is done at all three ports, sequentially ...... 47 Figure 31: S-parameters for the three-port photonic junction circulator with biasing field 2125 Gauss and input signal at Port 1...... 48 Figure 32: Representation of the E-fields for the circulator operation with input at Port 1. We observe very good transmission from Port 1 to Port 2 and good isolation at Port 3...... 49 Figure 33: S-parameters for the three-port photonic junction circulator with biasing field 2125 Gauss and input signal at Port 2...... 50 Figure 34: Representation of the E-fields for the circulator operation with input at Port 2. We observe very good transmission from Port 2 to Port 3 and good isolation at Port 1...... 51 Figure 35: S-parameters for the three-port photonic junction circulator with biasing field 2125 Gauss and input signal at Port 3...... 52 Figure 36: Representation of the E-fields for the circulator operation with input at Port 3. We observe very good transmission from Port 3 to Port 1 and good isolation at Port 2...... 53

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LIST OF TABLES

Table 1: Analysis of band gap size for different 푟푎 ratios of grid arrays...... 31 Table 2: Analysis for the Guided Mode Gap as a function of the number of dielectric elements on each side of the line-defect...... 35 Table 3: Comparison between a top of the line circulator (military grade) and our proposed design for a photonic circulator ...... 54

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1. INTRODUCTION

Communication networks are part of today’s knowledge society, in which optical fibers play an important role since they are widely used in communication networks and other the state of the art communication systems. However, the use of electronic devices in various electronic- optical networks still represents a major bottleneck for the full utilization of the large bandwidth and high-speed data rates provided by optical fibers. In order to avoid such congestion of network traffic, devices based on optical technologies are needed because they provide higher data transmission rates with fewer losses. Therefore, there is ongoing research on these devices based on photonic technologies, so we are able to improve their effectiveness, greater confinement and guiding of the optical signal, as well as reduce the insertion loss when it’s integrated with other devices. For all of the above, this is why it is very important for us to achieve a thorough understanding of photonic crystals and waveguide structures built with these technologies.

In Chapter 2, we present the Photonic Crystals, which in simple terms are periodic structures with high contrast ratios in their refraction indexes and obeying a certain symmetry, being the rectangular and triangular ones the most common. Lord Rayleigh has studied these crystals properties since the late nineteenth century, with the analysis of multilayers with different indices of refraction; however, their study has continued until nowadays because of the increasing need for high transmission rates. The main feature of these structures is to prohibit the propagation of electromagnetic waves of determined frequencies within the crystal structure, acting similarly to a microwave conductor (SIBILIA et al, 2008).

There are two major problems in transmissions of optical signals: (1) the miniaturization of the required circuitry and (2) the difficulty of obtaining guidance with low losses, since metallic materials used for guiding micrometric electromagnetic waves (microwaves), have high losses for optical frequencies. However, these problems can be solved by using waveguides made from photonic crystals. The dimensions of the elements in a photonic crystal are of the order of magnitude of the wavelength of the incident signal, that is, for optical signals they are of some micrometers, which is very useful for miniaturization of the devices. Also by confining a signal between two photonic crystal structures or removing some elements (forming a path) and then

focusing a signal in this region, one obtains a wave-guide with low loss. It is also possible to guide the desired signal in a path, with twists and turns, and building resonant cavities, as it is currently being done for conventional microwave signals.

In Chapter 3, we present the theory of ferrimagnetic components and some background on one of its most common applications: the ferrite circulator. Nowadays, microwave integrated systems include some of these nonreciprocal structures (circulators, isolators, phase shifters) operating over a broad frequency band (NOVA MICROWAVE, 2014). However, the potential drawbacks of these structures are high insertion losses associated with a ferrite slab, which it is often several wavelengths long. Recent research of ferrite-coupled devices focus on improvement of these parameters such as insertion losses and isolation. These devices are realized in finline, slotline or microstrip technology (MARYNOWSKI; MAZUR, 2009).

In Chapter 4, we present the numerical analysis we conducted for the design of the photonic grid, the photonic waveguides, the photonic splitter and the ferrite disk. These numerical analyses were conducted using software already developed for these applications (COMSOL MULTIPHYSICS, 2008), or in the case of the ferrite, using already known calculations presented in previous works.

In Chapter 5, we present the computer modeling of these photonic structures and conducted electromagnetic simulations on them (ANDONEGUI; GARCIA-ADEVA, 2013). The results we obtained match very well with the theory behind their design.

Finally, it is the intention of the present work, to achieve a greater understanding of photonic devices, photonic band gap techniques and magnetic effects, in order to construct an experimental device as means for a pedagogical approach and thorough understanding of these technologies. This pedagogical approach permits a hands-on experience for students and researchers on a complex subject, a kind of experimental implementation that usually demands sophisticated photonic fabrication resources. The microwave approach (YAO, 2012a; 2012b) allows one to construct conceptually equivalent but hand-manipulable prototypes.

2. PHOTONIC CRYSTALS

This chapter presents the analogy between the electrons in a semiconductor crystalline lattice and the photons in a photonic crystal. Then we present the Maxwell equations, which are used to analyze the structures of photonic crystals. Finally, some devices based on two-dimensional photonic crystals, which can be used in communication systems, are presented at the end of the chapter.

2.1 BASIC DEFINITIONS

In simple terms, a crystal is a periodic arrangement of atoms or molecules, and the pattern in which the atoms or molecules are repeated in space is the crystal lattice. Therefore the crystal presents a periodic potential to an electron propagating through it, and both the constituents of the crystal and the geometry of the lattice dictate the conduction properties of the crystal (JOANNOPOULOS et al, 2011).

The theory of quantum mechanics in a periodic potential explains that electrons propagate as waves, and waves that meet certain criteria can travel through a periodic potential without scattering (although they will be scattered by defects and impurities). Importantly, however, the lattice can also prohibit the propagation of certain waves, meaning that there may be gaps in the energy band structure of the crystal in which the electrons are forbidden to propagate with certain energies in certain directions. If the lattice potential is strong enough, the gap can extend to cover all possible propagation directions, resulting in a complete band gap. For example, a semiconductor has a complete band gap between the valence and conduction energy bands.

The optical analogue is the photonic crystal, in which the atoms or molecules are replaced by macroscopic media with differing dielectric constants, and the periodic potential is replaced by a periodic dielectric function (or, equivalently, a periodic index of refraction). If the dielectric constants of the materials in the crystal are sufficiently different, and if the absorption of light by the materials is minimal, then the refractions and reflections of light from all of the various

interfaces can produce many of the same phenomena for photons (light modes) that the atomic potential produces for electrons. One solution to the problem of optical control and manipulation is thus a photonic crystal, a low-loss periodic dielectric medium. In particular, we can design and construct photonic crystals with photonic band gaps, preventing light from propagating in certain directions with specified frequencies (i.e., a certain range of wavelengths, or “colors,” of light). We will also see that a photonic crystal can allow propagation in anomalous and useful ways.

Although the photonic crystal properties are well known since the end of the 19th century, in recent decades they have been widely studied because of their ability for presenting these band gaps for the propagation of electromagnetic signals in certain wavelengths, thus allowing the guidance and control of these signals. This is very important because the reflective materials commonly used for microwave frequencies (conducting metals), have very large losses in the optical frequencies. With photonic crystals, designs with steep curves, intersections and junctions can be built in the waveguides and still present very low losses. This added to the fact that they can be also integrated into microelectronics, since they can be fabricated on silicon wafers, which is the base material of such technology.

To develop this concept further, consider how metallic waveguides and cavities relate to photonic crystals. Metallic waveguides and cavities are widely used to control microwave propagation. The walls of a metallic cavity prohibit the propagation of electromagnetic waves with frequencies below a certain threshold frequency, and a metallic waveguide allows propagation only along its axis. It would be extremely useful to have these same capabilities for electromagnetic waves with frequencies outside the microwave regime, such as visible light. However, visible light energy is quickly dissipated within metallic components, which makes this method of optical control impossible to generalize. Photonic crystals allow the useful properties of cavities and waveguides to be generalized and scaled to encompass a wider range of frequencies. We may construct a photonic crystal of a given geometry with millimeter dimensions for microwave control, or with micron dimensions for infrared control.

The operating regime of photonic crystals is between two regimes of scattering. They have comparable dimensions to the wavelength of the propagating signal. Under this regime, when the dimensions are much smaller than the wavelength (Figure 1a), the medium can be considered homogeneous. Above this limit, when the dimensions are much larger than the wavelength (Figure 1b), there is an incoherent scattering, entering the regime of geometrical optics. However in the intermediate limit, when the dimensions are comparable to the propagating wavelength (Figure 1c), the regime becomes one of coherent scattering. In the intermediate situation, a periodic arrangement of elements can perform an addition of scattered fields, which leads to the emergence of the “band gap”.

Figure 1: Scattering regimes - (a) Homogeneous Medium (b) Incoherent scattering, as in geometrical optics (c) Coherent scattering as in a photonic crystal.(PASETTO, 2011)

Photonic crystals can be classified by the number of dimensions for which they exhibit symmetry in the network. As can be seen in Figure 2, there are crystals with symmetry in one dimension (1D - multilayers), with two dimensions of symmetry (2D) and three dimensions of symmetry (3D – woodpile). This following work will focus only on 2D photonic crystals.

Figure 2: Examples of photonic crystals a) one-dimensional (1D), b) two-dimensional (2D) and three-dimensional (3D). The different shades represent different materials with different dielectric constants (ROBINSON; NAKKEERAN, 2013).

A two-dimensional photonic crystal is periodic along two axes of symmetry (x-axis and y- axis as in Figure 3) and smooth on the other axis (z-axis in Figure 3). A typical arrangement consists of a set of cylinders of different dielectric constant than the background material (primarily high dielectric constant immersed in air or air holes in a material of high dielectric constant). Depending on the dielectric constant, the dimensions of the periodic arrangement (radius of the cylinders and spacing between their centers) and their distribution, the crystal would behave as having a rejection to certain frequencies (band gap) of the signals propagating in the x-y plane (JOANNOPOULOS et al, 2011).

Figure 3: A two-dimensional photonic crystal with square lattice arrangement of dielectric columns. The material is homogeneous along the z-axis and periodic along the x and y-axis.

An example of a widely used optical device is a multilayer dielectric mirror, such as a quarter-wave stack, consisting of alternating layers of material with different dielectric constants. Light of the proper wavelength, when incident on such a layered material, is completely reflected. The reason is that the light wave is partially reflected at each layer interface and, if the spacing is periodic, the multiple reflections of the incident wave interfere destructively to eliminate the forward-propagating wave. This well-known phenomenon, first explained by Lord Rayleigh in 1887, is the basis of many devices, including dielectric mirrors, dielectric Fabry-Perot filters, and distributed feedback lasers. All contain low-loss dielectrics that are periodic in one dimension, and by our definition, they are one-dimensional photonic crystals. Even these simplest of photonic crystals can have surprising properties. We will see that layered media can be designed to reflect light that is incident from any angle, with any polarization (an omnidirectional reflector) despite the common intuition that reflection can be arranged only for near-normal incidence.

If, for some frequency range, a photonic crystal prohibits the propagation of electromagnetic waves of any polarization traveling in any direction from any source, we say that the crystal has a complete photonic band gap. A crystal with a complete band gap will obviously be an omnidirectional reflector, but the opposite is not necessarily true. Usually, in order to create a complete photonic band gap, one must arrange for the dielectric lattice to be periodic along three axes, constituting a three-dimensional photonic crystal, but there are exceptions. A small amount of disorder in an otherwise periodic medium will not destroy a band gap, and even a highly disordered medium can prevent propagation in a useful way (SOUZA, 2012).

2.2 MAXWELL’S EQUATIONS

The following Maxwell equations were used in the analysis of the periodic structures of the Photonic Crystals:

∇ ∙ 퐷⃗⃗ = 휌, (2.1)

휕퐵⃗ ∇ × 퐸⃗ + = 0, (2.2) 휕푡 ∇ ∙ 퐵⃗ = 0, (2.3)

휕퐷⃗⃗ ∇ × 퐻⃗⃗ − = 퐽 , (2.4) 휕푡

Where 퐸⃗ is the electric field, 퐻⃗⃗ is the magnetic field, 퐷⃗⃗ the electric flux density, 퐵⃗ the magnetic flux density, 휌 is the density of free charges and 퐽 is the electric current density. The following constitutive relations are also needed:

퐵⃗ = 휇퐻⃗⃗ (2.5) 퐷⃗⃗ = 휀퐸⃗ (2.6)

Considering a medium free of charges and currents, it is well known that 퐽 = 0 and 휌 = 0.

For periodic structures, we have that 휀(푟 ) = 휀0휀푟(푟 ) and since the materials used are not magnetic then 휇(푟 ) = 휇0. Substituting the above expressions and constitutive relations in Maxwell’s equations we obtain:

∇ ∙ [휀푟(푟 )퐸⃗ (푟 , 푡)] = 0 (2.7)

휕퐻⃗⃗ (푟 ,푡) ∇ × 퐸⃗ (푟 , 푡) + 휇 = 0 (2.8) 0 휕푡 ∇ ∙ 퐻⃗⃗ (푟 , 푡) = 0 (2.9)

휕퐸⃗ (푟 ,푡) ∇ × 퐻⃗⃗ (푟 , 푡) − 휀 휀 = 0 (2.10) 0 푟 휕푡

The 퐸⃗ and 퐻⃗⃗ fields are functions of time and space, however, since Maxwell's equations are linear, they can be written as the product of their spatial dependence and temporal dependence of the form of a complex exponential, like this:

퐸⃗ (푟 , 푡) = 퐸⃗ (푟 )푒−푖휔푡 (2.11) 퐻⃗⃗ (푟 , 푡) = 퐻⃗⃗ (푟 )푒−푖휔푡 (2.12)

To find the equation describing the modes at a given frequency, simply enter these equations in (2.9) and (2.10) as (2.7) and (2.8) only ensures the free space and without load.

∇ × 퐸⃗ (푟 ) − 푖휔휇0퐻⃗⃗ (푟 ) = 0 (2.13)

∇ × 퐻⃗⃗ (푟 ) + 푖휔휀0휀푟(푟 )퐸⃗ (푟 ) = 0 (2.14)

From equations (2.13) and (2.14), the following equation can be obtained:

1 휔 2 ∇ × ( ∇ × 퐻⃗⃗ (푟 )) = ( ) 퐻⃗⃗ (푟 ) (2.15) 휀푟(푟 ) 푐

Since it is known that 휀푟(푟 ) is a periodic function in space, we conclude that 퐻⃗⃗ (푟 ) is too. From equation (2.15), (2.7) and (2.8), we calculate the modes of 퐻⃗⃗ (푟 ) and their corresponding frequencies. From which we can determine the frequencies where propagation is not possible, which is called the Electromagnetic Band Gap (EBG).

Note that this equation is invariant to scale. Equation (2.15) will apply the same both for microwave frequencies as for optic frequencies. For optical or photonic frequencies, the band gap region is called the Photonic Band Gap (PBG).

Considering now a mode for 퐻⃗⃗ (푟 ), with a frequency 휔 and in a periodic dielectric grid given by 휀푟(푟 ), we use equation (2.15), to study what happens with the mode when you scale the ′ ′ grid by a factor of 푠. In this case the following changes will be made: 휀푟(푟 ) = 휀푟(푟 ⁄푠), 푟 = 푠푟

and ∇′= ∇⁄푠. Which after some mathematic manipulation and substitutions (JOANNOPOULOS et al, 2011) we arrive at the following equation:

′ 2 ′ 1 ′ ⃗⃗ ′ ′ 휔 ⃗⃗ ′ ′ ∇ × ( ′ ′ ∇ × 퐻 (푟 )) = ( ) 퐻 (푟 ) (2.16) 휀푟(푟 ) 푐

Thus, it can be concluded that by resizing the dielectric grid, there will be new modes of a frequency inversely proportional to the scale resizing, or in other words, for any variation in the scale of the grid, we also use the same scale for the wavelength of the modes. By using this property we can conclude that any result that we will obtain in optical frequencies (photonic) can also be verified in microwave frequencies, as long as we scale the grid accordingly to the escalation in wavelength, or vice versa.

2.3 CRYSTALLINE STRUCTURE

An ideal crystal can be defined as a regular repetition of a block that is filling all space. The smallest structure to be periodically repeated to build the crystal is usually called a unit cell. An ideal crystal can have one, two or three dimensions of crystalline symmetry. In three dimensions, a crystal is determined by three basic vectors of translation: 푎 1, 푎 2 푎푛푑 푎 3. Thus, for a point represented by 푟 , is equivalent to another point represented by 푟 ′:

′ 푟 = 푟 + 푙1푎 1 + 푙2푎 2 + 푙3푎 3 = 푟 + 푅⃗ (2.17)

Where 푙1, 푙2 푎푛푑 푙3 are integers and 푅⃗ , called the lattice vector, is any vector translating a point in one cell to an equivalent point in the crystal lattice. In Figure 4a and in Figure 4b, can be seen two examples of two-dimensional crystal structures: the square lattice and the triangular lattice, respectively. In both cases, the elements are separated from each other by a distance 푎 from the center of each other.

Schemes in Figure 5a and Figure 6a show a view of photonic crystal cross sections in a x- y plane, with cylinders arranged in square or triangular lattice with period 푎, and with depicted primitive vectors 푎1 푎푛푑 푎2.

Figure 5: A schematic view of a) a square lattice photonic crystal with primitive vectors; and b) its representation in reciprocal space with reciprocal primitive vectors. (KOBA, 2013)

Figure 6: A schematic view of a) a triangular lattice photonic crystal with primitive vectors; and b) its representation in reciprocal space with reciprocal primitive vectors. (KOBA, 2013)

Figure 5b and Figure 6b show the reciprocal lattices corresponding, respectively, to the real square and triangular lattices. In the described case, the nodes of a two-dimensional structure can be expressed by:

푥∥ = 푙1푎 1 + 푙2푎 2 (2.18)

Where 푎 1 and 푎 2 are primitive vectors, 푙1 and 푙2 are arbitrary integers, 푥∥ specifies the placement on the plane 푥∥ = 푥̂푥 + 푦̂푦, where 푥̂ and 푦̂ are unit vectors along x and y axis, respectively. Primitive vectors for square lattice are described by the expressions: 푎 1 = (푎, 0),

푎 2 = (0, 푎) and for the triangular lattice: 푎 1 = (푎√3⁄2 , 푎⁄2), 푎 2 = (0, 푎).

In general, the reciprocal vectors can be written in the following form:

⃗ ⃗ 퐺(ℎ) = ℎ1푏1 + ℎ2푏2 (2.19)

⃗ ⃗ Where ℎ1 and ℎ2 are arbitrary integers, 푏1 and 푏2 are the primitive vectors of the two- dimensional reciprocal space, which are expressed by the following equations (KOBA, 2013):

⃗ 2휋 (2) (2) 푏1 = (푎푦 , −푎푥 ) (2.20) 푎푐 ⃗ 2휋 (1) (1) 푏2 = (−푎푦 , 푎푥 ) (2.21) 푎푐

(푖) Where 푎푗 is the j-th Cartesian component (x or y) of the 푎 푖 vector (푖 = 1 표푟 2). The areas 2 of primitive cells are 푎푐 = |푎 1 × 푎 2| = 푎 in case of the square lattices and 푎푐 = |푎 1 × 푎 2| = √3푎2⁄2 in case of the triangular lattices.

From the above equations, and using the expressions for square and triangular lattice primitive vectors and primitive cell areas, the reciprocal primitive vectors are described by the following formulas:

⃗ ⃗ 푏1 = (2휋⁄푎 , 0), 푏2 = (0, 2휋⁄푎) --- Square lattice (2.22) ⃗ ⃗ 푏1 = (4휋⁄√3푎 , 0), 푏2 = (−2휋⁄√3푎 , 2휋⁄푎) --- Triangular lattice (2.23)

Finally, we present the concept of the Brillouin zone, which is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.

Taking surfaces at the same distance from one element of the lattice and its neighbors, the volume included is the first Brillouin zone. Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane.

There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly n − 1 distinct Bragg planes.)

A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice.

Figure 7: The square lattice. On the left is the network of lattice points in real space. In the middle is the corresponding reciprocal lattice. On the right is the construction of the first Brillouin zone (JOANNOPOULOS et al, 2011).

Figure 8: The triangular lattice. On the left is the network of lattice points in real space. In the middle is the corresponding reciprocal lattice, which in this case is a rotated version of the original. On the right is the Brillouin zone construction. In this case, the first Brillouin zone is a hexagon centered on the origin (JOANNOPOULOS et al, 2011).

2.4 PHOTONIC DEVICES BASED ON TWO-DIMENSIONAL PHOTONIC CRYSTALS

The ability to control the flow of electromagnetic waves, in particular for the microwave range, has been developed for some time. One of the most common devices used in this process is the metallic waveguide, whose main function is to guide electromagnetic waves from one point to another by following a particular direction of propagation.

However, these materials are not suitable for controlling the flow of electromagnetic waves with frequency higher than the microwave, like visible light for example, because the losses in this frequency range are large enough to cripple the process.

In order to overcome this difficulty, photonic crystals can be used, since the dielectric materials commonly used in the synthesis of these crystals do not suffer as much with the dissipative losses as metallic materials, at frequencies above the microwave range.

The following are some devices based on two-dimensional photonic crystals, which can be used in a communication system.

2.4.1 Waveguide

A waveguide has the function of transporting electromagnetic waves from one point to another in a system. A dielectric waveguide, based on two-dimensional photonic crystals, uses the property of such crystals to be able to reflect electromagnetic waves often located in a specific band, known as Photonic Band Gap (PBG).

Considering a 2D photonic crystal, a perfectly periodic one, like the ones shown in Figure 5 or Figure 6; you can create a simple waveguide by removing a row of dielectric cylinders (creating a linear defect), as we can see in Figure 9.

Figure 9: Line-defect waveguide formed by a missing row in a square lattice of dielectric rods in air. (JOANNOPOULOS et al, 2011)

Since the structure with a linear defect is excited with electromagnetic waves in the frequency range of the PBG, then it can be said that these waves are confined outside the crystal lattice, since the walls surrounding the defect are reflective to those frequencies. Also in Figure 9, it is presented the profile of the electromagnetic field along a dielectric waveguide, considering propagation in the range of the PBG.

It is also interesting to point out that a PBG waveguide may present sharp curves, which is not possible in optical fibers (due to the bending losses). For example in Figure 10, we present a waveguide with a 90° bend.

Figure 10: 퐸푧 field in a right-angle bend waveguide, showing 100% transmission at the PBG range. (JOANNOPOULOS et al, 2011)

2.4.2 Resonant Cavity

A resonant cavity can be accomplished through the creation of a point defect in a two- dimensional photonic crystal. This defect can be created from the change of parameters of a single dielectric cylinder, the simple removal of it or the alteration of the radius.

Considering the light with frequency located in the PBG range and located within the defect, it can be said that it is prevented from leaving the defect, given that in this situation the walls surrounding the defect reflect light. These cavities are fundamental in laser systems and can be used in the manufacture of filters.

In Figure 11, we have the profile of the electromagnetic field inside a cavity, which was designed using PBG properties.

Figure 11: Point-defect cavity formed by a single missing rod in a square lattice. The cavity

supports a single mode inside the TM band gap, whose electric field 퐸푧 is shown. (JOANNOPOULOS et al, 2011)

2.4.3 Waveguide Splitter

Another very interesting device based on the existence of the PBG in two-dimensional photonic crystal is the waveguide splitter. The basic function of this device is to divide the input power between two waveguides at the output. In Figure 2.6, we have an accurate representation of this device operating in the PBG range.

The design of the photonic circulator will be based mostly on the design of a waveguide splitter considering 120º angles in between the waveguides and a weak resonant cavity in the middle.

Figure 12: Left: 퐸푧 field in a “T” splitter for a waveguide, showing essentially 100% transmission (50% in each branch). Right: Abstract model, treating junction as weak resonance, predicting

100% transmission when 1⁄휏1 = 1⁄휏2 + 1⁄휏3. (JOANNOPOULOS et al, 2011)

3. FERRITES AND THE CIRCULATOR

3.1 THEORY OF FERRIMAGNETIC COMPONENTS

Usually most materials studied in electromagnetism are isotropic (meaning that they behave in the same way independent of the direction of the signal polarization), however in order to implement devices that present directional properties when a signal has different polarization (such as the circulator) we must use anisotropic materials.

The most practical anisotropic materials for microwave applications are ferrimagnetic compounds such as YIG (yttrium iron garnet)(M POZAR, 2005), or ferrites which are ceramic compounds consisting of a mixture between iron oxide and one or more other metals, such as aluminum, cobalt, manganese, and nickel (BRITUN; DANYLOV; ZAGORODNY, 2002) which as a result end up forming a cubic crystal structure. In contrast to ferromagnetic materials (e.g., iron, steel), ferrimagnetic compounds have high resistivity and a significant amount of anisotropy at microwave frequencies, which makes them suitable in high-frequency electrical components such as antennas.

The magnetic anisotropy of a ferrimagnetic material is actually induced by applying a DC magnetic bias field. This field aligns the magnetic dipoles in the ferrite material to produce a net (nonzero) magnetic dipole moment, and causes the magnetic dipoles to precess at a frequency controlled by the strength of the bias field. A microwave signal circularly polarized in the same direction as this precession will interact strongly with the dipole moments, while an oppositely polarized field will interact less strongly. Since, for a given direction of rotation, the sense of polarization changes with the direction of propagation, a microwave signal will propagate through a ferrite differently in different directions. This effect can be utilized to fabricate directional devices such as isolators, circulators, and gyrators. Another useful characteristic of ferrimagnetic materials is that the interaction with an applied microwave signal can be controlled by adjusting the strength of the bias field. This effect leads to a variety of control devices such as phase shifters, switches, and tunable resonators and filters.

We will begin by considering the microscopic behavior of a ferrimagnetic material and its interaction with a microwave signal to derive the permeability tensor. This macroscopic description of the material can then be used with Maxwell's equations to analyze wave propagation in an infinite ferrite medium, and in a ferrite-loaded waveguide.

3.1.1 The Permeability Tensor

The magnetic properties of a material are due to the existence of magnetic dipole moments, which arise primarily from electron spin. In most solids, electron spins occur in pairs with opposite signs so the overall magnetic moment is negligible. However, in a magnetic material, a large fraction of the electron spins are unpaired (more left-hand spins than right-hand spins, or vice versa), but are generally oriented in random directions so that the net magnetic moment is still small. An external magnetic field, however, can cause the dipole moments to align in the same direction to produce a large overall magnetic moment. The existence of exchange forces can keep adjacent electron spins aligned after the external field is removed; the material is then said to be permanently magnetized.

For this reason when a ferrite is subjected to a static magnetic bias field 퐻0, it exhibits anisotropic properties, which can be described by means of the magnetic permeability tensor. The anisotropy of the magnetic permeability is responsible for the non-reciprocal behavior found in ferrite circulators. Losses can be neglected, however, whereas the applied bias magnetic field is strong enough so that the ferrite is in magnetic saturation. Under this condition, the permeability tensor of a magnetized ferrite the positive direction of z-axis is given by (M POZAR, 2005):

휇 푗휅 0 [휇] = [−푗휅 휇 0 ] 푧̂ (3.1) 0 0 휇0

Where the elements of the permeability tensor are defined by:

휔0휔푚 휇 = 휇0 (1 + 2 2) (3.2) 휔0−휔

휔 휔푚 휅 = 휇0 ( 2 2) (3.3) 휔0−휔

Where 휇0 is the permeability of free space, 휔 is the angular frequency, 휔0 is the precession frequency or Larmor frequency for the electron spin in the presence of an applied magnetic field

퐻0 (휔0 = 휇0훾퐻0), 휔푚 is the Larmor frequency for the electron when the ferrite is under magnetization saturation 푀푠 (휔푚 = 휇0훾푀푠), and 훾 is the gyromagnetic radius of the electron.

3.2 CIRCULATORS

Circulators are devices that use the non-reciprocal properties of ferrimagnetic materials, which means that in the presence of a magnetic field, allow for the control and transmission of microwave energy.

This interaction between microwave signals and the electrons within the ferrimagnetic material is usually for a specific resonant frequency. This resonant frequency, which is determined by the atomic composition of a ferrite component, allows for non-reciprocal behavior in the ferrite that makes microwave energy travel in the transmission path in only one direction with a small loss, but also absorbs the energy when applied from the opposite direction (ZAHWE et al, 2009).

In this sense, a ferrite circulator can be a passive non-reciprocal three-port or four-port device, in which a microwave or radio frequency signal entering any port is transmitted to the next port in rotation (only one way). A port in this context is a point where an external waveguide or transmission line (such as a microstrip line, coaxial cable or PBG waveguide) connects to the device. For a three-port circulator, as shown in Figure 13, a signal applied to port 1 only comes out of port 2; a signal applied to port 2 only comes out of port 3; a signal applied to port 3 only comes out of port 1 (PHILIPS SEMICONDUCTORS, 1998).

Figure 13: Ferrite circulator, principle of operation.

Circulators fall into two main classes: four-port waveguide circulators based on Faraday rotation of waves propagating in a magnetized material; and three-port "Y-junction" circulators based on cancellation of waves propagating over two different paths near a magnetized material.

Waveguide circulators may be of either type, while smaller devices based on striplines are of the three-port type. Sometimes two or more Y-junctions are combined in a single component to give four or more ports, but these differ in behavior from a true 4-port circulator.

Circulators exist for many frequency bands, ranging from VHF up to optical frequencies, the latter being used in optical fiber networks. For example, radio frequency circulators are composed of magnetized ferrite materials in which a permanent magnet produces the magnetic flux through the waveguide, however for optical frequencies, ferrimagnetic crystals are used to make optical circulators. At frequencies much below VHF, ferrite circulators become impractically large. It is however possible to simulate circulator behavior all the way down to DC using op-amp circuits. Unlike ferrite circulators, these active circulators are not lossless passive devices but require a supply of power to run. Also the power handling capability and linearity and signal to noise ratio of transistor-based circulators is not as high as those made from ferrites. 25

3.3 FERRITE Y-JUNCTION CIRCULATORS

The ferrite junction circulator is a versatile microwave device, whose main characteristic is that in addition to its use as a circulator, it also can be used as an isolator or as a switch. The three- port version of the ferrite junction circulator, usually called the Y-junction circulator, it is the most commonly used, and it can be constructed to join striplines, metallic waveguides or, in our case study, PBG waveguides.

3.3.1 Description of the Ferrite Circulator Operation

The easiest case study for a ferrite circulator is the stripline Y-junction ferrite circulator, which consists of two ferrite cylinders filling the space between a metallic conducting center disk and two conducting ground plates (TEOH; DAVIS, 2001). Thus, presenting the simplest geometrical arrangement and, therefore, is the easiest to treat analytically. This circulator in its basic form is illustrated in Figure 14. The connections to the center disk are in the form of three strip-line center conductors attached to the disk at points 120° apart around its circumference. A magnetizing field is applied parallel to the axis of the ferrite cylinders.

From several experimental results (FAY; COMSTOCK, 1965), it was found that the Y- circulator had some of the properties of a low-loss transmission cavity. At its resonant frequency, it was well matched but slightly under coupled, and a standing wave existed in the structure. The maximum isolation occurred at the frequency at which the insertion loss was a minimum. The isolation at the third port and the return loss from the input port correspond quite well as the frequency is changed. The above experimental evidence suggests a resonance of the center disk structure as being an essential feature of the operation of the circulator. The lowest frequency resonance of the circular disk structure of Figure 14 is the dipolar mode in which the electric field vectors are perpendicular to the plane of the disk and the RF magnetic field vectors lie parallel to the plane of the disk. This mode, as excited at port 1, is illustrated in the unmagnetized case by the standing-wave pattern of Figure 15a.

Figure 14: Stripline Y-junction circulator

With ferrite cylinders on each side of the disk, the bottom cylinder behaves as a mirror image of the top one, and the analysis of the device need only be concerned with one cylinder. Ports 2 and 3, if open-circuited, will see voltages that are 180° out of phase with the input voltage, and about half of the value of the input voltage (LIN; AFSAR, 2006). If the standing-wave pattern is rotated, as in Figure 15b then port 3 is situated at the voltage null of the disk and the voltages at ports 1 and 2 are equal. The device is equivalent to a transmission cavity between ports 1 and 2, and port 3 is isolated.

Figure 15: a) Dipolar mode of a dielectric disk (퐻푖푛푡 = 0 표푟 ∞). b) Analogous pattern for a magnetized disk (퐻푖푛푡 푓표푟 푐푖푟푐푢푙푎푡푖표푛). The pattern for the magnetized disk has been rotated to isolate port 3.

3.3.2 Design of a Ferrite Junction Circulator

There are several sources (FAY; COMSTOCK, 1965), (M POZAR, 2005) that deals with circulator design by the use of electromagnetic theory. We start by solving the Maxwell equations (2.1), (2.2), (2.3) and (2.4) and applying the permeability tensor (3.1) and the boundary conditions. Since the geometry of the ferrite is disk shaped, the permeability tensor should be applied in cylindrical coordinates.

The resonant modes can be found by enforcing the electromagnetic boundary conditions. If the ferrite is not magnetized, resonance occurs when:

퐽 (푘푅) 퐽 (푘푅) − 1 = 퐽 (푘푅) = 0 (3.4) 0 푘푅 1

Where 퐽푛(푘푅) is the n-order Bessel function of the first kind. 푅 is the radius of the ferrite disk, 푘 is the wavenumber. The first root of (3.4) is at 푘푅 = 1.841. For a more detailed mathematical deduction of this equation, this can be found in (FAY; COMSTOCK, 1965).

In order to find the value of 푘, we need to take into account the material properties of the ferrite disk, in this case the value of 푘 is determined by:

휇2−휅2 푘2 = 휔2휀 휀 휇 휇 = 휔2휀 휀 휇 (3.5) 0 푒푓푓 0 0 0 휇

Finally, from (3.4) and (3.5), we find the radius 푅 of the ferrite disk. However, we need to adjust another important property: the magnetic biasing field. This outer magnetic field, which is perpendicular to the axis of the ferrite disk, has to be large enough to rotate the standing wave pattern in the gyromagnetic resonator by 30 degrees. This condition can be satisfied by different magnetic fields, either below resonance or above resonance operation; however, the most important condition that should be met, is that the ferrite should be magnetically saturated, since the formulas obtained in the analysis of the permeability tensor are only valid when magnetic saturation was considered.

Choosing the proper magnetic field can be facilitated by the use of the hysteresis loop of the applied ferrite material that gives advice for the smallest field needed for saturation.

4. NUMERICAL ANALYSIS AND PROJECT DESIGN

The development of this project consisted of two stages. On the first stage, we design and model the dielectric elements that will be used in the Photonic Array, for this we use the Photonic Band Gap analysis and with this information determine the size (radius), spacing and configuration of the array. Also as part of this first stage, we will design the ferrite circulator and thus choose the radius that will match the frequencies chosen for our project.

On a second stage, we will test our design on an Electromagnetic Analysis Software. For this, we chose one of the most reliable software’s in the market, which is the CST Microwave Studio 2013®. The results from this stage will assure us that the theoretical design was well projected, and thus allow us to test these results on an experimental basis, which will be the objective of a future paper.

4.1 DESIGN AND MODELING OF THE PHOTONIC GRID

For the design of two-dimensional photonic crystal array, first, we use a mathematical tool to calculate the band diagrams. Through the band diagrams it is possible to determine if the structure we are proposing presents a photonic "gap" (region of forbidden frequencies propagating in all directions of the plane), and to establish the amount and ranges of possible forbidden frequencies.

For the current work, the mathematical tool we will use is the MPB(JOHNSON, [s.d.]) or MIT Photonic-Bands package, which is a free program for computing the band structures (dispersion relations) and electromagnetic modes of periodic dielectric structures. With this tool, we are able to analyze the electromagnetic band gap of different materials (in our case dielectric rods) as well as the different configurations of them.

The geometry used to generate the 2D photonic crystal array consists of dielectric rods

(휀푟 = 10) surrounded by air (휀푟 = 1), forming an hexagonal arrangement (triangular lattice), we

choose this configuration since for our final design (three-port PBG circulator) it has the adequate geometrical setup so we can create line defects (for the waveguides) with 120 degrees of separation in between them. These dielectric rods were developed by the research group of Prof. Dr. Sombra (ALMEIDA et al, 2008), which he provided as collaboration for the experimental tests based on this design.

For the MBP tool analysis, we are required to input the lattice configuration (triangular lattice in this case), the dielectric rods relative permittivity to the surrounding material (휀푟 = 10), and finally the fill factor ratio 푟⁄푎. Since the dielectric rods we are using for this project already have a fixed radius 푟 = 9.75 푚푚, then the remaining parameter 푎 (the lattice constant) is the one we will use in order to determine the best setup for the design. We then proceeded to make several analysis for different values of 푎 (as shown in Table 1), which shows us that a value of 푎 = 45푚푚 or 푟⁄푎 = 0.2167, will be the one which will keep the grid as small as possible (dimension wise) as well as provide the largest band gap necessary for the guided mode.

Radius “r” Periodicity r/a Start End Band (mm) “a” (mm) ratio of Gap of Gap Gap 9.75 20 0.4875 None None None 9.75 25 0.3900 0.2232 0.2592 14.93% 9.75 30 0.3250 0.2373 0.3119 27.17% 9.75 35 0.2786 0.2530 0.3604 35.02% 9.75 40 0.2438 0.2698 0.4044 39.94% 9.75 45 0.2167 0.2870 0.4439 42.94% 9.75 50 0.1950 0.3040 0.4772 42.34% 9.75 55 0.1773 0.3210 0.5039 41.23% 9.75 60 0.1625 0.3373 0.5233 39.98%

Table 1: Analysis of band gap size for different 푟⁄푎 ratios of grid arrays.

r/a = 0.4875 r/a = 0.3900 r/a = 0.3250 r/a = 0.2786

r/a = 0.2438 r/a = 0.2167 r/a = 0.1950 r/a = 0.1773

Figure 16: Unit-cell array configurations for different 푟⁄푎 ratios, as viewed by the MPB tool

In Figure 16, we show the different unit-cell configurations for the different 푟⁄푎 ratios as shown in Table 1. Figure 17 shows the unit cell and the corresponding Brillouin zone for the hexagonal array.

Figure 17: (a) Unit cell (highlighted by the parallelogram) to a hexagonal arrangement and (b) respective Brillouin zone. The irreducible zone is a triangular wedge of vertices Γ, K and M.

As can be seen in Figure 18, a photonic band gap exists between the first and second band in all directions of the photonic crystal, which corresponds to the normalized frequency band between 0.28697 < 휔푎⁄2휋푐 < 0.44391. In this range, there are no propagation modes.

Figure 18: Band Gap diagram calculated for a hexagonal array of dielectric columns surrounded by air, with geometric ratio 푟⁄푎 = 0.2167, for the TM polarization parameters.

4.2 PBG WAVEGUIDE DESIGN

The main property of photonic crystals is the existence of forbidden frequency bands that impede the propagation of electromagnetic waves in those bands. However, for these crystals to be used on the construction of devices (both active and passive), it is necessary the insertion of defects. Within these defects, the electromagnetic wave, with a wavelength in the range of the PBG, is confined, so it cannot propagate into other regions of the photonic crystal. This way, small devices with very low losses can be fabricated, as well as waveguides (with very narrow bandwidths and curved sections).

In our proposed design, waveguides in photonic crystals are generated by a linear defect obtained by removing rows or columns of the dielectric material along the direction of propagation, as shown in Figure 9. The dispersion relation and the electric field distribution are also calculated, considering the hexagonal symmetry for TM polarization. The TM polarization corresponds to modes whose electric field vector is parallel to the z-axis.

The analysis of the waveguides and the guided modes are also conducted by using the MPB tool. For this part of the analysis, we run the MPB for different super-cells configurations as shown in Figure 19. From there we obtain the results shown in Table 1 and thus obtain the band gaps in normalized frequency.

Figure 19: From left to right: Super-cell configurations for five, four, three, two and one dielectric elements (on each side) respectively. Each super-cell has a line-defect along the middle to allow for the guided mode propagation.

# of elements Start End Guided (each side) of Gap of Gap Mode Gap 11 0.28452 0.34313 0.10078 9 0.28359 0.34307 0.10084 7 0.28259 0.34269 0.10122 5 0.28105 0.34022 0.10369 4 0.27639 0.34998 0.09393 3 0.27496 0.32230 0.12161 2 0.26679 0.39143 0.05245 1 0.50000 0.76376 No Gap

Table 2: Analysis for the Guided Mode Gap as a function of the number of dielectric elements on each side of the line-defect.

From the results in Table 2, we observe that the introduction of the line-defect has reduced the band gap (End of Gap) from the previous value of 0.4439 to about 0.3223, this for the super cell configuration of three elements, which means that a guided mode has appeared in the gap difference between the two band-gap ranges (the one without and the one with the line defect). In this case, we also observe that super-cell configurations of three or more elements have the guided mode well confined, then we choose to implement the waveguide using the lowest possible super- cell size so that we require fewer elements when we design the PBG circulator.

In Figure 20, we plot the guided mode corresponding to the super-cell of three elements in a normalized frequency scale. As we can observe, this mode is guided inside the PBG waveguide between 0.3222 < 휔푎⁄2휋푐 < 0.4439. This way, if we replace the value of our lattice period of 푎 = 45푚푚, the photonic waveguide we designed will have a guided mode that is confined in the frequency range of 2.33 GHz < f < 2.96 GHz. Frequencies above this range will be spreading waves inside the crystal lattice, while frequencies below this range will be completely reflected.

Figure 20: TM guided mode for the super-cell configuration of three elements per side.

4.3 DESIGN OF THE FERRITE CYLINDER AND CIRCULATOR

From the results of section 4.2, we know that the proposed waveguide will have a guided mode between 2.33 GHz < f < 2.96 GHz. If we consider the central frequency of this range as the value for which we want the best transmission/isolation in our three-port circulator, then we choose our frequency for the rest of the calculations as 푓푐 = 2.6 퐺퐻푧.

Below are the material properties for the ferrite component, which was also given to us by Prof. Dr. Sombra as collaboration for this thesis work:

Relative permittivity = 14.45 Saturation magnetization (Ms) = 1786.5 Gauss Dielectric loss (tan ) = 2.73 x 10-5 Resonance linewidth (H) = 60 Gauss Height of the Disk = 12mm

Applying the calculated frequency 푓푐 and the above data for the ferrite material to equations (3.4) and (3.5), we obtain that the radius of the ferrite disk needs to be 푅 = 8,9 푚푚. This value will allow for the best transmission/isolation at ports 2/3, when we input a signal around the 푓푐 = 2.6 퐺퐻푧 value.

5. RESULTS

5.1 COMPUTER SIMULATIONS

5.1.1 Straight Photonic Waveguide

For the electromagnetic simulations, we use the aid of the computer software CST Microwave Studio 2013® (COMPUTER SIMULATION TECHNOLOGY, 2013). We start our analysis by first modeling a straight photonic waveguide (no bends or corners), which was designed using the line-defect technique. This means the removal of one row of dielectric rods in a photonic grid based on triangular lattice. In Figure 21, we show such waveguide with two coaxial ports, which we are going to excite with frequencies between 2GHz to 3GHz.

Figure 21: Straight Photonic Waveguide with two coaxial ports

The first result displayed in Figure 22 and Figure 23, shows the S-parameters for a signal injected at Port 1 with a frequency range from 1.5GHz to 3.5GHz. In this case, the S21 parameter (transmission coefficient) indicates very good transmission from Port 1 to Port 2 (meaning less than -1dB in transmission losses) in the frequency range from 2.35GHz to 2.90GHz. This result agrees very well with the numerical result we obtained in Section 4.2 where we predicted that the guided mode in this photonic waveguide would be between 2.33 GHz < f < 2.96 GHz.

Figure 22: S-parameters for the straight photonic waveguide. Input signal sweeps from 1.5GHz to 3.5GHz.

Figure 23: Zoom in on the S21 parameter (transmission coefficient). In this case it shows good transmission (under -1dB in losses) from 2.35 GHz < f < 2.90 GHz.

5.1.2 Photonic Waveguide Y-Splitter

Next, in our computer-aided design using CST Microwave Studio 2013®, we model a three- port photonic waveguide Y-Splitter, with waveguides at 120º from each other, as shown in Figure 24.

In this case, all three ports can be inputs for the test signal, which will also make the other two ports as outputs. This way each output port should receive 50% percent of the input signal (theoretically). However, since it is our objective to compare the obtained results with experimental ones to be done in later work, we are simulating this model considering possible losses in the materials (metal and dielectric components). In addition, we are modeling the standard SMA coaxial inputs in order to feed the test signal, which means there will be reflection losses for possible impedance mismatches that we will also try to minimize by designing impedance match connectors to be used in the experimental work too.

Figure 24: Three-Port Photonic Waveguide Y-Splitter

The results displayed in Figure 25, show the S-parameters for a signal injected at Port 1, with frequencies from 2.2GHz to 3GHz. In this case, the S21 and S31 parameters (transmission coefficients) are equal and overlapped, and show a -3.78dB loss in each output port. The ideal Y- splitter (with no losses) will output -3dB on each output port (50% of the input signal power), which means that in this case, the remaining -0.78dB are losses due to energy absorption in the materials, scattering and reflections due to impedance mismatch. From it, we can infer that our Photonic Y-Splitter is working as designed since it is splitting the energy of the input signal by half and delivering it to the output ports with minimal losses (-0.78 dB loss at each port).

5.1.3 3-Port Photonic Junction Circulator

Next in Figure 26, we present the schematic for the 3-Port Photonic Junction Circulator. As seen in the previous section, this circulator is modeled by adding a Ferrite Circulator in the junction that exists in the waveguide Y-splitter. This way, and after adjusting the right magnetic biasing field, we can control the transmission/isolation from an input signal in any port.

Figure 26: 3-Port Photonic Junction Circulator

From the Junction Circulator theory presented in Section 3.3.2, we know that in order to achieve circulation on the ferrite component we must operate with a magnetically saturated ferrite. From the ferrite material properties shown in section 4.3, we know that the minimum value for Saturation magnetization (4Ms) is 1786.5 Gauss. In Figure 27, we verify that with a value of 1500 Gauss of biasing field we still not see a transmission/isolation behavior, meaning we need more biasing field to achieve circulator operation.

Figure 27: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 1500 Gauss. Signal input is done only at Port-1 since all ports are symmetric

In Figure 28, we show the S-parameters for input at Port 1 and outputs at Port 2 (transmission) and Port 3 (isolation). In here, the value of the biasing field is of 1700 Gauss, which is just below the ferrite magnetization saturation. In addition, we start seeing the circulation behavior we expected, although not at the frequency we designed the circulator. This means we must increase the biasing field in order to adjust the dipole rotation for our desired frequency.

Figure 28: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 1700 Gauss. Signal input is done only at Port-1 since all ports are symmetric

In Figure 29, we adjust the biasing field of the ferrite for 2300 Gauss. The results below show that our operation frequency has shifted to the right and it is now at 2.63GHz. Since we want the operation to be as close as possible to 2.60GHz then we adjust the biasing field one more time and in Figure 30, we present our final results (for all 3 ports) in which the biasing field was at adjusted for 2125 Gauss and the circulator operating frequency stands at 2.60GHz.

Figure 29: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 2300 Gauss. Signal input is done only at Port-1 since all ports are symmetric

Figure 30: S-Parameters for the 3-port Junction Circulator with a ferrite-biasing field of 2125 Gauss. Signal input is done at all three ports, sequentially

In Figure 31 and Figure 32, we present the results for the circulator operation with input signal at Port 1. We verify that we have a very good transmission coefficient S21 (Port 1  Port 2) with only -0.212dB of losses, which is a very good value compared to commercial circulators available for the same frequency range. Signal isolation at Port 3 is below -20dB, same with the reflection losses on Port 1, which means we have good impedance match.

Figure 31: S-parameters for the three-port photonic junction circulator with biasing field 2125 Gauss and input signal at Port 1.

Figure 32: Representation of the E-fields for the circulator operation with input at Port 1. We observe very good transmission from Port 1 to Port 2 and good isolation at Port 3.

In Figure 33 and Figure 34, we have similar results for circulator operation with input signal at Port 2. We verify that we have a very good transmission coefficient S32 (Port 2  Port 3) as well, with only -0.211dB of losses. Signal isolation at Port 1 is below -20dB, same with the reflection losses on Port 2, which means we have good impedance match on this port as well.

Figure 33: S-parameters for the three-port photonic junction circulator with biasing field 2125 Gauss and input signal at Port 2.

Figure 34: Representation of the E-fields for the circulator operation with input at Port 2. We observe very good transmission from Port 2 to Port 3 and good isolation at Port 1.

Again, in Figure 35 and Figure 36, we have similar results for circulator operation with input signal at Port 3. We verify that we have a very good transmission coefficient S13 (port 3  port 1) with only -0.214dB of losses. Isolation is yet again below -20dB on port 2, and the reflection losses on Port 3 are about the same, which means we have again good impedance match.

Figure 35: S-parameters for the three-port photonic junction circulator with biasing field 2125 Gauss and input signal at Port 3.

Figure 36: Representation of the E-fields for the circulator operation with input at Port 3. We observe very good transmission from Port 3 to Port 1 and good isolation at Port 2.

Finally, we performed measurements of the bandwidth of our designed circulator. For comparison with these measurements, we obtained some parameters from data sheets available from a top of the line circulator (military specifications) (RENAISSANCE ELECTRONICS, 2006), which also operates in our microwave band (S-Band). From these measurements, we obtain the following Table 3.

Parameters Commercial Circulator Photonic Circulator Bandwidth Bandwidth Insertion Loss (-0.25dB Max) 200 MHz 103 MHz Isolation (-20dB Min) 200 MHz 48 MHz Reflection “VSWR” (-23 dB Min) 200 MHz 43 MHz

Table 3: Comparison between a top of the line circulator (military grade) and our proposed design for a photonic circulator

6. FINAL CONLUSIONS AND CONSIDERATIONS

The work presented on this thesis aimed for a pedagogical, but yet thorough, approach at understanding photonic crystals, photonic devices and magnetic effects. This pedagogical approach at the theoretical concepts, computer aided modeling, electromagnetic simulations and subsequent experimental implementation; allows for an easier introduction on a complex subject both for students and researchers alike. In that sense, the microwave approach into photonics, allows for the fabrication of a conceptually equivalent but hand-manipulable prototype, which would be very difficult to fabricate and manipulate if we were to use optic frequencies, for that approach usually demands sophisticated photonic fabrication resources and techniques.

The grid distribution or photonic lattice for the dielectric elements was chosen as a triangular one, since that grid arrangement was the one that provided the best band gap and was easily configurable when designing the photonic waveguides (separated by 120º) for the design of the three-port splitter and later three-port circulator. The size and periodicity of the grid elements were chosen by calculating the PBG diagrams with the aid of the MPB® mathematical tool created by MIT, which we used for computing the band structures (dispersion relations) and electromagnetic modes of our periodic dielectric structure. Finally, we proceeded to create a photonic crystal waveguide by inserting a linear defect in the crystal structure.

The ferrite dimensions were designed based on the frequency of operation for our structure, as well as the material properties for the ferrite component. The electromagnetic relations for that design were already summarized in Chapter 3.

On a next step, we proceeded to model the entire structure using CST Microwave Studio 2013®. In the modeling process we based the design on the dimensions and grid periodicity obtained from the numerical analysis, these dimensions were final and were not changed or fine- tuned later to provide better results, on the contrary, the electromagnetic simulation was performed, keeping in mind the need to fabricate a prototype and later replicate these results experimentally for subsequent work.

After the computer assisted modeling was completed, we proceeded to perform electromagnetic simulations to verify if our numerical analysis and design were correct. The simulation results shown in Chapter 5 conclude that the numerical analysis we performed was well thought of, and in every case was consistent with the theory we used to design and project the PBG Circulator.

Finally, we compared the performance of our designed PBG circulator, obtained from the analysis of the electromagnetic simulation results (reflection, insertion loss and isolation), with the performance of a commercially available circulator (military specs). From this comparison, we learned that our design still needs a few improvements in order to achieve the same performance as this top of the line microwave device, but still demonstrates the feasibility of constructing efficient microwave devices using photonic band gap techniques.

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