Optomechanics in photonic crystal cavities : Optomecânica em cavidades de cristal fotônico

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Instituto de F´ısica Gleb Wataghin

Rodrigo da Silva Benevides

Optomechanics in photonic crystal cavities

Optomecˆ

anica em cavidades de cristal fotˆ

onico

Campinas 2016

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Optomechanics in photonic crystal

cavities

Optomecˆ

anica em cavidades de cristal fotˆ

onico

Dissertation presented to the Physics Institute Gleb Wataghin of the University of Campinas in par-tial fulfillment of the requirements for the degree of Master in Physics.

Disserta¸c˜ao apresentada ao Instituto de F´ısica Gleb Wataghin da Universidade Estadual de Campinas como parte dos requisitos para obter o t´ıtulo de Mestre em F´ısica.

Advisor/Orientador: Prof. Dr. Thiago Pedro Mayer Alegre

ESTE EXEMPLAR CORRESPONDE À VERS ÃO FINAL DA DISSERTAÇ ÃO DEFENDIDA PELO ALUNO RODRIGO DA SILVA BENEVIDES E ORIENTADA PELO PROF. DR. THIAGO PEDRO MAYER ALEGRE

Campinas, SP 2016

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Ficha catalográfica

Universidade Estadual de Campinas Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174

Benevides, Rodrigo da Silva,

B435o BenOptomechanics in photonic crystal cavities / Rodrigo da Silva Benevides. – Campinas, SP : [s.n.], 2016.

BenOrientador: Thiago Pedro Mayer Alegre.

BenDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.

Ben1. Optomecânica de cavidade. 2. Cristais fotônicos. 3. Nanofotónica. I. Alegre, Thiago Pedro Mayer,1981-. II. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Optomecânica em cavidades de cristal fotônico Palavras-chave em inglês:

Cavity optomechanics Photonic crystals Nanophotonics

Área de concentração: Física Titulação: Mestre em Física Banca examinadora:

Thiago Pedro Mayer Alegre [Orientador] Leonardo de Souza Menezes

Paulo Clóvis Dainese Júnior

Data de defesa: 08-07-2016

Programa de Pós-Graduação: Física

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MEMBROS DA COMISSÃO JULGADORA DA DISSERTAÇÃO DE MESTRADO DE

RODRIGO DA SILVA BENEVIDES – RA: 151836 APRESENTADA E APROVADA

AO INSTITUTO DE FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 08/07/2016.

COMISSÃO JULGADORA:

- Prof. Dr. Thiago Pedro Mayer Alegre – (Orientador) – DFA/IFGW/UNICAMP - Prof. Dr. Leonardo de Souza Menezes – Departamento de Física/UFPE - Prof. Dr. Paulo Clóvis Dainese Júnior – DEQ/IFGW/UNICAMP

A Ata de Defesa, assinada pelos membros da Comissão Examinadora, consta no processo de vida acadêmica do aluno.

CAMPINAS 2016

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`

A minha mãe e ao meu pai, por seu incondicional apoio, apesar de não terem bem claro do que se tratavam meus estudos. Aos meus irmãos, Ana Luiza e Wellington, que sempre me inspiraram a querer fazer o meu melhor. A minha avó Julieta que, mesmo já tendo sa´ıdo desta vida, me traz sua tranquilidade e paz. A todos meus outros familiares, que sabem da enorme importância que tiveram e têm na minha vida.

Ao meu orientador, Professor Thiago Alegre, por toda a paciência e dedica¸cão despendida comigo nos últimos 2 anos e meio. Hoje vejo que sua minúcia e perfeccionismo foram cru-ciais para a qualidade deste trabalho e para o meu desenvolvimento. Ao Professor Gustavo Wiederhecker, por sempre estar dispon´ıvel para esclarecer minhas dúvidas. Aos meus amigos Felipe e Gustavo Luiz, por sempre me ajudarem muito com os trabalhos do laboratório, sem os quais esta disserta¸cão não teria ido tão longe.

A todas as amigas e amigos de lab - Guilherme, Débora, Yovanni, Lais, Jorge, Marvyn e Pedro - e demais colegas do IFGW, por tornarem minha rotina acadêmica muito mais divertida. Aos meus grandes amigos de gradua¸cão - Anne, Afonso, Fer, Gi, Guazelli, Neto e Paiva. Pelas nossas muitas noites em claro decifrando a f´ısica juntos.

Aos grandes amigos com quem eu dividi não só casa, mas também anseios e felicidades durante esse tempo. Briane, Carol, Henrique, Liana, Luciana, Susan, Ta´ıs e Tiago. Mais do que amigos, minha fam´ılia. A diversos outros amigos que me acompanharam ao longo dos anos: Bruna, Denise, Vivi, Edy, Daniel, Augusto, Vixuz, Jessyka, Janine, Rob, Paulo, Carol Azevedo, Jane, Will, Dani Souza, Renan, Ari, Ana Carla, Gabi, Rodrigo Cruz e todos as outras pessoas cuja a memória não me permitiu lembrar agora, mas que ao lerem esta dedicatória saberão que têm um lugar aqui. À minha companheira canina, Pitty, por estar comigo há 12 anos, sempre pronta para me acalmar nas minhas crises de ansiedade.

Agrade¸co `as agˆencias de fomento FAPESP, CAPES, CNPq e OSA por financiarem meu projeto e permitirem seu desenvolvimento.

Devo agradecer também a todas as pessoas que lutaram por melhorias sociais, tornando minha trajetória mais fácil. Em particular, agrade¸co ao movimento LGBT - por me dar apoio em diversos momentos de dificuldade e me mostrar que é poss´ıvel ir além do que a sociedade tenta oferecer a nossa classe - e aos movimentos pró-cotas - por me ajudarem a acessar o ensino superior público.

Por fim, agrade¸co fortemente àquele que já me acompanha há quatro anos. Tantas histórias, tantas vivências e quantas outras ainda por vir. Por isso tudo, obrigado, Serginho.

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A área de optomecânica de cavidades passou por um grande desenvolvimento na última década. O crescente interesse nesta área foi impulsionado principalmente pela interessante conexão entre movimentos mecânicos e campos ópticos. Tal acoplamento é amplamente explorado em diversos experimentos, com escalas variando de interferômetros quilométricos a cavidades ´

opticas microestruturadas. O principal desafio em todos estes experimentos é criar um dispo-sitivo optomecânico com um longo tempo de vida óptico e mecânico, ao mesmo tempo em que mantém um grande acoplamento. Neste contexto, as cavidades de cristal fotônico surgiram como fortes candidatas já que elas são capazes de confinar campo óptico em um volume modal muito reduzido e por um longo tempo de vida. No regime clássico, estes pequenos dispositi-vos, que podem oscilar mecanicamente com frequências de alguns poucos MHz até dezenas de GHz, permitem detectar for¸cas, massas e deslocamentos muito pequenos. Elas também são usadas para produzir osciladores mecânicos de alta qualidade, que podem ser sincronizados por intermédio do campo óptico. No regime quântico, a optomecânica quântica de cavidades tem sido usada para ajudar na compreensão do fenômeno de decoerência em uma escala me-soscópica, criando estados não-clássicos fortemente acoplados entre campo óptico e movimento mecânico, intermediado pela intera¸cão optomecânica. Entretanto, até agora, foram realizados poucos estudos sobre a possibilidade de produ¸cão destes dispositivos em larga escala, um passo necessário para massivas aplica¸cões tecnológicas e cient´ıficas destes dispositivos.

Neste trabalho, descrevemos um estudo detalhado de cavidades optomecânicas baseadas em cristais fotônicos produzidos numa fábrica de dispositivos compat´ıveis com indústria CMOS. Nós demonstramos a viabilidade desta plataforma explorando três geometrias distintas de cristais fotônicos. Primeiramente, nós mostramos como atingir fatores de qualidade muito elevados usando uma geometria consistente com as limita¸cões de fabrica¸cão. Nossos fatores de qualidade são os maiores já reportados usando cavidades de cristal fotônico fabricadas com litografia óptica. Em seguida, investigamos uma cavidade do tipo fenda, possibilitando a produ¸cão de alto acoplamento optomecânico usando um movimento mecânico planar. Por fim, desenhamos um escudo acústico, com dimensões variadas, para restringir o modo mecânico para dentro da região óptica. Essa estratégia foi usada de forma bem sucedida para produzir altos fatores de qualidade mecânicos e acoplamentos optomecânicos, permitindo a observa¸cão de resfriamento e amplifica¸cão de modos mecânicos à baixa temperatura.

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The field of cavity optomechanics has experienced a rapid growth in last decade. The increasing interest in this area was mostly driven by the intricate interface between mechanical motion and the optical field. Such coupling is widely explored in a variety of experiments scaling from kilometer long interferometers to micrometer optical cavities. The challenge on all these experiments is to create an optomechanical device with long-living optical and mechanical resonances while keeping a large coupling rate. In this context photonic crystal cavities have emerged as a strong candidate since they are able to produce very small optical mode volume and long optical lifetime. In the classical regime, these tiny devices, which can mechanically oscillate from frequencies ranging from couple MHz up to tens of GHz, allows for highly sensitive small forces, masses, displacements and acceleration detectors. They are also used to produce high quality optically driven mechanical oscillators which can be synchronized via an optical field. In the quantum regime, cavity quantum optomechanics is being used to understand decoherence phenomena in a mesoscopic scale by creating nonclassical states between light and mechanical modes intermediated by optomechanical interaction. However up to now, few studies have been done concerning the possibility of large scale production of these devices, a necessary step towards massive technological and scientific application of these devices.

In this work, we describe a detailed study of optomechanical cavities based upon photonic crystal cavities fabricated in a CMOS-compatible commercial foundry. We prove the feasibil-ity of this platform exploring three photonic crystal designs. First, we show how to achieve ultra-high optical quality factors using a design resilient to the fabrication constrains. Our demonstrated quality factors are the largest ever reported using photonic crystal cavities man-ufactured by optical lithography. Secondly, we investigate a slot type optical cavity, able to produce very large optomechanical coupling using a simple in-plane motion. Finally, we design a trimmable acoustic shield to restrict the mechanical motion inside the optical region. Such strategy was successfully used to produce high mechanical quality factor and optomechanical coupling which enabled the observation of cooling and amplification of mechanical modes at low temperature.

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1 Theory of photonic crystals 13

1.1 Calculation of bands in a 1D photonic crystal . . . 13

1.2 General calculation of bands - full plane-wave expansion method . . . 17

1.3 Electromagnetism as an eigenvalue equation . . . 18

1.3.1 Variational principle in electromagnetic theory . . . 19

1.3.2 Perturbation theory in electromagnetism . . . 19

1.3.3 Scalability of Maxwell’s equations . . . 20

1.4 Symmetries and the eigenvalue problem . . . 21

1.4.1 Continuous translational symmetry . . . 22

1.4.2 Discrete translational symmetry . . . 23

1.4.3 Rotational symmetry . . . 24

1.4.4 Time-reversal symmetry . . . 24

1.5 Simulation of photonic band diagrams . . . 25

1.5.1 Finite element method . . . 25

1.5.2 Bragg-reflector . . . 27

1.5.3 Bi-dimensional crystal - square lattice . . . 28

1.5.4 Bidimensional crystal - hexagonal lattice . . . 29

2 Photonic crystal slabs 31 2.1 Boundary conditions . . . 31

2.2 Phase matching condition . . . 32

2.3 Dielectric slab waveguide . . . 34

2.3.1 TE solutions . . . 34

2.4 Photonic crystal slabs . . . 37

2.4.1 1D-Photonic crystal slab . . . 37

2.4.2 2D-Photonic crystal slab . . . 39

2.4.3 Defects in photonic crystal slabs: waveguides and optical cavities . . . 39

3 CMOS-compatible optical microresonators 41 3.1 Microcavities . . . 41

3.1.1 Fabry-P´erot resonator . . . 41

3.1.2 Input-output formalism . . . 43

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3.2.3 Line-defect waveguide photonic crystal cavity . . . 48

3.3 Optical measurements . . . 49

3.3.1 Experimental setup . . . 49

3.3.2 Ultra-high optical quality factor . . . 50

3.3.3 Scalability of the problem . . . 52

3.3.4 Statistics of the modes . . . 52

4 Optomechanics in photonic crystal cavities 55 4.1 Introduction . . . 55

4.2 Optomechanical coupling . . . 55

4.2.1 Laser sidebands . . . 58

4.2.2 Dynamical backaction . . . 58

4.2.3 Mechanical noise power spectral density . . . 61

4.3 Slotted-optomechanical cavity . . . 62

4.3.1 Slot cavity design . . . 62

4.3.2 Optical characterization . . . 63

4.3.3 Mechanical characterization . . . 65

5 Optomechanical crystals 67 5.1 Phononic crystals . . . 67

5.1.1 Acoustic shield . . . 69

5.2 Optomechanical crystal characterization . . . 70

5.2.1 Optical cavity . . . 70

5.2.2 Mechanical spectrum . . . 71

5.2.3 Mechanical decay . . . 73

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Introduction

Since the first lenses made by the Assyrians, as the Nimrud lens [1] from ∼ 750 BC, to modern photonic memories [2] and optical interconnects [3] almost 3,000 years of evolution in optical devices and science has been paved. Several important steps were given, as the fundamental treatise Book of Optics in the 11th century of the father of modern optics Alhazen, or Maxwell’s equations in the year 1873 [4], or even the laser advent back in 1960 [5], leading our understanding of optical properties of materials further away.

Today optics is present in the everyday life as never before. Optical fibers, lasers, optical coherence tomography, CCD cameras are just some of the achievements through optical science and industry, changing completely the way we interact with the world. Moreover, it accounts now to a huge industry, with a global market of around US$400 billions, projected to reach over US$700 billions by 2020. Therefore, a deep comprehension of different branches of optics corresponds to society expectations.

Thinking about this evolution of optical science, we must look at what point we are. Basic effects, as refraction, guidance, polarization etc., are mostly understood [6]. However, more complex details of optical interaction and how they can lead to completely new effects are the state-of-art in optical science. Quantum optical effects [7], confinement in sub-wavelength regions [8], non-linear effects [9], to cite a few, are some examples of the extraordinaries topics related to advances now in top of scientific interests.

In this context this work is inserted. We were profoundly interested on understanding optomechanical interactions, a recent realm of optics dealing with the possibility of coupling between optical and mechanical modes [10–12]. In order to do so, we have chosen a modern platform, based on photonic and phononic crystal slab cavities [13, 14]. Such platform results in a variety of different interactions that will be covered throughout this dissertation. But two question arise...

Why optomechanics?

Very important in this timeline of optical science was the first suggestion of radiation pressure of light, made by Kepler in the 17th century, in order to explain the tilt of comet tails through an orbit around the sun. However experimental realization that could correctly account for this effect appeared just in the beginning of 20th century, made by Nichols and Hull [15].

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speed of light. However, because of the large value of c, this force is very small. For a typical laser pointer with 5 mW, the generated force is ∼ 10−11 N. A typical lamp would have even smaller forces. Consequently, this effect could not be detected before laser advent. The first application of radiation pressure was done by pioneers like Ashkin only in 1970’s, who trapped and manipulated small neutral particles with laser beams [16].

The possibility of using this force to modify the mechanical behavior of macroscopic objects was theoretically predicted by Bragisnky in the context of interferometers [17], used then to construct huge experiments as the Laser Interferometer Gravitational-Wave Observatory (LIGO), where it was recently possible to detect the first experimental data of the existence of gravitational waves [18]. Through squeezed light below quantum limits they could smear significantly detection noise, achieving high precision regime - smaller than 1/10,000th the width of a proton [19].

Beyond setting detection limits and manipulating small particles, radiation pressure can be used to actively change the dynamics of structures, through optomechanical interaction. In the 90’s several theoretical studies explored the optomechanical interaction at the classical and quantum regimes. From these studies a myriad of effects where predicted spanning from squeezing of light [20], quantum non-demolition measurements [21], non-classical coupling between light fields and mechanical modes [22] to cite a few.

However, only in the 2000’s microfabrication techniques for optical devices evolved enough [23] to allow the fabrication of structures which could support both optical and mechanical modes with long lifetime. In last decade the optomechanics field observed a boom of experiments, achieving several of the regimes theoretically predicted [10].

There are various reasons that drive this rapidly growing interest in cavity optomechanics. It promises to provide tools to generate and manipulate quantum signals [24–26], through non-classical states of light and mechanical motion. Its importance has become so clear that the magazine Nature has designed cavity optomechanics as the newest ”Photon Milestone”1_.

Why photonic crystal?

In 1987 Yablonovitch [27] proposed the use of structures which have a periodically modu-lated dielectric constant, in order to manipulate the density of states of the light in an emitter system, avoiding spontaneous emission. The periodicity of these materials has strong similar-ities to electronic crystals, where a periodic structure leads to electronic band structures [28]. Therefore, the name chosen for structures with periodicity in dielectric constant, yielding non-trivial band structures was photonic crystals.

Similar to electronic crystals, the photonic band diagrams can have several interesting properties, for example the ability to change group velocity [29], direction of propagation [30], the manipulation of the density of states [27] and can even generate bandgaps [31] which avoid the existence of modes inside the crystal. This can produce optical cavities having higher

1_{”...cavity optomechanics might allow quantum behavior to be observed in a macroscopic system...”, in the} magazine words.

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photon-lifetime [32, 11], measured through their quality factor Qopt, and modal volume as low

as (λ/n)3 _{[33], where λ is the wavelength in free space and n is the refraction index.}

Most of nonlinear optical interaction in cavities depend on the rate Qopt and they are

there-fore increased, allowing the production of ultrafast lasers [34], high-sensitivity sensors [35], channel-drop filters [36] and others. More recently, photonic crystal cavities have shown ex-treme utility in cavity optomechanics, due to their flexibility and the possibility of strong interaction between confined electromagnetic radiation and mechanical modes. Several phe-nomena were detected using this kind of structure. In [12] Chan et al. have demonstrated ground state cooling, in which a coupled nanoscale optical and mechanical resonator is used to cool the mechanical motion down to its quantum ground state, through laser radiation pres-sure. Optomechanically induced transparency [37] was demonstrated by Safavi-Naeini et al. with coherent cancellation of the loss channels in dressed optical and mechanical modes. More recently Cohen et al. have shown phonon counting through coherent coupling between optical and mechanical modes [38], utilizing single-photon measurement techniques, just to cite a few.

Dissertation organization

In this dissertation, we will show how we can merge optomechanics and photonic crystals to explore explore individual properties and enhance their coupling. In order to do so, we have the following structure.

In chapter 1 we study the general theory of photonic crystals, going through their basic concepts up to the creation of photonic crystal microcavities. Simulation are used to illustrate some of their properties for complex structures.

Chapter 2 will explore a specific kind of photonic crystal used very often along this work: the photonic crystal slab. We will study the specificities of these structures, focusing in the out-of-plane guiding phenomena dominated by total internal reflection, which turns out to be extremely important in these slabs.

In chapter 3 we develop a general theory of optical microcavities and introduce our fabri-cated micro-structures. We will then discuss all the relevant parameters to fabricate an optical micro-cavity with large quality factor and report a photonic crystal optical cavity fabricated through standard optical lithography technique, with the largest optical quality factor ever reported to our knowledge.

Chapter 4 shows one of our devices used for studying the optomechanical interaction. The basic theory of optomechanics is presented with all the relevant calibration procedures using a slot-type optomechanical cavity.

Finally, chapter 5 shows a quasi-2D optomechanical crystal cavity based upon an optical and mechanical crystals. This structure supports mechanical frequencies as high as ∼ 3.4 GHz which allows for the optomechanical cavity to enter the resolved sideband regime where efficient cooling and excitation of the mechanical modes through the optical field is allowed.

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Chapter 1 Theory of photonic crystals

In 1978 Hill et al. [39] have demonstrated that a periodicity in dielectric constant could be used to improve reflectivity in a narrow band, generating the first optical filter with a photonic crystal in an optical fiber. Nine years after it, Yablonovitch has proposed [27] that a three-dimensional periodical structure could be used to completely avoid the propagation of electromagnetic waves in a given bandwidth, in analogy with electronic crystals, opening the field of photonic crystals.

A basic property of photonic crystals is their possibility of generating strongly non-trivial dispersion curves, called photonic bands. Working properly the periodicity in dielectric constant, it is possible to generate several new-phenomena, as superprims [40], slow-light [29] and photonic crystal fibers [41], just to cite a few.

In this chapter we will explore the concept of photonic bands and the formation of bandgaps in an optically periodic structure. Its similarities and differences with the canonical electronic band-structure will be highlighted. First we will explore the simplest of all periodic systems composed by infinity 1D periodic planes. Some symmetry properties of periodic structures will be explored to extend our knowledge to higher dimensions, and then we will generalize this treatment using finite element solutions.

1.1. Calculation of bands in a 1D photonic crystal

The most fundamental method used to obtain the photonic bands of a periodic system is based upon an expansion of the solution in plane waves [42]. We will show how we can obtain a relation between coefficients in this expansion, considering the structure periodicity.

A way to understand how this method works is using it to solve the most simple photonic crystal - a 1D Bragg mirror, shown in figure 1.1a. This theoretical lattice is made by infi-nite layers in the x- and y-directions of dielectrics A and B with distinct values of dielectric constants, A and B. Let the widths be wA and wB for A and B respectively. The lattice

parameter, which defines the periodicity parameter of the structure, will be a = wA+ wB.

Moreover, we consider these materials don’t change in directions x and y. We can limit our-selves to the wavevector directed in the z direction. Then both electric and magnetic fields will be perpendicular to z direction. Let us consider, without loss of generality, the electric

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k

_ε

_A

_ε

_B w_aw_b z y x a) b) dielectric band air band

Figure 1.1: a) One dimensional Bragg reflector. b) Bands obtained by numerical calculation using the plane-wave expansion method. The coefficient p runs −5 ≤ p ≤ 5 [42].

field polarized in the x direction and the magnetic field in y direction, i.e., E(z) = E(z)ˆx and H(z) = H(z)ˆy. Therefore it is possible to write the wave equation in a scalar form as

∂2_{E(z, t)} ∂z2 − 1 c2 ∂2 ∂t2[(z)E(z, t)] = 0 (1.1)

Since the dielectric constant is periodic we can write it as an infinite Fourier series as follows [43]: (z) = ∞ X p=−∞ pei 2πp a z (1.2)

where the coefficients of this expansion are p = 1 a Z a 0 (z)e−i2πpa zdz (1.3)

The expansion coefficients can be easily computed for this Bragg mirror, yielding the values

p = ( Aw_aA + Bw_aB for p = 0 i 2πp (A− B) e−i2πaz− 1 for p 6= 0 (1.4)

We can turn our attention back to the wave equation (1.1). We can consider our trial solution as a harmonic temporal part times a spatial distribution, i.e., E(z, t) = e−iωtE(z). Moreover, as the dielectric constant is periodic, we suppose the spatial solutions are composed of plane waves with periodically modulated amplitude (Bloch modes).

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E(z) = ∞ X p=−∞ cpei(k+p 2π a)z (1.5)

We then replace the Fourier solutions for dielectric constant (1.4) and the plane wave expansion (1.5) in the wave equation (1.1), leading to

cp k + 2πp a 2 − ∞ X p0_=−∞ ω c 2 p−p0c_p0 = 0 (1.6)

In other words, we obtain a system of linear coupled equations for the coefficients cp. It is

possible to find a solution only when ω satisfies the secular equation built from the coefficients of these equations. Furthermore, it is important to note the periodicity of the solutions in k, due to assumption of Bloch solutions. The wave number k and k + p(2π/a), for any integer p, yield the same solutions and one can just calculate the bands in a specific region, −π/a < k < π/a, a region known as first Brillouin zone.

Despite the periodically assumption these equations give the exact solutions only if one considers a set of infinite coupled equations. One can easily solve for an approximated problem as long as one truncates the series. This can be translated in the size of the matrix of the linear equation problem. In figure 1.1b it is shown the results obtained for this plane-wave expansion for a 11 × 11 matrix, which is a result of truncating p between −5 and 5. It was plotted the frequencies ωn(k) against k obtaining the dispersion curves or, as known in crystal theory, the

(photonic) band structure of the crystal. The subscript n is usually called a band index. Several important features of photonic band structures, shared also by the 2D and 3D structures, are observed in figure 1.1b. The first band, with the lowest frequency, has an approximately linear dispersion close to the center of the Brillouin zone, where k ≈ 0. This is expected since the wavelength in this region is very large, such that the plane wave solution effectively interacts with an homogeneous medium with uniform , with a value between A

and B.

On the other hand, as k approaches the Brillouin zone edges, the dispersion curves are gradually curved. In this point the state is a mixture of eikx and e−ikx, such that the wave does not propagate to any direction and we get a standing wave. Therefore, the group velocity of the photons goes to zero

vg ≡ _∂k∂ω1(k) = 0 at k = π_a (1.7)

This effect can be observed in some other regions, especially in points of high symmetry where there are anti-crossings between bands. In 2D and 3D structures this effect is more pronounced, since there are more possibilities to form standing waves, which create null group velocity even inside the Brillouin zone.

The two lower bands are named: dielectric and air bands. These names come from the fact that most of the electric field are confined in a higher dielectric constant region for the first band and in a lower dielectric constant (in general, air) for the second band. An easy

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way to understand this behavior it is considering the wave equation (1.1) and adding the term (ω/c)2_{E(z) on both sides}

− ∂ 2 ∂z2E(z) − ω2 c2 ((z) − 1)) E(z) = ω2 c2E(z) (1.8)

In this way, we can compare this equation with the Schr¨odinger equation for quantum wave functions [44]. We see that the term

− ω

2

c2 ((z) − 1) (1.9)

plays the role of an effective potential. Therefore as > 1, we have an attractive potential relative to free space. This is why a band with the electric field confined mostly in the higher dielectric constant region is found to have lower frequency. Looking for the last term in equation (1.8) we observe that the quantity ω2_/c2 _{plays the role of eigenvalue in this equation.}

But this is a positive quantity, therefore it is not possible to have completely bound states in the photonic case. A complete photon confinement can not be achieved in this way and a leakage of electromagnetic energy will always be present. However, there are several ways of improving the confinement of these photons, as it will be explained later.

Finally, we can analyze the density of states ρ(ω) in a photonic band. This quantity is defined as the number of states N (ω) within an infinitesimal frequency region [ω, ω + ∆ω]

ρ(ω) = N (ω) ∆ω (1.10) But ∆ω = ∂ ∂kωn(k)∆k (1.11) or ∆k = ∂ ∂kωn(k) −1 ∆ω (1.12)

Dividing ∆k by the spacing 2π/L of the quantization of 1D phase space1 _{yields N (ω).}

Therefore, we get the density of states as

ρ(ω) = 2L 2πv

−1

g (ω). (1.13)

where the prefactor 2 accounts for the two polarization of photons propagating in the z direc-tion. What we learn from relation (1.13) is that the density of states is related to the inverse of the group velocity. The steeper a photonic band, the smaller the density of states in this k

1_{In the 1D system of lattice constant a and length L, used for the quantization of k, the allowed values of} k are k = 2πp L = p N 2π a

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region. The flatter the band, the larger the number of states. This can strongly increase some effects. For instance, the emission probability of photons of frequency ω from an atom in a photonic crystal is proportional to ρ(ω) and then it is observed an increase in light emission from atoms at the peaks of the density of states [45].

On the other hand, there are regions with null density of states, due to complete lack of states. These regions are called the photonic bandgaps and are one of the most important features of photonic crystals. This has led Yablonovitch to propose the use of photonic crystals to modify spontaneous emission in lasers [27].

In short, most of optical applications of photonic crystals are based on the increase or suppression of the number of states due to the photonic band structure. We will now study periodic structures in 2D and 3D and how this change in density of states can be improved.

1.2. General calculation of bands - full plane-wave

ex-pansion method

In order to calculate the photonic band structure for a general case, we need to utilize the vectorial of Maxwell equation. Combining the curl equations of electric and magnetic fields [46] we can get the equation2

∇ × ∇ × E(r) −ω

2

c2(r)E(r) = 0 (1.14)

where it was assumed that the fields are harmonic with a well-defined frequency ω such that

E(r, t) = E(r, ω)e−iωt (1.15)

The spatial solution, as in the 1D case, is expressed as a superposition of plane waves. In this way, we can expand the spatial solutions of the electric field as

Ek(r) =

X

h

ek(h) exp[i(k + h) · r] (1.16)

with ek(h) the amplitude of the plane wave, which needs to be found such that the equation

(1.16) is a solution of (1.15). Note that (1.16) satisfies Bloch’s theorem, i.e., Ek(r + R) =

eik·r_E

k(r), with R a lattice vector.

Furthermore, we are interested in photonic crystals, so the medium is supposed to have a periodical structure. Then, the dielectric constant can once again be expanded as a Fourier series

(r) =X

h

heih·r (1.17)

2_{To get the equation in this form one needs to assume that we have a non-magnetic and isotropic medium,} in a linear regime. Moreover, we neglect any dependence of the dielectric constant on frequency.

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We follow the same steps as in section 1.1 and substitute equations (1.16) and (1.17) into (1.15). After a few simplifications, we obtain the following expression

X h " −(k + h) × (k + h) × ek(h) − ω2 c2 X h0 h−h0e_k(h0) # = 0 (1.18)

This is a secular equation for the coefficients ek(h) of the plane-wave expansion. It is not

an easy equation to solve and there exist several numerical methods to deal with it.

1.3. Electromagnetism as an eigenvalue equation

The main goal of this section is to show that most of the important features of photonic crystals can be obtained looking to the problem by another perspective. If we look to the equation (1.14) obtained in the section 1.2, we see that it has a very known format in physics: it is an eigenvalue equation [47]. We can write an equivalent result for the magnetic field, where we get ∇ × 1 (r)∇× H(r) = ω 2 c2H(r) ˆ ΘH(r) = ω 2 c2H(r) (1.19)

That is, we can rewrite our electromagnetic problem into an eigenvalue problem, with a hermitian operator 3 _{Θ. This kind of operators was already extensively studied, specially due}ˆ

to its role in quantum mechanics.

Some of the most important properties are listed below:

• The operator ˆΘ is positive semi-definite. This means that its eigenvalues are all real and nonnegative. Therefore, the eigenfrequencies ω are real quantities.

• The eigenmodes are orthogonal. The hemiticity of ˆΘ forces any two harmonic modes H1(r) and H2(r) with different frequencies ω1 and ω2 to have an inner product of zero.

• If the system has well-defined boundaries, there is a quantization in the number of eigenvalues.

3_{A Hermitian operator is defined based on a given inner product. Our field space admits an inner product} in the following way

hF|Gi ≡ Z

F∗(r) · G(r)d3r

The hermiticity is the property that allows an operator act on both sides of an inner product yielding the same result. That means hF| ˆΘ|Gi = h ˆΘF|Gi = hF| ˆΘGi. Hermitian operators have real eigenvalues, orthogonal eigenfunctions, and the corresponding eigenfunctions form a complete orthogonal system.

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1.3.1 Variational principle in electromagnetic theory

Here we see under another point of view the conclusions concerning the photon localization explained in section 1.1. This tendency can be precisely explained when we write a variational principle for electromagnetism, in a similar way to what is done in quantum mechanics [44]. In this way the smallest eigenvalue ω2

0/c2 corresponds to the field patterns that minimizes the

functional

Uf(H) ≡

hH|Θ|Hi

hH|Hi (1.20)

We can call H0 the eigenmode that minimizes (1.20). So the next lowest-ω eigenmode H1

will be the one that minimizes again Uf, but within the space of functions orthogonal to H0.

The next one will be the field that minimizes Uf in the space of functions orthogonal to both

H0 and H1 and so on.

However the expression (1.20) brings more information when written as a function of E. We can use the Faraday’s law of induction

∇ × E = −µ∂H

∂t (1.21)

and substitute H fields by E fields. So we get the relation

Uf(E) = h∇ × E|∇ × Ei hE|(r)Ei = R |∇ × E| 2_d3_r R (r)|E(r)|2_d3_r (1.22)

From this expression we can see that to minimize the functional we must have both a small numerator and a big denominator. This means that we need an electric field with small curl, in other words, a solution with very little spatial changes. Moreover the field must be mostly concentrated in regions with higher dielectric constant. This explains the tendency of localization of photons in high- regions and the fact that, in general, the most fundamental modes change little in space.

1.3.2 Perturbation theory in electromagnetism

A perfect lossless, uniform and symmetrical material is a good idealization. However, real problems will generally need additional features to take into account possible deviations of ideality. In particular, in this work we are interested in modifications in the size of dielectric materials and how they modify the electromagnetic field in the structure. Additionally the approach below can be used to study the effect of small changes in dielectric constant or even in other properties of the medium.

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Let us use the equation (1.14) for the electric field subject to slight changes λ∆α , where α is a given property of the medium and λ is just a measure of the order of expansion. Let ˆQ be the operator of this eigenvalue equation 4

ˆ

Q|Ei ≡ ∇ × ∇ × |E(r)i =ω c

2

(r)|E(r)i (1.23)

Then, we assume that (r) is real and positive and take the approach usually done in perturbation problems [44]. The new eigensolutions |Ei and ω are expanded into powers n of ∆α: |Ei = P∞

n=0λ

(n)_|E(n)_{i and ω =} P∞

m=0λ

(m)_ω(m)_{. The terms |E}(0)_{i and ω}(0) _{are the}

non-perturbed values and λ(n) is the order of the expansion.

In this way, a perturbation ∆ in the dielectric constant yields a change in the eigenvalue equation in the following way

X n " ˆ Q − ω (0) c 2 ∆ # |E(n)i =X n X m,l ω(m)ω(l) c2|E (n)_i (1.24) Then we can use the orthogonality condition hE(n)|E(m)_{i = 0 and the equation for the}

non-perturbed fields ˆ Q|E(0)i = ω (0) c 2 |E(0)i.

With it we get an expression for the change in frequency due to a change in dielectric constant up to first order

ω(1) = −ω

(0)

2

hE(0)_|∆|E(0)_i

hE(0)_||E(0)_i (1.25)

With this expression, it is possible to evaluate how a small perturbation in the dielectric constant induces changes in resonance frequencies. These changes may be due to modification in the boundaries of the problem [48], as in a cavity, or they may be related to internal changes in the dielectric constant, as in the photoelastic effect [49].

1.3.3 Scalability of Maxwell’s equations

An important property of Maxwell’s equations that will be used further in this work is their scaling property, which means there is no fundamental length scale - since the system can be taken as macroscopic. This leads to a straightforward relationship between problems that are different just by a contraction or expansion of all distances.

Suppose we have a medium, with a given dielectric configuration (r), leading to eigenmodes H(r) and eigenfrequencies ω. We have again the equation (1.19)

∇ × 1 (r)∇× H(r) = ω 2 c2H(r)

4_{Although ˆ}_{Q is not an Hermitian operator, the problem becomes a generalized eigenproblem and the} expansions taken are still valid.

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Now assume we have a different dielectric configuration 0(r) that is just a rescale of the original configuration, that means, (r) → 0(sr), for a given scale factor s. Hence, we will have a rescale also in the eigenmodes H(r) → H0(sr). Consequently, the operator ∇ will change too ∇ = ˆid dx + ˆj d dy + ˆk d dz → ˆi d d(sx)+ ˆj d d(sy) + ˆk d d(sz) = 1 s∇ Putting everything together, we get a rescaled complete equation as

∇ × 1 (r/s)∇× H(r/s) = s 2_ω2 c2 H(r/s) (1.26)

Therefore as eigenvalues are rescaled by s2, the frequencies are just rescaled by the same factor s, in a linear relation. The main importance for this result in photonic crystals theory is that it doesn’t matter the exact size of the structure when calculating a photonic band diagram, only the relative size between the periodically dielectric regions. It is possible to use normalized quantities, for a given geometry, and then just resize the result for the real problems. Once one gets a different size of the same geometry, the same band diagram works. A final question that can arise in this thought is the assumption of a macroscopic domain to make this derivation. Would it still hold for microcavities, with such a tiny size? In order to confirm it, we can calculate approximately how many atoms there are in, for instance, a 10 µm of radius microdisk of silicon. If the width of the silicon layer is 250 nm, we get a total volume of V = π × (10 × 10−4)2_{× 250 × 10}−7 _{≈ 7.8 × 10}−11 _cm3_{. The density of silicon is approximately}

ρ = 2.33 g·cm−3. Therefore, the cavity weighs 2.33 × 7.8 × 10−11g ≈ 1.8 × 10−10 g. The molar mass of silicon is 28 g.mol−1. This leads then to a total amount of 6 × 1023× 1.8 × 10−10_{/28 ≈}

4 × 1012 _{atoms. It is a number large enough to lead to no doubts about the macroscopic}

behavior of the system.

1.4. Symmetries and the eigenvalue problem

When a dielectric structure has symmetric properties, this can be used to considerably simplify the problem of finding the eigenmodes and eigenfrequencies of the system. This is the case of photonic crystals, which have always certain symmetry characteristics. Below we approach some of these properties and how they are related to photonic crystal band diagrams and eigensolutions.

The first step is to understand what is considered a symmetry in a certain field. A general approach is to imagine a symmetry as some operation that leaves the system unchanged. But to quantify it, we can use an operator approach, where a symmetry is represented by the operator ˆO.

Now, we can compare two situations. First we can apply the electromagnetic operator ˆΘ, as defined in (1.19) over a given system. Similarly we can apply the symmetry operator ˆO, then the electromagnetic operator ˆΘ, then the inverse of the symmetry operator ˆO−1 over the

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same system. If ˆO is really a symmetry operator, both described situations will be equivalent, i.e.,

ˆ

Θ = ˆO−1Θ ˆˆO (1.27)

But if we apply this into a vector A, we can rewrite the equation as ( ˆΘ Ô − Ô ˆΘ)A = 0. We can define a relationship between two operators Ô and ˆP called commutator as

h ˆ_{O, ˆ}_Pi

≡ ˆO ˆP − ˆP ˆO (1.28)

With this definition, we see that a symmetry is found always when the commutator between an operator and the operator of the electromagnetic fields is null. But from linear algebra we know that when two operator commute, they have the same set of eigenvalues. This means that when we have a symmetry operator, we can easily obtain properties of the eigenvalues of the electromagnetic field operator by simply solving the eigenproblem for the symmetry operator, what is, in general, much easier.

1.4.1 Continuous translational symmetry

Suppose we have a system in which a translational operation ˆTd that changes the

co-ordinates by a given d leaves the system invariant. That means h ˆTd, ˆΘ

i = 0, because

ε

₂

ε

₁ 1 2 3 4 1 2 3 4

-k

d) c) a) b) a

k

R

_90º

Figure 1.2: Symmetry examples. a) Bulk with full continuous translational symmetry. b) Bragg reflector with discrete translational symmetry by vectors R = la, l integer. c) Example of system with rotational symmetry for integer multiples of 90o. d) Time-reversal symmetry. Waves propagating

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ˆ

Td(r) = (r − d) = (r). A typical example of such a system is a continuous bulk of

ho-mogeneous material.

Now, if the symmetry is specifically in the z direction, we can find the eigenfunctions of the translational operator with a functional form eikz in the z direction

ˆ

Tdˆzeikz = eik(z−d) = e−ikd eikz (1.29)

with the eigenvalue e−ikd. But, as said before, once we know the eigenfunction of the symmetry operator, we know the eigenfunction of the electromagnetic field operator too. Therefore, by simple assumptions and calculation, it was possible to find a general form of the eigenfunction of such symmetric systems.

Since we already know the general solution of a homogeneous medium, which has a con-tinuous translational symmetry in all three directions, the mode has to be eigenvector of a translational operator in three directions

Hk(r) = H0eik·r (1.30)

where H0 is a constant vector. These are the plane waves, the solutions obtained by solving

the Maxwell’s equations in a homogeneous medium.

On the other hand, for a slab with continuous translational symmetry in both x and y direction or for the 1D bragg reflector of figure 1.1 we would have

Hk(r) = ei(kxx+kyy)h(z) (1.31)

but the function h(z) needs to be found by other means, because the system has no symmetry in z direction. Nevertheless we still have a general form for the solution, which makes the problem easier.

1.4.2 Discrete translational symmetry

Discrete translational symmetry is utterly important in photonic crystals. As electronic crystals, they don’t have continuous symmetry in all directions, but their structures have symmetric laws related to discrete quantities.

If we look to a basic system, as the Bragg reflector of section 1.1 we can see that it still has a continuous symmetry in x and y direction, differently from the z-direction where it presents a discrete translational symmetry. The basic step length a = aˆz is called the lattice constant, and then we have (r) = (r + R), where R = la for a given integer l. The eigenmodes are still plane waves

ˆ

Tdˆxeikx = eikx(x−d) = e−ikxd eikx

ˆ

Tdˆyeiky = eiky(y−d) = e−ikyd eiky

ˆ

TReikzz = eikz(z−la) = e−ikzla eikzz

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Here we see an important feature. Not all the wavevectors kz yield different eigenvalues.

All of the wavevectors of the form kz+ m(2π/a), with an integer m, give the same eigenvalue

and are therefore degenerate.

We can write then the eigensolutions as a superposition of the possible solutions. So

H(r) = ei(kxx+kyy)X

m

c(z)ei(kz+mπ/a)z = eik·r·X

m

c(z)e(imπ/a)z = eik·r· ukz(z)

(1.33)

The result are plane waves multiplied by a periodic function in z, ukz(z). This is exactly what is known as Bloch’s theorem. So the eigenvalues that differ by integral multiples of 2π/a are identical from the physical point of view. Therefore we need only to consider a range of wavevectors −π/a < ky ≤ π/a, the first Brillouin Zone. Band diagrams usually are just

composed by this region, as the remaining k-space can be folded back to the first Brillouin zone.

1.4.3 Rotational symmetry

Suppose we have a photonic crystal that has a rotational symmetry, meaning that a rotation leaves the system invariant. Hence for a given rotation operator ˆOR, we have the commutation

relationh ˆΘ, ˆOR

i

= 0. Using this definition, we can look to the eigenvalue equation and apply this operator on it as follows:

ˆ Θ ÔRHk = ÔR ˆΘHk = ωn(k) c 2 ˆ_O_R_H_k (1.34) From this relation we see that ÔRHk

satisfies the eigenequation with the same eigen-values as Hk. This means that the rotated mode is analogous to original mode with the same

eigenvalue equation. Thus when there is rotational symmetry we have redundancies in the Brillouin zone and we can get rid of some of the redundant regions. The space within the Brillouin zone that has all the ωn(k) not redundant, i.e., not related by symmetry operations

is called reduced Brillouin zone. When dealing with bi-dimensional photonic crystals we will always plot the band diagrams just inside this region.

1.4.4 Time-reversal symmetry

From our assumption, the temporal dependence of our solution is oscillatory in time, that means

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Therefore, if there are no losses in the system, a conjugate operation in this kind of solution is equivalent of having the solution at time −t. This leads to the last symmetry important to photonic crystal structures, the time-reversal. If we take the complex conjugate of equation (1.19) and consider that the eigenvalues are real, we get

ˆ_ΘH_k,n∗ = ω 2 n(k) c2 H ∗ k,n ˆ ΘHk,n = ω2_n(k) c2 H ∗ k,n (1.36)

From these relations we notice as for H∗_k,n satisfies the same equation as for Hk,n, with the

same eigenvalue. However, from equation (1.33) we see that H∗_k,n is just a Bloch state with −k. Therefore, we have

ωn(k) = ωn(−k) (1.37)

We can then simplify further our bands calculation in systems with time-reversal symmetry by reducing the calculation to 0 < k ≤ π/a.

1.5. Simulation of photonic band diagrams

In section 1.1 we have seen an analytical approach to obtain the photonic band structure for a 1D Bragg reflector. However, this simple crystal geometry is not always sufficient to originate band diagrams convenient for some application. When one goes to a crystal with 2 or 3 dimensions with symmetry properties, this approach leads to equation (1.18), where the size of the determinant of the secular equation is increased due to higher dimensionality, which conversely leads to a complicated determinant equation, where problems of convergence can be observed. Hexagonal lattice crystals [32], nanobeams [12], snowflakes [14] and yablonovita [27] are just some examples of of systems that suffer from this complexity. Therefore another approach is necessary to derive the band diagrams of such structures.

In this work, we have used finite element method simulation to generate our band diagrams. In this section idealized examples of geometries are shown, such as infinite 1D and 2D crystals. From them we can infer how is the behavior of real geometries, such as the 2D photonic crystal slabs used in the experimental part of this work.

1.5.1 Finite element method

Finite element method is a numerical method that is widely used to solve differential equations with boundary conditions. It is based on subdividing a given structure in smaller parts, such that it is possible to approximate the solution for each part in a way that the set of all solutions is a close representation of the complete problem.

In general lines, the finite element method is a way of solving a typical problem of finding a function u which is solution for a differential equation problem. We can understand its basic

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behavior studying, for instance, a function u defined in a given closed interval [0, 1] where the function is solution to a equation of the type

d2u

dx2 + f = 0 (1.38)

with f : [0, 1] → IR a given function. There are different ways to fully defining this problem. The first is the strong formulation, that can be stated as

Strong formulation. Given f : [0, 1] → IR and constants g and h, find a function u : [0, 1] → IR such that d2_u dx2 + f = 0 u(1) = g −du dx = h (1.39)

This kind of formulation is used, for instance, in the finite difference method. Our case follows a slightly different form, equivalent to the strong formulation, the weak formulation. It is based on the use of two set of equations. The first of the trial solutions must be a set S where 5

S = {u|u ∈ H1, u(1) = g} (1.40)

and another set of weight functions in the set V which are the homogeneous equivalent.

V = {w|w ∈ H1, w(1) = 0} (1.41)

So, the weak formulation is

Weak formulation. Given f , g and h as before, find u ∈ S such that for all given w ∈ V, we have Z 1 0 dw dx du dxdx = Z 1 0 wf dx + w(0)h (1.42)

The usual way to solve this problem is to use the Galerkin method. It works with a given discretization of the space of the problem. Assuming that a given discretization with parameter h, we can see that the sets of solutions for S and V are identical, up to a constant gh_{. Then, we can reformulate the weak form in a different way}

Z 1 0 dwh dx dvh dxdx = Z 1 0 whf dx + wh(0)h − Z 1 0 dwh dx dgh dx dx (1.43)

This formulation leads to a set of coupled equations, that can be resumed as the matricial equation

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Kd = F (1.44) where K is usually called the stiffness matrix, d the displacement vector and F the force vector. Despite the names, this quantities can be associated with several physical properties. This kind of equation is solved by a typical software of finite element method and based on it it was possible to get the diagrams shown in next sections.

1.5.2 Bragg-reflector

The first problem I have worked with was the Bragg reflector. The simulated problem can be seen in figure 1.3a. All the finite element method simulations of this work were done in a commercial software with multiphysics platform [50].

X

Γ

Fr

equenc

y

ω

a/c

2.5 -1.5 0.5 107_(V/m) 2 -2 0 107_(V/m) 1st mode 2nd mode E_y E_y air (n=1) silicon (n=3.4) 1600 nm

a)

b)

c)

Si Si

0

0.4

1.2

0.8 bandgap

Figure 1.3: a) Simulated Bragg reflector, with dielectric constants and size. b) Dominant component of the electric field for 1st and 2nd bands at X-point of Brillouin zone. c) Band diagram for TE polarization in the simulated Bragg reflector. The green dots represent where the mode profiles of b)

were calculated.

To solve it, we have used a 2D geometry with out-of-plane wavevector kz = 0. Moreover,

Bloch-Floquet boundaries were used, to obtain the desired Bloch solutions. In this way, we could simulate the equivalent to an infinite Bragg reflector with propagation just in the plane of the image.

In figure 1.3b we can see the first interesting result. The simulated modes respect the variational principle. The first mode concentrates most of its field in the region with higher dielectric constant (silicon). On the other hand, the second mode is orthogonal to the first mode, where we can see that there is little overlap between both. In figure 1.3c we see the band

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Γ X M Brillouin zone Frequency (ω a/c)

a)

b)

c)

Г X M Г silicon air a=450nm 2r=220nm x y z max min Norm of the electric field 0 0.4 0.8 1.2 1.6 2

Figure 1.4: a) Bi-dimensional crystal of square lattice. The size of the simulated crystal is shown. b) Brillouin zone corresponding to the square lattice crystal. In gray we see the reduced Brillouin zone. c) Band diagram for TE-polarization in a bi-dimensional square lattice crystal. The insets are

the norm of the electric field in some points of bands, shown in green.

structure obtained from this simulation to this geometry. We can see that some bandgaps arise, avoiding the propagation of electromagnetic radiation with some frequencies. The format of the bands obtained are very similar to the ones obtained in the analytical approach in figure 1.1.

1.5.3 Bi-dimensional crystal - square lattice

As said before, the properties of photonic crystals can be better manipulated when one uses bi- or three-dimensional crystals. This has led to a continuous rise of scientific researches with this kind of structures. However, the fabrication techniques for quasi-bidimensional crystals (in form of slabs) have successfully been developed only in the last twenty years.

In order to understand the behavior of bi-dimensional crystals, I have worked firstly on simulations of square lattices, as the one showed at figure 1.4b. In this kind of lattice, the Brillouin zone is also square, as we can see on 1.4b. Due to rotational symmetry we can split this region into four sub-zones with identical solutions. Moreover, by time-reversal symmetry, we can split this region in two, yielding the reduced Brillouin zone Γ − X − M , as shown in figure 1.4b. These points have wavevectors k||= 0, k|| = π/aˆx, k|| = π/aˆx + π/aˆy, respectively.

We have again simulated a two-dimensional problem, with kz = 0. Moreover we have

defined the simulation to obtain only modes with inplane components for the electric field -this is equivalent to a TE solution for the problem. Appropriate boundary conditions were chosen, in order to yield the Bloch solutions for the problem.

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Brillouin zone, for the geometry of figure 1.4a. In general the maxima and minima of the bands appear in the edge of this region.

Some information can be obtained from the figure 1.4c. Firstly we can see that there are some forbidden regions for certain wave propagation directions in the structure. However, for every frequency there is always a possible wavevector in the band diagram, i.e., we have not a full bandgap for this polarization in this structure. This implies that incident waves in every frequency can couple to a guided mode in the crystal. Furthermore, we can see the confirmation of the variational principle presented in section 1.3.1: the fundamental mode localizes its energy mostly in regions with higher refractive index, with less spatial oscillations than other modes. We can also see that the second order mode has little overlap with the first mode, in particular point X - where the highest interaction between bands occurs - respecting the orthogonality of non-degenerate modes.

1.5.4 Bidimensional crystal - hexagonal lattice

It was seen in section 1.5.3 that the band structure of a square photonic crystal generates some regions with null density of states, but there is not a full bandgap. This makes some applications more difficult, as confinement of photons - one of the main goals of this work. Therefore, it is necessary to find another geometry that could allow for systems with a full bandgap. 500 400 300 200 100 0 Fr equency (T H z) Photonic bandgap Γ K _M _Γ d a Γ K M a) b) c) Real Space Brillouin Zone

107_(V/m) 7 4 1 9 5 1 x 107_(V/m) 1st mode 2nd mode y

Figure 1.5: a) Real space, Brillouin zone and reduced Brillouin zone for a bi-dimensional crystal of hexagonal lattice, composed by a periodical array of holes in a infinite silicon medium ( = 11.4). In simulation, a = 410 nm and d = 234 nm. b) Module of electric field for 1st and 2nd modes at K-point of Brillouin zone. c) Band diagram for TE polarization in a bi-dimensional crystal of

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An hexagonal lattice is such a geometry with bi-dimensional symmetry that can present full bandgaps. In figure 1.5a we can see the real space and the Brillouin zone of a geometry of this type. By rotational symmetry, we can split the problem in six regions with identical solutions. By temporal reversal symmetry, we can take just one half of a sixth of the Brillouin zone under consideration. Thus we end up with the reduced Brillouin zone Γ − K − M with k||= 0, k|| = (4π/3a)ˆx, k|| = π/aˆx − π/(

√

3a)ˆy respectively.

The simulation was done in the same way as shown in 1.5.3. The only difference was the edge of the Brillouin zone, that is different for the two cases. In figure 1.5b we can see the electric field for the first and second bands of TE-polarization at point K. We can see that, as expected by variational theorem of section 1.3.1, the fundamental mode is localized in regions with high refractive index (silicon). The second mode must be orthogonal to the first. Hence, it is localized mostly in the air, with little overlap to the first mode. Moreover, another crucial feature is shown in figure 1.5c. We can see the rise of a regions with a null density of states, forming a full bandgap in the region between approximately 150 and 190 THz. Due to this property, in this work we have used this geometry to create our optical cavities. Further details are shown in following chapters.

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Chapter 2 Photonic crystal slabs

Here we will particularize our previous solution to the actual geometry used on our work: a photonic crystal slab cavity. We begin with the analysis of the total internal reflection in the boundaries of a dielectric material. Then we turn our attention to dielectric waveguides and the dispersion curves in such structures. After that we study what is the difference when we have a periodic system and how light is guided and confined in 2D photonic crystal slabs with just two periodic axes. Finally we show how to build optical cavities using the band properties in this kind of element.

2.1. Boundary conditions

In chapter 1, we have always dealt with infinite periodical structures without borders. We have looked to their band structures, in particular considering how symmetry operations can lead to special properties. But we have not mentioned anything related to their boundaries and their interaction with an exterior medium. We can now focus on this important point, that will significantly change the behavior of real structures when compared to theoretical infinite ones.

We begin by analyzing the constraints imposed by Maxwell’s equations in the boundaries of a dielectric media. In figure 2.1a we see an example of possible boundary, where we have a piecewise change in dielectric constant when traveling from a medium to other. Considering we have a source-free and current-free region, we can use Maxwell’s equations in the integral form as I C E · ˆldl = −∂/∂t Z Z S B · ˆndS I C H · ˆldl = ∂/∂t Z Z S D · ˆndS (2.1)

where the line integral is closed along the curve C, which bounds the surface S. The curve and the differential surface element dS are shown in figure 2.1a. If we take the limit of ∆h → 0, we can see that the right side of equations 2.1 goes to zero, as the area approaches zero. However,

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Δl Δh dS C ε₁ ε₂ a) θ_i θ_r θ_t ε₁,μ₁ ε₂,μ₂ x z b) ε₁ ε₂ refraction critical angle total internal reflection c) S

Figure 2.1: a) Boundary between dielectric materials. b) Incident, reflected and transmitted waves in a dielectric surface. c) Difference between usual refraction, the angle where refraction stops to

occur and total internal reflection.

E · ˆl and H · ˆl goes to their tangential values Etan and Htan in interface. This leads to the

conditions of interface of these fields

Etan,1∆l − Etan,2∆l = 0

Htan,1∆l − Htan,2∆l = 0

(2.2) Since ∆l is arbitrary, we conclude that the tangential components of E and H must be continuous across the dielectric interface at all points along the boundary.

2.2. Phase matching condition

We can now see what happens with a wave that is incident in such a boundary. In figure 2.1b we see this interface, where we have regions 1 and 2, with their respective dielectric constant and magnetic permeability. The waves can be described by their amplitude and wavevector in the following way

Ei(r) = E0,ie−iki·r

Er(r) = E0,re−ikr·r

Et(r) = E0,te−ikt·r

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where the indexes i, r and t stand for incident, reflected and transmitted wave. The boundary condition imposes continuity of the tangential field at the interface. According to figure 2.1b this leads to

[Ei(0+, y, z) + Er(0+, y, z)]tan = [Et(0−, y, z)]tan (2.4)

which implies that

[E0,ie−i(kyy+kzz)+ E0,re−i(kyy+kzz)]tan = [E0,te−i(kyy+kzz)]tan (2.5)

This condition must be satisfied at any point of the surface. Therefore, we obtain the only nontrivial solution as a condition to the wavevector components

ki,y = kr,y = kt,y = ky

ki,z = kr,z= kt,z = kz

(2.6) These equations are known as phase-matching conditions. This conservation of the k component parallel to the surfaces is equivalent to the situation obtained by considering the continuous translational symmetry, obtained in section 1.4.1. We can rotate our system such that the plane of incidence be the x − z plane, as in figure 2.1b. From that we see that

ki,x = k1cos θi ki,z = k1sin θi

kr,x = k1cos θr kr,z = k1sin θr

kt,x = k2cos θt kt,z = k2sin θt

(2.7)

where we have that kn= ω

√

µnn, with n = 1, 2 the dispersion relation for each medium. The

requirement that the tangential components are equal leads to some relations. First, we have ki,z = kr,z =⇒ sin θi = sin θr, the basic relation of reflected waves, obtained by ray optics.

Secondly, we have k1sin θi= k2sin θt (2.8) sin θi sin θt =r µ22 µ11 = n1 n2 r µ1 µ2 (2.9) with n1 = √ 1 and n2 = √

2 the refractive indexes for the media. Equation 2.9, when

µ1 = µ2 is simply the Snell’s law. An important phenomenon for photonics arises from this

relation. When we have an incident wave with an angle θi> arcsin n1/n2 propagating through

a medium with refractive index n2 > n1, the law says that sin θt > 1. However this is not

possible with real angles. Therefore, there is not refraction and the ray is totally reflected. This phenomenon is called total internal reflection and is the base of dielectric waveguides. The critical angle θc = arcsin n1/n2, in which the effects starts to happen, exists only when

n1 < n2, that means, when the light is propagating through a more optically dense medium.

When we have a medium with higher dielectric constant inside a medium with lower dielectric constant, the propagation of a wave is possible just through this phenomenon, in a sequence of total internal reflections, in a way called index guiding.

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a) b) z y x d ε₁,μ₁ ε₂,μ₂ x ki E_i H_i TE wave TM wave z x z H_i E_i ε₁,μ₁ ε₂,μ₂ ε₁,μ₁ 0 -d/2 d/2 0 -d/2 d/2 k_i

Figure 2.2: a) Slab dielectric waveguide. b) Difference between TE and TM waves, for waves prop-agating inside a slab waveguide.

2.3. Dielectric slab waveguide

We can now turn our attention to a more complete example, depicted in figure 2.2: the dielectric slab. It consists of a uniform medium with dielectric constant d and thickness d,

immersed in a medium with lower dielectric constant, which here is assumed as air. This problem is important because it explains the vertical confinement of light in photonic crystal slabs, as explained later in section 2.4.2. We start working with the two possible polarizations separately. In figure 2.2b we can see the difference between them. The TE (TM) polarization is composed of a wave with the electric (magnetic) component perpendicular to the plane of incidence. Then we can analyze firstly the solutions of the slab in a TE-polarization. The TM case is analogous.

2.3.1 TE solutions

To find the solutions of the TE polarization, we can make some assumptions about our desired mode. First we can assume it is guided. This implies that we want to have most of electromagnetic energy close to dielectric waveguide. A possible solution is to have a wave that exponentially decays away from the dielectric slab, creating an evanescent field. Inside the dielectric, we assume we have a standing-wave resulting from the superposition of upward and downward propagating wave solutions. Then we have the solution for the electric field in the following way

Ey(x, z) =        E1e−αxx E2 cos (k2xx) sin (k2xx) ±E1eαxx        e−ikzz if x > d/2 if |x| ≤ d/2 if x < −d/2 (2.10)

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to a horizontal plane passing through x = 0. Moreover, the constants k2x and αx ≡ ik1x are

obtained from the dispersion relation in the slab and air. They are given by

k2x = p ω2_µ d− k2z αx = p k2 z − ω2µ0 (2.11)

Note that we are looking for modes that are evanescent. This means that the wavevector in directions away from the waveguide, k1x, is supposed to be an imaginary quantity. Therefore,

we have a real positive constant αx leading to the decay observed in solution 2.10.

However we still don’t know what are the possible k2x in the waveguide. To find it out, we

can use Maxwell’s equation and derive the magnetic component of this field from the electric ones. We used Hz(x, z) = i ωµ ∂ ∂xEy(x, z) (2.12) yielding Hz(x, z) = −i ωµ        αxE1e−αxx ±k2xE2 sin (k2xx) cos (k2xx) ∓αxE1eαxx        e−ikzz if x > d/2 if |x| ≤ d/2 if x < −d/2 (2.13)

We have two equations that define the desired fields we want in our solutions. The next step is to use the boundary conditions to see what the allowed solutions are. We use the solution at x = d/2 for the even modes, such that we get

Etan → E1e−αxd/2 = E2cos (k2xd/2)

Htan → αxE1e−αxd/2 = k2xE2sin (k2xd/2)

(2.14) We should now find common solutions to both equations in (2.14). A way to do it is to divide one equation by the other, such that both restrictions are respected. This process gets rid of the amplitude dependences, leading to the guidance condition for the TE-even modes:

tan k2xd/2 =

αx

k2x

(2.15) We can rewrite this equation as a function of k2xd/2, in a way that facilitates further

analyzes tan k2xd/2 = p(∆k)2_{− (k} 2x)2 k2x (2.16) with ∆k ≡ ω2µ( − 0). Therefore, we have found a relation between the geometrical structure,

its electromagnetic constants and the allowed wavevectors in x direction. This is a transcen-dental equation and needs either numerical or graphical analysis to be solved, by plotting both sides of it and comparing them to see which solutions are allowed for both, in the crossing