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Citation Zhang, Yinan. 2012. Manipulating Light on Wavelength Scale.Doctoral dissertation, Harvard University.

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Manipulating Light onWavelength Scale

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T S E A S

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esis advisor: Marko Loncar Yinan Zhang

Manipulating Light onWavelength Scale

A

Light, at the length-scale on the order of its wavelength, does not simply behave

as “light ray”, but instead diffracts, sca ers, and interferes with itself, as governed

by Maxwell’s equations. A profound understanding of the underlying physics has

inspired the emergence of a new frontier of materials and devices in the past few

decades. is thesis explores the concepts and approaches for manipulating light

at the wavelength-scale in a variety of topics, including anti-re ective coatings, on-

chip silicon photonics, optical microcavities and nanolasers, microwave particle

accelerators, and optical nonlinearities.

In Chapter , an optimal tapered pro le that maximizes light transmission be-

tween twomedia with different refractive indices is derived from analytical theory

and numerical modeling. A broadband wide-angle anti-re ective coating at the

air/silicon interface is designed for the application of photovoltaics.

In Chapter , a reverse design method for realizing arbitrary on-chip optical l-

ters is demonstrated using an analytical solution derived fromChapter . Example

designs are experimentally veri ed on a CMOS-compatible silicon-on-insulator

(SOI) platform. Among this device’s many potential applications, the use for ul-

trafast on-chip pulse shaping is highlighted and numerically demonstrated.

InChapter , the concept of tapering is applied to the design of photonic crystal

cavities. As a result, the sca ering losses of cavities are suppressed, and light can

be localized in a wavelength-scale volume for a long life-time.

iii

esis advisor: Marko Loncar Yinan Zhang

In Chapter , photonic crystal cavity-based nanolasers with low power con-

sumptionaredemonstratedwith twodifferentprototypes -photonic crystal nanobeams

and photonic crystal disks. e use of graphene is also explored in this chapter for

the purpose of electrically-driven nanoscale light-emi ing devices.

In Chapter , photonic crystal cavities at millimeter wavelength for particle ac-

celeration applications are developed.

In Chapter , a novel design of dual-polarized mode photonic crystal cavities,

and its potential for difference-frequency generations are examined.

iv

Contents

O. Introduction to impedance matching . . . . . . . . . . . . . . .. Derivation of Maxwell’s equations . . . . . . . . . . . . . . . .. Optimal taper function . . . . . . . . . . . . . . . . . . . . . .. Design and performance . . . . . . . . . . . . . . . . . . . . .

A -

I -Q/V -

. Ultrahigh-Q/V cavities based on nanowires . . . . . . . . . . .

. Ultrahigh-Q/Vmicropillar cavities . . . . . . . . . . . . . . . .

P. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .. Lasing threshold of photonic crystal lasers . . . . . . . . . . . .. Photonic crystal nanobeam lasers . . . . . . . . . . . . . . . .. Photonic crystal disk lasers . . . . . . . . . . . . . . . . . . . .. Graphene-contacted micro-LED . . . . . . . . . . . . . . . . .. Conclusion and outlook . . . . . . . . . . . . . . . . . . . . .

P. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

. Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Fabrication and measurement . . . . . . . . . . . . . . . . . .

. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D - -

. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Application for nonlinear optics . . . . . . . . . . . . . . . . .

. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

Included publications

Chapter includes:

Y. Zhang, C. Li, M. Loncar, “Optimal broadband anti-re ective taper,”Optics Let-ters Vol. , pp. ( )

Chapter includes:

I. Frank*, Y. Zhang*,M. Loncar, “Arbitrary on-chip optical lters for ultrafast pulseshaping,” in preparation to submission ( ) (*Equal contribution to the work)Chapter includes:

Y. Zhang, M. Loncar, “Submicrometer diameter micropillar cavities with highQuality factors and ultrasmall mode volumes,”Optics Le ers, Vol. , ( )[Selected for the April , issue of the Virtual Journal of Nanoscale Science andTechnology]

Y. Zhang, M. Loncar, “Design and simulation of nanowire-based highQuality fac-tor nanocavities,” Proc. SPIE, Vol. , W ( )

Y. Zhang, M. Loncar, “Ultra-high quality factor optical resonators based on semi-conductor nanowires,”Optics Express, Vol. , pp. - ( )

vii

Chapter includes:

Y. Zhang,M. Loncar, “Photonic crystal lasers,” inAlexei Baranov andEricTournie,Semiconductor lasers: fundamentals and applications, Cambridge, Woodhead Pub-lishing ( ).

Y. Zhang, M. Khan, Y. Huang, J. H. Ryou, P. B. Deotare, R. Dupuis, M. Lon-car, “Photonic crystal nanobeam lasers,” Applied Physics Le ers, Vol. ,( ) [Selected for the August , issue of the Virtual Journal of NanoscaleScience and Technology]

Y. Zhang, C. Hamsen, J. T. Choy, Y. Huang, J. H. Ryou, R. Dupuis, M. Loncar,“Photonic crystal disk lasers,”Optics Le ers, Vol. , pp. - ( )

Chapter includes:

Y. Zhang, I. Bulu, T. Bo o, W.M. Tam, B. Levi , M. Loncar, “HighQ/V air-modephotonic crystal cavities at microwave frequencies,” Optics Express, Vol. , pp.

- ( )

Chapter includes:

Y. Zhang, M. W. McCutcheon and M. Loncar, “Ultra-high-Q dual-polarized pho-tonic crystal nanocavities,” Optics Le ers, Vol. , ( ) [Selected for theSeptember , issueof theVirtual Journal ofNanoscale Science andTechnology]

I. B. Burgess*, Y. Zhang*, M. W. McCutcheon*, A. W. Rodriguez, J. Bravo-Abad,S. G. Johnson, and M. Loncar, “Efficient terahertz generation in triply resonantnonlinear photonic crystal microcavities,” Optics Le ers, Vol. , ( )(*Equal contribution to the work)

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Listing of gures

. . Comparison of different window functions p(u) [ ] for anti-re-ective coatings at silicon/air interface, and their respective re-ectance R = |r( )| predicted by the Fourier model. . . . . . .

. . Comparison of power re ectance between that predicted by theFourier model and that calculated by solving Maxwell’s Equa-tions. e Dolph-Chebyshev function in this Figure is optmizedfor a cutoff frequency of L/λmax = and has a sideband re-ectance of Rsb = − dB. . . . . . . . . . . . . . . . . . . . .

. . Comparison of different taper functions’ performance, for sili-con/air interface as an example. . . . . . . . . . . . . . . . . .

. . Example of a broadband wide-angle anti-re ective coating be-tween air and silicon. (b)(c) Re ectance dependence on inci-dent angle, at different wavelengths across the solar spectrum, forTE-and TM-polarized light. . . . . . . . . . . . . . . . . . . .

ix

. . (a) A cartoon representation of the lter in action. e red lightis transmi ed through the width modulated region, whereas theblue light is re ected back. (b) An SEM micrograph of a fabri-catedwaveguide showing theW(x)pro le. (c)Anexample targetR(λ). (d) ewidth pro leW(x) that is obtained by applying theinverse Fourier transform obtained from Eq. . to the spectrumfrom (c). (e) e solid lines are target amplitudes of labeled val-ues A. e dashed lines show the resulting re ectance when theW(x) pro le is checked by solving the exactMaxwell’s equationsnumerically. For small values ofA the agreement is excellent, butincreasingly larger values lead to distortion of the shape and dis-crepancies in the amplitude. . . . . . . . . . . . . . . . . . . .

. . (a) Time domain Gaussian input pulse. (b) e wavelength do-main re ectance lter shapes. Eq. . is used turn these ltershapes intoW(x) for the waveguides. (c) Time domain readoutof the input pulse re ected off the lters. e results are a Ham-ming and linear pulse shape, respectively. . . . . . . . . . . . . .

. . (a) SEM micrograph of example device; the inset shows a mag-ni cation of the width modulated region. Cartoons show owof experiment. (b) A set of ve target spectra. e intensity isin a linear saw-tooth pa ern. (c) Normalized, measured re ec-tions from fabricated devices. e dashed lines indicate the un-certainty in the normalization. . . . . . . . . . . . . . . . . . .

. . (a) Schematic of nanowire and mode pro le (Ex components)for fundamental HE mode with d = nm and nclad = .(b) Re ectance of nanowire facets with air and PMMA cladding(HE mode). . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . (a)Schematicof a semiconductornanowirewith DPhCde nedat its end. (b)Transmi ance and re ectance spectra for nanowirewith PhC consisting of PMMA/air pairs. . . . . . . . . . . .

x

. . (a) Schematic of guided-mode cavity. (b) Schematic of Bloch-mode cavity. (c) Dispersion line of Bloch mode with periodic-ity of . a (blue solid), Bloch mode with periodicity of a (pinksolid), and guided mode of nanowire embedded in PMMA (reddash-dot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . (a) Schematic of photonic band tapering. (b) Quality factor andmode volume as a function of number of taper segments. In allcases, the cavity was designed to support one resonance positionat the mid-gap wavelength of nm. (c) Mode pro le of cav-ity modes (Eφ component) with taper segments and mir-ror pairs. Con guration of the tapered gratings is also mapped asbackground. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . Fourier transform of Eφ along wire axis. k-space zones within thelight line are shown in green (light greenwithinPMMA light line,dark green within air light line). . . . . . . . . . . . . . . . . . .

. . Quality factor (red-square) as a function of imaginary part of re-fractive index (κ). e Q value with lossless cladding is indi-cated in black line. e dash lines represent estimation ofQ usingEq. . , while η = . (blue) and (magenta), respectively. . . .

. . (a) Schematic of hexagonal cross-section nanowire embedded inair/PMMA grating. (b)Mode pro le of Ex component of hexag-onal cross-section nanowire embedded in PMMA cladding. (c)Mode pro le of cavity modes (Ex component) with taper seg-ments and mirror pairs. . . . . . . . . . . . . . . . . . . . .

. . (a) Traditional design of micropillar cavities and (b) modi eddesign where the center segment is substituted by titania/silicapairs. e lateral mode pro le of Er component for cavity modeand evanescent Bloch mode that exists inside DBRs are shownon the right of the cavity layout. Improved mode-matching canbe seen in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . (a) Schematic of a -taper-segment micropillar cavity. (b)(c)Electric eld density pro le of the rst and second order mode,respectively. (d) Electric eld density pro le of the third ordermode of the -taper-segment micropillar cavity. (e) Mode dia-gram as a function of taper segment number. . . . . . . . . . . .

. . (a) Mode volume as a function of micropillar diameter. Here allthe modes are rst-order HE modes resonating at nm. (b)Lateral electric eld density pro les of HE (λ = nm), TE(λ = nm) and TM (λ = nm) cavity modes. . . . . . . .

. . (a) Schematic diagram of the rst reported PhC laser [ ]. It isbased on a D-PhC suspended membrane that contains four as-grown semiconductorQWs. (b)(c) Schematic diagramand scan-ning electron micrograph of the rst electrically-injected PhClaser [ ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . (a)(b) Schematic diagram and scanning electron (SEM) micro-graph of the buried heterostructure PhCL. e active region isembedded in an InP layer [ ]. (c)(d) Schematic diagram andSEM of the PhCL bonded on silicon-on-insulator wafer [ ]. .

. . Layout of thematerial system of the semiconductorQWs sampleused to realize nanobeam laser. e energy band of the semicon-ductor QWs are sketched on the right hand side. . . . . . . . . .

. . (a) Energy banddiagramof electrons, light holes (LH) andheavyholes (HH) bands of the semiconductor QWs. (b) e PL emis-sion spectrum of the quaternary QWs peaks at about . μm. . .

. . Layout of the photonic crystal nanobeam cavity design. . . . . .

. . (a)Modepro leof the fundamental cavitymodeof thenanobeamlaser. (b)(c) Spatial Fourier transform of the electric eld com-ponent Ex and Ez at y = plane. (d)Mode pro le of the second-order mode, resonating at a higher wavelength, outside the gainspectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . Scanning electronmicrographs of the fabricated photonic crystalnanobeam lasers. . . . . . . . . . . . . . . . . . . . . . . . . .

. . Illustration of the characterization setup. . . . . . . . . . . . . .

. . (a) Laser emi ed power as a function of the incident pumppower. e emission pro les obtained from the camera at dif-ferent pump levels are shown on the right. (b) e spectrum ofthe emi ed light near the threshold. (c) Output lasing poweras a function of the pump beam position. e pa ern of thenanobeam is superimposed as the background of the picture. (d)Polarization dependence of the lasing mode. . . . . . . . . . .

. . Log-log plot of the L-L curve (dots), with predictions from therate equations using different β factors (solid lines). . . . . . . .

. . (a) Schematic of photonic crystal disk laser and (b) fabricatedphotonic crystal disk laser. e device can be viewed as a hybridbetween (c) microdisk laser and (d) photonic crystal nanobeamlaser with photonic crystal folded back tominimize the transmis-sion losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . (a) Bangedge wavelength as a function of the radius of holes onphotonic crystal disk (black), with thebandgap shaded inbronze.

e green curve shows the corresponding normalized bandgapwidth. (b)(c) Resonantmode pro les at bandedge ofHz compo-nent, for dielectric-band (b) and air-band mode (c), respectively.

. . (a)(b) Images of photonic crystal disk and microdisk lasers withdifferent scaling factors. (c)(d) Electric eld density pro les ofphotonic crystal disk modes. (e) Experimental results of lasingwavelength dependence on diameter of photonic crystal disks(black-dot and red-dot) and microdisks (blue-dot and green-dot). e solid curves show the mode wavelength dependenceobtained using simulations. (f)(g) Electric eld density pro lesof microdisk modes. . . . . . . . . . . . . . . . . . . . . . . .

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. . (a) Images of fabricated photonic crystal disk from scanning elec-tron microscope. (b) Light-in light-out curve for photonic crys-tal disk laser and microdisk laser, respectively. Inset shows thespectrum of photonic crystal disk lasers at . × lasing thresh-old. (c) Log-log plot of the photonic crystal disk laser’sL-L curve(black-dots). e solid curves show L-L curves deduced fromrate equations with different β factors. (d) Lineshape of the las-ingmode above threshold (red-dot), ed with a Lorentzian linefunction (red-solid). Inset shows the emission pro le taken froman infrared camera. . . . . . . . . . . . . . . . . . . . . . . . .

. . (a) Optical microscopic image of a graphene sheet transferredon a SiO substrate. (b) Confocal Raman spectrum of mono-layer graphene. (c) Optical micrograph of a photo-ligrography-pa erned graphene a er O plasma treatment. (d)(e) ConfocalRaman mapping of the pa erned graphene. e Raman signal isspectrally integrated at G line and G’ line resepctively. e whitesquare in Fig. . . (c) indicates the spatial scanning range. . . . .

. . (a) Optical micrograph of pa erned graphene stripe for resistiv-ity measurement. (b) Graphene resistance of various lengths.

e width of the stripe is xed at μm. e black dash lineshows the linear ingof themeasureddata, resulting in agrapheneresistivity of . kΩ/square, and a contact resistance to be . kΩ.

. . (a) e fabricationproceduresof thegraphene-injectedmicrodisklaser. (b)(c) Scanning electronmicrographs of the fabricatedmi-crodisk lasers, a er wet etching (b), and graphene transfer (c). .

. . I-V characteristics of a microdisk device. Inset shows the emis-sion image taken from an IR camera. . . . . . . . . . . . . . . .

. . (a) Schematic of the device. (b) Diagram of tapered photonicbandgap. (c) Transverse mode pro le of the fundamental TM-polarized mode. (d)Mode pro le of the cavity mode. . . . . . .

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. . (a) Images of the fabricated alumina cavity with slant sections atboth ends to facilitate coupling tometallic waveguides. (b) Setupfor transmi ance measurement. (c) Coupling components be-tween the metallic waveguide and the dielectric rod. . . . . . . .

. . (a)Amplitude spectrumwith a large frequency range from GHzto GHz, showing the bandgap of the structure. (b) Amplitudeandphase spectraof the cavitymode fromthe transmi ancemea-surement. e dashed curves are ed with Eq. . . . . . . . . .

. . (a) Re ectancemeasurement of the center-fed antenna, with andwithout the cavity. (b) Re ectance depth as a function of the z-position. Also shown (dashed line), the simulated electric- eldamplitude along the resonator. . . . . . . . . . . . . . . . . . .

. . (a) Schematic of the nanobeam design, showing the nanobeamthickness (dy) andwidth (dx), and the hole spacing (a). (b)TEand TM transverse mode pro les for a ridge waveguide withdy = dx. (c) Transmission spectra for the TE (red) and TM(blue) Bloch modes. e shaded areas indicate the bandgaps forboth modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . (a) Schematic of the D photonic crystal nanobeam cavity, withthe tuning parametersRk andwk in the -segment tapered design.(b,c) Mode pro les of the electric eld components ETE,x andETM,y for the cavity design with dx = a, dy = a. (d,e) SpatialFourier transform of the electric eld component pro les (ETE,x

and ETM,y) in the xz plane (y = ). . . . . . . . . . . . . . . . .. . (a) TE (red) and TM (blue) cavity mode resonant frequen-

cies (do ed lines) as a function of the nanobeam thickness. ebandgap regions of the two modes are shaded. e frequencyseparation (δω) of the two modes with the TE-like mode wave-length xed at . μm by scaling the structure accordingly is plot-ted in green. (b,c) Dependence of the Q factor and nonlinearoverlap factor γ on the nanobeam thickness. . . . . . . . . . . .

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. . Parameters of the higher-order cavity modes for the design withdx = a, dy = a. . . . . . . . . . . . . . . . . . . . . . . . . .

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T .

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Acknowledgments

Today is September , . I am typing the rst few words of my doctoral the-sis; a gleam of late-summer sunshine, calm and serene, illuminates my keyboard.I lean away from my laptop, pause, and rejoice. I cannot help but recall the rstday when I arrived in Cambridge, ve years ago, also a day with sunshine. I waswalking aroundHarvard Square. An oldmanwas playing guitar in front of Au BonPain, calm and serene. I stopped in the midst of the pedestrian ow, paused, andrejoiced. In re ection, that very moment was likely a harbinger of the ve-yearjourney that was to follow.

I recall, the big smileMughees had on his face, a er I told him the nano-pa ernhe made with e-beam lithography the previous night vanished, because I decidedto “wash” the sample with sonication; I recall, the truce handshake with a pat onmyshoulder fromTancredi, followingaheateddiscussionbetweenusondesigninghigh-Qmicrowave cavities; I recall, the huge hug with Birgit at eight in the morn-ing inside the cleanroom, when I was completely powered-off a er an overnightfabricationwork; I recall, Edgar charting our secretive career plan on awhiteboardfor the next decade, as if he were composing a music score of two orchestratedinstruments.

At moments like these, I pause in the pursuit of happiness, and rejoice. Forexperiencingmoments like these, I wake up with happiness and work with a whis-tle in my soul. Because of moments like these, the scienti c research includingthis thesis becomes possible and more meaningful. erefore, I am profoundlygrateful for everyone in the past ve years, who made me pause and rejoice, who

xviii

I learnt from, whom I viewed as role models, who instructed me with insight andwisdom, who did cleanroom fabrication with me shoulder-to-shoulder, whom Ishared laughter and tears with, whom I could speak straightforwardly to withoutbeing judged. ese people are my advisor, my professors, my mentors, my col-leagues, my friends, and my family, whom I thank here.

I thankMarko Loncar, my advisor.Markoofferedme anopportunity that Iwill treasure for a lifetime. I still remem-

ber the day when I rst spoke to Marko, I was at my home in Beijing watching afootball game. RealMadridwas leading, thephone rang,mymompicked it up, andshe could not understand awordon the line. “Probably it was for you,” she toldme.In ve seconds, I realized the call was a phone interview withMarko. It was one ofthe few moments in my life I was petri ed (or “stoned”). I quickly went throughthe formulation ofMaxwell Equations three times inmy head (whichwas the onlyscienti c information I could remember at that moment). Fortunately, Marko didnot test my scienti c knowledge at all. Instead, he told me what his research inter-ests were, and kindly invited me to come to Harvard. I did not understand muchabout the research part, but I did not hesitate long in accepting the Harvard of-fer. Growing up in a middle-class Chinese academic family, I had li le freedom ofchoice anyway. If I said no to Harvard, my grandparents would probably kill me.

Marko has invested invaluably hardwork inmentoringme, which I really appre-ciate. I am not a conventional Asian student: not very low-key, and difficult to bemicro-managed; on the other hand, I have inherited the Asian nature of showingcomplete deference to professors. Even right now, I am still intimidated to openmy mouth when meeting with Marko. Additionally, I had almost no experiencein scienti c research when I joined the group. Hence, from every perspective, itdoes not seem to be a simple job to instruct a student like myself. But Marko hasbeen consistently passionate about and devoted tomy research andme, despite hisinsanely busy schedule. He has thrown a million ideas at me, and realizing a tinyportion of them has been sufficient to graduate. He has always had faith in me,

xix

even when things were not going well for an extended period of time. He neverallowed me to be concerned about nancial support. “Just go and do it,” was hismo o. And, moreover, he le room for me to learn, to grow, and to mellow, as ascientist.

I also thank Marko for being an exemplary group manager. He recruited a fab-ulous “ rst-generation” team, and I was honored to be one of them. We joined thegroup with diverse backgrounds, started the lab from scratch, and worked hard(many times, I found the entire CNS cleanroom occupied by the Loncar group),all inspired byMarko who worked harder than any of us.

I thank the entire Loncar group, a group that never lacks for positive dynamics,close collaborations, and great friendships. And within the group, specially,

I thank Mughees Khan, who did e-beam lithography for the photonic crystalnanobeam laser project. Mughees taught me all the fabrication techniques, hand-by-hand, when I started as a rookie. I will never forget his patience with all my am-ateur mistakes, being a fabrication guru himself. Mughees is an extremely humbleperson, which makes himmy role model.

I thank Irfan Bulu, who collaborated with me in the microwave high-Q cavityproject. Irfan was my rst officemate, and taught me to perform FDTD simula-tions. Irfan is one of those rare people, who derive purely joy from solving a quan-tum optics problemwith pencil and paper on a Saturday a ernoon, with the aid ofthreeDiet Cokes, of course. Irfan has also givenmemany valuable pieces of adviceon becoming a quality scientist, which he most certainly is.

I thank Murray McCutcheon, who co-authored ve academic papers with me,and taught me a great deal about nonlinear optics. Additionally, Murray is my rolemodel of professional ethics. Murray once told me that he rejected an invitationto co-author a paper to be published on Nature Photonics, because he was notconvinced that he contributed enough to the work.

I thank Parag Deotare, who helped build and align the optical setup for testingthe nanolasers. Moreover, he is the one, whom I bothered the most for help in

xx

debugging problems in my research. He has been like an academic elder brotherto me. We also went to many conference trips together, in California, in Sydney,and in Spain, with many good memories brought back as souvenirs. I am sure hewill be a brilliant professor in the near future, because he is already brilliant.

I thank Raji Shankar, an encyclopedia of knowledge. I have learnt a lot fromRaji, with knowledge ranging from mid-infrared silicon photonics to geography,Shakespeare English, British politics, and many other “irrelevant” elds. Raji hasalso been a very generous and faithful friend from the very rst time I met her,freely sharing her judgment and wisdom with me. Our friendship, like a supportgroup, has carried us through ups anddowns in our respective PhDs. And togetherthrough this friendship, we have seen each other grow up.

I thank JenniferChoy. Tome, Jen is beyond a diligent and competent scientist. Iknew Jen before she joined the group, and ever since then, I have been worshipingher unmatched kindness, her extreme humility, and her altruism. Blessed withsuch virtues, Jen has made people around her, including me, be blessed throughher.

I thank Birgit Hausmann. Many moments I spent with Birgit, reminded me ofthe movie “Le fabuleux destin d’Amélie Poulain”, a movie we both like a lot. einnocent joy of life, which she has brought to my andmany others’ lives, is forevertreasured. e pursuit of PhD is not always like being in a comedic movie though.

ere were also some rainy days, and even tsunamis. When a tsunami took placein my life, Birgit was there, helping me overcome the storm, lending me a shelter,and delivering her Amélie optimism to me, for which I cannot thank her enough.

I thank Ian Frank, who is currently collaborating withme on the programmableintegrated lter project. Every time we have a discussion, Ian can grasp my pointfrom the rst sentence, which makes our meeting never last long. To me, it hasalways been a pleasure to sit next to a colleague like Ian, who is much smarter thanme.

I thank omas Babinec. Tomand Iworked together on the diamond nanowireproject. I am also grateful for his efforts in organizing the lab, when the lab juststarted up. In addition, Tom was one of my guides to American culture.

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I thank Ian Burgess, a prodigy, who worked together with me on the TE/TMdual-polarizedcavityproject. Wehave also collaboratedondevelopinghigh-Q cav-ities based on low-index polymers. I have admired Ian from the beginning. As ayoung graduate student, he published like Virgil, assembled a team like Scipio, andpresentedhis results likeCicero. Iwas convincedhewas able to start his owngroupbefore he graduated.

I thank Haig Atikian. Haig and I have been working together on the graphene-contacted microdisk project for the past two years. He has demonstrated a widespectrum of expertise, spanning electronics and photonics. When he started hisPhD, he already had more experience than most of the senior PhD graduates, in-cludingme. I also thank him tobe a really fun person, as a colleague and as a friend.

I thank Mehmet Dundar, Christoph Hamsen, Eric Graves, and Changlin Li,who have worked closely with me, as research interns. Mehmet, a good friendof mine, helped with e-beam lithography for making nanobeam cavities. Unfortu-nately we did not produce results by the time he le . Chris was a catalyst for thegraphene-contactedmicrodisk project, as well as a catalyst for me to grow up. Erichelped transfer graphene lms on semiconductor chips. Changlin provided somevaluable positive feedback to the theory of broadband anti-re ective coatings.

Iwould also like to thankmanypeople outside the group. Without their supportand help, this thesis would not become possible.

Prof. Russell Dupuis (Georgia Institute of Technology) and his students YongHuang and Jae-Hyun Ryou helped grow the III-V semiconductor quantum wellwafers for the nanolasers.

Tancredi Bo o (Schlumberger Doll Research Center) was a close collaboratoron the microwave high-Q cavity project. He is a serious engineer I truly respect,and has also offered me a lot of personal guidance, inside and outside the project.I am also a big fan of his straightforwardness. Wai-Ming Tam and Ben Levi at thesame institute also provided assistance in measuring the device.

Prof. JingKong (MIT) andher studentsYi Song andHyesungParkhelped grow

xxii

and transfer graphene lms for the graphene-contacted microdisk project. eyhave also provided much constructive advice on my research.

Prof. Steven Johnson (MIT), Alexandro Rodriguez, and Jorge Bravo-Abadhelped develop the nonlinear optics theory based on the TE/TM dual-polarizedcavities.

Prof. Vladimir Bulovic (MIT) offered his valuable insight on the high-Q mi-cropillar cavity project, for its applications to coupling to J-aggregate polymer.

Prof. FedericoCapasso (Harvard) kindly allowedus the use of his spectrometerfor testing the photonic crystal disk lasers.

Again, I thank Prof. Federico Capasso, Prof. Vladimir Bulovic, and Prof.Michael Aziz to serve on my thesis commi ee, and for providing useful advice.

Most of the devices in the thesis have been fabricated at Harvard Center forNanoscale Systems (CNS). I would like to acknowledge the support provided byHarvard CNS staff -Jiangdong Deng (JD), Steve Paolini, Steven Hickman, YuanLu, Jason Tresback, Ling Xie, and EdMacomber.

e work in this thesis was nancially supported, in part by the National Sci-ence Foundation (NSF) grant, and the NSF Nanoscale Science and EngineeringCenter at Harvard University. I was also supported by the Graduate Consortiumon Energy and Environment Fellowship at Harvard University for one year.

Last but not least, a long list of people have supported this thesis, in very valu-able ways not directly related to the scienti c results.

I thank the “the family” -Edgar Barroso, Rosario Hubert, Viridiana Rios, andNicolas Chevrier, who I met at the graduate school orientation. Since then, wehave been friends, and their friendships have reshaped me, profoundly. I learntthe secret of living a meaningful and happy life, and the wisdom of oceanism fromEdgar; I learnt the value of empathy with a compassionate soul, and a classy senseof humor fromRosa; I started to examinemy timemanagement, andworkedmoreefficiently, since I worked together with Viri at the library; and Nico’s a itude tolife is a model to live my own life.

xxiii

I need to specially thank Edgar, “the guey”, for the uncountable time we havespent together in the past ve years, through which I was enlightened, in manydifferent ways.

I thank Prof. Arakawa, Prof. Iwamoto, YasutomoOta (TokyoUniversity), Prof.Baba (YokohamaNationalUniversity), andProf. XueFeng (TsinghuaUniversity),who have invited me to present my work to their group, and have given me usefuladvice on my thesis project.

I thank Mikhail Kats (Capasso group, Harvard), whose enthusiasm and devo-tion in science have consistently inspired me.

I thank Jonathan Lee (Hu group, Harvard), who offered me a great amount ofmoral support and encouragementwith his loyal friendship, andurgedme to nishmy thesis before it was too late.

I thank many other friends at Harvard, Daniel Floyd, Daniel Ramos, MichaelBurek, Ray (Jia Hong) Ng, Kirsten Smith, Dongwan Ha, Fatih Degirmenci, NanNiu, John Joo, and Vikas Lonakadi. I also thank the following people, Chong Liu,Yuan Yang, Ji Cheng, Guan Pang, Xi Chen, and Chenhui Li, with whom I havebeen friends since high school.

I thank Uncle Peter who has given me lots of life and career advice as a mentorto me, and who has also invited me to many gourmet meals.

I thank Mrs. Tam for her kindness in providing nutritional supplies across thePaci c Ocean.

Finally, I thank my family -my grandpa and grandma, my uncles and aunts, mycousins, and last,

I thank mymom and dad, to whom this thesis is dedicated.

xxiv

He who knows the truth is not equal to him who loves it; hewho loves it is not equal to him who delights in it.

Confucius, the Analects

1Optimal taper for impedancematching

. I

W , a portion of thepower is re ected because of impedance mismatch. is is a familiar concept indisciplines as diverse as electromagnetism, acoustics, and seismology. Minimizingthe re ectance causedby impedancemismatch is therefore anextremely importantissue in many elds: anti-re ective coatings are used on lenses (ranging from eye-glasses to telescopes); impedance matching in electrical transmission lines max-imizes the power transfer between the source and the load [ ]; microwave anti-re ective components are used in concealingmilitary targets from radar detectionas well as in construction of anechoic chambers for antennameasurements [ ]; inacoustics, horns and megaphones amplify the sound coupling of human voice or

musical instruments to open space.e impedance-matching components in general can be realizedwith the aid of

interference. e quarter-wave coating on optical lenses offers a classic example: asingle-layer coating with a thickness of one-quarter wavelength of light minimizesre ection of that particular wavelength (λ) by canceling out re ection from thefront and the back of the coating [ ]. e quarter-wave impedance transformerused in transmission line employs the same mechanism. However, such interfer-ence-based devices, by their nature, only operate within a narrow bandwidth, andin the case of optics over limited range of incident angles.

In contrast, a broadband impedance-matching component can be achieved byadiabatically coupling the wave from one medium to another through a graded-impedance taper. e tapered layer whose impedance varies continuously canbe realized by a textured interface. When the feature size of the structured inter-face is much smaller than the wavelength, the effective impedance of the blendedmedium is determined by theweighted average of the impedance of the twomedia[ ]. Such structured interface has been inspired by natural biological organisms(e.g. dragon y wings, moth eyes) [ – ], and has been fabricated with differentnanofabrication technologies, including focused ion beam [ ], dry etching withself-assembledmask [ ], colloidal lithography [ ] and interference lithography[ ], and nanoimprint from bio-template [ ].

e performance of the taper is dependent on its tapered length (l), as well asthe tapered impedance pro le [ – ]. As the tapered length increases with re-spect to λ, re ection can be reduced because of the smoother adiabatic conversion.However, in practical realization, the tapered length is o en limited by fabricationand/or other implementation constraints.

In this section, we seek the optimal impedance pro le, whichminimizes the re-ectance across a broad frequency band for a given taper length. While we focuson the applications in optics, our approach is general and the results are applicableto a wide class of physical phenomena that involve wave propagation. For exam-ple, this work could lead to a signi cant improvement of performance in stealthtechnology, and photovoltaics [ , ].

. D M ’

e behavior of electromagnetic wave propagating inside a medium with variedindex can be described by DMaxwell’s equations,

− ∂

∂xE(x, t) = μ(x)

∂

∂tH(x, t)

− ∂

∂xH(x, t) = ε(x)

∂

∂tE(x, t)

( . )

where E and H are the electric and magnetic components and μ and ε are themedium permeability and permi ivity. e telegrapher’s equations of transmis-sion lines and acoustic equations follow the same form. In an electrical transmis-sion line, E andH are replaced by voltage V and current I respectively, while μ andε are replaced by characteristic inductance L and capacitance C respectively. Inacoustic theory E andH are replaced by acoustic pressure p and the acoustic uidvelocity vector v respectively, while μ and ε are replaced by mass density ρ and theinverse of bulk modulus κ− respectively.

For further insight into Eq. . , we perform the following normalization: wede ne the optical length of the tapered section to be L =

∫ l n(x)dx, where n(x)is the material’s refractive index. Next, we normalize the x-axis with respect to itsoptical length. e normalized unit u ∈ [ , ] is de ned by

u =L

∫ x

n(x′)dx′ ( . )

With this normalized unit u, Eq. . (in the time-harmonic form) can be re-wri en as,

dEdu

= i πLλZ(u)H(u)

dHdu

= i πLλ Z(u)

E(u)

( . )

where Z =√

μ/ε is the material impedance. From Eq. . , the electromagneticwave can be separated into a forward propagating wave and a backward propagat-ing wave with a coefficient r,

E(u) = A[exp(i πLλu) + r exp(−i π

Lλu)]

H(u) =AZ[exp(i π

Lλu)− r exp(−i π

Lλu)]

( . )

Importantly, r( ) describes the percentage of light in amplitude that re ects offfrom the u = interface, and r( ) = because we assume there is no backwardpropagating light incident from u = interface.

From Eq. . we can derive

r(u) =E(u)− Z(u)H(u)E(u) + Z(u)H(u)

( . )

Combining Eq. . and Eq. . leads to a nonlinear ordinary differential equa-tion of r(u).

r′ + i πLλr = − ( − r )p(u) ( . )

where p = d lnZ/du contains the information of the taper pro le.e goal is to nd the optimal taper function p(u) that results in minimal r( ),

given a certain bandwidth [λmin, λmax]. First, we employ an approximation to solvethis problem semi-analytically, and then use numerical approaches to nd the ex-act solutions. For a small r, using an approximation r ≪ , Eq. . can be reducedto a linear ordinary differential equation, which has the exact solution,

r( ) =

∫p(u′) exp(i π

Lλu′)du′ ( . )

Note that Eq. . is equivalent to Fourier transform (FT),

r( ) = P( L/λ) ( . )

where P(u) is the Fourier transform of p(u), and L/λ corresponds to the fre-quency of the Fourier transform. It is important to note that P( ) is constrainedto constant for any tapered function p(u).

P( ) =

∫p(u′)du′ = lnZ − lnZ ( . )

whereZ andZ are the impedance of the twomedia. is is because P( ) approx-imates the re ection between the two media without taper.

erefore the problemof nding the optimal taper pro le that hasminimum re-ectance at bandwidth [λmin, λmax] given the optical length L, is equivalent to nd-ing the optimal window function p(u), con nedwithin [ , ], whose Fourier trans-form’s sideband level R∗ above cutoff frequency fc = L/λmax is minimal. R∗(fc) isde ned as,

R∗(fc) = max{| P(f)P( )

|f>fc} ( . )

. O

Interestingly, the above-mentioned problem is analogous to the side-lobe suppres-sion problem in signal-processing in order to minimize so-called ”spectrum leaks”of digital signals that result in the cross-talk between different frequency bands[ ]. Among themanywindow functions utilized in signal-processing, theDolph-Chebyshevwindow function satis es the above-mentioned requirements forp(u):it minimizes the sideband level R∗(fc) for given cutoff frequency fc [ ]. Histor-ically, Dolph-Chebyshev window has been used to optimize the directionality ofphase antenna [ ], and to design tapered section in electrical transmission line(Klopfenstein taper) [ ]. In the la er case, the refractive index has been treatedas constant throughout the tapering, which is only valid for TEMmode in co-axialmetal waveguide. Here, we provide a generalized model for designing broadbandanti-re ective device.

In Fig. . . we compare different tapering pro les logZ(u), their respective

Figure 1.3.1: Comparison of different window functions p(u) [21] for anti-reflective coatings at silicon/air interface, and their respective reflectance R =|r( )| predicted by the Fourier model.

window functions p(u), and their re ectance spectra (in decibels). As a concreteexample, we choose to maximize the transmission (minimize the re ection) be-tween air (n = ) and silicon (n = . ). In the case of a quarter-wave coating, thestep function in the impedance pro le results in two Dirac delta functions witha spacing of unity in p(u). Its Fourier transform is the result of beating betweentwo constant-amplitude functions with a frequency difference, leading to zero re-ectance at L/λ = / + m/ for m ∈ N, which is consistent with the phe-nomenon in quarter-wave coating. At all other frequencies where this condition isnot satis ed, the re ectance level remains high ( %,− dB).

On the other hand, when tapering is applied, for example, re ectance side-lobescan be suppressed over a wide normalized frequency range (the height of themain

lobe remains the same as in the previous case). In addition, for all tapering pro-les except for the Dolph-Chebyshev one, the height of the side-lobes decreases

as the normalized frequency increases. is is expected since for givenwavelengthof incident wave, larger normalized frequencies mean longer structure with moreadiabatic tapering. e Dolph-Chebyshev taper, specially, can have signi cantlysmaller and frequency-independent side-lobes Rsb. As an example, in Fig. . .we plot a Dolph-Chebyshev function that is optimized for a cutoff frequency ofL/λmax = and has a sideband re ectance of Rsb = − dB.

Figure 1.3.2: Comparison of power reflectance between that predicted bythe Fourier model and that calculated by solving Maxwell’s Equations. TheDolph-Chebyshev function in this Figure is optmized for a cutoff frequency ofL/λmax = and has a sideband reflectance of Rsb = − dB.

. D

Having identi ed Dolph-Chebyshev tapering pro le as the most promising one,in Fig. . . we evaluate the validity of our modeling. We compare the analyticalsolution predicted by the Fourier model (with the r approximation in Eq. . ),with the numerical solution to Eq. . . In Fig. . . , the result derived from theFourier transform is plo ed in solid line, while that produced by numeric solutionis plo ed in dots. Here, different from the one in Fig. . . , the Dolph-Chebyshev

function is optmized for a cutoff frequency of L/λmax = and has a sidebandre ectance of Rsb = − dB. Fig. . . shows that the Fourier model prediction isin excellent agreement with the simulation.

Figure 1.4.1: Comparison of different taper functions’ performance, for sili-con/air interface as an example.

Next, in Fig. . . we compare the re ectance level R∗(fc) of Dolph-Chebyshevtaper to other taper functions, calculated with numerical solution. It can be seenthat the Dolph-Chebyshev taper outperforms the other impedance pro les. edifference in performance becomes increasingly dramatic as desired R∗ decreases.For example, in order to reach a re ectance level of− dB, implementing with alinear taper requires an optical length L longer than λ; the same re ectance canbe achieved with a Dolph-Chebyshev taper of an optical length L of only . λ.

Finally, weprovide anexampleof abroadbandwide-angle anti-re ective coatingwith Dolph-Chebyshev taper using our theory. We aim to minimizing the re ec-tion loss between air and silicon across the solar spectrum from nm to nm,with R∗ = − dB at normal incidence. We choose L = . λmax = . μm to sat-isfy this condition, which results in a coating thickness of l = . μm. Fig. . .

Figure 1.4.2: Example of a broadband wide-angle anti-reflective coating be-tween air and silicon. (b)(c) Reflectance dependence on incident angle, at dif-ferent wavelengths across the solar spectrum, for TE-and TM-polarized light.

shows the re ectance spectra as a function of the incident angle at different wave-lengths, for TE-and TM-polarized light respectively. e result is obtained withnite-difference time-domain method (FDTD) code. As shown in Fig. , our de-sign not only demonstrates high anti-re ective property at different wavelengths,as expected, but also performs well within a large incident angle range [ ]. epower loss from re ectance can remain below % within an incident angle of o

across the entire solar spectrum for both polarizations.In summary, starting with DMaxwell equations, we found semi-analytical so-

lution of the optimal taper function, that minimizes the impedance mismatch (re-ection) between two materials that light propagate in. We have demonstratedthat Dolph-Chebyshev taper can achieve the same anti-re ective performancewith a much shorter cones, compared with other taper functions. We believe this

workwill shed light ondesigns of broadband anti-re ective components in variousareas.

2Arbitrary on-chip optical lters for

ultrafast pulse shaping

E -, humankind has been actively developing new

ways of controlling the wavelength and direction of re ected light. People havestrived for increasingly ner control over light using anything frommirrors to pho-tographic lters. e advent of photonic band-gap materials enabled the imple-mentation of structural color as in the scales of a bu er y’s wings [ ]. Additionalelds such as plasmonics and metamaterials [ , ], alongside photonic band-gapmaterials, provide general rules for designing structural color but o en requireheavy computation in order to achieve precise wavelengths; even then there lackssufficient exibility that allows for arbitrary lter response. We present a method

for easily designing a structure that will re ect an arbitrary spectrum. While thismethod can be generally applied to any structure where the refractive index pro-le can be controlled, we have concentrated on implementing arbitrary re ectivelters in compact, on-chip waveguides. ese integrated lters have myriad appli-

cations from on-chip signal routing to compact, ultra-fast pulse shaping [ ].In our recent work we have studied a region with a refractive index modulation

in onedimension [ ] andderived theFourier transform relationship between there ectance spectrum and the refractive index pro le. We show here that this prin-ciple can also be extended to optical waveguides on the SOI platform, where therefractive index pro le is represented by the SOI waveguide-mode’s (fundamentalTE mode) effective index. e modulation of the effective index is controlledby the variation in the width of the silicon waveguide [W(x)].

We previously detailed [ ] the derivation of the ordinary differential equationthat describes the re ection coefficient, r, as a function of the index modulation[n(x)] in time harmonic form. Eq. . shows a variation on this result,

r(λ) = J(λ)∫ l dW

dxexp(i π

neffλ

x)dx ( . )

where J(λ) = −neffdneffdW takes into account the wavelength dispersion of the

optical waveguide as well as the material dispersion. e former dispersion is cal-culated through an eigenfrequency analysis of the waveguide’s cross-section. eintegration in Eq. . is performed over the length of the width-modulated regionof the waveguide. Note that Eq. . is of the form of a Fourier transform. ( eintegration limits can be extended to in nity as the integrand evaluates to zero ev-erywhere outside the width modulated region.) is gives us a powerful methodfor solving forW(x) -from a target r(λ) spectrum -simply by inverting the Fouriertransform. Fig. . . (c)(d) show a target spectrum and the corresponding widthpro le, andFig. . . (b) shows theSEMmicrographof thewidth-modulationpro-le realized on the SOI platform.It is important to note that Eq. . is only an approximate solution -its deriva-

tion relies upon small amplitudes of the re ection coefficient -and the target r(λ)

spectra are not exactly reproduced. e efficacy of the method can be tested bysolving the exact Maxwell’s equations numerically. As expected, our reverse de-sign method’s results start to diverge from the target spectra as the magnitude of rincreases. Fig. . . (e) shows this trend and extends the target spectra into non-physical values greater than unity.

Figure 2.0.1: (a) A cartoon representation of the filter in action. The redlight is transmitted through the width modulated region, whereas the bluelight is reflected back. (b) An SEM micrograph of a fabricated waveguideshowing the W(x) profile. (c) An example target R(λ). (d) The width profileW(x) that is obtained by applying the inverse Fourier transform obtained fromEq. 2.1 to the spectrum from (c). (e) The solid lines are target amplitudesof labeled values A. The dashed lines show the resulting reflectance when theW(x) profile is checked by solving the exact Maxwell’s equations numerically.For small values of A the agreement is excellent, but increasingly larger valueslead to distortion of the shape and discrepancies in the amplitude.

When transferring a continuousmodulation of a waveguide width onto an SOIsample through electron-beam (e-beam) lithography, two problems occur. First,due to the nite length of the lter there is a truncation effect: if an r spectrumwithlarge ’tails’ in its Fourier transform is chosen, there will be signi cant degradationin the resulting spectrum because a large proportion of the Fourier componentswill be lost. Second, errors arise from the nite resolution of e-beam lithography.

is leads to a washing out of the ner features in the r spectrum. Viewed throughtheprismof classical digital signal processing, these two issueswouldbe equivalentto not sampling the data for a sufficiently long period in the time domain, and toobtaining an analog to digital conversion with insufficient bits to properly resolvethe amplitude of the signal.

Figure 2.0.2: (a) Time domain Gaussian input pulse. (b) The wavelengthdomain reflectance filter shapes. Eq. 2.1 is used turn these filter shapes intoW(x) for the waveguides. (c) Time domain readout of the input pulse re-flected off the filters. The results are a Hamming and linear pulse shape, re-spectively.

An important application for this method is in shaping ultra-fast pulses. Bulkyapparatus is currently used for ultra-fast shaping, and it requires precision align-ment [ , ]. By using the SOI waveguide platform our lters allow us to gen-erate arbitrary pulse shapes in an integrated, on-chip fashion. e small footprintof the lters additionally permits a single external pulse to excite many different

pulse shapes in parallel. e key to pulse shaping is the control of amplitude andphase over a wide wavelength range [ ]. As Eq. . solves for r (the re ectioncoefficient) rather than R (the re ectance, or |r| ), the necessary conditions forultra-fast pulse shaping are met.

We now present nite difference time domain simulations showing a singleGaussian pulse is launched into two different width-modulated SOI waveguides,and is converted into three distinct Hamming pulses, and a linear (saw-tooth)pulse, respectively. Fig. . . shows the time domain representation of the inputand the two simulated re ected pulses. All simulations are three dimensional withthe mesh grid size of nm, which is similar to the e-beam lithography resolutionavailable to us.

For experimental demonstration, we have selected ve arbitrary spectra, asshown in Fig. . . (b). e spectra were subsequently translated to differentwaveguides’ widthmodulation shapes [W(x)] using Eq. . . e devices were fab-ricated on SOI wafers (SOITEC) with a nm device layer and a μm buriedoxide layer. e waveguides were wri en using a negative resist (XR - )and kV electron-beam lithography (Elionix ). e exposure windowwasa μm square with a dot-pitch of . nm. A er development (TMAH . )the pa ernwas transferred to the device layer using reactive-ion etching (C F andSF ). SU- Polymer waveguides were de ned using e-beam lithography for spotsize conversion. Finally, the device was capped using PE-CVD deposited silicondioxide to enable facet polishing. Fig. . . (a) shows an SEM micrograph of anexample device prior to PE-CVD deposition.

Filter characterization was performed with a scanned tunable laser (Agilent). Light was coupled onto the chip through a tapered, lensed ber (Oz op-

tics). An on-chip, dB directional coupler was used to extract the re ected sig-nal to an output arm and back to an SU- waveguide as depicted in Fig. . . (a).Fig. . . (b) and (c) show example target spectra alongside experimentally mea-sured spectra; the agreement between the two is excellent. e signal can be dis-torted by Fabry-Perot resonances in the system as well as truncation effects due

Figure 2.0.3: (a) SEM micrograph of example device; the inset shows a mag-nification of the width modulated region. Cartoons show flow of experiment.(b) A set of five target spectra. The intensity is in a linear saw-tooth pattern.(c) Normalized, measured reflections from fabricated devices. The dashedlines indicate the uncertainty in the normalization.

to the nite size and resolution of the width-modulated lters. e absolute val-ues of measured re ectance are based on average values for transmission throughun-modulated waveguides with the uncertainty in the measurement arising fromvariations in the polymerwaveguide facets and insertion and extraction losses thatarise from this coupling method.

In conclusion, we have demonstrated a reverse method for designing arbitrarylters with the footprint of an on-chip waveguide. In addition to allowing the de-

sign of arbitrary phase and amplitude lters these structures show great promise inshaping ultrafast pulses as demonstrated through FDTD simulations. We believethis system provides a novel and feasible platform for control of ultra-fast pulses

with vastly improved footprints and greatly reduced experimental complexity. Inthe future we will look to implementing ultra-fast pulse shaping as well as usingdynamic methods to de ne the lters.

3Impedancematching for designing

ultrahigh-Q/V nanocavities

. U -Q/V

. . I

Semiconductor nanowires have recently emerged as novel light sources for inte-grated photonics. Lasers and electrically-driven light emi ing diodes have beenreported in various material systems (CdS, ZnO, GaN, etc) [ – ]. In all ofthese studies, a semiconductor nanowire is used both as the active medium and

the Fabry-Perot optical cavity; the nanowire body serves as the optical waveguide,while its end facets serve as themirrors bounding the optical cavity. However, dueto the small diameter of a nanowire, signi cant evanescent eld exists outside thenanowire body. is reduces the re ection of the nanowire facets and introducessigni cant losses, thus limiting the Quality factor, Q, of the cavity to∼ [ –]. In addition, the large evanescent eld results in the small modal gain of the

nanowire [ ]. Both of these effects can increase the lasing threshold of nanowirelasers. At the same time, the Fabry-Perot nature of the optical cavity can result inmulti-mode lasing, with lasing wavelengths dependent on the length of nanowire.In many applications single-wavelength emission with well-de ned lasing wave-length is desired. e properties of nanowire lasers and LEDs could be improvedby embedding nanowires into optical structures including photonic-crystals andrace-track resonators [ ], metallic gratings [ ], and micro-stadium resonators[ ]. However, to the best of our knowledge all previously reported structuresbased on nanowires hadQ ≥ , .

In addition to their promise as low-threshold andhigh switching-speednanolasers,semiconductor nanowires offer an a ractive platform for the realization of electri-cally-injected, on-demand, single-photon sources. A reliable and bright source ofsingle photons would nd immediate applications in spectroscopy, quantum in-formation processing and quantum cryptography. Solid-state version of single-photon sources based on self-assembled epitaxially-grown quantum dots (QDs)have been demonstrated in different microcavity con gurations [ – ]. In all ofthese cases, a highQ and small mode volume (V) of the microcavity were instru-mental for achieving single-photon emission. Recently, there have been severalproposals to achieve electrically-driven single-photon sources based on QDs em-beddedwithin semiconductor nanowires [ – ]. However, these reports didnotconsider the use of an optical cavity to improve the performance of such a source.

In thiswork,wepropose anapproach to signi cantly improve theQofnanowire-based optical resonators, and we demonstrate cavities with Q = × andV < . (λ/n) . Our approach is based on engineering a cavity in a one-di-mensional ( D) photonic crystal (PhC) [ , ], which is pa erned around the

nanowire. We demonstrate using numerical modeling that our platform coupledwith aQD is well-suited for operation in the so-called strong-coupling limit of cav-ity quantum electrodynamics (QED), in which there is a coherent interaction be-tween the photons con ned to the optical nanowire cavity and excitons trapped inthe QD [ ].

. . I

In this work, we consider nanowires with a refractive index nwire = . and emis-sion wavelength of λ ≈ nm (e.g. CdS nanowires). However, our approach isgeneral and applicable to different material systems. We assume a circular cross-section of our nanowires [Fig. . . (a)]. is allows us to take advantage of theradial symmetry of the system and signi cantly simplify the analysis. e moretypical hexagonal cross-section of a nanowire is taken into account in the later sec-tion, and good agreement with our simpli ed model is found.

Figure 3.1.1: (a) Schematic of nanowire and mode profile (Ex components)for fundamental HE mode with d = nm and nclad = . (b) Reflectance ofnanowire facets with air and PMMA cladding (HE mode).

First, we model nanowire as a cylindrical optical waveguide using Maxwell’sequations in cylindrical coordinates [ , ], considering the nanowire with air(nclad = ) and low-index material cladding (e.g. polymer, nclad = . ). We ndthat nanowires with air (polymer) cladding support only the fundamental HEmode for d < nm (d < nm). is single-mode regime of operation is pre-cisely the region that we are interested in for this work. Next, the re ectance of

the nanowire facet was studied using the nite-difference time-domain (FDTD)method (grid size< nm), and taking advantage of radial symmetry of the system.

e fundamentalHE nanowiremode is launched towards the nanowire end, andpower re ected from the facet is monitored [Fig. . . (b)]. It can be seen that thenanowire facet re ectance is smaller than % ( %) for single-modenanowires inair (polymer). Similar results have been found previously by other authors [ –]. Such a poor facet re ection is responsible for large mirror losses and small

quality factor (Q ∼ ) of the optical cavity formed by the nanowire.

Figure 3.1.2: (a) Schematic of a semiconductor nanowire with 1D PhC de-fined at its end. (b) Transmittance and reflectance spectra for nanowire withPhC consisting of 30 PMMA/air pairs.

In order to increase facet re ection and overall Q of the nanowire-cavity, weconsider the structure shown in Fig. . . (a) [ ]. e system consists of ananowire embeddedwithinpolymer cladding,with DPhCde nedat thenanowireend. One particularly appealing approach is to use Poly(methyl methacrylate)(PMMA) electron-beam lithography resist as the cladding material. In this case,fabrication of the D PhC structure is very simple, and can be accomplished usingelectron-beam lithography, only.

In Fig. . . (b) we show the re ectance (R) and transmi ance (T) spectra forthe d = nm nanowire with a grating of periodicity a = nm. It can beseen that within the bandgap λ ∈ ( nm, nm), the re ectance can be as highas %, which is almost a -fold improvement over the facet re ection of barenanowire. InFig. . . (b)wealso show the sca ering loss, de ned asL = −R−T

(absorption losses of the nanowire are neglected). is loss can be a ributedto the sca ering at the nanowire - D PhC interface, due to a mismatch betweenthe propagating fundamental HE mode inside the nanowire and the evanescentBloch mode that exists inside the grating section [ ]. In the next section wewill show that this sca ering loss can be signi cantly minimized using techniquessimilar to the ones developed by Lalanne and colleagues [ , ]. Outside thebandgap, at the short-wavelength (high-frequency) end, the loss increases signi -cantly due to coupling to the leaky modes that exist inside the mirror section (theBloch modes of the photonic crystal cross the light line and radiate energy).

It is important tomention that the position and width of the photonic bandgapwill depend strongly on the nanowire diameter. erefore, in an experimentalrealization of our platform, it is important that nanowires are straight and with-out signi cant diameter variations. Sophisticated growth techniques that result instraight and uniform nanowires have recently been demonstrated [ , , , ,]. Finally, wenote that photonic bandgap can closewhen the nanowire diameter

is larger than nm, due to the presence of higher order modes.

. . S

In this section we provide a detailed design of photonic crystal nanowire cav-ities taking advantage of the D PhC mirrors concept introduced in the previ-ous section. We start with the Fabry-Perot cavity shown in Fig. . . (a) wherethe nanowire section of length s is sandwiched between two D PhC mirrors.We call such cavity guided-mode cavity. We select the same parameters used inFig. . . (b) (d = nm, a = nm, pairs of PMMA/air grating at eachside), in order to assure single-mode behavior of the nanowire, as well as to posi-tion the emission wavelength of CdS ( nm) at the midgap of the PhC mirror.

is provides the smallest penetrationdepth into thePhCmirror leading to a smallmode volume, as well as minimizes the mirror transmission loss thus maximizingthe overall Q. By tuning the cavity length (s), cavity modes with different sym-metries can be formed and positioned precisely at the mid-gap frequency. e Q

factors of these modes are , and , for s/a = . , . , . , respectively.We note that using this simple guided-mode approach it is possible to realize cavi-ties with higherQ at the expense of an enlargedmode volume, by tuning the cavityresonance closer to the air-band edge. However, these modes are not of interest inthis work due to the reducedQ/V ratio.

e totalQ factor of the D PhC cavity can be separated into transmission lossdue to the nite length of the mirror, and the above-mentioned sca ering:

Q=

Qsc+

Qw( . )

In our case, layers of PhCmirror result in quality factor from waveguide lossof Qw ∼ , which is signi cantly larger than the Q of guided-mode cavities ob-tained above. is implies that thedominant lossmechanism is sca ering atmirrorinterfaces, as noted previously in Ref. [ ]. In that work it was suggested that themodepro lemismatchbetween guidedmode at the cavity region andBlochmodeis the main reason for large sca ering losses. is mismatch can also be viewed asthe effective impedancemismatchbetweencavitymodeand theBlochmodeprop-agating in the mirror. From Fig. . . (c) we can see that the cavity guided modehas a mode index of ncavity = . , while the evanescent Blochmode positioned atthe midgap frequency has a mode index of nmid−gap = . .

To eliminate this index difference and the resulting mode mismatch, we substi-tute the uniform cavity region with a PMMA/air grating with the same duty cyclebut a smaller period [Fig. . . (b)]. By choosing the period of this cavity sectionto be w = . a, we tune the cavity resonance to the mid-gap frequency of themirror. Fig. . . (c) shows the photonic bands of the cavity segment as well as thePhC mirror. e propagating dielectric-band Bloch mode supported in the cav-ity region couples to the evanescent Bloch mode that exists within the bandgapof the PhC mirror to form the cavity mode. In contrast with the former design,the cavity mode here is a Blochmode (instead of guidedmode), and therefore thecavity is referred to as Bloch-mode cavity. eQ factor of our Bloch-mode cavity

Figure 3.1.3: (a) Schematic of guided-mode cavity. (b) Schematic of Bloch-mode cavity. (c) Dispersion line of Bloch mode with periodicity of . a (bluesolid), Bloch mode with periodicity of a (pink solid), and guided mode ofnanowire embedded in PMMA (red dash-dot).

is , and the mode volume is . (λ/n) . As expected, the increase in Q isdue to the reduced mode pro le mismatch between the Bloch mode of the cavityand evanescent Blochmode of the mirror. Recently another group have proposedsimilar cavity design in order to realize high-Q cavities [ ].

Further Q enhancement can be realized by tapering the mode pro le from thedielectric band tomid-bandgapbyadding taper segments, as shown inFig. . . (a).Two degrees of freedom are available to achieve the transition between the cavityBlochmode and the evanescent Blochmode of the mirror, namely the period (w)and the duty cycle of each segment. Here we keep the duty cycle xed at . , aswe did in the cavity segment. Linear interpolation of grating constant ( π/w) ofeach segment is used to carry out the tapering process. Mid-bandgap resonanceis achieved by altering the length of the central segment w . Similar tapering tech-

niqueswere previously used to realize high-Qheterostructure cavities basedon DPhC waveguides [ – ].

Figure 3.1.4: (a) Schematic of photonic band tapering. (b) Quality factorand mode volume as a function of number of taper segments. In all cases,the cavity was designed to support one resonance position at the mid-gapwavelength of nm. (c) Mode profile of cavity modes (Eφ component) with

taper segments and mirror pairs. Configuration of the tapered gratings isalso mapped as background.

In Fig. . . (b) we present the dependence of the Quality factor on the num-ber of segments placed in the taper region (one-segment structure corresponds toBloch-mode cavity). For the case of mirror pairs at each end (not including ta-per segments), Q rst increases logarithmically as the number of taper segmentsincreases, and then levels off when the number of tapers is larger than . Whenwe increase the number of PhC mirror pairs to , the logarithmic dependenceof Q on number of segments is recovered. erefore, we conclude that for largenumber of taper segments, transmission losses become dominant, and more mir-ror pairs are required. In Fig. . . (b), we also present the mode volume of the

cavity calculated using Eq. . .

V =

∫V ε (r) |E (r)| dVmax [ε (r) |E (r)| ]

( . )

As expected, the mode volume increases as the number of taper segments in-creases. However, the increase ofV is modest, especially for large number of tapersegments. Moreover, in all cases the mode volume is smaller than . (λ/n) . isis due to the fact that we deliberately positioned the cavity mode precisely at themidgap frequency, and therefore the cavity eld decays rapidly inside thePhCmir-ror. In the case shown in Fig. . . (c), Q/V can be as high as . × per cubicwavelength in material.

In previous paragraphs, we explained the ultra-high Q of our nanowire-basedcavities using the mode-matching arguments. e high Q factor can also be ex-plained by looking at the distribution of the cavity eld energy in momentumspace (k-space, spatial Fourier transform space) [ , – ]. In Fig. . . , weshow the Fourier transform of the electric components Eφ of the cavity mode.kz = ncladω/c de nes the light line of the cladding. Components with kz smallerthan the light line support plane waves leaking energy radially into the cladding,and are responsible for sca ering losses. e light cones in PMMA and air arecolored in light green and dark green, respectively. We can see that as we includemore taper segments, the Fourier spectrumof themode pro le concentratesmoretightly around the edge of the Brillouin zone kz = π/a, reducing the energy of themode within the light cones, thus reducing the sca ering. For comparison, thespatial spectrum of the guided-mode cavity, shown in black, extends signi cantlyinside the air and polymer light line, indicating large sca ering losses.

Our system is very similar to micropost (micropillar) optical microcavities thathavebeen extensively used in vertical-cavity surface-emi ing lasers [ ] and singlephoton sources [ , – ], and more recently proposed in the context of semi-conductor nanowires [ ]. However, the big advantage of our approach that com-bines bo om-up synthesized nanowires with top-down fabricated photonic-crys-

Figure 3.1.5: Fourier transform of Eφ along wire axis. k-space zones withinthe light line are shown in green (light green within PMMA light line, darkgreen within air light line).

tals (only electron-beam lithography step), is simple fabrication procedure [ ].Complicated epitaxial growth of Bragg mirror, typical for conventional micropostcavities, is not required. Our technique is therefore fully compatible with differ-ent nanowire growth approaches including solution-based synthesis, vapor-solid-liquid, etc.

Next, in Fig. . . we show how the overall quality factor depends on thematerial losses of the cladding (red-square), in the case of the optimized cavity(Q = . × ). Refractive index of the cladding is assumed to be of the formnclad + iκclad, resulting in loss ( /Q) of the form η × κclad/nclad, where the co-efficient η takes into account the overlap between the mode and lossy cladding(η < ). us the overallQ factor can be derived using Eq. . .

Q=

Qlossless+ η

κcladnclad

( . )

UsingFDTDwe found that η = . (Fig. . . ). In the caseofPMMAcladding,κclad is smaller than × − for wavelength of nm, that is the absorption co-efficient of PMMA is smaller than α = . mm− [ ]. As shown in Fig. . . ,

Figure 3.1.6: Quality factor (red-square) as a function of imaginary part ofrefractive index (κ). The Q value with lossless cladding is indicated in blackline. The dash lines represent estimation of Q using Eq. 3.3, while η = .(blue) and (magenta), respectively.

this loss results in small reduction of overall quality factor from Q = × toQ = × . It is important to note that absorption coefficient of PMMA is evensmaller (almost an order of magnitude) at red, near-infrared and telecom wave-lengths (with the exception of two bands around nm and nm [ ]) andtherefore even higher quality factors can be obtained with nanowires emi ing atthese longer wavelengths.

Depending on the crystal structure of the nanowire material and the preferredgrowth direction, the nanowire cross-section can be triangular, square, hexagonaland so on. Herewe consider nanowireswith hexagonal cross-section embedded inour D PhC cavity [Fig. . . (a)]. First, the waveguide mode [Fig. . . (b)] andeffective mode index of a nanowire with hexagonal cross-section is found. Next,the diameter of the equivalent cylindrical nanowire is chosen, so that it supportsthe mode with the same effective mode index. en, a high-Q cavity is designedfor the cylindrical nanowire by taking advantage of radial symmetry and usingthe optimization procedure described above. e same cavity design is then ap-

plied to the nanowire with hexagonal cross-section. In this way the optimization isdone using D-FDTDwith radial symmetry, which is signi cantly faster than per-forming a D-FDTD computation, which would be necessary for nanowires withhexagonal cross-section. Using this approach, we designed cavity for a nanowirewith dhex = nm, using mirror pairs and taper segments. e resonance atλ = nm had a Q = . × and V = . (λ/n) [Fig. . . (c)]. Furtheroptimization of the structure using DFDTDcould result in even higherQ values.

Figure 3.1.7: (a) Schematic of hexagonal cross-section nanowire embeddedin air/PMMA grating. (b) Mode profile of Ex component of hexagonal cross-section nanowire embedded in PMMA cladding. (c) Mode profile of cavitymodes (Ex component) with taper segments and mirror pairs.

. . L -

As rst noted by Purcell [ ], the emission rate of a radiating dipole can be mod-i ed by placing the dipole inside an optical cavity. e enhancement of the ra-diative emission rate into the cavity mode, when compared to the spontaneousemission rate without the cavity, can be described by the Purcell factor where Vis mode volume of the cavity and εM is the dielectric constant at the eld inten-sity maximum point. If F ≫ , the dipole will emit much faster into the cavitythan into free space. is increase in the radiative recombination rate is bene cialfor the reduction of nanowire lasers threshold, and could result in the realization of

threshold-less lasers [ ]. In the case of a single-photon source based on aQDem-bedded within the semiconductor nanowire, the large Purcell factor means a highphoton-production rate and a smaller probability for nonradiative recombination.Moreover, photons are preferentially emi ed into the well-de ned cavity modesand thus can easily be coupled out, using for example proximal optical waveguides,thereby increasing the overall collection efficiency of generated photons. In oursystem we nd that Purcell factor (F = F /nwire) can be as high as . × when-segment taper is used with material losses considered. It drops to . × and. × when four and two taper segments are used, respectively. Large Pur-cell factor is due to the ultra-high Q and very small V in our cavity. e cavitymay even enter the strong-coupling regime of light-ma er interaction [ , ], inwhich coherent exchangeof energybetweenphoton trapped in the cavity and exci-ton trapped in theQDexists. is happens when cavity eld decay rate κ = ω/ Qand exciton decay rate γ (exciton loss due to the emission into non-cavity modesand non-radiative recombination channels) are smaller than exciton-photon cou-pling parameter g [ , ]. For the cavity de ned precisely around the quantumdot positioned in the center of the nanowire (at the electric eld maximum), andtransition dipole moment aligned with the electric eld dipole, the coupling pa-rameter can be expressed as g = Γ

√V / V, where g is the Rabi frequency of the

system on resonance and V = ( c λ ε )/( πΓεM). Here Γ = nwireω μ / πε ~cis the spontaneous emission rate in the material. Assuming radiative life time ofexciton without cavity to be ns, that is its radiative rate of Γ = . GHz we getcoupling parameter g = GHz. Typical values for non-radiative decay rates ofexcitons are bellow this value [ ], and therefore the limiting factor for strong-cou-pling regime is cavity eld decay rate. erefore, we conclude that in our system,g > κ, γ when Q > (κ < GHz), and the system is well into the strongcoupling regime even when only three taper segments are used.

Finally, we note that themost promisingmaterial systems used for realization ofheterostructure QDs within nanowires are based on semiconductors with refrac-tive index larger than nwire ≈ . [ – ]. For example, in Ref. [ ], the nanowireis basedonGaPplatformwithnwire = . . Our hybridnanowire- DPhCplatform

is general, and we con rmed that it can be used to realize cavities with highQ andsmall Vwith these nanowires with larger refractive index.

. . C

Designof anultra-highQoptical nanocavity consistingof a semiconductornanowireembedded in D photonic crystal has been demonstrated. e mechanism of ef-fectively suppressing cavity losses has been theoretically analyzed, and a cavitywithQ = × andmode volume smaller than . (λ/n) has been designed. Ul-tra-high Purcell factor, and operation in the strong-coupling regime are predictedin the proposed platform. HighQ/V cavities based on nanowires with hexagonalcross-section have also been designed. Our system is similar to micropost (mi-cropillar) optical resonators that have been used in vertical-cavity surface-emi inglasers and single photon sources. However, our approach requires simple fabrica-tion procedure that combines bo om-up nanowire synthesis with top-down sin-gle-step e-beam lithography.

. U -Q/V

. . I

Micropillar optical cavities, typically used in low-threshold vertical-cavity surface-emi ing lasers (VCSELs) [ – ], have recently a racted considerable a entionas a promising platform for solid-state implementations of cavity quantum elec-trodynamics (cQED) experiments [ , ]. ese applications bene t from largequality factor (Q) that can be obtained in micropillar cavities, which in turn re-sults in a long photon life time κ = ω/ Q (ω is the radial frequency of the cav-ity mode). For example, high-Q micropillar cavities (Q ∼ , ) have beendemonstrated recently for large diameter (d = ”μm) pillars, resulting in a rel-atively large mode volume [V > (λ/n) ] [ ]. However, for applications incQED, coupling between an emi er and a photon localized in the cavity requiresa large Rabi frequency g that is proportional to /

√(V). In order to decrease the

mode volume, it is necessary to decrease the pillar diameter and thereby improvethe radial con nement of light. However, this can lead to a signi cant reductionin the Q factor in the traditional micropillar designs [ ], and the best Q factorsreported in the case of sub-micron diameter micropillars have been theoreticallylimited to approximately , [ ].

In this section, we theoretically demonstrate sub-micron diameter micropillarcavities with an ultra-highQ/V that is three orders of magnitude larger than previ-ously reported. is is achieved by simultaneously increasing theQ (Q ∼ × )and reducing the mode volume [V ∼ . (λ/n) ], using a bandgap taperingmethod developed recently in D photonic crystal structures [ , ]. While theproposed approach is general and can be applied to a range ofmaterial systems andapplications, our cavities are designed to operate at nm wavelength and there-fore are suitable for coupling to nitrogen-vacancy (NV) color centers embeddedwithin diamondnanocrystals positioned at themiddle of the cavity. NV color cen-ters have recently a racted signi cant a ention as promising quantum emi ers[ ]. NVs emission is broad band ( nm- nm) with stable zero-phonon lineat nm visible even at room temperature.

. . D

Our micropillar is based on two distributed Bragg mirrors (DBR) that consists ofTiO (nTiO = . ) and SiO (nSiO = . ) alternating layers, and a TiO spacerof thickness s sandwiched between them [Fig. . . (a)]. e micropillar cavityQis inversely proportional to the cavity losses that in turn can be separated into twocomponents: the transmission losses due to the nite length of the DBRs, and thesca ering losses at the spacer/mirror interfaces. e former can beminimized (fora given number of TiO /SiO pairs) using a ”quarter stack” DBR that consists ofTiO /SiO layers with thickness λ/ neff, where λ = nm. is also maximizeslight con nement along the pillar axis, resulting in aminimizedmode volume. Forexample, in the case of a micropillar with diameter d = nm, the thicknesses ofTiO and SiO layers in DBR are . nm and . nm, respectively, resulting in

Figure 3.2.1: (a) Traditional design of micropillar cavities and (b) modifieddesign where the center segment is substituted by titania/silica pairs. Thelateral mode profile of Er component for cavity mode and evanescent Blochmode that exists inside DBRs are shown on the right of the cavity layout. Im-proved mode-matching can be seen in (b).

total DBR periodicity of a = nm. Next, the thickness of the spacer is chosen(s = nm) in order to position the cavity resonance at nm, and the qualityfactor (Q) of such cavity is obtained using nite-difference time-domain method(FDTD). We nd Q ∼ which is consistent with previous reports [ ]. elow Q is a ributed to sca ering losses arising from the mode pro le mismatchbetween the localized cavity mode and evanescent Bloch mode inside the DBRs[ ]. Fig. . . (a) illustrates the pro le mismatch for Er component. To suppressthismodemismatch and the resulting sca ering losses, we use themodematchingtechnique previously developed for D photonic crystal cavities [ ]. We substi-tute the uniform center segment with a single TiO /SiO pair with the same as-pect ratio as in the DBR but a smaller thickness w. When w = . a, the cavityresonates at λ = nm with Q = , , a -fold improvement over the con-ventional design.

In order to increase the Q factor further, we incorporate more TiO /SiO seg-ments with varying the thicknesses wi (i is the segment number). is can also

Figure 3.2.2: (a) Schematic of a -taper-segment micropillar cavity. (b)(c)Electric field density profile of the first and second order mode, respectively.(d) Electric field density profile of the third order mode of the -taper-seg-ment micropillar cavity. (e) Mode diagram as a function of taper segmentnumber.

be seen as a ”tapered DBR” approach, where each taper section further reducesthe mode mismatch in order to suppress the sca ering losses. In Fig. . . (a), weuse tapered segments and DBR pairs at each side. In order to set the resonat-ing wavelength at nm, the thickness of each taper segment is precisely tunedto w = . nm, w = . nm, w = . nm, and w = . nm. e re-sulting mode has aQ factor of , and a mode volume of . (λ/n) , whichrepresents at least three orders of magnitude enhancement of Q/V compared toany previous micropillar designs. As shown in Fig. . . (b), this high-Qmode hasan anti-node at the central TiO segment and therefore is ideally suited for cou-pling to quantum emi ers, such as an NV center in diamond or a semiconduc-tor nanocrystal, embedded within this layer. We also nd a second-order cavity

mode [Fig. . . (c)] at wavelength of nm with a respectable Q = , .eQ factor of the fundamental mode can be enhanced by increasing the tapering

process. For instance, we obtain Q = , , and V = . (λ/n) withtaper segments. However, increasing the cavity length pulls higher-order modesfrom the dielectric band into the bandgap, as shown in Fig. . . (d) and (e). esehigher ordermodes can potentially couple to the emi er placed at the center of thecavity and the resultingmulti-mode cavity is notdesirable. erefore fromhereon,we only consider -taper-segment cavities, which limits ourQ to , .

Figure 3.2.3: (a) Mode volume as a function of micropillar diameter. Hereall the modes are first-order HE modes resonating at nm. (b) Lateralelectric field density profiles of HE (λ = nm), TE (λ = nm) andTM (λ = nm) cavity modes.

Next, we optimize the diameter of our micropillar cavity to minimize its modevolume. It can be seen in Fig. . . (a) that the smallest mode volume is obtainedat d = nm. For d < nm, V increases due to the reduced con nementin the axial direction: the effective impedance contrast between TiO and SiOis reduced for small diameters, and therefore the width of the bandgap decreases,resulting in deeper penetration of the cavity mode into the DBRs, thus increasing

V. For d > nm, however, the mode is almost completely con ned within thepillar and the neff of each segment approach the refractive index of the material.

erefore, the width of the bandgap remains approximately constant as the pillardiameter increases and the axial con nement remains the same. However, V in-creases due to the larger mode cross-section (radial con nement increases). etrade-off between radial and axial con nement results in an optimized diameter ofd = nm. For the -taper-segment cavity, we also nd the cavity modes withTE and TM pro les at wavelengths of nm and nm, respectively. Both ofthese modes peak at the central TiO segment. However, the lateral electric elddensity pro les shown in Fig. . . (b) indicate that these modes have a node atthe center of the micropillar and therefore will not couple efficiently to the nano-emi er placed at the center. Moreover, these modes are detuned from the emis-sion spectrumof anNVcenter and therefore are not of interest. It is also importantto note that if the pillar diameter increases further, additional higher-ordermodes,with the same azimuthal order as HE (e.g. EH ), are allowed. ese modes cancouple to the fundamental HE cavity mode and thus introduce additional lossmechanism [ ], and reduce theQ factor of the fundamental cavity mode.

. . C

In conclusion, we have demonstrated that high Quality factor micropillar cavitiescan be realized with sub-micron diameter pillars. We have engineered the cavi-ties with a record low mode volume of V = . (λ/n) and a Quality factorof , . We expect, however, that realistic fabricated structures will have re-duced quality factors due to fabrication-related imperfections, including surfaceroughness, slanted walls and material absorption. One possible approach to over-come these problems is based onoxide-aperture design [ ]. Wepredicted that byembedding a diamond nanocrystal with an NV color center at the middle of thecavity, the strong coupling limit of light-ma er interaction can be achieved. Ourmethod can be easily adapted to different material systems and enable realizationof an ultra-highQ/V cavity in AlAs/GaAs platform suitable for realization of low-

threshold VCSELs, for example.

4Photonic crystal lasers

. I

Photonic crystal (PhC) [ , ], material with a periodic variation of refractiveindex, is a versatile platform for manipulating the propagation, re ection and re-fraction of light. Speci cally, light can be localized within the photonic bandgap(PBG), where the propagation of light is prohibited via Bragg Sca ering. is isof great interest for realization of functional optical devices including nanocavi-ties and waveguides. Empowered by numerical simulation methods, such as thenite-difference time-domain (FDTD) and the nite element method (FEM),and state-of-the-art nano-fabrication techniques (e-beam lithography, reactive-ion etching, scanning electron microscopy, etc.), photonic crystal cavities can bedesigned and fabricated with ultra-high Quality factors (Q) of over a million andsmall mode volumes (V) close to the diffraction limit [∼ (λ/ n) ]. Current tech-

nologies enable production of one-dimensional ( D) [ , , ] and two-dimen-sional ( D) [ , , – ] PhC cavities with high integration capacity on asemiconductor chip. High-quality three-dimensional ( D) photonic crystal cav-ities have also been demonstrated [ ], yet scalable productions of D-PhC de-vices still remains challenging.

Photonic crystal lasers (PhCLs) are lasers that utilize photonic crystal cavitiesto achieve the optical feedback. e history of developing lasers con ned withphotonic bandgap can be traced back to the vertical-cavity surface-emi ing lasers(VCSELs) [ ], where distributed Bragg re ectors (DBRs) are used to con nelight in the vertical direction: in fact, the concept of Bragg re ector can be seenas an one-dimensional photonic crystal. On the other hand, typical VCSELs havetheir device diameters orders of magnitude larger than the operating wavelengths,resulting inweakphotoncon nement in the lateral directions andcorrespondinglya large mode volume.

Figure 4.1.1: (a) Schematic diagram of the first reported PhC laser [106]. Itis based on a 2D-PhC suspended membrane that contains four as-grown semi-conductor QWs. (b)(c) Schematic diagram and scanning electron micrographof the first electrically-injected PhC laser [107].

e rst claimed photonic crystal laser was demonstrated in Axel Scherer’sgroup at Caltech in year [ ] using a D-PhC slab cavity. In this work, asuspended InGaAsP membrane perforated with a D triangular la ice of holeswas used to localize a cavity mode within mode volume V ∼ . (λ/n) usingDBR in the lateral directions, and total internal re ection (TIR) in the vertical di-rection, as shown in Fig. . . (a). e suspended slab contained four as-grownquantum wells that provided the optical gain at telecommunication wavelength(∼ nm). e laser was pumped optically, using a pump laser operating atlower wavelength ( nm). Furthermore, the pumped laser was pulsed with %

duty cycle and the substrate was cooled to K to avoid excessive heating of thedevice. Owing to their planar nature, large arrays of PhCLs can be easily fab-ricated and lithographically tuned to operate at different wavelengths. is is aclear advantage over VCSELs, for example, which require epitaxial growth of alarge number of Bragg-mirror layers using metal-organic chemical vapor deposi-tion (MOCVD) or molecular beam epitaxy (MBE).

Following this rst demonstration, the PhCL operating at room temperaturewith continuous wave (CW) optical excitation was demonstrated by the Leegroup at IST in year [ ]. In this work, a D photonic crystal slab con-taining semiconductor QWs was fabricated on a low-index Al O substrate to fa-cilitate heat dissipation. e same group at IST later demonstrated the rstelectrically-driven PhCL in year [ ]. e main difficulty associated withrealization of electrically-driven PhCL was placement of electrodes close to thecavity to enable efficient carrier injectionwithout inducing large optical losses dueto metal absorption which can signi cantly reduce the cavityQ and be detrimen-tal for the laser performance. In their work, the IST group solved the problemby leaving a small post at the center of the cavity supporting the D PhC slab, asshown in Fig. . . (b) and (c). e post acts as an electronic wire that deliverscarriers directly in the center of the cavity without degrading theQ signi cantly.

Owing to their planar nature, PhCLs can easily be integrated with passive opti-cal components. ere are two main approaches demonstrated so far. e rst

Figure 4.1.2: (a)(b) Schematic diagram and scanning electron (SEM) micro-graph of the buried heterostructure PhCL. The active region is embedded inan InP layer [109]. (c)(d) Schematic diagram and SEM of the PhCL bondedon silicon-on-insulator wafer [110].

is based on the so-called buried heterostructure [ ]: the as-grown InGaAsPquantum wells are rst pa erned into a quantum box and a passive InP layer isthen re-grown on top of them, and the sample is planarized. e D-PhC cav-ity is then de ned using e-beam lithography so that it spatially overlaps with thequantum wells. An optical waveguide is also realized in the same lithography stepto enable efficient in-coupling of pump light and out-coupling of generated lasersignal, as shown Fig. . . (a) and (b). is compact structure not only greatly im-proves the pumping and collection efficiency through direct waveguide coupling,but also avoids the excessive heating. e second example makes use of planaris-ing BCB polymer to bond III-V active materials onto a silicon-on-insulator (SOI)wafer [ ]. e process starts by de ning anoptical waveguide in the device layerof SOI, followed by planarization of the SOI wafer and bonding with III-V mate-rial. A er the bonding, the PhCLs is fabricated in the activematerial, on top of theSi waveguide. In this way, PhC cavity is evanescently coupled to the siliconwaveg-

uide, as shown in Fig. . . (c) and (d), which facilitates in-and out-coupling oflight. is approach has reduced coupling efficiency than the rst, direct waveg-uide coupling, method, but in turn enables PhCL integration with silicon photon-ics.

e advantages of photonic crystal lasers go well beyond their small footprintsand integration capacity. e most prominent advantage of PhCLs is that they re-quire low threshold powers, mostly a ributed to their small mode volumes andhighQuality factors. Optical pumping thresholds have been reported to be on theorder of one microwa for semiconductor quantum well lasers [ , ] and aslow as tens of nanowa s for semiconductor quantum dot (QD) lasers [ , ].In the case of electrical pumping, threshold currents as low as nA have beenreported, which is orders of magnitude smaller than other laser devices [ ].In fact, theory predicts that threshold-less lasing is achievable in an ideal single-mode PhC cavity fabricated in perfect D photonic crystal with omni-directionalbandgap. In this case, radiative emission coupled to all non-lasing optical modesis prohibited [ , ]. A detailed theoretical analysis of lasing thresholds of Ph-CLs is presented in the next section.

In addition to low thresholds, PhC lasers can also be drivenwith a highmodula-tion speed: the highQ/V ratio of PhC cavities result in cavity Purcell effect, whichextensively reduces the radiative lifetime of carriers [ ]. A modulation speedthat exceeds GHz has been demonstrated based on a D PhC laser [ ]. isfeature is especially important for information processing applications. Moreover,the design exibility of PhC cavities allows tailoring the lasers’ polarizations [ ]and far- eld pa erns [ ]. Finally, the wavelength of PhCLs can be controllablytuned, by employing opto-mechanic effects [ ] or liquid crystals [ ].

. L

It has been rst predictedbyYamamoto et al. that a high spontaneous emission fac-tor (β), de ned as the fraction of spontaneous emission that couples to the lasingmode, is responsible for decreasing the lasing threshold [ ]. In the ideal sce-

nario where the emi er does not suffer from non-radiative decay, a system witha β = can achieve a fascinating effect: threshold-less lasing. is limit of βequals unity can be achieved using two vary different approaches, (i) by increasingthe rate of spontaneous emission into the lasing mode, or (ii) by suppressing thespontaneous emission into all other non-lasing modes. Both approaches will bediscussed in this work.

In conventional semiconductor lasers based on Fabry-Perot cavities for exam-ple, β < . due to a large number of cavity modes that are supported by thelarge cavity [ , ]. Larger β factors can be achieved inmicrodisk lasers due tolarger free-spectral range (FSR) of this geometry. For instance, for a small μ m-diametermicrodisk lasing at . μmwavelength, β factor as high as∼ . wasmea-sured [ ]. In contrast, as we will see, PhC cavity can be designed to have veryfew (or only one) cavitymodes localizedwithin the photonic bandgap. is exten-sively decreases the number of non-lasingmodes coupled to the gainmedium andincreases β factor. Furthermore, PhC bandgap can decrease the optical density ofstates within the gain spectrum, therefore reducing the coupling to other radiativemodes.

. . P β

First consider a basic model: a quantum emi er coupled with a single opticalmode through cavity-induced interaction. Predicted by Purcell [ ], the radia-tive lifetime of a quantum emi er located within a cavity can be altered. edipole-cavity coupled systemcanbedescribedusing the Jaynes-CummingsHamil-tonian [ ],

H = ~ωσz + ~ω(a+a+ ) + i~g(σ−a+ + σ+a) ( . )

where the rst and second term represent the dipole and photon energy respec-tively, and the third term represents the coupled energy between the dipole andthe photon. g is the Rabi frequency. Two decay channels can induce decoherenceto the system: emi er’s non-radiative decay (γ) and photon’s cavity loss (κ). e

eigen-frequencies of the system can be derived as

ω± =ωc + ωe ±

√g + (

δω − iκ − γ

) − i(κ + γ) ( . )

To simplify the analysis, one canmake the assumption that there is no frequencydetuning between the emi er and the cavity (δω = ), and the loss rate inducedby the cavity overtakes the loss of the emi er (γ ≪ κ). en in the strong damp-ing limit (g ≪ κ), the emi er loss rate can be evaluated as

Γsp = Im{ω+} =gκ

( . )

where g is inversely proportional to V, and κ is inversely proportional toQ.Twomore assumptions aremade: (i) the emi er is located at the opticalmode’s

eldmaximum, and (ii) the dipole’s oscillating direction is co-directional with theelectric eld. e Purcell factor can now be deduced by comparing Eq. . to thespontaneous emission rate of an emi er in a homogeneousmediumwith refractiveindex nref.

F =Γsp

Γsp,free=

πQ

V/(λ/nref)( . )

Note that all the foregoing derivations are under the assumption that (γ ≪ κ).However, formost solid-state emi ers (bulk semiconductor, semiconductorQWs,etc.), non-radiative intraband transition is considerable at room temperature. Forinstance, the InGaAsPQWshave a homogeneous broadening of∼ . nmat roomtemperature [ ]. erefore, for high-Q cavitywithQ factor larger than , thecavitymode linewidth ismuchnarrower than the electric transition spectrum. ePurcell factor should thus be modi ed to the more general expression:

F =Γsp

Γsp,free=

πλ

ΔλM V/(λ/nref)( . )

where ΔλM = max{Δλe,Δλc}.Eq. . applies to allmodelswhere the emi er is coupled to a single cavitymode.

In reality, more than one optical mode can interact with the emi er. For instance,considering an ideal Dphoton gasmodel, where the photon is con ned bymetal-lic boundaries in all the three dimensions, the frequency spacing between twoneighboring modes is calculated as,

ΔωFSR =π c

nrefVω( . )

Suppose the emi er spectrally overlaps with one of the cavity modes (modewith double degeneracy) at ω, then Eq. . is only valid under the single modecondition when ΔωFSR ≫ Δω. If V is large enough, the mode spacing allowsthe emi er to couple to multiple modes, and the alteration ratio of the total spon-taneous emission rate (F) should include the sum of all these interactions, i.e.F′

=Γsp

Γsp,free=

∑N−i= Fi. As V approaches in nity, the number of modes within

the broadening ΔωM can be calculated as N = ΔωMΔωFSR

= π ΔλMλ

V(λ/nref)

, and thetotal spontaneous emission ratio F′ ≈ NF approaches unity, as expected [ ].

For a dielectric cavity, in addition to the discrete cavity modes, the emi er alsocan couple to a continuum of radiative modes, which cannot be neglected. us,Eq. . should be corrected as

F =Γsp

Γsp,free=

N−∑i=

Fi + ζ ( . )

where ζ denotes the ratioof the emi er’s emission rate coupled to these radiativemodes with respect to the emission rate in the homogeneous medium. If modedenotes the lasing mode, the spontaneous emission factor β can be expressed as

β =Γsp,Γsp

=F∑N−

i= Fi + ζ=

Fα

( . )

where α is de ned as α =∑N−

i= Fi+ζ, which represents the spontaneous emissioncoupled to all the non-lasing modes.

. . R

In this subsection, the lasing threshold is analyzed from the laser rate equations[ ],

dNdt

=γinFinV

− ΓG(N)P− Nτr

− Nτnr

dPdt

= ΓG(N)P− Pτc

+ βNτr

( . )

whereN and P are the carrier and photon density that are con ned in themodevolume V. ΓG(N)P represents the stimulated emission, where Γ is the con ne-ment factor and G(N) is the gain coefficient. τr and τnr are the carrier radia-tive and non-radiative lifetime respectively. τc is the photon’s cavity lifetime. Finis the pump ux (unit: s− ) of either injected carriers or pumped photons, andFout = P/τout×V is the lasing output ux, where τout is the photon’s out-couplinglifetime. De ne γin as the pumping efficiency, and γout = τc/τout as the outputcoupling efficiency. At steady state, Eq. . can be wri en as,

γinFinV

= ΓG(N)P+ (F + α)N

τr,free+

Nτnr

FoutγoutV

=Pτc

= ΓG(N)P+ FN

τr,free

( . )

Solving Eq. . leads to

P = FN

τr,free[τc

− ΓG(N)]− ( . )

It is evident from Eq. . that, at steady state, the photon generation rate fromthe stimulated emission cannot exceed the photon cavity loss rate. Here de nethe saturation carrier density (Ns) that satis es the following equation, ΓG(Ns) =

τc= ω

Q . As the carrier density approachesNs, the photon gain rate approaches thecavity loss rate, and the stimulated emission term overtakes other contributions in

the rate equations. In this saturation limit, all the pumped carriers predominantlyrecombine and emit to the lasingmode through stimulated emission, which resultsin a differential internal quantum efficiency of unity, or a linear light-in light-out(L-L) curve. In classical laser theory, this effect is also referred as ”gain clamping”or ”gain saturation” effect [ ]. e rate equations in the saturation limit can bemodi ed to

γinFinV

= ΓG(Ns)P+ (F + α)Ns

τr,free+

Ns

τnr

FoutγoutV

=Pτc

= ΓG(Ns)P+ FNs

τr,free

( . )

e classical lasing threshold is de ned as the pump level of the kink in L-Lcurve. From Eq. . , the classical lasing threshold can be deduced as

Fth,classical = [(N−∑i=

Fi + ζ)Ns

τr,free+

Ns

τnr]V/γin ( . )

From Eq. . , it is evident that the lasing threshold is dependent on multiplefactors: ( ) mode volume: the threshold scales linearly with mode volume; ( )saturation carrier density: Ns is inversely proportional to the Q factor. A high-Qcavity is desired to reduce Ns, though an ultrahigh-Q (∼ ) is not necessary;( ) coupling to other cavity modes: this term can be eliminated by properly de-signing a single-mode PhC cavity; ( ) coupling to radiative modes: ideally oneneeds an omni-directional PBG, that is a D-PhC cavity, to eliminate this term[ ]. However, an incomplete PBG has also been shown to be able to largelysuppress this radiative coupling [ ]; ( ) non-radiative recombination: most ofthe non-radiative recombination in a PhCL arises from surface recombination andAuger recombination. For QW-based PhCLs, Surface recombination is especiallyprominent because of the many perforated holes in PhC structures; and ( ) ex-ternal pump efficiency. From Eq. . , it is interesting to notice that, the classicallasing threshold is independent of the cavity mode’s Purcell factor F , though a

high F results in a large β factor.It is also important to mention that there is another de nition of the lasing

threshold [ ], which corresponds to the pump level at which the number ofstimulated emi ed photons starts to exceed the number of spontaneously emi edphotons. In contrast to the classical threshold de nition, this threshold is referredas ”quantum threshold”. From Eq. . , the quantum threshold can be deduced,

Fth,quantum = [( F +N−∑i=

Fi + ζ)Ns

τr,free+

Ns

τnr]V/γin ( . )

where Nq satis es ΓG(Nq) = τc= ω

Q . It has a similar form as Eq. . ,with the exception that thequantumthreshold increases asPurcell factor increases.

is counter-intuitive effect can be explained as follows: at the quantum thresh-old condition, there are an equal number of photons emi ed by spontaneous asby stimulated emission. e Purcell effect accelerates the spontaneous emissioninto the cavity mode. In order to reach the threshold condition for lasing in thepresence of this effect, it is thus necessary to also increase the rate of stimulatedemission.

To conclude, the ultimate low-threshold laser requires a cavity with small modevolume, relatively high Q factor, single-mode operation, and high pumping effi-ciency. PhC cavities, with their exibility in engineering the photonic band, offerthe possibility to approach this ultimate goal.

. P

Photonic crystal nanobeam cavity is a type of D-PhC cavities. It is based on aridge dielectric waveguide, with an array of perforated holes on top that form aPBGmirror [ ]. Photonic crystal nanobeam lasers [ – ] recently have at-tracted a lot of interests because they can achieve highQ/V factors while occupy-ing smaller footprints than D-PhC [ , ] and D-PhC lasers [ ]. Further-more, nanobeam cavities can be designed to have nomode degeneracy, compared

with other designs [ , , ]. is is crucial for nanolasers to operate in thesingle-mode regime,which is the key to achieving a large β factor and the reductionof lasing threshold.

. . M

ere are many promising ways to realize a nanobeam photonic crystal laser. eone described here features semiconductor quantumwells (QWs) la ice-matchedto an InP wafer. e QWs provide the optical gain. ey contain four nm-thickcompressively-strained In . Ga . As layers, which are sandwiched in a nmthick In . (Al . Ga . ) . As slab. Below the semiconductor QW slab, there is aμm thick InP sacri cial layer for selective wet etching. All the layers are grownwithmetal-organic chemical-vapor deposition (MOCVD) technique tominimizethe crystalline defects [ ].

Figure 4.3.1: Layout of the material system of the semiconductor QWs sam-ple used to realize nanobeam laser. The energy band of the semiconductorQWs are sketched on the right hand side.

e ratio of III-V components in the semiconductor QWs results in a . %

in-plane compressive strain while satisfying the la ice-matching condition. iscompressive strain shi s the heavy-hole (HH) state to lie on top of the light-hole(LH) state, which leads to a dominantly TE-polarized gain. e TE-polarizedmodes favor vertical light emissions compared to the TM-polarized modes, andtherefore are more suitable for surface-emi ing lasers. Fig. . . (b) shows thephotoluminescence (PL) spectrum of the sample. e emission spectrum ranges

Figure 4.3.2: (a) Energy band diagram of electrons, light holes (LH) andheavy holes (HH) bands of the semiconductor QWs. (b) The PL emissionspectrum of the quaternary QWs peaks at about . μm.

from . μm to . μm, and peaks at . μm.

. . C

e photonic crystal nanobeam design is based on a suspended ridge waveguidewith an array of identical holes, which forms a PBGmirror. e refractive index ofthe dielectric is . (the index of InAlGaAs/InGaAs QWs at . μm). e ridge is

nm wide and nm thick, the periodicity of the holes is a = nm, and thehole’s radius is set to be r = . a to optimize the bandgap.

Figure 4.3.3: Layout of the photonic crystal nanobeam cavity design.

Introducing a la ice grading to the periodic structure creates a localized poten-

tial for the fundamental TE-polarizedmode. To optimize the mode’sQ factor, thebandgap-tapering technique [ , ] is employed for suppressing the sca eringlosses that take place at the interface between the cavity section and themirror sec-tion [ ]. is design contains a four-segment tapered section with holesR −Rand a -segment mirror section at each side of the cavity, as shown in Fig. . . .Two degrees of freedom are available to modify each tapered segment: the widthwk and the radius rk. We keep the ratio rk/wk xed at each segment and imple-ment a linear interpolation of the grating constant π/wk. When the central seg-ment w is set to . a, a cavity mode is obtained to resonate at . μm with anultrahighQ factor above , , , and a very small mode volume of . (λ/n)[Fig. . . (a)]. e ultrahighQ factors can also be explained by looking at themo-mentum space of the cavity mode. Fig. . . (b) and (c) demonstrate the spatialFourier transformation (FT) of the electric eld components Ex and Ez in the xzplane (y = ), with the light cone indicated by the white circle. It can be seen thatboth modes’ Fourier components are localized tightly at the bandedge of the Bril-louin zone on the kz axis: kz = π/a. is minimizes the amount of mode energywithin the light cone that is responsible for sca ering losses [ ].

In addition to this fundamental mode, the cavity supports another mode at alonger wavelength . μm with a larger mode volume of . (λ/n) . is is anextended mode with a node at the center of the cavity, as shown in Fig. . . (d).

is mode resonates at a wavelength outside the gain spectrum of the quantumwells, and therefore does not affect the single-mode operation of our laser.

. . F

e nanobeam pa ern is de ned using a negative e-beam lithography resist [Hy-drogen Silsesquioxane (HSQ)]. e e-beam resist is spun on the surface with aspinning speed of rpm resulting in a thickness of ∼ − nm. e re-sist is subsequently cross-linked using a STS Elionix ELS- e-beam writer atan acceleration voltage of kV and a beam current of pA.

Next, the pa ern is transferred to the substrate with inductively coupled plasma

Figure 4.3.4: (a) Mode profile of the fundamental cavity mode of thenanobeam laser. (b)(c) Spatial Fourier transform of the electric field com-ponent Ex and Ez at y = plane. (d) Mode profile of the second-order mode,resonating at a higher wavelength, outside the gain spectrum.

reactive ion etching (ICP-RIE) using BCl /CH /Ar/HBr chemistry at oC.is anisotropic etching creates a mesa extended to the sacri cial layer. Hydro u-

oric acid (HF) is then used to dissolve the e-beam resist. Finally the InP sacri ciallayer is selectively wet-etched by HCl:H O = : solution. e crystal orien-tation dependent etch rate is about − nm/s at oC, which leads to an etchtime of s for a completely suspended nanobeam structure. Fig. . . shows thescanning electronmicrographs of an array of fabricated nanobeam lasers. e twopads are designed to support the suspended PhC structures.

Figure 4.3.5: Scanning electron micrographs of the fabricated photonic crys-tal nanobeam lasers.

. . C

e devices are optically pumped at room temperature using a nmpulsed laserdiode. e pulse width is ns at a kHz repetition rate, which corresponds to aduty cycle of . %. e pump beam is focused to the sample surface via a Xobjective lens, and the emi ed light is collected via the same objective lens andanalyzedusingoptical spectrumanalyzer (OSA), near-infrared (NIR)camera, anda InGaAs detector.

Figure 4.3.6: Illustration of the characterization setup.

In Fig. . . (a) we show the L-L (Light-in, Light-out) curve for one of themea-sured devices. It can be seen that there is no pronounced kink near the threshold.

is so turn-on of the laser indicates a large β factor. e lasing images taken atdifferent pump levels, using the NIR camera, show that the emission spot is well

con ned to the center of the nanobeam, which unambiguously proves that the las-ing is from the localized defectmode. Fig. . . (b) shows the lasing spectrumnearthe threshold. e spectrum is ed with a Lorentzian function with a full-widthhalf-maximum (FWHM) of . nm, which corresponds to a Q factor of , .

eQ factor is likely limited by the resolution of the OSA used.

Figure 4.3.7: (a) Laser emitted power as a function of the incident pumppower. The emission profiles obtained from the camera at different pump lev-els are shown on the right. (b) The spectrum of the emitted light near thethreshold. (c) Output lasing power as a function of the pump beam position.The pattern of the nanobeam is superimposed as the background of the pic-ture. (d) Polarization dependence of the lasing mode.

In addition, Fig. . . (c) shows the dependence of the output power on thepump beam position by scanning the sample in xy plane using a piezo-actuatedstage with a spatial resolution of nm. As the pump beam is moved away from thecenter of the cavity, the beam intensity decreases rapidly and nally vanishes. isis a further con rmation of emission from the localizedmode of the cavity insteadof extended band-edge emission. Here the pump spot is much bigger than thecavity mode, so the cavity mode samples the pump beam. erefore, Fig. . . (c)shows the shapeof thepumpspot as opposed to thepro leof the lasingmode. isalso allows us to evaluate the effective pumppower, that is, the overlap of the pump

beamwith the nanobeam,which is reported inFig. . . (a). An effective thresholdof μW is evaluated for this nanobeam laser, whereas the total power thresholdmeasured behind the objective lens is μW. e threshold power is even smaller,considering that not all the pump power incident onto the nanobeam is absorbedby the cavity. Finally, Fig. . . (d) shows the polarization dependence of the laseremission. e emission is polarized along x-axis as expected, and exhibits a largepolarization ratio of∼ .

Next, thebehaviors of this photonic crystal nanobeam laser is analyzedby ingto the rate equation that is similar to Eq. . ,

γinFinV

= ΓG(N)P+ AN+ (F + ζ)BN + CN

FoutγoutV

=Pτc

= ΓG(N)P+ F BN

( . )

whereAN, (F + α)BN andCN represent the carrier loss rate through surfacerecombination, radiative recombination and Auger recombination, respectively.For this nanobeam cavity, ζ is evaluated as . through FDTD simulation. elasing mode’s Purcell factor is estimated to be∼ , with taking into account theQW’s homogeneous broadening at room temperature and the spatial overlap be-tween the active medium and the optical mode. For other parameters, typical val-ues for InP-based quantum wells at room temperature are applied [ ].

Fig. . . shows the measured L-L in log-log scale along with curves obtainedfrom rate equations for different Purcell factors and spontaneous emission factors(β). It can be seen that the experimental data are in excellent agreement with thetheoretical prediction of F ∼ and β = . . is is in good agreement withthe theoretical estimation of Purcell factor.

Figure 4.3.8: Log-log plot of the L-L curve (dots), with predictions from therate equations using different β factors (solid lines).

. P

PhCLsbasedon D-PhCnanobeamsand D-PhCslabshaveproved tomakegoodsurface-emi er nanolasers with low threshold and high modulation speeds, butthey have a few drawbacks. First, the footprint of the photonic crystal lasers islimited by themultiple periodic Bragg layers used to con ne the cavitymode. Sec-ond, most photonic crystal cavities rely on suspended semiconductor membranesto provide a good index contrast, which makes efficient electrical injection of car-riers into the cavity mode difficult. Microdisk lasers [ ], on the other hand, relyon whispering gallery mode pinned via total internal re ection to the boundaryof the disk. ey offer a platform for very compact electrically-driven lasers, withfootprint on the order of optical wavelength [ – ]. However, the whisperinggallery mode travels through the whole circular edge of the microdisk, and there-fore results in a relatively large mode volume [Fig. . . (c)]. Furthermore, thewhispering gallery mode does not emit vertically, which makes the collection ofphotonsdifficult [ ]. Normally evanescent coupling via a tapered ber is used tocollect the emi edphotons efficiently [ ]. ismakes integrationof large arrays

of microdisk lasers problematic. To overcome this, vertically-emi ing microdisklaser uses a second-order metallic grating atop to extract the light out [ ].

Figure 4.4.1: (a) Schematic of photonic crystal disk laser and (b) fabricatedphotonic crystal disk laser. The device can be viewed as a hybrid between (c)microdisk laser and (d) photonic crystal nanobeam laser with photonic crystalfolded back to minimize the transmission losses.

In this section, we demonstrate nanolasers operated at room temperature basedon a novel type of nanocavities, which combines the properties of microdisk andphotonic crystal lasers. It incorporates an array of holes at the perimeter of themi-crodisk, in order to con ne the whispering gallerymode in a limited angular rangewithin a smallmodevolume. edesign canalsobeunderstoodas aphotonic crys-tal nanobeam cavity [Fig. . . (d)] [ – ] bent into a disk [Fig. . . (a)]. Inthis way, Bragg mirrors at each end of the nanobeam are combined in one curvedBragg mirror. erefore, the number of holes can be decreased because the trans-mission losses are cycled through the disk, which reduces the device footprint.

We start our design with a . μm diameter, nm thick microdisk that sup-ports a resonance of TE , mode at nm. Next, we add holes with equalangular spacing around the perimeter of the disk. is causes the two degener-ate TE , modes, which propagate in clockwise and counter-clockwise direction,to split into two standing-wave modes: one mode with its eld concentrated inthe dielectric whereas the other in the hole region, as shown in Fig. . . (b) and(c). ese modes are similar to the modes of microgear cavities [ ]. In anal-

Figure 4.4.2: (a) Bangedge wavelength as a function of the radius of holeson photonic crystal disk (black), with the bandgap shaded in bronze. Thegreen curve shows the corresponding normalized bandgap width. (b)(c) Reso-nant mode profiles at bandedge of Hz component, for dielectric-band (b) andair-band mode (c), respectively.

ogy to photonic crystals, these twomodes correspond to the dielectric band-edgeand air band-edge. Between these bandedges, the propagation of TE , mode isforbidden. In order to maximize the bandgap width, and thereby optimize the az-imuthal con nement, we place the center of the holes at electric eld maximum.Fig. . . (a) shows the band-edge wavelength and normalized bandgap width (ra-tio of the gap width to the midgap wavelength) as a function of the hole radius. Itcan be seen that the bandgap is maximized at nm, and the center of the bandgapis at∼ nm. We emphasize that the bandgap is not complete: there are higher-ordermodeswithdifferent radial elddistribution thatmayexist in thewavelengthof interest.

Next, we introduce the defect region to the disk, by modifying the hole-to-hole angular spacing as well as the holes’ radii. e nal structure is shown inFig. . . (b), where there are identical holes on the bo om half acting as Braggmirrors, and tapered holes on the top half to localize the mode while suppress-ing the sca ering losses [ ]. e cavity mode resonates at nm, and, withoutconsidering material losses, has a Q factor of . × , and a modal volume of. (λ/n) . e mode volume is on the same order of photonic crystal nanobeamcavities [ – ].

Figure 4.4.3: (a)(b) Images of photonic crystal disk and microdisk laserswith different scaling factors. (c)(d) Electric field density profiles of photoniccrystal disk modes. (e) Experimental results of lasing wavelength dependenceon diameter of photonic crystal disks (black-dot and red-dot) and microdisks(blue-dot and green-dot). The solid curves show the mode wavelength de-pendence obtained using simulations. (f)(g) Electric field density profiles ofmicrodisk modes.

Ourcavities are fabricatedoncommercial InP substrate. A nmthick In . (Al .

Ga . ) . As layer is epitaxially grown atopusingmetal-organic chemical vapor de-position. It contains four compressively strained In . Ga . As quantum wells,which support TE-polarized gain covering the wavelength range from nm to

nm. e pa ern is de ned with electron-beam lithography. e pa ern issubsequently transferred to In . (Al . Ga . ) . As slab and InP substrate with in-ductively coupled plasma reactive ion etching. e disk structure is nally realizedby selectively wet etching the mesa with : HCl:H O solution [Fig. . . (a)].We also fabricate microdisk without perforated holes with the same diameter tocompare the results [Fig. . . (b)]. e two arrays are scaled linearly in size tovary the cavities’ resonant wavelengths.

e devices are optically pumped at room temperature using a nm semi-conductor laser, with ns pulses and kHz repetition rate. e pump beam isfocused to a μm diameter spot using a X objective lens. e effective pumppower is estimated with power measurement a er the objective, while taking in

account the spatial overlap between the pump beam and the lasing mode. eemission beam is collected through the same objective lens from the top, and an-alyzed with an InGaAs detector ltered by a monochromator.

For both photonic crystal disks and microdisks, Fig. . . (e) depicts the lasingwavelength as a function of the diameter of the disk, where modes (c) and (d)are photonic crystal disk modes, and modes (f) and (g) are microdisk modes. Itshows good agreement with simulation results plo ed in solid lines, which veri esthe lasingmode (c) is of the designed defectmode. Wenote that by controlling theposition of the pump spot, lasing from two different photonic crystal disk modescould be obtained in some structures [Fig. . . (e)]. e nanolaser, however, doesoperate in single-mode regime in both cases (only one mode lases at one time).We also note that the two microdisk lasing modes are not the fundamental TE ,m

modes, but the higher-order TE , , and TE , modes. ese modes have a node ofelectric eld in the radial direction. e fundamental modes cannot be collectedfrom top, because it emits in in-plane directions [ ].

Next, we study the properties of the designed photonic crystal disk mode fromone single device [Fig. . . (a)]. Fig. . . (b) shows the lasing power as a func-tion of the effective pump power (also known as L-L curve), in comparison with amicrodisk laser emi ing at the samewavelength. e injection efficiencies are esti-mated to be . % and . % for photonic crystal diskmode andmicrodiskmode,respectively. From Fig. . . (b), the photonic crystal disk lasers have much be erextraction efficiencies thanmicrodisk lasers. We believe that this efficiency can befurther boosted with subtle far- eld engineering of photonic crystals [ ]. Insetplots the spectrum at . times the threshold power of photonic crystal diskmode,which shows clear single-mode lasing emission. In Fig. . . (c) we plot the L-Lcurve for the photonic crystal disk laser in log-log scale (black dots), along withthe L-L curves obtained from rate equations for different spontaneous emissionfactors (β). e experimental data show good agreement with a β factor of . .More than ten-fold reduction in β-factor of this laser compared to nanobeam laser[ ] can be a ributed to the existence of higher-order modes. In Fig. . . (d)

Figure 4.4.4: (a) Images of fabricated photonic crystal disk from scanningelectron microscope. (b) Light-in light-out curve for photonic crystal disk laserand microdisk laser, respectively. Inset shows the spectrum of photonic crystaldisk lasers at . × lasing threshold. (c) Log-log plot of the photonic crystaldisk laser’s L-L curve (black-dots). The solid curves show L-L curves deducedfrom rate equations with different β factors. (d) Lineshape of the lasing modeabove threshold (red-dot), fitted with a Lorentzian line function (red-solid).Inset shows the emission profile taken from an infrared camera.

we show the lasing spectrum of the photonic crystal disk lasers slightly above thelasing threshold ( . ×threshold). It has a full-width half-maximum (FWHM) of. nm, which corresponds to a Quality factor of∼ . e Quality factor islimited by the resolution of the monochromator. Free-carrier absorption is alsoknown to decrease the Q factor extensively below its passive value [ ]. Wealso show the lasing emission pro le of the photonic crystal disk laser in the in-set of Fig. . . (d), which is taken from a near-infrared camera. Finally, we notethat linewidth narrowing effect above threshold could not be observed, due to thestrong heating effects in nanolasers, as previously reported [ ].

In summary, we have demonstrated a novel type of photonic crystal lasers,

which takes advantage of bothmicrodisk and photonic crystal geometries, and hasa small footprint, small mode volume, and high extraction efficiency.

. G - -LED

. . I

Developing an electrically driven nanolaser is critical to making nanolaser a com-petitive technology. However, most of the nanolasers previously reported wereoptically pumped with another laser diode, with few exceptions [ , ]. ebig obstacle lies in con guring the electrical contacts to the optical cavity in or-der to effectively inject carriers into the cavity mode, but the large absorption ofconventional metal or indium tin oxide (ITO) contacts will degrade the cavity’sQuality factor dramatically, and prevent the lasing behavior.

Graphene, the one-atomic-thick layer of carbon atoms that are aligned in a hon-eycomb crystal la ice, on the other hand, has both high optical transparency overvisible and infrared wavelengths and high electrical conductivity, and thus is anideal material candidate for transparent conducting electrodes. Graphene has al-ready been applied in devices, such as light emi ing diodes [ ], photovoltaiccells [ ], and liquid crystal displays [ ]. In addition, graphene has extremelyhigh mechanical exibility, and can conform to a pa erned surface over large ar-eas. erefore, we believe, graphene could provide a promising solution to electri-cally driven nanolasers. In this section, we rst demonstrated graphene-contactedLEDs based on microdisk structures.

. . P

Our graphene is grown on single-crystalline copper foil with chemical vapor de-position [ ]. A thin layer of PMMA ( -A ) is spun onto the graphene, andthe PMMA/graphene lm is then released in a copper etchant (FeCl ) bath, andsubsequently transferred onto the receiving substrate in deionized water environ-

Figure 4.5.1: (a) Optical microscopic image of a graphene sheet transferredon a SiO substrate. (b) Confocal Raman spectrum of monolayer graphene.(c) Optical micrograph of a photo-ligrography-patterned graphene afterO plasma treatment. (d)(e) Confocal Raman mapping of the patternedgraphene. The Raman signal is spectrally integrated at G line and G’ line re-sepctively. The white square in Fig. 4.5.1(c) indicates the spatial scanningrange.

ment. Finally, the PMMA layer could be gently removed with acetone vapor.In Fig. . . (a), we show the optical microscopic image of a graphene sheet

transferred on a SiO substrate. e substrate area with graphene atop shows aslightly different color compared to the bare substrate area. e quality of thegraphene is tested using confocal Raman spectroscopy. e Raman signal is ob-tained with a pump laser of λ = nm focused on the substrate. Fig. . . (b)shows the Raman spectrum of the transferred graphene on SiO substrate, withtwo pronounced peaks identi ed as the G band at cm− and the G’ band at

cm− . e lowG toG’ peak intensity ratio (∼ . ) and the narrow linewidthof the G’ line (∼ cm− ) are signatures of monolayer graphene [ ].

e graphene lm can be pa erned with lithography and oxygen plasma treat-ment ( sccm, wa , minute), as shown in Fig. . . (c). Confocal Ramanmapping of the pa erned graphene, demonstrated in Fig. . . (d) and (e), veri esthe graphene exposed toO plasma is etched away. e Raman signal is spectrallyintegrated atG line andG’ line respectively. e spatial scanning range is indicated

by the white square in Fig. . . (c).

Figure 4.5.2: (a) Optical micrograph of patterned graphene stripe for resis-tivity measurement. (b) Graphene resistance of various lengths. The widthof the stripe is fixed at μm. The black dash line shows the linear fitting ofthe measured data, resulting in a graphene resistivity of . kΩ/square, and acontact resistance to be . kΩ.

Next, we characterize the electronic properties of the graphene lm. e receiv-ing substrate is a silicon substrate covered with a nm thick SU- photoresistthat acts as the insulating layer. In order tomeasure the resistivity of graphene, thegraphene is pa erned into a μmwide stripe, as shown inFig. . . (a). e resis-tance of the graphene stripe whose length varies from μm to μm ismeasuredwith palladium / gold ( nm / nm) contact. By ing the resistance measuredwith various length / width ratio, as shown in Fig. . . , the resistivity of grapheneis evaluated to be . kΩ / square, and the contact resistance to be . kΩ.

. . F - -LED

We adopt the simple prototype of microdisk structures to demonstrate graphene-contacted light-emi ing devices. Fig. . . (a)-(d) shows the fabrication proce-dure of the proposed graphene injected micro-LED. e heterostructures havebeen grown epitaxially on n-doped InP substrates by metalorganic chemical va-por deposition, in order to provide the optical gain of the laser. A heavily p-doped

InP/InGaAs layer caps the quantum wells to provide efficient hole injection.First, themicrodiskwith a radius of . μm is de nedwith negativeHSQe-beam

resist (Dow Corning Co., XR- ). e pa ern is transferred to the substratewith reactive ion etching. e etching depth is measured by scanning electronmicroscopy to be ∼ nm. Next, a nm thick SiO layer and a nm thickTi( nm) / Au( nm) / Pd( nm) metal contact are deposited around the mi-cropost, to provide the top contacts that are insulated from the substrate. e dis-tance between themicrodisk and the SiO /metal sidewall is kept to be of − μm.At this distance, the metal does not affect the optical mode. Subsequently, the mi-crodisk is nalized with selective wet etching in HCl/H O = : bath at oC forminutes. e thin post supporting the microdisk also provides the current path

for electrical injection. Finally, a PMMA/graphene lm is transferred to the sam-ple, and pa erned by e-beam lithography and oxygen plasma. Fig. . . (e) and (f)show the scanning electron micrograph and optical micrograph of the microdiska er wet etching. Fig. . . (g) shows the con guration of the nal sample withpa erned graphene on top.

We characterize the I-V characteristics of the fabricated microdisk structure.DC voltage is applied between the Au contact and the bo om of the substrate.

e I-V curve of the the microdisk structure, as shown in Fig. . . , is in consis-tencewith thep-i-n diode. Wecollect light emission from themicrodiskwith Xobjective lens, andmonitor with an IR camera. e emission images are shown inthe inset of Fig. . . . e turn-on voltage of the electroluminescence is measuredat . volts, and the intensity of the electroluminescence increases as the voltageincreases. e edge of the SiO /Au sidewall at the vicinity of the microdisk canalso be seen from light sca ering.

. C

In this chapter, we introduced the basic concepts needed for development of pho-tonic crystal lasers. As illustrated by the rate equation analysis, PhC cavities are an

Figure 4.5.3: (a) The fabrication procedures of the graphene-injected mi-crodisk laser. (b)(c) Scanning electron micrographs of the fabricated mi-crodisk lasers, after wet etching (b), and graphene transfer (c).

ideal platform for the realization of the ultimate low-threshold nano-lasers, sincethey can support a single mode operation with a high Quality factor and a smallmode volume across the gain spectrum. Furthermore, the highQ/V factor leads tostrong cavity Purcell effect, which enables high-speed operation of directly-mod-ulated lasers. Two examples of photonic crystal lasers, which operate with pulsedoptical pumping at room temperature, are then demonstrated. e photonic crys-tal nanobeam laser shows a high spontaneous emission factor of . due to itssingle-mode nature; the photonic crystal disk laser has a small device footprintand higher collection efficiency than microdisk lasers.

During the last decade we witnessed rapid progress in the development of pho-tonic crystal lasers: the reduced lasing threshold a ributed to its PhC cavity’ssmall mode volume and highQ factor [ , ] and the nano-scale footprint, the

Figure 4.5.4: I-V characteristics of a microdisk device. Inset shows the emis-sion image taken from an IR camera.

boosted modulation speed resulting from Purcell effect [ , ], the control-lable laser polarizations [ ], the wavelength tunability integrated with optome-chanics [ ], and so forth. Extensive research has revealed interesting underlyingphysics, including the strong coupling limit of cavity QED [ ], and has also in-spired applications such as those in bio-chemical sensing [ ].

Nevertheless, several challenges still remain tobe solved. First, the current state-of-the-art technologies cannot produce photonic crystal lasers that operate at thedesigned wavelength, due to fabrication imperfections. is is especially impor-tant when the gain medium consists of quantum emi ers with a very narrow gainspectrum (semiconductor QD, for instance). Second, while a couple of electri-cally driven PhCLs have been reported so far [ , ], the reliable productionof the large-scale electrically pumped PhCL array calls for more research efforts.Finally, PhCLsneed to nd aniche in the commercialmarket for applications rang-ing from bio-chemical sensors to on-chip interconnects.

5Photonic crystal cavities at microwave

frequencies

. I

Photonic crystal cavities with high Quality factors (Qs) are capable of extendingthe lifetime of cavity photons con ned within mode volumes (V) of the order ofa cubic-wavelength, and hence can greatly enhance the interaction between elec-tromagnetic elds and ma er. Recently, high Q/V photonic crystal cavities havebeen extensively investigated with various geometries at optical frequencies, act-ing as a powerful platform for studying cavity quantum electrodynamics (cQED)[ ] and developing amyriad of nano-photonic devices. Yet few photonic crystalcavities have been reported at microwave frequencies [ – ].

A high Q/V microwave cavity is demanded for a number of applications. In

atomic physics, a single Rydberg atom coupled to a single photon is the funda-mental system for cavityQED studies, especially when one reaches the interestingstrong coupling regime where energy can be coherently exchanged between thetwo. For strong coupling to occur, the atom-photon coupling rate g (inversely pro-portional to square root ofV)must exceed both the photon leakage rate (inverselyproportional toQ) and the atom decoherence rate. Since the frequency differenceof Rydberg states is in the microwave region [ ], a highQ/Vmicrowave cavityis crucial to achieve this type of coupling. Strong coupling in Rydberg atoms hasbeen previously observed in large-scale Fabry-Perot cavities and superconductingcircuits [ , ]. Photonic crystal cavities, in contrast, offer an alternative plat-form that is more compact and can be operated at room temperature.

Secondly, given the cavity’s intrinsic eld enhancement factor, there are alsosigni cant advantages in practical applications such as tunable microwave ltersand antennas, where the efficiency of a radiator can be signi cantly improved bycoupling with the cavity mode [ ]; uorescence microscopy for biological sys-tems, where a highQ/Vmicrowave cavity can be used to control the temperatureand drive the biological reactions [ ]; resonance-enhanced microwave detec-tors [ ]; particle accelerators [ , ]; and refractive index sensors.

In this chapter, we have designed and experimentally demonstrated an all-dielectric photonic crystal microwave cavity. e cavity can be fabricated with avariety of materials using conventional machining techniques. We report a cav-ity mode at . GHz with a very small mode volume limited to about one cubic-wavelength and a record-high Q-factor of , at room temperature. Besidesthe high Q/V factor, the defect mode has a TM-polarized electric eld concen-trated in the air region [ ], as opposed tomost photonic crystal cavities designswith TE-polarized electric eld concentrated in the host dielectric [ ]. esefeatures are crucial for effectively coupling microwaves to ma er in a number ofapplications. Moreover, when a center-fed antenna is placed inside the cavity act-ing as a radiating dipole, we observe a strong signature of the cavity’s Purcell en-hancement factor. In summary, we believe this type of device offers great promisefor studying cavity QED phenomena as well as for enabling novel applications at

microwave frequencies.

. D

e structure we propose starts from a so-called rectangular dielectric rod waveg-uide with a periodic array of circular holes, as shown in Fig. . . (a) [ , , ,

, ]. We choose a refractive index of . for the rod, which corresponds tothe refractive index of alumina (Al O ) at microwave frequencies. e rectangu-lar waveguide has an aspect ratio (thickness/width) of : , and supports a fun-damental TM-polarized mode with its major component aligned along the y-axis[Fig. . . (c)]. From our previous work [ ], introducing a suitable periodic ar-ray of holes in the bulk materials of this high aspect-ratio waveguide results in arelatively wide bandgap for TM-polarized modes. Here we select the thickness tobe w = . mm, depth h = . mm, and hole periodicity a = . mm. eradius of the holes is chosen as r = . mm.

Next, we introduce a defect region into the cavity by gradually increasing theperiodicity (hole-to-hole distance) and hole diameter for each segment startingfrom a pair of outer holes and symmetrically moving towards the center. Whenthe feature size of a segment is enlarged, the band-gap is red-shi ed, resulting in agraded photonic band, as shown in Fig. . . (b). is allows con ning an air-bandmode in the defect region: the air-band mode is coupled to the evanescent Blochmodes within the band-gaps at each end, effectively trapping it between a pair ofBragg mirrors.

A three-dimensional nite-difference time-domain (FDTD) code was used tocalculate the resonant frequency and theQ factor of the cavitymode. On each sideof the cavity we introduce segments with progressive tapering of the hole diam-eter and spacing, in order to adiabatically couple the cavitymode to the evanescentBloch mode. is adiabatic process is tuned to suppress the sca ering loss result-ing from the effective index mismatch between the two modes [ , ]. Withoutmaterial losses, the cavitymodehas anultra-highQ factor of , , , limited by

Figure 5.2.1: (a) Schematic of the device. (b) Diagram of tapered photonicbandgap. (c) Transverse mode profile of the fundamental TM-polarized mode.(d) Mode profile of the cavity mode.

sca ering alone (Q = Qsc). e cavity mode pro les are shown in Fig. . . (d),where the Ey component is plo ed at themirror plane of y-axis and x-axis, respec-tively. e tapered section’s parameters are tailored to localize an air-band moderesonating at . GHz bymaximizing con nement along the z-axis, which resultsin a small cavity mode volume. As expected from the general features of air-bandmodes, the electric eld concentrates in the air-region. However, themaximumofthe electric eld density [ε |E| ] is not in the air region, so we de ne the effectivecavity mode volume as,

Veff =

∫ε |E| dV

|Emax,air|( . )

where Emax,air is the electric eld maximum in the air region, located at the centerof the cavity. For our cavity we found an effective mode volume of . λ .

e cavity mode is normally excited from an external waveguide port. eamount of energy stored (U) is proportional to the launched power (P ) [ ],

U =ω

QQw

P ( . )

In Eq. . Qw denotes the quality factor responsible for the eld energy leakagefrom the cavity section via the dielectric waveguide. e total cavity quality factorQ, including material losses, can be wri en as

Q=

Qsc+

Qm+

Qw( . )

whereQsc accounts for sca ering losses due tomodemismatch in the tapered sec-tion and Qm for material losses in the dielectric. Substituting Eq. . into Eq. . ,we have

U = Umax =Pω

Q ( . )

when Qw = QscQmQsc+Qm

. erefore, for given Qsc and Qm, it is important to tune Qw

by xing the number of Bragg mirror pairs on each side of the resonator so as toachieve optimal critical coupling conditions by satisfying Eq. . .

. F

e resonatorwas fabricatedwithultra-highpurity alumina fromCoorstek (ADPlasmaPure). ismaterial has a relatively high refractive index of . atmicrowavefrequencies and very small dielectric losses. Dielectric losses are important asthey ultimately limit the quality factor of the resonator. e material that weuse has a nominal loss factor of tan(δ) = . × − at GHz, which results inQm = , . For our device, we put Bragg mirror pairs at each side, whichcorresponds to Qw = , . erefore, the total Q factor is expected to be, . Fabricationwas donewith standard computer-numerical-control (CNC)

milling techniques, with a nominal positional accuracy of∼ μm for each hole.e fabricated structure is shown in Fig. . . (a), and the setup used for our

two-port transmi ance measurements in Fig. . . (b). e device was connected

Figure 5.3.1: (a) Images of the fabricated alumina cavity with slant sectionsat both ends to facilitate coupling to metallic waveguides. (b) Setup for trans-mittance measurement. (c) Coupling components between the metallic waveg-uide and the dielectric rod.

to a network analyzer via conventional WR microwave waveguides placed ateach end. Because of the geometrical mismatch between the metallic waveguide( . mm× . mm) and our dielectric waveguide resonator ( . mm× . mm),we designed slant sections at both ends of the resonator to allow partial inser-tion into the WR waveguide and thereby facilitate coupling [Fig. . . (c)]. InFig. . . (a), we show the amplitude spectrum from the transmission measure-ment. It demonstrates a large bandgap from . GHz to . GHz, which is con-sistent with the theory. e transmission coefficient of the cavity mode can begenerally ed using a Fano model [ , ],

t = A

[ηexp (iϕ) +

+ i ω−ωΔωFWHM/

]( . )

where A is a constant representing system loss, η denotes the ratio of power cou-pled to other channels (direct transmission, higher-order modes, etc.), ϕ is thephase difference,ω is the resonant frequency, andΔωFWHM is the cavity full-widthhalf-maximum linewidth.

Figure 5.3.2: (a) Amplitude spectrum with a large frequency range fromGHz to GHz, showing the bandgap of the structure. (b) Amplitude and

phase spectra of the cavity mode from the transmittance measurement. Thedashed curves are fitted with Eq. 5.5.

Unlike most photonic crystal cavity experiments at optical frequencies, hereboth amplitude and phase of the transmi ance can be obtained. By varying theposition of the resonator with respect to the waveguide ports, both η and φ can bemodi ed, whereas ω and ΔωFWHM stay constant. Here we select a set of data withminimized coupling to direct, non-resonant channels (negligible η), which resultsin a Lorentzian resonance lineshape. We plot the resulting spectra in Fig. . . (b),where the cavity resonance is clearly seen at . GHz, in good agreement withthe design value. By ing the measured data with Eq. . we obtain a full-widthhalf-maximum linewidth ΔωFWHM of . MHz, corresponding to a Q factor of, , which is slightly smaller than the expected Q value of , . With the

resonator end-coupled to a larger WR waveguide, we observed Q values up to, albeit with a less symmetrical line-shape. e observed discrepancy of Q

factor can be a ributed to higher dielectric losses in the bulk material at this fre-quency, to fabrication tolerances on the holes position and size and to resonant

losses at the transition between waveguide and dielectric. In general, a higher Qfactor can be obtained with less lossy materials or at cryogenic temperatures.

Next, we insert a small antenna inside the cavity in order to study the interac-tion between the cavity mode and a radiating dipole. e antenna is constructedby stripping both the outer conductor and the dielectric core of a coaxial cable(Megaphase Corp., model ClearPath-A ), exposing the inner conductor for a to-tal lengthof mm. e relatively small diameter ( . mm)of the inner conductorwas chosen to minimize any perturbation of the cavity mode.

In Fig. . . (a), we plot the re ectance spectrummeasured with a network an-alyzer connected to the above mentioned center-fed antenna, when this is placedin free space (red) and in the center hole of cavity (blue). A clearly visible re-ectance dip with a . dB depth at∼ . GHz is observed, which we interpretas a convincing signature of the Purcell effect of the cavity. When the antennais coupled to the cavity mode, the radiation rate is enhanced, and thus the loadimpedance is modi ed. is results in a variation of the re ectance at the ana-lyzer port, due to the impedance mismatch between the antenna load and the ca-ble. From Fig. . . (a), we also note that there is a change in the resonant fre-quency of the system, from . GHz to . GHz. is is due to the small butnon-negligible perturbation of the metallic antenna, which blue-shi s the reso-nance. e free-space background in Fig. . . (a) (red-solid curve) arises fromthe small re ectance between the network analyzer/cable connection, and the ca-ble/antenna connection. is background prevents us from accurately quantify-ing the load impedance variation and subsequently extracting a numerical valuefor the Purcell enhancement factor.

Next, we scanned the antenna in z-direction along the length of the dielectricrod to probe the electric eld distribution of the cavity mode. is is analogous tonear- eld scanning optical microscopy (NSOM) [ , ]. In Fig. . . (b), weplot themeasured re ectance depth as a function of the z-position (blue-solid). Incomparison, we also plot the electric eld amplitude (black-dash). It can be seenthat the two curves follow the same z-dependence, which veri es that the S dip

Figure 5.3.3: (a) Reflectance measurement of the center-fed antenna, withand without the cavity. (b) Reflectance depth as a function of the z-position.Also shown (dashed line), the simulated electric-field amplitude along the res-onator.

we observe is due to coupling to the cavity mode.

. S

A high-Qmicrowave resonator with a mode volume smaller than one cubic wave-length has been designed and fabricated. A record high Quality factor (Q =

, ) for photonic crystal cavities atmicrowave frequencies has beenmeasuredat room temperature. In addition to its ultra-highQ/V factor, the cavity is uniquelydesigned to have its electric eld concentrated in air. Coupling to this cavitymodeis enhanced by the Purcell factor. We believe that this device is well suited for con-ducting microwave cavity QED experiments and for developing a variety of novelmicrowave devices.

6Dual-polarized photonic crystal cavities

for nonlinear applications

. I

Ultra-high Quality factor (Q) photonic crystal nanocavities, which are capable ofstoring photons within a cubic-wavelength-scale volume (V), enable enhancedlight-ma er interactions, and therefore provide an a ractive platform for cavityquantum electrodynamics [ , ] and nonlinear optics [ – ]. In mostcases, high Q/V nanocavities are achieved with planar photonic crystal platformbased on thin semiconductor slabs perforated with a la ice of holes. ese struc-

tures favor transverse-electric-like (TE-like) polarized modes (the electric eld inthe central mirror plane of the photonic crystal slab is perpendicular to the airholes). In contrast, the transverse-magnetic-like (TM-like) polarized bandgap isfavored in a la ice of high-aspect-ratio rods [ , ]. TM-like cavities have beendesigned in an air-hole geometry, as well [ – ], but theQ factors of these cav-ities were limited to the order of . In addition, the lack of vertical con nementof these cavities results in large mode volumes [ ]. ough it is possible to em-ploy surface plasmons to localize the light tightly in the vertical direction, the lossynature of metal limits theQ to about [ ].

In this chapter, we report a one-dimensional ( D) photonic crystal nanobeamcavity design that supports an ultra-high-Q (Q > ) TM-like cavity mode withV ∼ (λ/n) . is cavity greatly broadens the applications of optical nanocavities.For example, it iswell-suited for photonic crystal quantumcascade lasers, since theinter-subband transition in quantum cascade lasers is TM-polarized [ – ].We also demonstrate that our cavity simultaneously supports two ultra-high-Qmodes with orthogonal polarizations (one TE-like and one TM-like). e fre-quency difference of the two modes can be widely tuned while maintaining thehigh Q factor of each mode, which is of interest for applications in nonlinear op-tics.

Figure 6.1.1: (a) Schematic of the nanobeam design, showing the nanobeamthickness (dy) and width (dx), and the hole spacing (a). (b) TE and TMtransverse mode profiles for a ridge waveguide with dy = dx. (c) Transmissionspectra for the TE (red) and TM (blue) Bloch modes. The shaded areasindicate the bandgaps for both modes.

. D

Our design is based on a dielectric suspended ridge waveguide with an array ofuniform holes of periodicity, a, and radius, R, which form a D photonic crystalBragg mirror [ ], as shown in Fig. . . (a). e refractive index of the dielec-tric is set to n = . (similar to Si and GaAs at ∼ . μm). We rst start with aridge of height:width:period ratio of : : (dx = a, dy = a) and R = . a.Fig. . . (b) shows the transverse pro les of the fundamentalTM-like andTE-likemodes (TM and TE ) supported by the ridge waveguide. e TM mode hasits major component (Ey) lined along the hole axis, whereas the TE mode’s ma-jor component (Ex) is perpendicular to the air holes. Using the three-dimensional( D) nite-difference time-domain (FDTD) method, the transmi ance spectraare obtained of the TM and TE modes launched towards the Bragg mirror.Fig. . . (c) shows the TM andTE bandgaps, respectively. In contrast to two-dimensional ( D) photonic crystal slabs, where the photon is localized in the xzplane via Bragg sca ering, here we only require Bragg con nement in the longitu-dinal (z) direction, as light is transversely con ned in the other two dimensions bytotal internal re ection. It has also been shown experimentally that D photonic

crystal nanobeam cavities have comparableQ/V ratios to D systems [ , ].

Figure 6.2.1: (a) Schematic of the 1D photonic crystal nanobeam cavity,with the tuning parameters Rk and wk in the -segment tapered design. (b,c)Mode profiles of the electric field components ETE,x and ETM,y for the cavitydesign with dx = a, dy = a. (d,e) Spatial Fourier transform of the electricfield component profiles (ETE,x and ETM,y) in the xz plane (y = ).

Introducing a la ice grading to the periodic structure creates a localized po-tential for both TE- and TM-like modes. To optimize the mode Q factors, weapply the bandgap-tapering technique that is well-developed in previous work[ , , , , ]. We use an -segment tapered section with holes (R -R )and a -period mirror section at each side. Two degrees of freedom are availablefor each tapered segment: the length (wk) and the radius (Rk). We keep the ra-tio Rk/wk xed at each segment, and then implement a linear interpolation of thegrating constant ( π/wk). When the central segment w is set to . a, we ob-tain ultra-highQs and low mode volumes for both TE- and TM-polarized modes(QTE = . × , QTM = . × , VTE = VTM = . (λ/n) ), with free-space wavelengths . a and . a, respectively. Fig. . . (b) and (c) show themode pro les of the major components of the two modes in the xzmirror plane.

e ultra-highQ factors can also be interpreted inmomentum space [ , , ].Fig. . . (d) and (e) demonstrate the Fourier transformed (FT) pro les of theelectric eld components ETE,x and ETM,y in the xz plane (y = ), with the lightcone indicated by the white circle. It can be seen that both modes’ Fourier com-ponents are localized tightly at the bandedge of the Brillouin zone on the kz-axis(kz = π/a). is reduces the amount of mode energy within the light conethat is responsible for sca ering losses. It is also worthwhile to note that higher-longitudinal-order TE and TM cavity modes with different symmetry withrespect to the xy mirror plane exist [ ]. For example, the second-order TEmode, which has a node at the xymirror plane, resonates at a wavelength of . a.It has a higherQ factor of . × , but a larger mode volume of . (λ/n) .

For a number of applications of interest, control of the frequency spacing be-tween the two modes is required. Examples include polarization-entangled pho-ton generation for degenerate modes [ ], and terahertz generation for . −THzmode spli ing [ ]. We tune the frequency separation of the twomodes

by varying the thickness of the structure while keeping the other parameters con-stant. In Fig. . . (a), the cavity resonances of the TE and TM modes aretraced as a function of the nanobeam thickness (dy/a). e TM-likemodes have amuch larger dependence on the thickness than the TE-likemodes. e modes aredegenerate at dy = . a, and for thicknesses beyond this value, ωTE is larger thanωTM. As dy increases, the spli ing increases until it saturates when the system ap-proaches the D limit (structure is in nite in the y-direction). In this limit, we ndthat λTE = . a and λTM = . a. e frequency separation (δω = |ωTE − ωTM|)of this design ranges from THz to THz, with the TE-like mode wavelengthxed at . μm by scaling the structure accordingly. Fig. . . (b) shows the thick-

ness dependence of theQ factor for the xz design speci cations listed above. It canbe seen that theQ factors of both TE- and TM-polarizedmodes stay above forthe ωTE > ωTM branch.

Decreasing dy causes thewidth of theTMbandgap to sharply decrease, whereasthewidth of theTEbandgap remains almost constant. e narrowedTMbandgap

Figure 6.2.2: (a) TE (red) and TM (blue) cavity mode resonant frequen-cies (dotted lines) as a function of the nanobeam thickness. The bandgapregions of the two modes are shaded. The frequency separation (δω) of thetwo modes with the TE-like mode wavelength fixed at . μm by scaling thestructure accordingly is plotted in green. (b,c) Dependence of the Q factorand nonlinear overlap factor γ on the nanobeam thickness.

results in a reduced Bragg con nement, which increases the transmission lossesthrough the Bragg mirrors. is is evidenced by the Q factor of the TM mode,which drops to , when the thickness:width ratio is : . ough this leak-age can be compensated for, in principle, by increasing the number of periods ofthe mirror sections, the length of the structure also increases, which makes fabri-cation more challenging for a suspended nanobeam geometry. A narrow bandgapalso leads to large penetration depth of the mode into the Bragg mirrors, therebyincreasing the mode volume.

. A

Next, we examine the application of our dual-polarized cavity for the resonanceenhancement of nonlinear processes. To achieve a large nonlinear interaction inmaterials with dominant off-diagonal nonlinear susceptibility terms (e.g. χ( )

ijk , i =j = k), such as III-V semiconductors [ , , ], it is bene cial to mix twomodes with orthogonal polarizations. As shown in our previous work [ ], thestrength of the nonlinear interaction can be characterized by the modal overlap,which can be quanti ed using the following gure of merit,

γ ≡ εr,d

∫d d r

∑i,j,i=j ETE,iETM,j√∫

d rεr|ETE|√∫

d rεr|ETM|. ( . )

where∫d denotes integration over only the regions of nonlinear dielectric, and

εr,d denotes the maximum dielectric constant of the nonlinear material. Note thatwe have normalized γ so that γ = corresponds to the theoretical maximumoverlap. For the TE and TM modes we studied, the two major components(ETE,x and ETM,y) share the same parity (have anti-nodes in all the three mirrorplanes), and only two overlap components, ETE,xETM,y and ETE,yETM,x, in Eq. .do not vanish. is allows a large nonlinear spatial overlap. We obtain γ = .

for the cavity shown in Fig. . . . e overlap approaches γ = . in the limitdy → ∞. We nd that the overlap factor, γ, stays at a reasonably high value

(> . ) across the full range of the frequency difference tuning (for ωTE > ωTM

branch) [Fig. . . (c)].

Figure 6.3.1: Parameters of the higher-order cavity modes for the designwith dx = a, dy = a.

Finally, it is important to note that thick nanobeams can support higher-ordermodeswith adifferent numberof nodes in thexyplane, aswell. esehigher-ordermodes are also con ned in the tapered section within their respective bandgaps,with theQ factors and wavelengths listed in Fig. . . for the dx = a and dy = acase. esemodes can offer a broader spectral range than the fundamentalmodes,which is of great interest to nonlinear applications requiring a large bandwidth[ ].

. S

In conclusion, we have demonstrated that ultra-high-Q TE- and TM-like funda-mental modes with mode-volumes ∼ (λ/n) can be designed in D photoniccrystal nanobeam cavities. We have shown that the frequency spli ing of thesetwo modes can be tuned over a wide range without compromising the Q factors.We have also shown that these modes can have a high nonlinear overlap in mate-rials with large off-diagonal nonlinear susceptibility terms across the entire tuningrangeof the frequency spacing. Weexpect these cavities to have broad applicationsin the enhancement of nonlinear processes.

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Author List

e following authors contributed to Chapter :Changlin Li.

e following authors contributed to Chapter :Ian Frank.

e following authors contributed to Chapter :Mughees Khan, Parag Deotare, Yong Huang, Jae-Hyun Ryou, Russell Dupuis,

Yi Song, Jing Kong.

e following authors contributed to Chapter :Irfan Bulu, Tancredi Bo o, Wai-Ming Tam, Ben Levi .

e following authors contributed to Chapter :Ian Burgess, MurrayMcCutcheon.