Cavity optomechanics in silicon disks and nanostructured disks : Optomecânica de cavidades em discos de silício e discos nanosestruturados de silício

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Instituto de Física Gleb Wataghin

Felipe Gustavo da Silva Santos

Cavity optomechanics in silicon disks and

nanostructured disks

Optomecânica de cavidades em discos de silício

e discos nanoestruturados de silício

Campinas

2017

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Felipe Gustavo da Silva Santos

Cavity optomechanics in silicon disks and

nanostructured disks

Optomecânica de cavidades em discos de silício

e discos nanoestruturados de silício

Thesis presented to the Gleb Wataghin Insti-tute of Physics of the University of Campinas in partial fulfillment of the requirements for the degree of Doctor of Science.

Tese apresentada ao Instituto de Física Gleb Wataghin da Universidade Estadual de Campinas como parte dos requisitos para a obtenção do título de Doutor em Ciências.

Supervisor/Orientador: Thiago Pedro Mayer Alegre Co-supervisor/Coorientador: Gustavo Silva Wiederhecker

Este exemplar corresponde à versão final da tese defendida pelo aluno Felipe Gustavo da Silva Santos e orientada pelo Prof. Dr. Thiago Pedro Mayer Alegre

Campinas

2017

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ORCID: http://orcid.org/0000-0002-5712-8107

Ficha catalográfica

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

Santos, Felipe Gustavo da Silva,

Sa59c SanCavity optomechanics in silicon disks and nanostructured disks / Felipe Gustavo da Silva Santos. – Campinas, SP : [s.n.], 2017.

SanOrientador: Thiago Pedro Mayer Alegre. SanCoorientador: Gustavo Silva Wiederhecker.

SanTese (doutorado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.

San1. Optomecânica de cavidade. 2. Nanofotônica. 3. Bandgap fonônico. I. Alegre, Thiago Pedro Mayer,1981-. II. Wiederhecker, Gustavo Silva,1981-. III. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. IV. Título.

Informações para Biblioteca Digital

Título em outro idioma: Optomecânica de cavidades em discos de silício e discos

nanosestruturados de silício

Palavras-chave em inglês:

Cavity optomechanics Nanophotonics

Phononic bandgap

Área de concentração: Física Titulação: Doutor em Ciências Banca examinadora:

Thiago Pedro Mayer Alegre [Orientador] Lino Misoguti

Christiano José Santiago de Matos José Antonio Roversi

Lucas Heitzmann Gabrielli

Data de defesa: 09-06-2017

Programa de Pós-Graduação: Física

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MEMBROS DA COMISSÃO JULGADORA DA TESE DE DOUTORADO DE

FELIPE GUSTAVO DA SILVA SANTOS – RA 076237 APRESENTADA E

APROVADA AO INSTITUTO DE FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 09/06/2017.

COMISSÃO JULGADORA:

- Prof. Dr. Thiago Pedro Mayer Alegre - (Orientador) DFA/IFGW/UNICAMP - Prof. Dr. José Antonio Roversi - DEQ/IFGW/UNICAMP

- Prof. Dr. Lucas Heitzmann Gabrielli - DC/FEEC/UNICAMP - Prof. Dr. Lino Misoguti - IFSC/USP

- Prof. Dr. Christiano José Santiago de Matos – Mackenzie

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

CAMPINAS 2017

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tone, “it means just what I choose it to mean — neither more nor less.”

Lewis Carroll’s (in many ways) fantastic book, “Through the Looking Glass”.

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Agradecimentos

Como em todo trabalho longo, o apoio de muitas pessoas foi importante para a conclusão desta tese. Embora eu defitivamente não combine com protocolo, faço questão de seguir este de demonstrar por escrito e publicamente meus agradecimentos a essas pessoas.

Agradecimentos muito especiais aos meus pais. Quanto mais velho fico, mais claro me é a noção de que mais da metade das oportunidades de crescimento intelectual que me foram concedidas jamais seriam possíveis sem a dedicação e até mesmo sacrifício deles. Isabel e Paulo, muito obrigado.

Agradeço também ao meu orientador, Thiago Alegre, pelo compartilhamento de conheci-mento e pela orientação em si. Agradeço ainda pelo esforço (que, vale dizer, foi recíproco) em entender meu, digamos, heterodoxismo quanto a temas complexos como “o que é a boa vida” ou “o que é física fundamental”, visando a boa convivência e o bom desenvolvimento da pesquisa que culminou nesta tese. Agradeço também aos professores Gustavo, Newton e Felippe pelas discussões (da física, do universo e tudo mais), churrascos e bares compartilhados.

Aos que trabalharam comigo o tempo todo, Débora, Gustavo, Rodrigo, Guilherme, Mário, Yovanny, Celso, Totó, Jorge e Laís. A interação com vocês puxou os limites de qualidade para o lado certo. Agradeço ainda ao Felipe (Saci) e ao Luís Barea pela paciência nos primeiros experimentos e várias reflexões acerca de óptica e física aplicada.

Não posso deixar de agradecer aos parceiros de fabricação: Marcão, Lucão, Emilio, Alessan-dra — obrigado pelas discussões frutíferas, tanto na sala limpa quanto na mesa de bar. Agradeço também ao CCS pela infraestrutura.

Um agradecimento especial à Laís pelos útlimos meses e por me ajudar a respirar quando sufocava, recuperar o equilíbrio quando me desestabilizava e me (re-)habituar a enxergar bem além do horizonte em dias tempestuosos.

Com ênfase, agradeço a Mariana Gonçales Gerzeli Santos e Rodrigo de Oliveira Alahmar pelo indispensável tratamento de minha saúde mental e física, respectivamente. É impossível conceber a finalização dessa tese sem a ajuda de vocês dois.

Também agradeço ao incrível pessoal da BIF e aos demais funcionários do IFGW, em especial aos das secretarias da CPG, do DFA e da diretoria.

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discente. Um agradecimento especial ao pessoal envolvido nesses “extras”, que para mim foram tão essenciais.

Agradeço a todos os outros amigos que fiz e que mantive ao longo desses anos de doutorado, felizmente numerosos demais para que eu cite todos sem cometer uma injustiça.

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Resumo

A optomecânica de cavidades se transformou numa área de estudos muito rica, com apli-cações em interferômetros de ondas gravitacionais, fundamentos de mecânica quântica, simu-lações quânticas, sincronização, filtros ópticos reconfiguráveis por luz, memórias ópticas — entre diversas outras. Dos muitos dispositivos relatados na literatura, microcavidades integradas em

chips são uma alternativa promissora para o estudo de efeitos dinâmicos devido à interação

entre ondas ópticas e ondas mecânicas confinadas. Entre as microcavidades, discos e cristais optomecânicos (baseados em confinamento por bandgaps fotônico e fonônico) são dispositivos especialmente promissores e frequentemente estudados, cada um tendo vantagens únicas.

Nesta tese, unimos a versatilidade dos discos ao confinamento por bandgap numa nova proposta de dispositivo optomecânico, o bullseye (inglês para “alvo”). De um lado, produzimos conhecimento local em fabricação e caracterização de discos optomecânicos de silício, chegando a larguras de linha óptica menores que 1 GHz (fator de qualidade da ordem de 105_{). De outro,}

mostramos que o bullseye pode superar algumas limitações dos discos simples com o intuito de alcançar o chamado regime de banda lateral resolvida, no qual a frequência de ressonância mecânica é maior que a largura de linha óptica. A partir de simulações pelo método dos elementos finitos, compreendemos em profundidade as propriedades optomecânicas do bullseye, prevendo modos mecânicos de alta frequência com taxa de acoplamento optomecânico (medida do desvio da frequência óptica devido a flutuações do estado fundamental mecânico) de até 200 kHz — valor igual ao dispositivos considerados estado-da-arte. Por fim, demonstramos experimentalmente as propriedades optomecânicas do bullseye em amostras fabricadas com processos industriais CMOS, um resultado importante que abre o caminho para aplicações massivas, tanto comerciais quanto em pesquisa, de cavidades optomecânicas.

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Cavity optomechanics has proven to be a very rich field of study with applications reaching gravitational wave interferometers, fundamentals of quantum mechanics, quantum simulation, synchronization, all-optical tunable optical filters, optical memories — to cite a few. Among the many devices reported in the literature, microcavities integrated on chips offer a very promising alternative for studying dynamical effects due to the interaction between confined mechanical and optical waves. Among microcavities, disks and optomechanical crystals (based on photonic and phononic bandgap confinement) are very promising and frequently studied devices, each having unique advantages.

In this thesis, we allow for disks to meet bandgap confinement in a novel optomechanical design, the bullseye. On one hand, we developed know-how on the fabrication and characteri-zation of optomechanical silicon disks, reaching optical linewidth smaller than 1 GHz (quality factor of order 105_{). On the other hand, we show how the bullseye design can overcome some}

limitations of simple disk cavities in order to achieve the so called resolved sideband regime, in which the mechanical resonance frequency is larger than the optical linewidth. We used finite elements method simulations to deeply understand the bullseye’s optomechanical properties, predicting high frequency (larger than 8 GHz) mechanical modes with vacuum optomechan-ical coupling rate (measure of the optoptomechan-ical frequency shift induced by the mechanoptomechan-ical ground state’s fluctuations) as high as 200 kHz — a value comparable to the current state-of-the-art. Finally, we experimentally demonstrate the bullseye’s optomechanical properties in samples fab-ricated by CMOS-compatible processes, thus paving a new way towards massive commercial and research-related applications of cavity optomechanics.

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B Summary of fabrication parameters 126 C Phase-matching and intermodal scattering suppression 128 D Threshold for optomechanical bistability 131 E The linear crystal approximation 133 F Solving eigenmodes through the Finite Element Method 137 F.1 Key ideas . . . 137

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F.3 Perfectly matched layer parameters in FEM simulations . . . 141

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Chapter 1 Introduction

Cavity optomechanics has become a very fruitful field of research. Its rich dynamics enables new studies in optical cooling of mechanical modes towards the ground state [1], quantum-coherent coupling between optical and mechanical excitations [2], non-classical photon-phonon correlations [3]; self-sustained mechanical oscillations and synchronization [4,5]; electromagnet-ically induced transparency and slow light [6], optical memories [7,8], all-optical tunable optical filters [9]; topological physics [10–12]; among many others [13].

Most of these applications rely on a feedback loop: while mechanical deformation induces optical resonance frequency shifts on one hand, optical forces drive such mechanical deformation on the other. The dynamical effects arising from this feedback loop are often called dynamical backaction, whose consequences were first predicted in the late 60s by Braginski˘ı [14] in the con-text of interferometric measurements. Analyzing a Fabry-Perot cavity (two mirrors facing each other), Braginski˘ı claimed that the radiation pressure force experienced by the mirrors would change if one of the mirrors was slightly displaced, meaning that the optical force effectively modifies the rigidity of the mechanical system (in this case, the mirror mounting); this is the so called optical spring effect. The sign of the optical spring effect would depend on the slope of the resonance to which a laser is tuned, as illustrated in Fig. 1.1(a): for a negative slope, the optical force is restorative and hence contribute to a negative spring effect (and vice-versa).

According to Braginski˘ı, because the optical resonator has a finite time constant, such variations in the optical force would be retarded (phase delayed) with respect to the mirror’s motion. Therefore, while the in-phase component of the optical force is responsible for the optical spring effect, the out-of-phase component would lead to an additional optically induced modification of the mechanical damping. He also noted that both the optical spring and damping effects become more pronounced for smaller values of the cavity’s linewidth, since the radiation pressure would then vary more rapidly with the mirror displacement. Surprisingly, the optical damping can be either positive, which is equivalent to cooling down the mechanical mode, or negative, which is equivalent to amplification of the motion.

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Chapter 1. Introduction 14 a) b) c) d) Cavity length (or laser wavelength) Equilibrium position St or ed ener gy Frequency Laser Enhanced sideband Suppressed sideband Cavity’s response Amplification Laser sb Cooling Laser sb Gravitational wavevector

Figure 1.1: Optomechanical feedback. a) Optical energy stored in a Fabry-Perot cavity (inset) as a function of the cavity length (or pump laser wavelength); the equilibrium point describes the state in which the optical force is compensated by the elastic force (e.g. due to the mirror mounting). If one mirror moves away from the other, the stored energy (hence the optical force) decreases; effectively, the mirror behaves as if its rigidity is smaller (negative optical spring effect). If the laser wavelength is tuned to the positive slope side the effect is reversed and the mirror becomes effectively more rigid (positive optical spring effect). b) As the mirror vibrates, the intracavity optical field becomes frequency modulated, i.e., light is scattered to lower and higher sidebands. From energy conservation, the mirror gives (takes) vibrational energy to produce the higher (lower) sideband, being damped (amplified) in the process. If the pump laser is red (blue) detuned from resonance, the mirror’s motion is optically damped (amplified). c) In a quantum picture, a laser photon is scattered by the moving mirror into a sideband (sb) photon, generating (amplification) or absorbing (cooling) an elementary mechanical excitation (a phonon) in the process. d) LIGO’s gravitational wave detector is a Michelson interferometer with Fabry-Perot cavities on both arms. Because of the enormous optical power in each arm, radiation pressure may lead to important optomechanical effects which affect the detection.

The optical damping effect can be better understood in a scattering picture, although such analysis was not in Braginski˘ı’s first work. If a fixed frequency laser pumps the cavity, mechan-ically induced frequency shifts modulate the light inside the cavity. In the frequency domain, it means that the light inside the cavity will have a component (the carrier) at the exact laser frequency and two small components (the sidebands), one slightly higher and the other slightly lower than the carrier frequency (Fig. 1.1(b)); the distance between the carrier and each side-band turns out to be exactly one mechanical frequency. In a scattering picture, the appearance of these sidebands can be attributed to mechanically-induced light scattering from the carrier frequency to the sideband frequency. If we now consider a semi-classical picture, such scat-tering events can be reinterpreted as the following: whenever a pump photon is scattered to the lower (or higher) sideband, its energy becomes lower (higher), which then implies, by en-ergy conservation, that the mechanical resonator is gaining (losing) enen-ergy. Therefore, light

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scattering events to the lower sideband contribute to amplification of the motion (negative opti-cal damping) whereas scattering to the higher sideband contributes to cooling (positive optiopti-cal damping). Optical cooling or amplification therefore depends on which of the sidebands receives more scatterred pump photons. This asymmetry in the scattering rates can be controlled by the detuning between the cavity’s resonant frequency and the laser frequency as exemplified in Fig. 1.1(b). Note that this asymmetry can be made really large if the distance between the sidebands is larger than the cavity’s linewidth. In other words, optical damping may lead to very efficient optical cooling or amplification if the natural frequency of mechanical oscillations is much larger than the optical loss rate; this important limit is usually called resolved sideband

regime.

Braginski˘ı could later experimentally demonstrate these phenomena in a 10 m long mi-crowave resonator (resonance frequency ≈ 8 GHz, linewidth ∼ 400 kHz) whose termination was attached to a macroscopic pendulum weighting a few milligrams [15]. The pendulum’s oscillation would then change the cavity’s length by 120 − 150 µm — a very weak perturbation producing astonishing new phenomena.

Among the many implications of Braginski˘ı’s pioneering work, there is the recent inter-ferometric detection of gravitational waves by the LIGO and Virgo Collaborations [16]. The detectors used by these observatories are a highly enhanced version of the Michelson interfer-ometer (Fig. 1.1(d) shows a simplified schematic), which is sensitive to strains produced by gravitational waves propagating through the interferometer. To achieve the necessary strain sensitivity, the interferometer’s arms must be very long and have a huge circulating optical power. This is accomplished by inserting an extra mirror within each arm, thus transforming it into a Fabry-Perot cavity; cavity optomechanics then becomes important. It turns out that a negative optical damping may completely compensate the intrinsic damping of the recycling mirror, leading to self-sustained oscillations (also called “parametric instability” in the inter-ferometric gravitational wave community). Furthermore, the mirror’s motion adds noise to the laser beam even at a quantum level. Even though these recycling mirror’s are 40 kg heavy, the circulating power is so huge (100 kW [16]) that dynamical backaction effects arising from cavity optomechanics have to be (and indeed are) somehow mitigated in the gravitational wave detectors to improve their strain sensitivity [17].

On the other hand, dynamical backaction in optomechanical cavities offers a wide range of applications — instead of just being something to avoid. Advances in micro- and nanofabrica-tion techniques have allowed for the development of very small on-chip optical cavities [18] that enabled exploration of cavity optomechanics at low input optical powers. Miniaturization offers two advantages in this scenario. First, it enhances radiation pressure effects in the mechanical dynamics due to the small motional masses involved (10−₁₆₋₁₀−₇ _{kg [13]). Second, the optical}

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Chapter 1. Introduction 16 Nevertheless, low loss Fabry-Perot cavities are no longer viable, instead being replaced by a huge variety of new geometries [18]. These new micro-sized devices allowed for rather extreme manifestations of dynamical backaction, such as radiation-pressure-induced self-sustained (or regenerative) oscillations and ground-state cooling of the mechanical mode.

Radiation-pressure-induced self-sustained mechanical oscillations in micro-sized cavities were first demonstrated for toroidal silica microcavities, whose dimensions were ∼ 100 µm [19–21]. Further than being highly susceptible to optical forces due to their small masses (∼ 10−₁₁ _kg),

the microtoroid’s ultrahigh optical Q-factors (∼ 108_{, linewidth ∼ 1 MHz) were also important}

to place it in the resolved sideband regime (the mechanical frequencies were ∼ 10 MHz), for which regenerative mechanical oscillations with an input laser power of less than 1 mW were observed. These experiments also showed that the input power threshold for such oscillations could be made even lower than 100 µW in these devices by improving the mechanical Q-factor, since this would mean a smaller intrinsic damping to be compensated by the negative optical damping effect.

Radiation-pressure-induced cooling of on-chip mechanical modes down to the ground state was first accomplished in RF cavities [22]. In the optical domain, the first demonstration of such effect was performed in photonic crystal nanobeam cavities [1, 23] for mechanical modes weighting only ∼ 10−₁₆ _{kg. In these structures, the resolved sideband regime could be reached}

by a combination of high Q-factors (∼ 105_{, linewidth ∼ 1 GHz) and very high mechanical}

frequencies (∼ 5 GHz).

It should be noted that the origin of the optomechanical coupling in toroidal and nanobeam cavities is not the same. In the microtoroids, the interaction is similar to the Fabry-Perot example: mechanical deformations change the round-trip length which then shifts the optical resonance; no effect on the material’s refractive index is relevant for the coupling. On the other hand, the optomechanical interaction in nanobeams relies exactly on changes in the material’s refractive index induced by mechanical strain; round-trip length variations due to deformations are irrelevant. Nevertheless, the dynamics of both cavities can be modeled similarly, thus leading to similar dynamical effects.

Dynamical backaction in optomechanical cavities lies at the heart of their versatility ranging from promising technological applications to fundamental physics research. It is instructive to consider two of such examples in more detail: mechanically-mediated optical wavelength conversion [24] and non-classical correlations between photons and phonons [3]. Both effects take advantage of the coherent interaction between photons and phonons enabled by these optomechanical cavities [2,25]. These two experiments were conducted in silicon optomechanical crystals similar to those used for the ground-state cooling demonstration.

In a quantum picture, the optomechanical interaction can be represented by the diagrams shown in Fig. 1.1(c). We then see that the optical cooling process also converts phonons to

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side-mech

C*

C

a)

b)

input output mech to pump 1 pump 2 C* C (0,0) write pulse read pulse (0,1) readout mech pulse read write pulse Dt mech herald A _C (1,0) (1,1) (1,0) (1,1) (2,0)

Figure 1.2: Applications of dynamical backaction. a) Wavelength conversion can take place in a system with two optical modes coupled to the same mechanical one (left). The process is more easily understood in a scattering picture (right): the reciprocal (i.e., time-reversed) cooling interaction (C∗_{) converts an input photon} into a phonon (and a photon at the mode’s pump frequency), which is later converted back into a new photon through the direct cooling interaction (C). b) Non-classical photon-phonon correlations are expected from the amplification process (A), since it converts a blue detuned input photon (write pulse) into a quantum correlated photon-phonon pair. Such correlation can be measured by transforming the phonon of this pair into a second photon using a red detuned (read) pulse to trigger the cooling interaction (C) and later performing optical interferometric measurements between the heralded and readout photons.

band photons; and because time-reversal symmetry holds, photons at the sideband frequency coming from elsewhere can also be converted to phonons. These two conversion processes were used by Hill et al to demonstrate on-chip wavelength conversion by putting a single mechanical mode to interact with two different optical modes in the same structure [24]. The principles of this application are illustrated in Fig. 1.2(a). A light signal traveling through one of the optical modes is transformed to mechanical excitations (process C∗_{) and then converted back to light}

(process C), but in another optical mode with a different resonant wavelength. The conversion efficiency benefits from intense optomechanical couplings. Low optical and mechanical losses are also important in order to keep the external noise (coming through the loss channels) from contaminating the signal. On top of that, the optically-induced mechanical amplification must be suppressed, so the resolved sideband regime is crucial.

The reason for avoiding optical amplification in optomechanical wavelength conversion is better understood from the amplification diagram in Fig. 1.1(c). While cooling is equivalent

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Chapter 1. Introduction 18 to photon-phonon conversion, optical amplification leads to the creation (or destruction) of correlated sideband-photon-phonon pairs. Such process cannot be explored for wavelength conversion, instead effectively adding noise to the process.

Nevertheless, pair-creation (the “Amplification” process in Fig. 1.1(c)) can be explored for different applications. Indeed, Riedinger et al used both the pair-creation and the conversion processes in a very clever experiment that demonstrates non-classical correlations between light and sound [3] in a device prepared at its ground state. A correlated photon-phonon pair would first be created by the pair-creation interaction by using a blue detuned writing pulse (see Fig. 1.2(b)). The photon-phonon would then be measured by first using a red detuned reading pulse to convert the phonon into a second photon, which would interfere with the first one. The resulting correlations were higher than any classical description, thus demonstrating that the device’s normal mode of vibration can behave quantum-mechanically — but in a device more than 10 orders of magnitude heavier than an electron!

Again, the resolved sideband regime is crucial, otherwise it is impossible to select either pair-creation or conversion at will. Moreover, large optomechanical coupling is also important to boost the processes’ efficiency, and low losses are very important in order to reduce decoherence effects that can destroy quantum correlations.

From the discussion above, it should be clear that dynamical backaction effects are improved for

1. Mechanical frequencies larger than the optical linewidth — the resolved sideband regime; 2. Low optical losses (high optical Q) — the narrower the optical resonance, the more the

radiation-pressure varies for a given deformation of the cavity;

3. Low mechanical losses (high mechanical Q) — cooling or amplification of the movement can be made more efficient;

4. Strong optomechanical interaction — the larger the optical frequency shift induced by deformations, the more pronounced dynamical backaction effects will be;

5. High circulating optical power — enhances the radiation pressure force.

Towards this optimum combination of parameters in integrated optics, two main classes of devices have been studied so far: (i) bandgap-based optomechanical crystals [23, 26]; and (ii) circular cavities, such as disks [27,28], rings [29] and toroids [19–21].

Although essentially any effect discussed above could in principle be observed in both classes of devices, optomechanical crystals used to be uniquely versatile as its bandgap confinement strategy for both optical and mechanical modes allows for wide tailorability. These devices were fabricated using the traditional approach of direct write electron beam lithography. However,

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for massive fundamental studies and applications it is desirable to migrate the fabrication process of these devices to commercial CMOS-compatible facilities, where the photo-lithography process considerably reduces time and cost when compared to the traditional approach.

In this context, our group within the Device Research Laboratory has conducted studies regarding many different cavity designs, such as paddle resonators [30], slotted photonic crys-tals [31], line-defect optomechanical cryscrys-tals [32], double disks [33–35], coupled disks [36,37] and the object of this thesis, the bullseye disk [38]. This design variety turns out to be very useful to investigate different aspects of cavity optomechanics, since each device design highlights a different set of properties underlying the optomechanical interaction.

In this thesis, we allow for circular cavities to meet tailorability in a CMOS-foundry com-patible device. We present a novel optomechanical cavity that merges the simple and effective optical confinement of circular geometries (the whispering gallery effect) to the versatility of phononic crystals. The device is based on a disk with a bullseye-shaped grating in its center, hence its name “bullseye”.

The resonator is designed to enhance light-sound interactions, prevent optical and mechan-ical losses and to be in the resolved sideband regime. Moreover, our design is a novel approach to cavity optomechanics, presenting features unique to circular cavities: the co-existence of degenerate clockwise and counter-clockwise optical modes, the co-existence of several families of optical modes possessing similar transverse profiles, and the completely different mechanisms leading to optical (whispering gallery effect) and mechanical (phononic bandgap) confinement, which allow for independently tailoring the optical and mechanical properties. These features make the bullseye a promising and unique new platform for optomechanical experiments and applications.

This thesis is organized as follows. In Chapter 2, we describe a general theory of cavity optomechanics to derive the system’s dynamical equations as well as an expression for the computation of the optomechanical coupling rate in any cavity design; figures-of-merit are also indicated along the chapter. Chapter 3 then focuses on how to translate the dynamical equations into experimentally accessible quantities.

Next, we apply our general theory for cavity optomechanics to disk cavities in Chapter 4, a geometry we used to develop important instrumentation for in-house fabrication and charac-terization of optomechanical cavities. Chapter 5 describes such instrumentation development in the context optomechanical disks made of silicon.

We finally go into the details of the bullseye design in Chapter 6, highlighting physical insights and presenting numerical simulations in order to precisely reach an optimized bullseye geometry in silicon. In Chapter 7, we demonstrate the main features of the bullseye disk in samples fabricated at a CMOS photonics foundry, thus also demonstrating the scalability of our approach and paving the way towards large-scale, both commercial and research related,

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Chapter 1. Introduction 20 applications of optomechanical cavities. Finally, Chapter 8 is devoted to final considerations as well as some perspectives.

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Chapter 2 Fundamentals of Cavity Optomechanics

In this chapter, we shall describe the theoretical framework necessary for understanding cavity optomechanics, beginning by a phenomenological approach that leads to a very intuitive picture of the optomechanical interaction. We then move to a derivation of these dynamical equations from field theory considerations (classical electrodynamics and continuum mechan-ics) which helps in understanding how the optomechanical coupling arises in more general geometries. On top of rigorously deriving the optomechanical equations, our treatment is in-tended to be easily generalizable to cavities subject to more general optical nonlinearities, as phase-matching considerations naturally arise along our reasoning.

Next, we solve the optomechanical dynamical equations for the limit of strong pump fields, illustrating how dynamical backaction effects benefit from low optical and mechanical losses, intense optomechanical coupling and the resolved sideband regime.

2.1 Preliminary considerations

Let us begin from the simple model shown in Fig. 2.1: a Fabry-Perot optical cavity (in vacuum, refractive index equal to 1) in which one of the mirrors is actually a mass-spring oscillator with resonance frequency Ωm. Assuming that the displacement x of the moving

mirror (relative to its equilibrium position) is much smaller than the cavity length L, this resonance condition may be written as

ω(x) = mπc L+ x ≈ mπc L 1 − x L (2.1) where ω(x) is the cavity’s resonance frequency, m an integer number describing the mode order and c the speed of light in vacuum; this equation immediately follows from matching the optical phase accumulated along a round-trip (2Lω(x)/c) to an integer multiple of 2π. In other words, equation (2.1) states that the bare optical frequency ωc = mπc/L is perturbatively shifted by

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Chapter 2. Fundamentals of Cavity Optomechanics 22 ki Wm g wc wL ke _G Laser L PD a) b) _Dw_c = Gx wc

Figure 2.1: Optomechanical toy model. a) Fabry-Perot cavity (length L, resonance frequency ωc) attached to a mass-spring oscillator (frequency Ωm). If the mirror moves, the cavity length becomes L + x and the optical resonance shifts by ∆ωc = Gx (x L). The optical and mechanical properties can be interrogated by a pump laser whose reflection is measured at a photodetector (PD). The intrinsic loss rates κi (optical) and γ (mechanical) as well as the cavity-laser coupling rate (extrinsic optical loss) κe are shown; all optomechanical dynamics can be understood from these lumped parameters. b) As the optical resonance frequency bounces back and forth due to the oscillator motion, the intracavity laser field acquires frequency modulation which becomes intensity modulation as the field leaves the cavity towards the detector. This is used for optical readout of mechanical motion.

∆ωc = Gx, where G = −ωc/L is an optomechanical transduction coefficient describing how

strongly a given displacement x shifts the optical resonance from its bare value.

On the other hand, radiation pressure may drive the mirror’s motion for high enough fields within the cavity. Such force can be easily quantified for our simplified model in a semiclassical perspective. Let us assume that a pump laser of frequency ωL is delivering photons to the

Fabry-Perot cavity as in Fig. 2.1. Each photon reflected by the moving mirror transfers a momentum δp = 2~ωL/c to it, where ~ is the reduced Planck constant; the force by such

photon scattering would then be δp/τrt, τrt = 2L/c being the cavity round-trip time. The

net optical force felt by the mirror is then given by this single photon force multiplied by the number of photons confined to the cavity, nc = Uc/~ωL, where Ucis the electromagnetic energy

stored in the cavity. Therefore, the radiation pressure force is given by

F_opt = δp

τ_rtnc= − G

ω_cUc (2.2)

where the substitution L = −ωc/Ghas been performed. Although derived for a specific model,

equation (2.2) along with the frequency shift expression ∆ωc= Gx are very general expressions

(see Section 2.4).

It is worth noting that, according to equation (2.1), the mirror’s motion can interact with many distinct optical modes. However, in many cases we can focus on the dynamics of a single optical mode as far as it is the only populated one. Such condition can be readily achieved if the optical modes are spectrally resolved by tuning the pump laser to the desired mode’s frequency (see Section 2.2) as far as no other effect scatters light to modes not being pumped (see Section 2.4).

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This analysis also leads to a very important feature: the emergence of dynamical backaction in optomechanical cavities; the mirror’s motion changes the system’s optical response while the stored light drives the mirror’s motion. This feedback loop allows for a plethora of interesting dynamical phenomena [13].

In the remainder of this chapter, we shall derive the dynamical equations from classical electrodynamics and continuum solid mechanics, but for more general systems than the Fabry-Perot example. Such approach will also help in understanding the processes that contribute to mechanically induced optical frequency shifts, and therefore are relevant in quantifying G.

2.2 Electromagnetic fields and optical modes

In this section, we derive the optical properties of optomechanical cavities from Maxwell’s equations. We choose to work with the electric field ~E and magnetic intensity field ~H, such

that Maxwell’s equations take the form

∇ × ~E = −µ₀∂ ~H ∂t (2.3a) ∇ × ~H = ∂ ∂t ₀E~ + ~P (2.3b)

where the polarization field ~P is the sum of a linear and a nonlinear response,

~

P = ₀χ(1)E~ + ~PNL (2.4)

Here, the linear susceptibility χ(1) _{relates to the material’s dielectric constant = 1 + χ}(1)_{; we}

henceforth assume to be piecewise constant as that is the case of our devices. Along with these curl equations, the following (sourceless) divergence ones

∇ · ~D= 0 (2.5a)

∇ · ~B = 0 (2.5b)

where ~D = 0E~ + ~P and ~B = µ0H~, also hold; these later equations are accompanied by a few

boundary conditions at every interface where changes value, namely continuity of the normal (tangential) components of the fields ~D and ~B ( ~E and ~H).

The nonlinear polarization ~PNL _{describes any possible nonlinearity, including mechanically}

induced ones; these nonlinearities are assumed to be small enough to allow for perturbative methods to hold — we shall return to the details regarding such nonlinearities in Section 2.4. Under these circumstances, it is reasonable to choose the ansatz (“c.c.” means complex

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conju-Chapter 2. Fundamentals of Cavity Optomechanics 24 gate) ~ E(~r, t) =X α aα(t)~eα(~r) + c.c. (2.6a) ~ H(~r, t) =X α aα(t)~hα(~r) + c.c. (2.6b)

where each pair (~eα(~r),~hα(~r)) describes the field distribution of an unperturbed cavity harmonic mode of frequency ωα, i.e.

∇ × ~eα(~r) = iωαµ0~hα(~r) (2.7a)

∇ × ~hα(~r) = −iωα0~eα(~r) (2.7b)

and the coefficients aα(t) are the product between the harmonic field dependence e−iωαt and slowly-varying envelopes ãα(t) such that dãα/dt ωαãα. We choose dimensions such that

nα= a∗αaα gives the total photon number occupying the mode labeled by α. By substituting (2.6) and (2.7) into (2.3), we get

X

α

(˙aα+ iωαaα)µ0~hα+ c.c. = 0 (2.8a)

X α (˙aα+ iωαaα)0~eα+ c.c. = − ∂ ∂t ~ PNL (2.8b)

where ˙a is a short-handed notation for da/dt. Integrating (2.8a)·~h∗

α0+(2.8b)·~e∗_α0 over the whole space then gives (˙aα+ iωαaα) Z dV µ₀~hα· ~hα∗ + 0~eα· ~e∗α + o.r.t. = −Z dV ∂ ∂t ~ PNL· ~e∗_α (2.9) where “o.r.t” denotes off-resonance terms that negligibly contribute to the system dynamics. We have also used the orthogonality condition [39]

h~eα|0|~eαi= Z dV ₀~e∗_α0· ~e_α = δ_αα0 Z dV ₀~e∗_α· ~eα = δαα0 Z dV µ₀~h∗_α· ~hα = Z dV µ₀~h∗_α0· ~h_α= 1 2~ωα (2.10) where the last equality immediately follows from requiring nα = a∗αaα to be the number of photons in the mode α.

In order to predict and analyze the response of real systems, it is desirable to introduce losses and sources in eq. (2.9); we shall do both phenomenologically.

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net effect of doing so would be to include a lossy term to (2.9)’s RHS (right hand side), giving1 ˙aα = −iωαaα− κ_i,α 2 aα− h~eα|∂tP~NLi 20h~eα||~eαi (2.11) where the intrinsic loss rate κi,α gives the photon number (or energy) decay rate of the mode

α.2 In eq. (2.11), we borrow Dirac’s notation from quantum mechanics [42] to denote

h~eα0||~e_αi= Z

dV ~e∗_α0 · · ~e_α (2.12)

Sources, on the other hand, can be included as a combination of current and charge densities in Maxwell’s equations, which is analogous to including an extra polarization term ~Pin to the

total polarization vector ~P ; it suffices then for ~Pinto carry the source’s time dependence, being

independent of the intracavity field.

The net effect of this rather sophisticated phenomenological step would be to introduce an extra term on the RHS of eq. (2.11), resulting in

˙aα = − iωα+ κ_i,α 2 aα− h~eα|∂tP~NLi 20h~eα||~eαi + h~eα|∂tP~ini h~eα||~eαi (2.13) However, this approach does not yet completely describes the source. That is because the same port that brings energy from the source to the cavity also allows energy to scape from the cavity. In the spirit of our phenomenological considerations, it means that the source also modifies the ’s imaginary part — or, equivalently, that ~Pinhas an imaginary part that depends

linearly on the intracavity field and does not follow the source’s time dependence. Therefore, the dynamical equation (2.11) becomes

˙aα = − iωα+ κα 2 aα− h~eα|∂tP~NLi 20h~eα||~eαi +h~eα|∂tP~ini h~eα||~eαi (2.14) where the total loss rate κα = κi,α+ κe,α is composed by the cavity’s intrinsic loss rate κi,α and an extrinsic loss rate κe,α.

Finally, from either time-reversal considerations [43] or input-output theory [44], it can be shown that the feeding term can be written as √κ_e,αa_in(t), where a∗_ina_in gives the incoming

photon flux. Note that the time dependence of ain(t) then becomes that of h~eα|∂tP~ini, i.e., the

1_∂

tis a short-handed notation for _∂t∂.

2_{That is most easily observed by disregarding the nonlinear term and solving the resulting dynamical equation} to obtain the transient solution for nα= a∗αaα.

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Chapter 2. Fundamentals of Cavity Optomechanics 26 source’s time dependence. Therefore, the equation of motion becomes

˙aα = − iωα+ κα 2 aα− h~eα|∂tP~NLi 20h~eα||~eαi +√κ_e,αa_in (2.15)

It should be noted that another measure of the losses in a system is the quality (or Q) factor. It corresponds to the ratio between the resonance frequency and the loss rate. As we have intrinsic, extrinsic and total losses, it is useful to define 3 Q-factors as well: the intrinsic, Qi,α = ωα/κi,α, the extrinsic, Qe,α = ωα/κe,α, and the total (or loaded) quality factor,

Qα = ωα/κα. From the above definitions, it can be shown that Q−α1 = Q

−₁

i,α + Q−e,α1.

Under typical experimental conditions (i.e., laser sources) ain corresponds to a very narrow

spectral distribution centered at some frequency ωL, so ain ∝ e−iωLt. It is then common to

rewrite eq. (2.15) in a rotating reference frame that turns the source term into a constant, which means interchanging a → ae−iωLt, in (2.15), giving

˙aα = − i∆α+ κα 2 aα− h~eα|eiωLt|∂tP~NLi 20h~eα||~eαi +√κ_e,αa_in (2.16)

where the quantity with ∆α = ωα− ωL is called the detuning.

It is worth noting that if we again disregard the optomechanical interaction term, eq. (2.16) gives the unperturbed population of each optical mode as a function of pump frequency. The steady state result is

aα = √ κ_e,αa_in i∆α+ κα/2 ⇒ nα = κ_e,α|a_in|2 ∆2 α+ (κα/2)2 (2.17) It becomes then clear that only modes in the vicinity of ωα, such that ∆α . κα, are relevant for describing the unperturbed optical dynamics.

2.3 Acoustic field equations: mechanical modes

Now we move to the analysis of acoustic waves, for which we shall use an approach very similar to the one of Section 2.2. The relevant acoustic field is the displacement ~U(~L, t) =

~

`(~L, t) − ~L, where ~`(~L, t) is the displaced position of a particle whose equilibrium position is ~

L within the solid. The ~U field is interesting because their differentials are always zero for

undeformed motion of the solid (as translations and rotations), but are always non-zero if the body is deformed [45]. Newton’s law then takes the form

~ f = ρ∂

2_U~

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where ρ is the material’s density and ~f is the force per unit volume upon an infinitesimal volume element lying at ~L. This force can be written as [45]

~

f = ∇ · T + ~f_rad (2.19)

where T is the 2nd-order symmetric stress tensor field and ~frad( ~E, ~H; ~r, t) is a perturbative

force density exerted by the optical field (whose detailed analysis is left to Section 2.4). In the elastic regime, a continuum version of Hooke’s law applies,

T= c : S = X kl cijklSkl ! 3x3 (2.20) where c is the material’s stiffness (4th-rank) tensor and S is the symmetric strain tensor field given by

S= 1 2

∇~U + (∇~U)T (2.21)

where the superscriptT _{denotes transposition.}

Substituting (2.21) and (2.20) into (2.18), we get the field equation

ρ∂_t2U~ = ∇ · (c : S) + ~f_rad (2.22)

The ∇ operators here are usually defined through the vector ~L. The tensor ∇~U is defined through the displacement differential

d ~U = ∇~U · d~L =

3

X

i,j=1

(∇~U)ijˆLidLj (2.23)

where dLi are the components of d~L for some orthogonal coordinate system with unit vectors ˆLi; in other words, the volume infinitesimal can be expressed as dV = dL1dL2dL3.

Using cartesian coordinates (x, y, z), for example, we have

d~L= dxˆx + dyˆy + dzˆz (2.24)

so the length differentials can be chosen as dL1 = dx, dL2 = dy and dL3 = dz. The strain

expression then becomes

Sij = 1 2 ∂Ui ∂xj + ∂Uj ∂xi ! (2.25) with the xi running over x, y and z.

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Chapter 2. Fundamentals of Cavity Optomechanics 28 In analogy to the vector divergence

∇ · ~U = ∂(~U · ˆx) ∂x + ∂(~U · ˆy) ∂y + ∂(~U · ˆz) ∂z (2.26)

the tensor divergence is defined as ∇ · T= ∂(T · ˆx) ∂x + ∂(T · ˆy) ∂y + ∂(T · ˆz) ∂z = X ij ∂ ∂xi Tijxˆj (2.27)

So, equation (2.22) takes the form

ρ∂_t2Ui = (∇ · (c : S))i = X jkl ∂ ∂xj (cijklSkl) = X jkl cijkl ∂2 ∂xkxk ∂j∂kUl+ frad,i (2.28) where c’s general symmetry properties in cartesian coordinates [45]

cijkl = cijlk = cjikl = cklij (2.29)

have been used.

For cubic crystals (as silicon), few components of the stiffness tensor are non-zero due to symmetry [45]; these are ciiii, ciijj and cijij, with i 6= j. Also because of symmetry, these components are actually independent of the indices, being often referred to as c11, c12 and c44,

respectively.

It is worth considering the case of cylindrical coordinates (r, ϕ, z) as well, as the devices mainly discussed in this thesis are cylindrically symmetric. The length differentials then are

dL₁ = dr, dL₂ = rdϕ and dL₃ = dz, but caution must be taken when deriving the expressions

for ∇~U and ∇ · T as the unit vectors ˆr and ˆϕ both depend on ϕ. Therefore, the displacement differential is

d ~U = ∂Ur ∂r ˆr + ∂Uϕ ∂r ˆϕ + ∂Uz ∂r ˆz ! dr+ 1 r ∂Ur ∂ϕ − Uϕ ! ˆr + ∂Uϕ ∂ϕ + Ur ! ˆϕ + ∂Uz ∂ϕ ˆz ! (rdϕ)+ ∂Ur ∂z ˆr + ∂Uϕ ∂z ˆϕ + ∂Uz ∂z ˆz ! dz (2.30)

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Also in analogy to the vector divergence in cylindrical coordinates, the tensor divergence is ∇ · T= 1 r ∂(rT · ˆr) ∂r + 1 r ∂(T · ˆϕ) ∂ϕ + ∂(T · ˆz) ∂z =         1 r + ∂ ∂r − 1 r 0 0 ∂ ∂z 1 r ∂ ∂ϕ 0 1 r ∂ ∂ϕ 0 ∂ ∂z 0 2 r + ∂ ∂r 0 0 ∂ ∂z 1 r ∂ ∂ϕ 1 r + ∂ ∂r 0                       Trr Tϕϕ Tzz Tϕz Trz Trϕ               (2.31)

where Tij = ˆxi· T ·ˆxj symmetry upon transposition has been used.

Now we head back to the to the wave equation (2.22). Following Section 2.2, we propose the ansatz

~

U(~r, t) =X

α

xα(t)~uα(~r) (2.32)

where xα(t) are real amplitudes and ~uα(~r) are the real-valued unperturbed harmonic solutions of frequency Ωα obeying

− ρΩ2_α~u(~r) = ∇ · (c : S) (2.33)

where the material’s properties ρ and c are assumed to be piecewise constant just like the dielectric constant . The harmonic mechanical modes turn out to be orthogonal [45], i.e.

Z

dV ~uα0· ~u_α = δ_αα0 Z

dV ~uα· ~uα (2.34)

Following the optical field analysis, substituting this ansatz in the field equation (2.22) yields X α ρ(¨xα+ Ω2αxα)~uα = ~frad (2.35)

Multiplying by ~uα0 and integrating over the whole space then gives, using (2.34),

m_eff,α(¨xα+ Ω2αxα) = Fα (2.36) where Fα and meff,α (usually called the mode’s effective mass or motional mass) are given by

Fα = Z dV ~uα· ~frad (2.37a) m_eff,α= ρ Z dV ~uα· ~uα (2.37b)

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Chapter 2. Fundamentals of Cavity Optomechanics 30 have a more accurate description, a loss rate γα may be included to yield

m_eff,α(¨xα+ γα˙xα+ Ω2αxα) = Fα+ Fth,α (2.38) where we have also included a stochastic force Fth,α to account for the inevitable thermal

fluctuations that accompany the damping according to the fluctuation-dissipation theorem [13, 44, 46]. In analogy to the optical case, a mechanical quality factor can also be defined as

Q_m,α = Ωα/γα. The thermal force must obey

hF_th,α(t + τ)F_th,α(t)i = 2m_eff,αγαkBTbδ(τ) (2.39)

k_B being Boltzmann’s constant, T_b the surrounding environment’s absolute temperature and δ(τ) Dirac’s delta function. The average in the LHS (left hand side) of equation (2.39) is to

be understood as a statistical average, i.e., an average over all possible realizations of Fth(t)

weighted by the respective probability for each realization to occur. This autocorrelation then ensures that x(t) obeys the equipartition theorem (as shown in Section 3.1). For more details regarding stochastic calculus, we refer the reader to the very comprehensive book by Kurt Jacobs [47] and to the more advanced book by Roy Howard [48]; for a more physically appealing introduction, see [49].

2.4 Interaction between optical and mechanical modes

Now we turn into the details of the optomechanical interaction, starting with the nonlinear polarization. From the formalism developed in Section 2.2, it suffices to calculate

− h~eα|∂tP~

NL_i

20h~eα||~eαi

(2.40) as this term alone describes any mechanically induced modification to the optical field dynamics in eq. (2.16).

In order to account for mechanically induced light scattering, the lowest order contribution to ~PNL _{should have the form}

0∆(~U; ~r, t) · ~E, where the tensorial nonlinear susceptibility ∆

does not depend explicitly on the electric field, but depends on the mechanical displacement field ~U(~r, t) to which the system is submitted. Therefore, the interaction term in (2.16) becomes (neglecting rapidly rotating terms)

− h~eα|∂tP~NLi 20h~eα||~eαi ≈X α0 iωα0 2 h~eα|∆|~eα0i h~eα||~eαi aα0 = X α0 gαα0a_α0 = X α0₆_=α gαα0a_α0+ ∆ω_αa_α (2.41)

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where we have neglected (∂t∆)aα and ∆ ˙aα with respect to ωαaα∆ because of the huge difference in time-scales (ωα Ωβ,∆α).

The last equality in eq. (2.41) is useful to highlight two possibilities: while the first term describes mechanically induced light scattering from a mode α0 ₆_{= α to the mode α with}

scattering amplitude gαα0, the second term describes a mechanically induced optical frequency shift of ∆ωα= gαα (or light scattering within the same optical mode). Now, in order to evaluate the scattering rates gαα0 for both inter (α0 6= α) and intramodal (α0 = α) light scattering processes, we must understand ∆’s functional form.

Two main effects contribute to the modulation of the dielectric constant. One of them comes from the movement of the cavity’s boundaries, which changes the optical resonance frequency by modifying the electromagnetic field’s boundary conditions; this is a generalization of the optomechanical coupling in the Fabry-Perot model of Fig. 2.1. The inner product h~eα|∆|~eα0i then becomes a surface integral over the unperturbed surface where changes value. Further-more, only displacements orthogonal to the unperturbed boundary surface contribute to the inner product since these are the only components of ~U that indeed cause the electromagnetic boundary conditions to change.

However, this surface contribution has to be computed with caution: in order to have a well defined integral, the integrand must be written as a function of the continuous components, the tangential electric field ~eα,k and the normal displacement field ~dα,⊥ = ~eα,⊥. To lowest order, the moving boundary (MB) contribution becomes [50]

h~eα|∆|~eα0i_MB= Z dS(~U · ˆn)h(₂ − ₁)~e_α,k∗ · ~eα0_,k−(−1 2 − −11)~d∗α,⊥· ~dα0_,⊥ i (2.42) where the indices 1 and 2 label opposite sides of the surface and ˆn is the unit vector normal to the surface from medium 1 to medium 2.

Now the other contribution comes from strain-induced modifications of the dielectric con-stant, the so called photoelastic effect. Because the strains are usually small, the change in dielectric constant is accurately described by a linear relation [23,51]

∆_PE = 2p : S (2.43)

where the photoelastic (4th rank) tensor p is a material property having the same symmetry properties of the stiffness tensor shown in eq. (2.29). The inner product regarding ∆PE is

then simply

h~eα|∆|~eα0i_PE = Z

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Chapter 2. Fundamentals of Cavity Optomechanics 32 Using equation (2.32), the sum of (2.42) and (2.44) becomes

h~eα|∆|~eα0i= X β xβh~eα|∆β|~eα0i= X β xβ Z dV e∗_α,i2pijkl∂kuβ,leα0_,j+ Z dS(~uβ·ˆn) h (2− 1)~e∗α,k· ~eα0_,k−(−1 2 − −11)~d∗α,⊥· ~dα0_,⊥ i (2.45)

Note that this expression can be evaluated from knowledge of the material properties and device geometry (from which the unperturbed optical and mechanical fields can be calculated). Conversely, the material properties and device geometry can be explored in order to design optomechanical cavities. The dynamical equation for the optical amplitudes (2.16) becomes

˙aα = − i∆α+ κα 2 aα− i X α0_β G(β)_αα0x_βa_α0 +√κ_e,αa_in (2.46)

where the optomechanical coupling constants G(β)αα0 are given by

G(β)_αα0 =

ωα0 2

heα,i|2pijkl∂kuβ,l|eα0_,ji+ h~e_α|(~u_β ·ˆn)(₂− ₁)|~e_α0i_boundary h~eα||~eαi

(2.47) and the last term in the numerator is a surface integral that must be calculated using continuous field components according to (2.42).

To complete our dynamical description as a function of aα and xβ, we just need to calculate the optical force ~frad. One route is to begin from Poynting’s theorem, thus deriving ~frad from

Poynting’s vector and Maxwell’s stress tensor (including the electrostriction force). However, this route has lead to some complications due to controversy regarding the correct expressions for Poynting’s vector and Maxwell’s stress tensor [52–54]. Even though the problem seems to have been recently solved by Wolff and coworkers [54], we choose to evaluate the optical force from the system’s free energy to avoid such complications. It turns out that the work extractable from the electromagnetic field is limited to the Helmholtz free energy [55]

F = 1 2 Z dV E · ~~ D+ ~B · ~H= 1 2 h ~E|₀| ~Ei+ h ~E| ~PNLi+ h ~H|µ₀| ~Hi (2.48)

If losses are small enough, temperature and entropy variations can be neglected and the free energy variation is equal to the potential energy variation. By expressing h ~E| ~PNL_i _{as a}

function of G(β)αα0 and x_β, we can directly calculate F_β (instead of ~frad) in (2.38), the result being

Fβ = − ∂F ∂xβ = −~₂X αα0 ωα ωα0 G(β)_αα0a∗_αaα0+ c.c. (2.49) where rapidly rotating terms were neglected and equation (2.10) was used. The dynamical

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equations for the mechanical amplitudes then become m_eff,β(¨xβ+ γβ˙xβ + Ω2βxβ) = −~ 2 X αα0 ωα ωα0 G(β)_αα0a∗_αa_α0 + c.c. ! + Fth,β (2.50)

Although equations (2.46) and (2.50) fully describe the dynamics of optomechanical cavities, not all of these equations are actually relevant under typical experimental conditions. It turns out that a reasonable optomechanical coupling G(β)αα0 is not easily accomplished due to the form of the overlap integrals in (2.47); it becomes even harder to have a relevant coupling for α0 ₆_{= α}

as it requires matching the overlap of the three distinct fields, two of which are orthogonal. The design of an optomechanical cavity then plays a crucial role for it determines which kind of scattering can be observed in a given device. Since the bullseye resonators explored in this thesis were developed to enhance the frequency-shift effects (or light scattering within the same optical mode) and to suppress intermodal scattering, we shall disregard intermodal light scattering from now on3_.

Moreover, under typical experimental conditions, the monochromatic laser source hardly excites more than a single optical mode at once. Since we are neglecting intermodal light scattering, it becomes reasonable to look only at the dynamics of the optical mode, of frequency

ω_c, being excited by the laser source.

Furthermore, because the interaction is weak, any correction on a mechanical amplitude

xβ0 coming from another amplitude x_β (β 6= β0) (mediated by the optical field) is necessarily a higher-order correction that can be neglected. Therefore, an accurate analysis can be done by looking separately at each mechanical mode (having frequency Ωm) interacting with the

pumped optical mode.

In summary, the dynamics of the optomechanical cavity is accurately described by the following set of equations:

˙a = −i∆ + κ

2

a − iGxa+√κ_ea_in (2.51a)

m_eff(¨x + γ ˙x + Ω2_mx) = −~Ga∗a+ F_th (2.51b)

where ∆ = ωc− ωL. Note that all the α and β indices could be dropped without altering the

physical meaning of the symbols.

Remarks on normalization

There is an ambiguity in the definition of G coming from an ambiguity in the definition of

xβ: if such variables are multiplied by arbitrary non-zero constants sβ, the physical deformation 3_{Once the device geometry is known, we can develop more rigorous arguments for neglecting intermodal} light scattering. See Chapter 6 and Appendix C.

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Chapter 2. Fundamentals of Cavity Optomechanics 34

~

U remains the same as long as the modal displacements ~uβ are correspondingly multiplied by

s−_β1. This ambiguity is reflected in the effective mass (rescaling x by s causes m_eff to be rescaled

by s−2_{) and in the optomechanical coupling strength (G would be rescaled by s}−1_).

Such arbitrariness obviously does not affect experimental predictions, as will become clear in Chapter 3. However, it poses a difficulty in comparing different optomechanical devices. It becomes even harder to compare devices having different geometries, such as ring- and Fabry-Perot cavities.

Two procedures are typically done to circumvent this problem. One of them is to normalize all mechanical modes to max |~u| = 1; this way, G becomes the “optical frequency shift per unit length of mechanical motion”. The other one is to refer to g0 = G/xzpf, where xzpf =

q

~/2meffΩm are the zero point fluctuations of the mechanical oscillator. Note that, although

x_zpf depends on normalization through m_eff, g₀ does not.

In the remainder of this chapter, we keep using both G and g0. In the next chapters,

however, we always refer to g0 in order to compare our design, the bullseye resonator, to other

devices in the literature.

2.5 Dynamical backaction in cavity optomechanics

From equations (2.51), we can now describe some of the most explored consequences of the optomechanical feedback. For such, we use a mean-value approximation for the optical amplitude a = α + δa, where δa describes fluctuations around the mean-amplitude α. Like-wise, the mechanical amplitude shall obey a similar approximation x = x0 + δx, δx being the

fluctuations around an equilibrium position x0. The equations of motion then become, keeping

only first-order fluctuating terms,

α= √ κ_ea_in i(∆ + Gx₀) + κ/2 (2.52a) Ω2 mx0 = −~G|α|2 (2.52b) ˙δa = − i∆ + κ 2 δa − iG(x₀δa+ δxα) (2.52c) m_eff(δ¨x + γδ ˙x + Ω2_mδx) = −~G(α∗δa+ αδa∗) + F_th (2.52d)

From equation (2.52a), we notice that α is the steady-state optical amplitude at a modified detuning ∆ + Gx0 that depends on the new equilibrium position x0. On the other hand, the

second equation states that such equilibrium position is modified by the optical force ~G|α|2_.

As a function of x0, these algebraic equations can be reduced to

Ω2

mx0+ ~Gκe|ain| 2

(35)

which is just the balance between the elastic and optical forces. For high enough pump powers, such force balance may lead to a bistable mechanical response which can be observed through optical measurements (see Appendix D).

Let us now focus on dynamical effects due to the fluctuation dynamics arising from this optomechanical interaction. We shall concentrate our discussion to steady-state solutions — the transient response is too fast to be observed by the experimental techniques discussed in Chapters 3, 5 and 7. To simplify the notation, we drop the δ symbols from now on and refer to a and x as the fluctuations around mean-values of the optical and mechanical fields, respectively. The equations of motion then become

˙a = −i∆ + κ

2

a − iGαx (2.54a) m_eff(¨x + γ ˙x + Ω2_mx) = −~G(α∗a+ αa∗) + F_th (2.54b)

where the detuning ∆ now incorporates the static frequency shift Gx0.

Because these equations are linear, a straightforward approach would be to solve them in frequency domain. However, we believe it is more instructive (and yet equivalent) to use the following ansatz

a= a₊e−iΩt+ a−eiΩt (2.55a)

x= x₁e−iΩt+ x∗₁eiΩt (2.55b)

Physically, this ansatz describes forced oscillations of the mechanical mode at some frequency Ω and amplitude 2|x1|, along with associated optical sidebands at frequencies both higher (a+)

and lower (a−) than the pump laser frequency. Substitution of this ansatz in (2.54) yields

a₊= Gαx1 Ω − ∆ + iκ/2 (2.56a) a− = − Gαx∗₁ Ω + ∆ + iκ/2 (2.56b) x₁ =nχxx(Ω)−1+ Σ(Ω) o−1 F_th(Ω) (2.56c) where χxx(Ω) = 1 m_eff(Ω2_m−Ω2− iγΩ) (2.57)

is the bare mechanical susceptibility and the thermal force Fth was Fourier-decomposed

accord-ing to4

F_th(t) =

Z

F_th(Ω)e−iΩtdΩ (2.58)

Cavity optomechanics in silicon disks and nanostructured disks : Optomecânica de cavidades em discos de silício e discos nanosestruturados de silício

Instituto de Física Gleb Wataghin

Felipe Gustavo da Silva Santos

Cavity optomechanics in silicon disks and

nanostructured disks

Optomecânica de cavidades em discos de silício

e discos nanoestruturados de silício

Campinas

2017

Felipe Gustavo da Silva Santos

Cavity optomechanics in silicon disks and

nanostructured disks

Optomecânica de cavidades em discos de silício

e discos nanoestruturados de silício

Campinas

2017

Agradecimentos

Resumo

Contents

Chapter 1

Introduction

side-mech

C*

C

a)

b)

Chapter 2

Fundamentals of Cavity Optomechanics

2.1

Preliminary considerations

2.2

Electromagnetic fields and optical modes

2.3

Acoustic field equations: mechanical modes

2.4

Interaction between optical and mechanical modes

Remarks on normalization

2.5

Dynamical backaction in cavity optomechanics