Nonlinear Properties of III-V Semiconductor Nanowaveguides

Home , Gallium phosphide, Indium phosphide

ELEONORA DE LUCA

Doctoral Thesis in Physics School of Engineering Sciences KTH Royal Institute of Technology Stockholm, Sweden 2019

Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan fram- lägges till oﬀentlig granskning för avläggande av teknologie doktorsexamen i fysik onsdag den 23 oktober 2019 klockan 10:00 i sal FA32, AlbaNova Universitetscent- rum, Kungliga Tekniska Högskolan, Roslagstullsbacken 21, Stockholm.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 "O frati," dissi, "che per cento milia perigli siete giunti a l’occidente, a questa tanto picciola vigilia d’i nostri sensi ch’é del rimanente non vogliate negar l’esperïenza, di retro al sol, del mondo sanza gente. Considerate la vostra semenza: fatti non foste a viver come bruti, ma per seguir virtute e canoscenza". Dante Alighieri. Commedia. Inferno – Canto XXVI.

"O brothers, who amid a hundred thousand Perils," I said, "have come unto the West, To this so inconsiderable vigil Which is remaining of your senses still, Be ye unwilling to deny the knowledge, Following the sun, of the unpeopled world. Consider ye the seed from which ye sprang; Ye were not made to live like unto brutes, But for pursuit of virtue and of knowledge" Dante Alighieri. Commedia. Inferno – Canto XXVI. Translated by Henry Wadsworth Longfellow.

Abstract

Nonlinear optics (NLO) plays a major role in the modern world: nonlinear optical phenomena have been observed in a wavelength range going from the deep infrared to the extreme ultraviolet, to THz radiation. The optical nonlinearities can be found in crystals, amorphous materials, polymers, liquid crystals, liquids, organic materials, and even gases and plasmas. Nowadays, NLO is relevant for applications in quantum optics, quantum computing, ultra-cold atom physics, plasma physics, and particle accelerators. The work presented in the thesis is limited only to the semiconductors that have a second-order optical nonlinearity and includes two phenomena that use second-order nonlinearity: second-harmonic generation (SHG) and spontaneous parametric down-conversion (SPDC). Among the many options available, the investigation presented concerns gallium phosphide (GaP) and gallium indium phosphide (Ga0.51In0.49P), two semiconductors of the group III-V with the 43¯ m crystal symmetry. However, some of the results found can be generalized for other materials with 43¯ m crystal symmetry. In the thesis, the fabrication of GaP nanowaveguides with dimensions from 0.03 µm and an aspect ratio above 20 using focused ion beam (FIB) milling is discussed. The problem of the formation of gallium droplets on the surface is solved by using a pulsed laser to oxidize the excess surface gallium locally on the FIB-milled nanowaveguides. SHG is used to evaluate the optical quality of the fabricated GaP nanowaveguides. Additionally, a theoretical and experimental way to enhance SHG in nanowaveguides is introduced. This process uses the overlap of interacting fields defined by the fundamental mode of the pump and the second-order mode of the SHG, which is enhanced by the longitudinal component of the nonlinear polarization density. Through this method, it was possible to obtain a maximum efficiency of 10−4, which corresponds to 50 W−1cm−2. The method can be generalized for any material with a 43¯ m crystal symmetry. Furthermore, SHG is used to characterize the nonlinear properties of a nanostructure exposed for a long time to a CW laser at 405 nm to reduce the photoluminescence (PL) of Ga0.51In0.49P. The PL was reduced by -34 dB without causing any damage to the nanostructures or modifying the nonlinear properties. The fabrication process for obtaining the nanowaveguide is interesting as well, since the fabricated waveguide in Ga0.51In0.49P, whose sizes are ∼ 200 nm thick, ∼ 11 µm wide and ∼ 1.5 mm long, was transferred on silicon dioxide (SiO2). This type of nanowaveguide is interesting for SPDC, since it satisfies the long interaction length necessary for an efficient SPDC. Finally, a configuration consisting of illuminating the top surface of a nanowaveguide with a pump beam to generate signal and idler by SPDC is presented. These fabricated nanostructures open a way to the generation of counter-propagating idler and signal with orthogonal polarization. By using a different cut of the crystal, i.e. [110], it makes possible to obtain degenerate wavelength generation, and in certain conditions to obtain polarization-entangled photons or squeezed states.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7

Sammanfattning

Icke-linjär optik spelar en viktig roll i det moderna samhället: icke-linjära optikfenomen har observerats i ett våglängdsområde som inkluderar det djupa infraröda till det extrema UV, till THz-strålning. De optiska icke-lineariteterna kan återfinnas i kristaller, amorfa material, polymerer, flytande kristaller, vätskor, organiska material och till och med gas och plasma. Några av de mest kända applikationerna inkluderar andra ordningens harmonisk generation, Q- switching och modlåsning, som är frekvent använda i kommersiellt tillgängliga lasrar. Dessutom uppfyller i allt högre grad icke-linjär optik kraven som ställs av kvantoptik, kvantberäkning, ultrakalla atomer fysik, plasmafysik och parti- kelacceleratorer. Arbetet som presenteras i avhandlingen är endast begränsat till icke-linjära optik i halvledare som har en andra ordningens icke-linjäritet och inkluderar två av de möjliga fenomenen: andra ordningens harmonisk generation och spontan paramterisk nerkonvertering. Bland de olika alterna- tiven avser i synnerhet arbetet som presenteras i avhandlingen en specifik kategori av halvledare: de så kallade III-V halvledarna med kristall symmetri 43¯ m. I kategorin ingår många material, för vilka några av de resultat vi hittat kan generaliseras från det specifika fallet med studien, som har som huvudfokus på galliumfosfid (GaP) och galliumindiumfosfid (GaInP). I den här avhandlingen kan läsaren lära sig mer om tillverkningen av GaP nanovågledare med dimensioner från 0.03 µm och höjd-till-breddkvot över 20 tillverkad med fokuserad jonstråleetsning och hur problemet med bildan- det av galliumdroppar på ytan löses genom att använda en pulsad laser för att oxidera överskottet av gallium lokalt på de fokuserad jonstråleetsade na- nostrukturerna. Andra ordningens harmonisk generationen används för att utvärdera den optiska kvaliten hos de tillverkade GaP nanovågledare. Dessutom introduceras ett teoretiskt och experimentellt sätt att förbättra den andra ordningens harmonisk generationen i nanovågledare genom att till- fredsställa modal fasmatchning. Denna process använder överlappet av de samverkande fälten definierade av den grundläggande pump moden och andra ordningens mod för det upperkonvertade ljuset. Processen utnyttjar också den längsgående komponenten av den olinjära polarisationen. Genom denna metod var det möjligt att uppnå en maximal effektivitet på 10−4, vilket mot- svarar 50 W−1 cm−2. Metoden kan i princip kan användas i vilket material som helst med kristallografisk symmetri på 43¯ m. Dessutom undersöks förstärkning. Den andra ordningens harmonisk generationen används för att karakterisera de olinjära egenskaperna hos en nanovåg- ledare som exponerats under en lång tid för en CW-laser vid 405 nm, med målet att minska materialets fotoluminiscens. Metoden var framgångsrik i re- duktionen av fotoluminiscen eftersom fotoluminiscen minskades med -34 dB utan att orsaka någon skada på nanovågledarna eller att modifiera de olinjära egenskaperna. Tillverkningsprocessen för att erhålla nanovågledaren också intressant eftersom den tillverkade vågledaren i GaInP, vars storlekar är ∼ 200 nm tjocka, ∼ 11 µ m bred och ∼ 1.5 mm lång, överfördes på kiseldioxid. Denna typ av nanovåg- ledare är intressant för spontan parametrisk nedkonvertering, som kräver en längre interaktionslängd än den som krävs för andra ordningens harmonisk

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 generation. Slutligen presenteras en intressant konfiguration, som består i att belysa den övre ytan på en nanovågledare med en pumpstråle för att generera en signal och en idler med genom spontan paramterisk nedkonvertering. De tillverkade nanostrukturernamedjer generering av motpropagerande idler och signal mo- der med ortogonal polarisering. Genom användning av en annan skärning snitt av kristallen, dvs [110], är det möjligt att erhålla degenererad våglängdsgene- rering, och under vissa förhållanden kan man få polarisationssammanflätade fotoner. I denna konfiguration, genom att lägga till en spegel i ena änden av vågledaren, kan systemet användas för att producera ett s.k. klämt tillstånd.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Acknowledgements

I wish to express all my gratitude to Dr. Marcin Swillo, who gave me the oppor- tunity to work on these fascinating topics as well as the continuous help to move forward in my research. I cannot quantify how much I learned from him! I would also thank my co-supervisors Prof. Gunnar Björk and Prof. Anand Srinivasan for their suggestions, help, and support. I am also grateful to Prof. Katia Gallo, who shares with me this intricate and passionate love for Italy and whose scientific passion gave me new nourishment when I felt doubtful about my research. I wish to sincerely thank Dr. Adrian Iovan and Dr. Anders Liljeborg, for their help inside the Albanova NanoFabLab cleanroom. I also would like to thank all the wonderful people that have been colleagues ("or sort of...", Ed.) or former colleagues. Special thanks to Amin, Saroosh, Mattias, and Alessandro, for the coffee chats and the office chats. A big thanks to Anne-Lise for being such a good travel companion and a lovely friend. I wish to thank all the long-time and long-distance friends, who spent endless time listening and cuddling me. Thank you, Lu, Isa, Mary, Ile, Giró, Dani, Teo, Mar- colino. Heartfelt thanks go to my "Swedish Family", the time spent with you has been precious. Last but not least, I would like to thank my parents and my sister Elisabetta for the support they provided along with my whole life, words are not enough to express my feelings for you. Vi voglio bene. E mi mancate ogni giorno. Finally, I wish to thank my partner (in life and science) Federico, a wonderful peer (our collaboration ended up being a paper, Ed.), my strongest supporter or faultfinder, depending on the need. I love you.

Eleonora De Luca, Stockholm, 2019-10-23

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7

Contents

Contents x

List of Figures xii

List of Tables xiv

List of Publications xix

1 Introduction. 1 1.1 III-V Semiconductors...... 2 1.2 Optical Waveguides in Semiconductors...... 5 1.3 Outline of the Thesis...... 6

2 Nonlinear Optical Properties of III-V Semiconductors. 7 2.1 The Electromagnetic Formulation of the Nonlinear Interaction. . . . 9 2.2 Optical Second-Harmonic Generation...... 10 2.3 Surface Contribution to Second Harmonic Generation...... 13 2.4 Spontaneous Parametric Down-Conversion...... 14

3 Fabrication of III-V Nanostructures. 19 3.1 Nanowaveguides in Gallium Phosphide...... 20 3.1.1 Focused Ion Beam...... 20 3.1.2 Electron-beam Lithography and Dry Etching...... 22 3.1.3 Maskless Photolithography for Planar Nanowaveguides. . . . 22 3.1.4 Dry Etching and Wet Etching...... 23 3.2 Nanostructures in Gallium Indium Phosphide...... 25 3.2.1 Maskless Photolithography for Planar Nanowaveguides and Grids...... 25 3.2.2 Dry Etching and Wet Etching for Transferring the Nanos- tructures on a Glass Substrate...... 26 3.2.3 Reducing the Width of the Nanostructures...... 29 3.3 Summary ...... 29

x 4 Nonlinear Optical Properties of Gallium Phosphide Nanostruc- tures. 31 4.1 Mode Analysis and Phase Matching for Second-Harmonic Genera- tion: Simulations...... 31 4.1.1 Mode Analysis for Arrays of Slab Waveguides...... 33 4.1.2 Mode Analysis for Arrays of Cylindrical Nanowaveguides. . . 36 4.1.3 Modal Phase Matching Vs. Quasi-Phase Matching...... 39 4.2 Wide Bandwidth Second-Harmonic Generation...... 41 4.3 Surface Nonlinearity as a Tool to Evaluate the Quality and the Width of the Nanowaveguides...... 44 4.4 Summary...... 46

5 Nonlinear Applications of Gallium Indium Phosphide Nanowaveg- uides. 47 5.1 Simulations and Analysis of Spontaneous Parametric-Down Conver- sion in Nanowaveguides...... 48 5.2 Phase-Matching Condition of the Fabricated Nanowaveguides. . . . . 55 5.3 Reducing the Photoluminescence by Laser Irradiation...... 57 5.4 Summary...... 59

6 Conclusions and Outlook. 61

A Rotation of the axis to evaluate the nonlinear components. 65

Bibliography 67

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 List of Figures

1.1 Refractive index of GaP...... 4 1.2 Refractive index of Ga0.51In0.49P...... 5

2.1 Schematics of energy and momentum conservation principles for SHG process...... 11 2.2 Schematics of OPA process and SPDC process...... 14

3.1 SEM images of one of the arrays of SWs in GaP made by FIB milling. . 21 3.2 SEM images of an array of NPs and an array of SWs made by EBL. . . 22 3.3 SEM images of the cross-section of one of the attempted fabrication of T-shaped waveguides in GaP...... 24 3.4 Scheme of the fabrication steps for a T-shaped waveguide...... 24 3.5 Schemes of GaInP epitaxial wafer and of the fabrication steps to obtain the nanostructures in GaInP...... 26 3.6 Nanostructures on Ga0.51In0.49P. View with optical microscope. (a) Lines of resist on Ga0.51In0.49P before etching. (b) Grid of resist on Ga0.51In0.49P before etching. (c) Enlarged part of one of the waveguides in Ga0.51In0.49P transferred onto glass. (d) Piece of one of the grids in Ga0.51In0.49P transferred onto glass. (e) Entire waveguide in Ga0.51In0.49P transferred onto glass...... 27 3.7 Nanostructures on Ga0.51In0.49P. View with optical microscope. (a) Entire waveguide 150 µm-wide in Ga0.51In0.49P transferred on glass with polymer on top. (b) Entire waveguide 10 µm-wide in Ga0.51In0.49P transferred on glass...... 28

4.1 Orientation of the array of SWs with respect to the plane xy and the crystallographic axis...... 33 4.2 Electric field profiles of the modes TE0 and TM0 in a GaP array of 5 SWs (205 nm width, 150 nm air) for the wavelength 1140 nm...... 34 4.3 Electric field profiles of the modes TE1 and TM1 in a GaP array of 5 SWs for the wavelength 570 nm...... 35 4.4 Effective refractive index of the interacting modes in SHG process. Nor- malized SHG intensity for an array of 5 GaP SWs...... 36

xii 4.5 Electric field profiles of the mode HE11 in a GaP array of 5 by 5 NPs for the wavelength 1140 nm...... 37 4.6 Electric field profiles of the modes excited in the SHG for the mode HE21 and TM01 in a GaP array of 5 by 5 NPs for the wavelength 570 nm. Effective refractive index of the modes and normalized SHG intensity (Ix). 38 4.7 Efficiency of SHG process for CW pump at the wavelength 1.2 µm in a single-slab waveguide...... 40 4.8 Optical setup used to measure SHG in the arrays of GaP NPs and SWs. 41 4.9 Power of the SHG light measured in an array of 5 by 5 nanopillars and in an array of 5 slab waveguides...... 42 4.10 Far field of SHG generated in array of SWs...... 43 4.11 Far field of SHG generated in array of NPs and simulation of its Fourier decomposition...... 43 4.12 Optical setup and measurements of SHG in the arrays of GaP SWs fabricated by FIB...... 44 4.13 Polarization plots of SHG light in an array of SWs (fabricated at 50 pA FIB current) ...... 45

5.1 Schemes (a) for SPDC process in a nonlinear waveguide with 43¯ m crystal symmetry for counter-propagating signal and idler. (b) Momentum conservation diagram presented in the reciprocal space ...... 49 5.2 Calculated dispersion inside a waveguide made of GaInP in a air cladding for diﬀerent widths of the waveguide for λp = 700 nm...... 51 5.3 Overlap among the modes for a waveguide made of GaInP in a air cladding for non-degenerate generation of photon-pairs...... 52 5.4 Schemes of a slab waveguide with 43¯ m crystal symmetry with the crystal axis rotated 45o with respect to xyz...... 52 5.5 Overlap among the modes for a waveguide made of GaInP in a air cladding for degenerate generation of photon-pairs...... 54 5.6 Nanowaveguide geometry including a mirror at the end of the waveguide. 54 5.7 PM condition for an asymmetric SW...... 56 5.8 Normalized PL vs. 405 nm pump power...... 58 5.9 Polarization measurement of the SHG light for a GaInP nanostructure. 59

A.1 Slab waveguide rotated 45o with respect to the crystallographic axis. . . 65

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 List of Tables

1.1 Comparison of materials used for applications in nonlinear optics. . . . 2

3.1 Comparison of diﬀerent objectives for the SPL...... 23

5.1 PM condition for an asymmetric SW...... 57

xiv Acronyms

(AlxGa1−x)yIn1−yP aluminium gallium indium phosphide

Al aluminium AlxGa1−xP aluminium gallium phosphide As arsenic

BOE buﬀered oxide etch BPM birefringent phase matching

CCD charge-coupled device CW continuous wave

DI-H2O deionized water

EBL electron-beam lithography

FIB focused ion beam

Ga gallium Ga+ gallium ions Ga2O3 gallium oxide GaxIn1−xP gallium indium phosphide Ga0.51In0.49P gallium indium phosphide GaAs gallium arsenide GaP gallium phosphide

H2O2 hydrogen peroxide H3PO4 phosphoric acid HCl hydrochloric acid

xv HF hydroﬂuoric acid

ICP inductively coupled plasma ICP-RIE inductively coupled plasma reactive ion etching In indium

KTP potassium titanyl phosphate

LED ligth emitting diode LiNbO3 lithium niobate

MPM modal phase matching

N nitrogen NLO nonlinear optics NP nanopillar

OPA optical parametric ampliﬁcation OPO optical parametric oscillator

P phosphorus PB photobleaching PD photodarkening PL photoluminescence PM phase-matching

QPM quasi-phase matching

RIE reactive-ion etching

Sb antimony SEM scanning electron microscope SHG second-harmonic generation Si silicon SiO2 silicon dioxide SOI silicon-on-insulator SPAD single-photon avalanche diode SPDC spontaneous parametric down-conversion SPL smart print lithography

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 SVEA slowly varying envelope approximation SW slab waveguide

UV ultraviolet

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List of Publications

Publications Included in the Thesis.

1. E. De Luca, R. Sanatinia, S. Anand, and M. Swillo, Focused ion beam milling of gallium phosphide nanostructures for photonic applications, Optical Mate- rials Express 6, 587-596 (2016). Author Contribution: Design of the experiments, simulations, sample fabrication, part of characterization, data analysis, writing the paper. 2. E. De Luca, R. Sanatinia, M. Mensi, S. Anand, and M. Swillo, Modal phase matching in nanostructured zinc-blende semiconductors for second-order nonlinear optical interactions, Physical Review B 96, 075303 (2017). Author Contribution: Part of the theoretical background, simulations, part of the characterization, data analysis, writing the paper collaboratively with M. Swillo. 3. E. De Luca, D.Visser , S. Anand, and M. Swillo, Gallium indium phosphide microstructures with suppressed photoluminescence for applications in nonlinear optics, Accepted for publication in Optics Letters (2019). Author Contribution: Sample fabrication, characterization and data analysis, writing the paper. 4. E. De Luca and M. Swillo, Degenerated spontaneous parametric down conversion in semiconductor waveguide with -43m Crystal Symmetry, Manuscript (2019). Author Contribution: Part of the theoretical background, simulations, writing the paper collaboratively with M. Swillo.

xix Other Publications not Included in the Thesis.

1. F. Ribet, E. De Luca, F. Ottonello-Briano, M. Swillo, N. Roxhed, and G. Stemme, Zero-insertion-loss optical shutter based on electrowetting-on- dielectric actuation of opaque ionic liquid microdroplets, Appl. Phys. Lett. 115 (7), 073502 (2019). Author Contribution: Data collection and analysis about the characterization in the visible range.

Conference Contributions

1. E. De Luca, R. Sanatinia, S. Anand, and M. Swillo, Focused Ion Beam Milling of Gallium Phosphide Nanowaveguides for Non-linear Optical Applications, in Advanced Photonics 2016 (IPR, NOMA, Sensors, Networks, SPPCom, SOF), OSA technical Digest (online) (Optical Society of America, 2016), ITu3A.5. 2. E. De Luca, R. Sanatinia, M. Mensi, S. Anand, and M. Swillo, Modal Phase Matching in Nanostructured Zincblende Semiconductors for Second-Harmonic Generation, in Conference on Lasers and Electro-Optics, OSA Technical Di- gest (online) (Optical Society of America, 2017), JTu5A.60. 3. E. De Luca, D. Visser, S. Anand, and M. Swillo, Gallium Indium Phosphide Nanostructures with Suppressed Photoluminescence for Applications in Non- linear Optics, in Frontiers in Optics / Laser Science, OSA Technical Digest (Optical Society of America, 2018), JTu3A.83. 4. D. Visser, R. Yapparov, E. De Luca, M. Swillo, S. Marcinkevicius and S. Anand, Top-Down Fabrication of High Quality Gallium Indium Phosphide Nanopillar/disk Array Structures, in 14th IEEE Nanotechnology Materials and Devices Conference (IEEE NMDC 2019).

Regional Conference Contributions

1. E. De Luca, R. Sanatinia, S. Anand, and M. Swillo, Focused Ion Beam Milling of Gallium Phosphide Nanostructures for Optical Nonlinear Applications, in Optics and Photonics in Sweden OPS, 2015, Sweden 2. E. De Luca, R. Sanatinia, M. Mensi, S. Anand, and M. Swillo, Modal Phase Matching in Gallium Phosphide Nanowaveguides, in Optics and Photonics in Sweden OPS, 2016, Sweden 3. E. De Luca, R. Sanatinia, M. Mensi, S. Anand, and M. Swillo, Modal Phase Matching in Nanostructured Zincblende Semiconductors for Second-Harmonic Generation, in Optics and Photonics in Sweden OPS, 2017, Sweden

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Chapter 1

Introduction.

Nonlinear optics (NLO) is the branch of optics that describes the behavior of light in a nonlinear medium, so all that media in which the polarization density responds nonlinearly to a high electric field of the light. The field started to have relevance after 1960, when two-photon absorption [1] and second-harmonic generation (SHG) generation were observed [2], following the construction of the first laser [3], as well as the theoretical work on optical wave-mixing was presented [4]. In the following decades, the field was subject to a prodigious growth, leading to the observation of other physical phenomena and giving rise to novel concepts and applications. Examples include frequency doubling of a monochromatic wave (SHG), the mixing of two monochromatic waves to generate a third wave at their sum or difference frequencies (frequency conversion), the use of two monochromatic waves to amplify a third wave (optical parametric amplification (OPA)), and the incorporation of feed- back in a parametric-amplification device to create an oscillator (optical parametric oscillator (OPO)), third-harmonic generation, self-phase modulation, self-focusing, four-wave mixing, and phase conjugation [5]. Most of the first achievements were made on bulk crystals, where cumbersome phase-matching (PM) condition (often based on the birefringence of the material) limits the efficiency of the nonlinear processes. However, recently, nonlinear optics moved towards miniaturization of the nonlinear media. Significant advancements in nano- and micro-fabrication technologies have widened the experimental and theoretical framework in which nonlinear optical processes are investigated, opening to new materials that do not have natural birefringence. In this context, materials with outstanding second-order nonlinear properties emerged as presented in Table 1.1. Although materials like potassium titanyl phosphate (KTP) and lithium niobate (LiNbO3) are widely used for several commercial application, semiconductors are receiving more and more attention for their large second-order nonlinear optical coefficients, their large refractive index, as well as the possibility to guarantee integration on wider platforms. Among those, one finds the III-V semiconductors alloys as well as silicon (Si).

1 (2ω) dijk Transparency Refractive Thermal Material [pm/V] at Range Index Conductivity Birefringent 1.064 µm [µm] at 1.5 µm [W/mK] GaP 70 0.5 - 11 3.1 110 No GaAs 170 0.9 -17 3.4 50 No GaInP 110 > 0.68 3.1 5.3 No Si 0 1.1 - 5 3.5 150 No LiNbO3 25 0.4 - 4.5 2.2 5 Yes KTP 17 0.35 - 4.5 1.7 13 Yes

Table 1.1: Comparison of materials used for applications in nonlinear optics. References from [6–11]

While a more detailed discussion about III-V alloys can be found further on in the chapter, a brief aside about Si is required. This material does not show any second-order nonlinear optical coeﬃcient in bulk (Table 1.1) but shows third-order, making it suitable for nonlinear eﬀects such four-wave mixing [12] or third-harmonic generation [13, 14] that will not be discussed further in the thesis. However, it has a second-order type of nonlinearity on the surface, due to the broken symmetry at the interfaces, making it interesting for second-order nonlinear processes when micro and nanostructures are involved [15,16]. Si is strongly exploited due to high technological knowledge on this material and its fabrication since it has been the main platform for the electronics industries, it is cheap and abundant on Earth and has also good thermal and mechanical properties. In this sense, most of the new technologies are realized with an eye to a possible integration on Si platforms as well as silicon-on-insulator (SOI) or oxides [17–24], which would allow electronic- photonic integration as well as reducing costs, which for most materials are very high if compared with Si.

1.1 III-V Semiconductors.

Compound semiconductors are another important category, which includes the so- called III-V semiconductors alloys, which are made of elements of group III (gallium (Ga), aluminium (Al), indium (In)) and elements of group V (arsenic (As), phosphorus (P), nitrogen (N), antimony (Sb)). They are generally divided in binary (e.g. gallium arsenide (GaAs), gallium phosphide (GaP)), ternary (e.g. gallium indium phosphide (GaxIn1−xP)), and quaternary (e.g. aluminium gallium indium phosphide ((AlxGa1−x)yIn1−yP)). III-V semiconducting compounds are broadly used as materials for optoelectronic devices such as ligth emitting diodes (LEDs), laser diodes and photo-detectors, as well as for field-effect transistors, high electron mobility transistors and heterojunction bipolar transistors [21,22,25–30]. More recently their use for nonlinear optical application became more relevant [18,31–45]. Among the binary compounds, it is possible to find material with direct bandgap

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 (GaAs) as well as with indirect bandgap (GaP). The ternary and quaternary alloys bandgap can be adjusted by changing the composition, generating materials whose bandgap can be either direct or indirect depending on the wavelength of emission [28]. However, defects cause a great limitation on the quality of the material. As a consequence, it is settled practice to grow the ternary or quaternary alloy on a material with the same lattice constant, thus limiting the presence of defects. Hence, in these alloys, the bandgap energy, and the lattice constant can not be selected independently [28]. Among the different III-V compounds, a very interesting material for nonlinear optical applications is GaP, because of its high nonlinear coefficient, and indirect bandgap in the visible range at 2.24 eV [6]. The latter characteristic is of particular interest due to the fact that in principle the material can be used for nonlinear optical processes exciting the material close to the bandgap, where the nonlinearity is larger. The refractive index of GaP is between 4.0 at 400 nm, 3.47 at 540 nm and 3.18 at 830 nm [6], as shown in Fig. 1.1. This enhances the rapid variation of the electric field normal to the surface, making the material of interest also for nonlinear optical applications involving the surface [38]. Furthermore, GaP shows a very strong nonlinear coefficient also for bulk, which is 70 [pm/V] ε0 at 1064 nm and 159 [pm/V] ε0 at 852 nm [7], which is larger than most of the material presented in Table 1.1. Moreover, GaP has a high thermal conductivity ( ≈ 110 W/mK [8]) as well as broad transparency range (from ≈ 0.5 to 11 µm [6]). GaP has been widely employed for photonics applications in recent years. To fully exploit the nonlinear properties of the material, it is necessary to use microstructure and nanostructures, hence several geometries have been investigated. These include nanopillars [35, 36, 38], and photonic crystals and membranes [31, 32, 46], and a recent techniques consisting in spatially inverting the nonlinear susceptibility during growth of GaP, known as orientation-patterned GaP (OPGaP) [39]. More recently, example of integration on Si and silicon dioxide (SiO2) have been presented [23, 24]. However, GaP has a main constraint due to the limited knowledge about the fabrication as well as the relatively poor availability as a commercial product. The material is has been used for the fabrication of LEDs [47, 48], but the requirements for nonlinear optical applications are different: impurities and defects have a large impact on the final quality. The techniques to obtain nanostructures and microstructures of high quality are more and more under investigation due to the recent interest of the nonlinear optics community in the material. All the works cited have in common the difficulties in producing high-quality structures, a problem that is intrinsic to the III-V alloys [49]. In the experimental work presented in the thesis, a wafer 400 µm thick was used. The fabrication procedures applied allowed us to obtain good quality nanostructures for SHG. The necessity to increase the aspect ratio of the fabricated nanostructures (i.e. increasing the length while keeping the cross-section constant) brought us to a turning point: working with a thin layer of GaP grown on another material or changing the material, with similar nonlinear properties but that could be easily

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 grown as a thin layer. One of the major issues was ﬁnding commercially available epitaxially-grown multi-layers of GaP of good quality (like grown on aluminium gallium phosphide (AlxGa1−xP) [31]). Diﬀerent approaches consisting of reducing the thickness of the 0.400 µm wafer or alternative geometries were tested. However, after not obtaining satisfactory results, the problem was bypassed by changing the material

Figure 1.1: Refractive index of GaP as from ref. [6, 11].

Gallium indium phosphide (Ga0.51In0.49P) due to its higher availability on the market as well as the similar optical properties (Fig. 1.2), and a larger nonlinear coefficient [9] than GaP was chosen. In particular, the composition chosen has the same lattice constant as the one of GaAs [28,49,51]: this allows a good quality material grown epitaxially on a GaAs substrate. However, there is one main difference with respect to GaP: the material, with this composition, has a direct bandgap, around 1.9 eV [52]. While this make the material advantageous for several applications, including biomed- ical imaging [53], LEDs [21,25], transistors [26,27] and photovoltaic cells [22,29,30], the direct bandgap can be a limitation since a very strong photoluminescence (PL) is excited when the material is excited close to the bandgap. To be able to use Ga0.51In0.49P close to the bandgap, reducing the excited PL was an important requirement to satisfy. To avoid this problem, most of the work for nonlinear optical application made on the material is for the so-called telecom wavelengths (1260 - 1625 nm) [18, 33, 34, 54]. The choice of using the material close to the bandgap, despite the indisputable second-order nonlinear properties is also suggested by the possibility of detection in the detection window of Si single-photon avalanche diode (SPAD), below 1 µm [55], this because of the larger efficiency, lower costs and the possibility of using it with-

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 1.2: Refractive index of Ga0.51In0.49P as from ref. [50] out a cryostat or bulky cooling systems. This is possible since the material is transparent above 0.68 µm (Table 1.1). As for GaP the material is relatively newly used for applications in nonlinear optics, so the fabrication techniques to obtain high-quality microstructures and nanostructures are under development.

1.2 Optical Waveguides in Semiconductors.

Optical waveguides are structures in which an optical signal can propagate without experiencing diffraction. It is usually made by a core material that has a higher refractive index than the surrounding environment (identified with the cladding, by likeness with optical fibers, or the substrate). The light is confined in the region of higher refractive index in the transverse plane and propagates along the longitudinal direction, orthogonal to the transverse plane. The confinement of the light inside the waveguide is dependent on the refractive index of the core and the contrast with the cladding, as well as from the geometry and the mode of propagation of the light inside the waveguide: all these elements will be investigated further in the thesis for the cases related to the material and the nanostructures realized1. However, it is worth mentioning that waveguides have a very interesting property called waveguide (or modal) dispersion: depending on the properties of the waveguide and the wavelength of propagation, a different effective refractive index (neff )

1For references to the theory of the optical waveguides, a complete description is presented in [56, 57] and will be omitted here.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 can be associated to a specific mode, which can be seen as an indication of the quality of the confinement inside the waveguide. The effective refractive index is not only a material property but depends on the whole waveguide design, making waveguide especially interesting when it is important to work with PM, as better described later. However, III-V semiconductors are overall good materials to fabricate nano- and microwaveguide from due to their large refractive index. For this reason, most of the work presented in the thesis is based on their use for nonlinear optical applications.

1.3 Outline of the Thesis.

This thesis is organized into six chapters as follows: • Chapter 2 contains a brief theoretical background for the work discussed in the thesis: there the reader can find more explanation about the nonlinear optical process called SHG, the conditions that have to be satisfied for an efficient generation and the surface contribution of SHG. This chapter, being far from exhaustive, include a brief summary of what was written by A. Yariv and P. Yeh in their ev- erlasting book Photonics [56]. Furthermore, a brief introduction to the concept of spontaneous parametric down-conversion (SPDC) based on the Schrödinger’s interpretation of quantum mechanics is also included. • Chapter 3 contains all the procedures and techniques used for the fabrication of the nanostructures. This chapter is divided in two sections, one related to GaP and one related to Ga0.51In0.49P; it includes the work from papers Paper 1, Paper 2 and Paper 3. • Chapter 4 contains the characterization of the nonlinear optical properties of the nanostructures made in GaP presented in chapter 3. In particular, it refers to the papers Paper 1 and Paper 2. Here you can find also a more theoretical part including the analysis of the modes, one of the fundamental tool to fully understand the work presented in the thesis. • Chapter 5 contains an introductory characterization of the nonlinear optical properties of the microstructures made in Ga0.51In0.49P presented in chapter 3. In this chapter is included the theoretical analysis and a newly proposed scheme to generate polarization-entangled photon-pairs and vacuum squeezed state from a slab waveguide. The work presented in the chapter is still work in progress in many aspects, and, hopefully, the reader will find some interesting insights. It also refers to Paper 3 and Paper 4. • Chapter 6 contains the conclusions and the outlook related to this work. At the end, one can find the bibliography, where all the references are included.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Chapter 2

Nonlinear Optical Properties of III-V Semiconductors.

According to the scientific journal Nature, Nonlinear optics is the study of how intense light interacts with matter [58]. In general, the optical response of a material scales linearly with the amplitude of the electric field, this can be expressed by the polarization density P = ε0χ¯E, where ε0, χ¯ and E are the electric constant, the electric susceptibility tensor of the medium and the electric field vector, respectively. At high power, typically as the one provided by the lasers, the material shows a different behaviour: this lead to the so-called nonlinear effects. As a matter of fact, in any real atomic system, the polarization induced in the medium by any electric field is not exactly proportional to the electric field but can be expressed in a Taylor series expansion:

(1) (2) 2 (3) 3 P = ε0(¯χ E +χ ¯ E +χ ¯ E + ...) = (ω) (2ω) (3ω) (ω) = P + P + P + ... = P + PNL, (2.1)

(n) where PNL is the nonlinear polarization vector and χ¯ is known as n-th-order nonlinear optical susceptibilities of the medium and the presence of such a term is referred to as an n-th-order nonlinearity [56]. The χ¯(n) is usually a tensor with (n + 1)-th rank since they include both the polarization-dependent nature of the interaction and the symmetries (due to the crystal structure) of the nonlinear material. One component of the second-order nonlinear polarization vector P(2ω) can be written as:

(2ω) (ω) (ω) (ω) (ω) (ω) (ω) Pi = dixxEx Ex + diyyEy Ey + dizzEz Ez (ω) (ω) (ω) (ω) (ω) (ω) + 2dizyEz Ey + 2dizxEz Ex + 2dixyEx Ey , (2.2) where i = x, y, z. (2) The second-order nonlinear-optical tensor d = ε0χ¯ has non-zero elements for all the materials that are non-centrosymmetric, those that do not show inversion-center

7 symmetry as, for example, silicon (Si) does. However, the second-order nonlinear- optical coefficients show a dispersion depending on the wavelength [9]. The different crystal symmetries have different second-order nonlinear optical tensors, and while it is possible to define it through a tensor with 27 components, the contracted notation is mainly used [56]:

 2  Ex 2  (2ω)    Ey  Px d d d d d d 11 12 13 14 15 16  E2   (2ω) d d d d d d  z  Py  =  21 22 23 24 25 26   , (2.3) (2ω) 2EyEz  P d31 d32 d33 d34 d35 d36   z 2EzEx 2ExEy where the indices jk from Eq. (2.2) can be substituted by xx = 1, yy = 2, zz = 3, yz = zy = 4, xz = zx = 5, xy = yx = 6 and i = 1, 2, 3, respectively [56]. In most of the crystals, the tensor d can be further simplified: many elements simplify to zeros while other are equal because the physical properties of a crystal must be invariant under certain symmetry operations. As mentioned in Chapter 1, two materials are investigated in this thesis: gallium phosphide (GaP) and gallium indium phosphide (Ga0.51In0.49P). Both materials exist in so-called zinc-blende crystal structure1(defined as 43¯ m in the Hermann - Mauguin notation [59]). For this reason, the d tensor can be simplified to:   0 0 0 d14 0 0 d = 0 0 0 0 d25 0  , (2.4) 0 0 0 0 0 d36 where it also holds that d = d14 = d25 = d36. In conclusion for those two materials, when the bulk is considered, the second-order nonlinear polarization along the different crystallographic axis are:

(2ω) Px = 2dEyEz, (2.5) (2ω) Py = 2dExEz, (2.6) (2ω) Pz = 2dExEy. (2.7)

Second-order nonlinearity in the material is responsible for several nonlinear eﬀects such as optical parametric ampliﬁcation (OPA) and spontaneous parametric down- conversion (SPDC) and three-wave mixing, including second-harmonic generation (SHG). While most of the discussion presented here is valid also when large crystals are considered, it is important to remember that the main focus of this work is on nanowaveguides.

1Both materials exist in another crystal structure known as wurtzite [60], which will not be investigated further.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 2.1 The Electromagnetic Formulation of the Nonlinear Interaction.

Before proceeding further, a brief overview of the electromagnetic formulation of the nonlinear interaction is required. The formulation used in this paragraph is adapted from [56]. Let’s consider Maxwell’s equations:

∂µ0H ∇ × E = − , (2.8) ∂t ∂D ∇ × H = J + , (2.9) ∂t as well as the following constitutive relations

D = ε0E + P, (2.10) (1) P = ε0(χ E + PNL), (2.11) J = σE, (2.12) where H, µ0, J, D, PNL and σ are the magnetizing field strength, the permeability of free space, the total electric current density, the displacement field, the nonlinear polarization density and the conductivity respectively. By rearranging Maxwell’s equations and the constitutive relations, it is possible to obtain the following wave equation: 2 2 2 ∂E ∂ E ∂ PNL ∇ E = µ0σ + µ0ε + ε0 . (2.13) ∂t ∂t2 ∂t2 If the problem is simplified to a one dimensional problem, with the wave propagating along the z direction and by defining the corresponding electromagnetic field to be a travelling wave, the following equation is obtained:

(ω1) 1 −i(ω1t−k1z) E (z, t) = (E1,ie + c.c.) (2.14) i 2 where the subscripts i = x0, y0, z0 are the Cartesian components in the principal coordinates of the nonlinear medium and the subscript 1 refers to the ﬁrst wave of the three involved in the process, ω1 and k1 are the frequency and the wavevector of the travelling wave and c.c. stands for complex conjugate. Since the processes described here are the one related to the second-order nonlinearity, i.e. including the interaction of three waves, it is possible to deﬁne similar equations for the other two waves:

(ω2) 1 −i(ω2t−k2z) E (z, t) = (E2,je + c.c.), (2.15) j 2

(ω3) 1 −i(ω3t−k3z) E (z, t) = (E3,ke + c.c.), (2.16) k 2 0 0 0 where the subscript j, k = x , y , z and ω2 (ω3) and k2 (k3) are the frequency and the wavevector of the second (third) travelling wave.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 After applying the slowly varying envelope approximation (SVEA), i.e. considering 2 ∂ El ∂El | ∂z2 | |kl ∂z | with (l = 1, 2, 3), and explicitly writing PNL(z, t) for ω1 = ω3 −ω2, which correspond to the process of the SHG when ω3 = 2ω1:

(ω1) ∗ −i[(ω3−ω2)t−(k3−k2)z PNL(z, t) = Ddijka2ja3kE2 E3e . (2.17) where D is a degeneracy factor, which is D = 2 if the fields Ek and Ej are distin- P guishable, or D = 1 otherwise and d = ijk dijka1ia2ja3k is the effective second- order nonlinear coefficient and a1i, a2j, a3k represent the polarization direction of the normal modes of propagation. After few steps of algebra it is possible to obtain the following equations: dE rµ rµ 1 0 0 ∗ −i(k3−k2−k1)z = −σ1 E1 − iω1 dE3E2 e , (2.18) dz ε1 ε1 dE∗ rµ rµ 2 0 ∗ 0 ∗ i(k3−k2−k1)z = −σ2 E2 + iω2 dE1E3 e , (2.19) dz ε2 ε2 dE rµ rµ 3 0 0 i(k3−k2−k1)z = −σ3 E3 − iω3 dE1E2e . (2.20) dz ε3 ε3

The Eqs. (2.18)–(2.20) can be simplified through the introduction of the field variable Al, corresponding to the photon flux amplitude: r nl Al = El with l = 1, 2, 3, (2.21) ωl q which is related to the so-called coupling parameter κ = d µ0 ω1ω2ω3 and the ε0 n1n2n3

q µ0 parameter αl = σl . It is worth noticing that αl = 0 when the nonlinear εl medium is transparent at the respective frequency ωl. If one wants to rewrite Eqs. (2.18)–(2.20) by using the deﬁnition in Eq. (2.21), the following general coupled equations are obtained: dA 1 ∗ −i(k3−k2−k1)z = −α1A1 − iκA A3, e (2.22) dz 2 dA 2 ∗ −i(k1−k3+k2)z = −α2A2 + iκA1A , e (2.23) dz 3 dA 3 −i(k1+k2−k3)z = −α3A3 − iκA1A2e ; (2.24) dz which will be further investigated in the two cases of SHG and OPA.

2.2 Optical Second-Harmonic Generation.

SHG is used in a variety of applications, including the generation of visible co- herent light [61], probing the quality of surfaces [36, 62], quantum information

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 2.1: (a) Schematic of energy conservation principle for SHG process. The energy is considered in units of ~. (b) Schematic of momentum conservation principle for SHG process. The linear momentum is considered in units of h. communication [63], identiﬁcation of crystal structure [64, 65], bio-sensing [66, 67], imaging in scattering media [68] or biological elements [69, 70], and nonlinear microscopy [70, 71]. Due to the second-order nonlinear susceptibility (χ(2)), the fundamental angular frequency ω generates a nonlinear polarization wave which oscillates with twice the fundamental frequency (2ω) (Fig. 2.1(a)). This simpliﬁes the coupled equations Eqs. (2.22)–(2.24) to:

(ω) dA ∗ −i∆kz = −iκshgA A2ωe , (2.25) dz ω (2ω) dA i 2 i∆kz = − κshgA e ; (2.26) dz 2 ω q d µ0 2ω3 where by definition, ω1 = ω2 = ω and ω3 = 2ω, with κshg = (ω) (2ω) and n ε0 n α = 0. The SHG process requires phase-matching (PM) to be satisfied for an effective nonlinear interaction. In fact, a proper phase relationship between the interacting waves has to be maintained along the propagation direction, satisfying the momentum conservation, i.e., 2k(ω) = k(2ω) where k(ω) (k(2ω)) represents the wavevector at a frequency ω (2ω). Therefore, the wavevector mismatch (or phase mismatch):

[n(2ω) − n(ω)] ∆k = k(2ω) − 2k(ω) = = 2π (2.27) λ is defined between the two waves. The wavevector mismatch sets the maximum crystal length that is useful in producing SH power (Fig. 2.1(b)) [56]. This length is usually considered as half of the so-called coherence length, which is defined: 2π 2π lc = = . (2.28) ∆k k(2ω) − 2k(ω) Different methods can be applied to reduce the phase mismatch: one of the most common, known as birefringent phase matching (BPM), consists in using the natural birefringence of the material or introducing an artificial birefringence [40].

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 By a different method, known as quasi-phase matching (QPM), the phase matching can be obtained by introducing a periodic modulation of the nonlinear coefficient [39, 41, 42, 72, 73]. The most common materials where QPM is used are ferroelectric materials such as lithium niobate (LiNbO3) and potassium titanyl phosphate (KTP). The most popular technique for generating QP-matched crystals is periodic poling of the ferroelectric nonlinear crystal materials. Here, a strong electric field is applied to the crystal for some time, using microstructured electrodes, so that the crystal orientation, and thus the sign of the nonlinear coefficient is perma- nently reversed only below the electrodes. The poling period determines the wavelengths for which certain nonlinear processes can be QP-matched [73]. Another approach, which has been widely applied to semiconductors to make use of their large nonlinearities is known as orientation-patterned growth. Orientation-patterning techniques have been mostly applied to gallium arsenide (GaAs) so far [41, 42, 72] but attempts in using it on GaP has been made as well [39, 74]. Both techniques present difficulties in obtaining high-quality crystals due to technological reasons. Lastly, modal phase matching (MPM) is a valuable method to obtain phase matching when working with waveguides. MPM is based on modal dispersion engineering of different interacting modes in the nonlinear process [43–45,75]. In this case, the matching can be achieved by propagating the SHG light in a higher order mode than the one used in the fundamental (pump) wave: this requires that the effective (2ω) (ω) refractive index (neff ) of the two modes satisfies Eq. (2.27), i.e. neff = neff . This is indeed possible due to the waveguide (or modal) dispersion of the waveguides. Examples of modal dispersion for the specific cases analyzed in this thesis are included in Chapter 4 and 5. It is possible to define the quality of the MPM by considering the coupling between the nonlinear polarization density and the mode that one wants to excite inside the waveguide [57]. The coupling is the additional important parameter to obtain efficient conversion when MPM is considered. To evaluate this parameter one can consider the following equation, which defines the change of amplitudes due to the presence of the nonlinear polarization density vector PNL [57]:

∂A ZZ ∞ µ ∗ iβµz = −iω dxdyPNL · Eµe , (2.29) ∂z ∞ where Eµ is the normalized electric field’s profile of the excited mode, and βµ is the propagation constant of the mode. Therefore, for MPM, it is important to find a trade-off between mode coupling and similar effective refractive indices to obtain efficient SHG. The last two methods are widely used for semiconductors, which generally lack natural birefringence. However, while QPM requires proper tools for controlling the epitaxial growth the structures showing the right properties, MPM can be implemented by using industrially-made wafers and proceed with top-down fabrication to obtain the waveguides, necessary for the PM and the mode overlap. Furthermore, more and more interest is direct towards processes that allow the

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 use of successively smaller optical components and integration with others, making microwaveguides and nanowaveguides an interesting approach to the problem. It is also interesting that MPM has a better efficiency than QPM when a waveguide designed in a 43¯ m crystal is considered: in Paper 2, an analysis of the efficiency of the two methods has been presented in case of a slab waveguide, considering the waveguide aligned 45◦ with respect to the crystallographic axes. This investigation shows that MPM is more efficient for second-order nonlinear processes in waveguides with significant longitudinal electric field component (Pz). It is worth noting that for such a coordinate system, i.e. 45◦ rotated with respect to the crystallographic axes of the crystal, the following components of the nonlinear polarization can be excited:

(2) (−i2β0z) Px = 20dExEze , (2.30)

(2) 2 (−i2β0z) Pz = 0dExe , (2.31) where β0 is the propagation constant of the guided mode for the fundamental frequency and Ex, Ey, Ez are the corresponding electric ﬁeld components. The Eqs. (2.30)–(2.31) can be obtained from Eqs. (2.5)–(2.7) by applying a rotation of the axis in the system (Appendix A). Finally, considering the non-depleted pump approximation and the equations presented in Eqs. (2.25)–(2.26), it is possible to extract the eﬃciency for SHG:

3 (2ω) 2 2 2 2 2 I 2ω d L µ0 sin [(∆k)L/2] η I(ω), SHG = (ω) = 2 2 (2.32) I n(ω) n(2ω) 0 [(∆k)L/2] where L corresponds to the length of the waveguide where the generation take place and I(ω) is the the intensity of the pump wave. In case the PM condition is satisﬁed, the next important parameter becomes the length of the crystal or waveguide where the generation happens. It is important to notice that the intensity of the SHG depends quadratically on the intensity of the pump beam.

2.3 Surface Contribution to Second Harmonic Generation.

Previously, the description of the SHG generated in a bulk material with a non-zero second-order nonlinearity was introduced. It is important to mention that SHG can be induced on the surface of any bulk material since the crystalline symmetry is broken at the interface between the two materials. The first surface-induced optical nonlinearity was reported in 1962 in calcite [76]: calcite is a centrosymmetric material, which does not show SHG otherwise. There- fore, surface SHG can be seen in any centrosymmetric materials, included Si [16]. The components of the SHG on the surface can be distinguished from the ones of the bulk [38]: the rapid variation of the electric field across the interface results in a strong gradient of electric field (electric quadrupole), especially important for semiconductors with high refractive indices, but also the structural discontinuity

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 at the interfaces causes an electric-dipole contribution, an additional contribution to the surface nonlinearity. Surface SHG is of great interest to enhance the bulk SHG, but can also be used for evaluating the surface quality of nanostructures [36]. For completeness, the second-order nonlinear susceptibility tensor of the surface of a crystal with 43¯ m symmetry [36] can be deﬁned as:

 11 12 13 14  dS dS dS dS 0 0 25 26 dS =  0 0 0 0 dS dS  . (2.33) 35 36 0 0 0 0 dS dS

11 However, the only component of our interest was dS . Surface SHG is discussed and is used as technique to evaluate the quality of nanostructured GaP in Paper 1.

2.4 Spontaneous Parametric Down-Conversion.

Figure 2.2: (a) Schematic of OPA process. (b) Schematic of SPDC process.

SPDC is a nonlinear, instantaneusly optical process that converts one pump photon into a pair of photons (namely, a signal photon, and an idler photon), in accordance with the law of conservation of energy and law of conservation of momentum (PM). This can lead to multiple solutions, each forming multiple idlers and signals, with specific frequencies and directions of propagation. Although SPDC was predicted in the 1960s [77], only in the 1970s it was first demonstrated [78]. Later on, it became one of the main processes used for several applications in quantum optics, including quantum cryptography [79], quantum computing [80], quantum metrology [81] as well as for testing fundamental laws of quantum mechanics [82]. To describe the SPDC process through the electromagnetic formulation, the eas- iest way is by using the OPA process as a reference. In OPA, a wave known as signal (or seed) can be amplified using a parametric nonlinearity and a pump wave (Fig. 2.2(a)). The signal beam (with angular frequency ωs) propagates through the crystal together with a pump beam of shorter wavelength (ωp). Pump photons are then converted into pairs signal photons and so-called idler photons. The photon energy of the idler is the difference between the photon energies of pump and signal

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 (the relation ωi = ωp − ωs has to hold because of the energy conservation principle) [56]. For the case of an OPA, the following coupled equations can be found from Eqs. (2.22)– (2.24) [56]:

dAs 1 2 i ∗ −i∆kz = − αsA − κopaA Ap, e , (2.34) dz 2 s 2 i ∗ dAi 1 2 i ∗ i∆kz = − αiA + κopaA Ap, e , (2.35) dz 2 i 2 s dAp 1 2 i i∆kz = − αpA − κopaAsAi, e , (2.36) dz 2 p 2 where rµ0 ωiωsωp κopa = d . (2.37) ε0 ninsnp In Eqs. (2.34)–(2.36), the idler, the signal and the pump are acting on the nonlinear medium at the point z = 0, so that As(0), Ai(0) and Ap(0) are non-zero. However, this is only an intermediate step of the calculation since in the OPA, Ai(0) = 0. Furthermore, it is possible to assume that the pump is in a regime of non-depletion, which enables to view Ap(z) as a constant. Additionally, the assumption of no losses, i.e. αs = αi = αp = 0 is made. With the assumptions stated above and ∆k = kp − (ks + ki), Eqs. (2.34)–(2.36) become:

dAs ig = − A∗e−i∆kz, (2.38) dz 2 i ∗ dAi ig i∆kz = Ase , (2.39) dz 2

q µ0 ωiωs where g = κopaAp(0) = dijk Ep(0). ε0 nins The solutions of the coupled equations, considering the boundary condition are:

i(∆k/2)z ∆k g ∗ Ase = As(0)[cosh(sz)] + i sinh(sz)] − i A (0) sinh(sz) (2.40) 2s 2s i ∗ −i(∆k/2)z ∗ ∆k g A e = A (0)[cosh(sz)] − i sinh(sz)] + i As(0) sinh(sz) (2.41) i i 2s 2s

p 2 with s = |g/2| − (∆k/2) which simplify if ∆k = 0 and Ai(0) = 0 to:

As(z) = As(0) cosh(γz), (2.42) ∗ g A (z) = i As(0) sinh(γz), (2.43) i |g|

g where γ = | 2 |. However, SPDC has neither a signal nor the idler at z = 0 (see Fig. 2.2(b)), causing ∗ As(0) = Ai (0) = 0. Furthermore, SPDC is always in a regime of non-depletion of

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 the pump, due to the very low efficiency, then it is possible to assume that power drained from the pump by signal and idler is negligible. Classical nonlinear optics doesn’t allow the SPDC to exist. For this reason, it is necessary to introduce a quantum description of the phenomenon. The representation in this section is the so-called Schrödinger picture [83]. As a first step, it is necessary to quantize the electromagnetic field in energy units of hν (the "photon"), where h corresponds to the Plank’s constant and ν is the frequency. Energy is an observable, which is associated with a Hermitian operator (whose eigenvalues are real) and with a complete set of eigenstates2. Since the energy of such a eigenstate can be written hν(n+1/2) (with the number of electromagnetic quanta in the mode n = 0, 1, 2, ...), the electromagnetic energy eigenstates are usually denoted with {|ni} and called number states or Fock states, with {|0i} being the ground state (or "vacuum") . When using the number-state basis {|ni} , it is possible to define the annihilation operator, aˆ, and, its conjugate, the creation operator aˆ† as follow:

aˆl|1li = 0, (2.44) √ aˆl|ni = n|(n − 1)li, (2.45) † √ aˆl |ni = n + 1|(n + 1)li, (2.46) † aˆl |0i = |1li. (2.47) In this context, quantum optics explains perfectly the possibility of spontaneously creating a photon of certain energy in a given optical mode, and so a photon of light can split into two photons of lower energies. To deﬁne the process, the Hamiltonian for a certain volume V for the SPDC can be written as:

† i∆k·r−i∆ωt † † −i∆k·r+i∆ωt Hspdc = i~κ(ˆaiaˆsaˆpe +a ˆi aˆsaˆpe ) (2.48)

3 1 1 q µ0 ωiωsωp with ∆ω = ωp − ωs − ωi and ∆k = kp − ks − ki and κ = d( ) 2 . The 3 V ε0 ninsnp ﬁrst term of the Hamiltonian accounts for the sum frequency generation, in which the annihilation of two photons generate a new one, while the second term accounts for the SPDC, since the annihilation of one photon generates two. To evaluate the evolution of the incoming pump photons (Np) impinging a nonlinear medium it is necessary to look at the eﬀect of the SPDC Hamiltonian from Eq. (2.48) 3 on the initial state |0s, 0i,Npi, by using the Schrödinger equation in time :

1 R t 0 0 i Hspdc(t )dt |ψ(t)i = e ~ 0 |0s, 0i,Npi. (2.49)

2The eigenvalues corresponds to sets of the possible result of a measurement: this allows to find a specific function, the eigenfunction, that describes that specific state (the eigenstate) [84]. 3The Schrödinger equation is an operation on the wave-function. Usually denoted with ψ, the normalised wave-function describes the instantaneous physical condition of the system and from there it is possible to define all the physical properties of the system. Any of the dynamical variable of the system is represented by a linear operator, in this case the Hamiltonian, which allows to evaluate the energy of the system. The solution to the Schrödinger equation for a given energy involves also finding the specific function (eigenfunction) which describes that energy state [84].

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Since κ is rather small, i.e. the SPDC is very ineﬃcient and the pump beam is unperturbed, it is possible to make a Taylor expansion of the exponential Hamilto- nian: Z t 1 0 0 |ψ(t)i ≈ C0|0s, 0i,Npi + C1 Hspdc(t )dt |0s, 0i,Npi + ..., (2.50) i~ 0 where C0 and C1 ≈ 1 − C0 are the coeﬃcients due to the normalization. By considering ∆ω ≈ 0, i.e. energy conservation among the three photons, then the integral in Eq. (2.50) is a Dirac function and by applying the operators on the initial state, it is possible to obtain:

−i∆k·r |ψ(t)i = C0|0s, 0i,Npi + κC1e |1s, 1i,Np − 1i. (2.51)

Finally, the intensity of the measured SPDC will be proportional to the square of the wave-function in the Eq. (2.51)4. From this definition of the intensity, it is possible to observe that the SPDC process is a linear function of the pump power, furthermore SPDC is by all means a probabilistic source of photons, whose probability is proportional to the factor κ2. To conclude, an efficient SPDC requires the PM condition to be satisfied by care- fully consider the momentum conservation, for the energies of interest. What is said in Section 2.2 concerning methods and conditions of PM is valid also in the case of SPDC, becoming even more fundamental due to the inefficient nature of the process. Moreover, the detection of SPDC requires special attention: the measurement of the spectrum requires advanced detectors, as well as low levels of noise and reduced bandwidth. A more common method to evaluate SPDC is by doing coincidence measurements, i.e. measuring the double-detection events on two separated detectors in a very short time-scale range [86]. The photons generated through SPDC can be used as a single photon sources, by detecting one photon (the idler) which heralds the arrival of the other photon (the signal) [20, 87], as well as source of indistinguishable photons [86], or as source of entangled photons [20, 37, 79]. Entanglement will be discussed more thoroughly in Chapter 5. However, it is important to remember that SPDC has a thermal distribution of generated photon-pairs, so the last consideration is valid only when a low mean number of photons is considered; vice-versa the probability of generation of multiple pairs simultaneously is not negligible.

4In Born’s statistical interpretation [85], the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle’s being detected at a speciﬁc place - or momentum - at a given time. The integral of this quantity, over all the system’s degree of freedom, must be R dζψ∗ψ = 1 in accordance with the probability interpretation [84].

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7

Chapter 3

Fabrication of III-V Nanostructures.

The III-V semiconductors form an important class of materials, which has been widely used in optical and high-speed electronic devices [49]. However, these materials hide a great potential for applications in nonlinear optics, due to few main characteristics: the large nonlinear coefficients, the large transparency windows, the large thermal conductivity, and possibility of integration on other platform, such as silicon (Si) or silicon dioxide (SiO2) [6–9, 11, 17, 18]. Among the III-V semiconductors, some share the 43¯ m (or zinc-blende) crystallographic symmetry, which can be very interesting for nonlinear optical applications. However, due to the lack of the birefringence, these materials can not be used as bulk materials, and, for most of the applications in nonlinear optics, some kind of fabrication of micro- and nanostructures is required. Several approaches to the fabrication have been tried to target the different materials. Since only two of these materials are investigated in this thesis, i.e. gallium phosphide (GaP), and gallium indium phosphide (Ga0.51In0.49P), a brief review on the fabrication techniques available for these materials is included at the beginning of each section dedicated to the specific material. The work done on GaP is reported in Paper 1 and Paper 2, whereas the fabrication of micro- and nanostructures in Ga0.51In0.49P is reported in Paper 3. Most of the technology presented in the chapter presents a certain degree of nov- elty. The technology to fabricate nanostructures and microstructures in these materials for nonlinear optical applications is still widely unexplored, making overall a challenge to work with these materials. Furthermore, the epitaxial growth (for Ga0.51In0.49P) and the bulk crystal growth (for GaP) are not trivial, due to the multi-components nature of the materials [49]. The presence of defects and impurities limits the optical properties of the materials as well as the possibility to replicate the fabrication process presented in the literature. In this thesis, a top-down approach was used for the fabrication of the nanostruc-

19 tures. The main building blocks of the processes consist in creating a design (a pattern) of the desired structures (by optical lithography or by using focused ion beam (FIB)) followed by a process to remove the excess material (etching or milling). Depending on the ﬁnal result that one wants to achieve, diﬀerent techniques have been used.

3.1 Nanowaveguides in Gallium Phosphide.

The fabrication of GaP nanowaveguides starts from an undoped [100] wafer, polished on both sides and 0.4 mm thick. This is an unusual starting point in the fabrication of nanowaveguides for nonlinear optical application: in the literature, many of the photonic crystals and membranes have as initial material GaP grown on aluminium gallium phosphide (AlxGa1−xP) [24, 31, 32, 46, 88]. However, nan- odisks on GaP substrate have been reported [89], as well as nanopillars [35, 38]. Since we wanted to investigate the optical nonlinearity in transmission conﬁgura- tion, the nanowaveguides were made on the vertical direction from the substrate. In our work, two preferential directions were used to pattern the nanowaveguides, either parallel to the crystallographic axis of the wafers, i.e. 45o rotated respect to the cleaving edge of the samples, or parallel to the cleaving edge. This is important for satisfying the phase-matching (PM) condition in the characterization and to test the interactions of the diﬀerent modes, as will be described in detail in Chapter 4.

3.1.1 Focused Ion Beam. FIB fabrication has been investigated in Paper 1 as a possible alternative to fabricate nanostructures with high aspect ratio. The tool is valuable for prototyping complex geometries [90] as well as small features (sub-5 nm) [91] and high aspect ratio structures [92]. The system used is a dual-beam FIB/scanning electron microscope (SEM) machine FEI Nova 200. The liquid-metal ion-source responsible for the generation of the ion beam was made of gallium ions (Ga+). The con- figurable acceleration voltage was set at 30 kV. Several FIB milling currents were tested but 10 pA and 50 pA allowed us to obtain the best results. The nominal beam diameters at 10 pA and 50 pA are 12 nm and 19 nm, respectively. The FIB current is a very important parameter to take into consideration since can affect the implantation characteristics, the amount of superficial gallium (Ga) generated, redeposition of the material and roughness of the surfaces. While a better quality of the nanowaveguides can be guaranteed by using lower currents, when shorter fabrication time and/or longer waveguides are required the most reasonable choice is to increase the current. Other parameters that could be set in the tool are the dwell time (20 µs in this case) and the scan mode (raster or serpentine, not influ- ential for this application). We fabricated slab waveguides (SWs) at 10 pA, which have dimensions that go from w = 0.040 to 0.450 µm wide, length around l = 5 µm and height ranging from h = 0.3 to 1.10 µm (Fig. 3.1(a)); the slab waveguides

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 3.1: SEM picture of one of the arrays of SWs in GaP made by FIB milling at 10 pA. (a) After the milling, it is possible to notice the droplets all over the milled structures. (b) After the exposure to the laser beam, it is possible to notice the Ga2O3 where the structure was exposed to the laser beam. (c) After the HF wet etching, the droplets and the oxide are removed.

fabricated at 50 pA have also similar dimensions. The best aspect ratio (h/w) achieved for this process is approximately 20. However, FIB milling creates superficial droplets of Ga (Fig. 3.1(a)) [93], which reduce the transmission of the nanowaveguides since Ga is strongly absorbing in the visible wavelength range [94]. This is a great limitation for any nonlinear optical application, especially for second-harmonic generation (SHG). Commonly this problem is addressed by thermal annealing, and so using a temperature above 200oC [95]. However, to address this problem, we exposed the nanowaveguide surfaces to an intense laser beam (150 mW), which converted the Ga-rich surface into gallium oxide. The transformation of the Ga droplets in gallium oxide (Ga2O3) is shown in Fig. 3.1(b). Ga2O3 is transparent starting from 300 nm [96]. The whole process had to be followed by a hydrofluoric acid (HF) wet etching to reduce the roughness of the surfaces and improve the optical properties of the surface (Fig. 3.1(c)). The nanowaveguides were characterize using SHG, with the gallium droplets first, after the creation of the layer of Ga2O3, and after the HF wet etching. The results of the characterization are presented in Section 4.3. This is an innovative method to approach the problem. It requires only little post-processing and allowed us to recover the waveguides transmission.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 3.2: SEM images of (a) an array of NPs and (b) an array of SWs in GaP made by EBL followed by dry etching.

3.1.2 Electron-beam Lithography and Dry Etching. Electron-beam lithography (EBL) and dry etching were used in the fabrication of GaP nanowaveguides. With this technique it was possible to realize arrays of nanopillars (NPs) and arrays of SWs. Since a detailed description of the process is reported in [35], only a brief overview is reported here, as well as in Paper 2. The array of NPs consisted of a square lattice of 5 by 5 NPs, with the diameter of dTOP = 220 nm and dBASE = 320 nm, at the top and at the bottom respectively, and the height h = 750 nm. The pitch of the array is p = 300 nm (Fig. 3.2). The array of SW consisted of ﬁve slabs of 205 nm width and 750 nm height; around each slab, there is an air gap of 60 nm. A hard mask made of SiO2 was used as the etch mask and was directly deposited on a double-side polished undoped [001] GaP substrate. This step was followed by spin coating of negative tone e-beam resist, and the structures were written by a Raith 150 EBL system. Afterward, the pattern was transferred to the SiO2 layer by reactive-ion etching (RIE). Finally, inductively coupled plasma reactive ion etching (ICP-RIE) was used to transfer the pattern into GaP, using Cl2/H2/CH4 chemistry. The residuals of the mask were removed by buﬀered oxide etch (BOE). The nanowaveguides fabricated using this process are characterized by SHG, and presented in Chapter 4.

3.1.3 Maskless Photolithography for Planar Nanowaveguides. Maskless photolithography by using smart print lithography (SPL) was used to fabricate planar waveguides. The tool, made by the French company Smartforce

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Objective Field of view, mm Precision, µm Min feature size, µm 10x 1.35 x 0.75 0.7 2 2.5x 5.4 x 3 2.8 8

Table 3.1: Comparison of diﬀerent objectives for the SPL.

Technologies, is an optimal tool for rapid-prototyping of microstructures since it does not require a different mask for each design. It allows to use g-line photoresists since the light for exposure has a wavelength from 430 nm to 470 nm. The exposure is defined by a grey-scale bitmap containing the design. It is reproduced as a single exposure from the objective, which is interchangeable, allowing one to tune the field of view, affecting the precision and the minimum feature size. In this work, mainly two objectives were used: 10x and 2.5x, which features are shown in Table 3.1. The drawback of this tool is the non-homogeneous exposure. However, this can be corrected by tuning a parameter called image correction. One kind of design was tested, namely lines of several µm of thickness and 1.5 mm of length. The scope of using SPL was to fabricate planar waveguides in GaP, for which a sort of T-shaped cross-section was required.

3.1.4 Dry Etching and Wet Etching.

To realize the T-shape waveguides, the first step consisted in testing a wet etching process. From the literature, we found a suggested GaP etchant to underetch that is a mixture made of hydrochloric acid (HCl), hydrogen peroxide (H2O2) and deionized water (DI-H2O), in proportion of 80:4:1 with a reported etch rate of 0.030 µm/s. The result of 35 s is shown in Fig. 3.3(a), the etching profile suggests that the etching is isotropic. The wet etching alone, would not allow us to obtain the required shape. For this reason, we thought about combining an initial dry etching process to obtain straight sidewalls, followed by a wet etching process to underetch the obtained structures. To perform the dry etching, an additional step before the lithography was required: a hard mask made of SiO2 was used as the etch mask and was directly deposited on an undoped [001] GaP substrate Fig. 3.4(b), with the procedure described above. This has been followed by a lithography with the SPL tool with 10x objective (Fig. 3.4(c-e)), followed by the dry etching process as presented in Section 3.1.2 (Fig. 3.4(f-i)); the result is shown in (Fig. 3.3(b)). At this point, a second lithography, with thicker lines to protect the edges, was made by using the SPL, aligning the new lithography with the previous lithography (Fig. 3.4(j-l)). Then, by using the polymer as a mask, a wet etching (Fig. 3.4(m)) was tested for different times, ranging from few s (Fig. 3.3(c)) up to 2 min (Fig. 3.3(d)). From the results shown in Fig. 3.3 (c-d), it is clear that the adopted approach requires improvements. However, minimal changes (temperature, humidity, mixture proportions) can improve the underetching required to obtain the T-shaped waveg-

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 3.3: SEM picture of the cross-section of one of the fabricated steps, attempt- ing to obtaining a T-shaped waveguide, in GaP made by (a) Wet etching, (b) Dry etching, dry etching followed by (c) a few s and (d) 2 minutes of wet etching.

Figure 3.4: Scheme of the fabrication steps for a T-shaped waveguide.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 uides. Another option that can be considered is increasing the height of the step created by the dry etching, and repeat the process. This second part is challenging due to the dry etching process used for GaP, which selectivity to the SiO2 mask is low. Changing the mask in the dry etching process need to be addressed in the future, since it would also improve the fabrication of longer vertical waveguides.

3.2 Nanostructures in Gallium Indium Phosphide.

The nanowaveguides were fabricated by a top-down approach starting from an epitaxial wafer, made by layers of Ga0.51In0.49P and gallium arsenide (GaAs) as a sacrificial layer, grown on a GaAs substrate, as illustrated in Fig. 3.5(a). This composition of Ga0.51In0.49P is lattice matched to GaAs, with a lattice mismatch ∆a/a ≈ 1.8 · 10−3, where a is the lattice parameter. However, the initial wafer used for the fabrication is not very homogeneous between the edges and the center, since the estimated bandgap energy in the center is ≈ 1.89 eV and at the very edge is ≈ 1.87 eV. This confirms that it is quite challenging to obtain a homogeneous and high-quality wafer, due to the nature of the epitaxial growth [49]. For what concern waveguides in GaInP for nonlinear optical applications, two main approaches are considered in the literature, but all starting from a III-V stack of epitaxially grown layers. The first one includes the deposition of a layer of SiO2 on the top of the GaInP and then bonding this top layer to the glass substrate (through some additional processing). Then, the sacrificial layer, usually GaAs, is completely removed [18, 33]. The second approach consists of creating a Bragg reflector with GaInP as gain medium [97]. This second approach, however, won’t be further explored.

3.2.1 Maskless Photolithography for Planar Nanowaveguides and Grids.

After removing the cap of GaAs (Fig. 3.5(b)), a thin layer of a positive photoresist was spin-coated onto the 200-nm-thick-Ga0.51In0.49P top layer (Fig. 3.5(c)), and then exposed to ultraviolet (UV) light through the maskless photolithographic tool, the SPL described in Section 3.1.3 (Fig. 3.5(d-e)). The light exposure trans- fers the pattern containing the nanostructures onto the photoresist. Some designs were tested, due to the necessity to optimize the quality and/or extension of the Ga0.51In0.49P layer that needed to be transferred onto a diﬀerent substrate than GaAs. In particular, lines and grids (Fig. 3.6(b-c)) were the preferred designs for our applications. The thickness of the layer as well as the possibility to realize more than 1-mm long structures are very promising for the generation of photon-pairs by spontaneous parametric down-conversion (SPDC), as discussed in Chapter 5 [24].

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 (a) In itia I Substrate (b) Cap removal (c) Spin Photoresist (d) Lithography

1.,.::1ac - -, � nm GalnP -200 nm GaAs-50 nm GalnP -250 nm GaAs-50 nm GalnP -150 nm (g) GaAs (f) GalnP Wet Etching (e) Development GaAs-50 nm Wet Etching GalnP -200 nm

Ga As

(h) Release of the (k) GalnP microstructures (i) Transfer on Glass (j) Resist Removal 200- in water 8 150- � -� 100 l I N 50

0 0 2 4 6 8 10 12 x - axis, [µm]

Figure 3.5: (a) Schematic of the epitaxial wafer used in the fabrication of the nanostructures in Ga0.51In0.49P.(b-j) Schematic of the fabrication steps to obtain the nanostructures in Ga0.51In0.49P. (k) Cross-sectional proﬁle of the slab waveguide obtained by an atomic force microscope image.

3.2.2 Dry Etching and Wet Etching for Transferring the Nanostructures on a Glass Substrate. Two etching techniques were tested on the nanostructures: dry etching and wet etching. The photoresist was used as a mask for both the etching processes. The first step was to obtain the nanostructures from the Ga0.51In0.49P layer. In this case, the wet etching process is based on a mixture of HCl and phosphoric acid (H3PO4), with proportion 1:1, having a etching rate of 10 nm/s (Fig. 3.5(f)). Af- terward, the GaAs sacrificial layer was removed completely by a wet etching based on H3PO4 and H2O2 and DI-H2O, with proportion 3:1:20, which has a etch rate of 5 nm/s (Fig. 3.5(g)). It is worth mentioning that the sacrificial layer of this epitaxially grown wafer was about 50 nm thick (Fig. 3.5(a)). This created some problems during the etching process of the sacrificial layer. A longer etching time

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 3.6: Nanostructures on Ga0.51In0.49P. View with optical microscope. (a) Lines of resist on Ga0.51In0.49P before etching. (b) Grid of resist on Ga0.51In0.49P before etching. (c) Enlarged part of one of the waveguides in Ga0.51In0.49P transferred onto glass. (d) Piece of one of the grids in Ga0.51In0.49P transferred onto glass. (e) Entire waveguide in Ga0.51In0.49P transferred onto glass. was required, creating some overall worsening of the cross-section of the waveguide, which thickness on the top is not uniform, as shown in Fig. 3.5(k). Once the sacrificial layer was removed, the structures were naturally released from the substrate as soon as they were submerged into DI-H2O (Fig. 3.5(h)). Here the waveguides and the different structures started floating, making it possible to move them onto a SiO2 substrate as in Fig. 3.5(i). Some examples of transferred structures on SiO2 as presented in Fig. 3.6(c-e). At this point, the resist is removed by acetone. After this process, most of the structures are left to dry, and subsequently, they are cleaned in a bath of acetone, followed by isopropyl alcohol and finally DI-H2O (Fig. 3.5(j)).

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 The technique allowed us to transfer structures that can be more than 1 mm long onto a new substrate, as in Fig. 3.6(e). Characterizing the nanowaveguide in Fig. 3.6(e) by ﬁtting the transmission function to the data of the scattered light recorded by a charge-coupled device (CCD) camera in top-view when a laser at 780 nm is launched into the waveguide, an attenuation coeﬃcient of 12 dBcm−1 was estimated. In Fig. 3.7, waveguides 150 µm-wide and 10 µm-wide are shown. These have been

Figure 3.7: Nanostructures on Ga0.51In0.49P. View with optical microscope. (a) Entire waveguide 150 µm-wide in Ga0.51In0.49P transferred on glass with polymer on top. (b) Entire waveguide 10 µm-wide in Ga0.51In0.49P transferred on glass.

fabricated from a different epitaxial layer, with the Ga0.51In0.49P top layer 250 nm-thick and the GaAs sacrificial layers 200 nm-thick. By using this multi-layers and the same process, it was possible to obtain larger structures of 1.5 mm by 0.150 mm (Fig. 3.7(a)) as well as replicate the process for 10 µm-thick waveguides (Fig. 3.7(b)). The thicker sacrificial layer reduced the etching time, improving the overall quality of the cross-section of the fabricated waveguide. Further work needs to be done with this last promising wafer. The advantage of the technique described above is that the same process, in principle, could work with any kind of substrate, without requiring any additional step. To improve the quality of the side-walls, and so the attenuation coefficient, an alternative approach was tested by using inductively coupled plasma (ICP) to etch the Ga0.51In0.49P layer: the process involves the same gases as the ones used for GaP, namely a chemistry based on Cl2/H2/CH4. However, the dry etching resulted in a permanent modification of the photoresist, and we could not see any release of the nanostructure. We believe that the elasticity of the photoresist plays an important role in the natural release that happens to the nanostructures. However, more investigation on the nature of the process is required to confirm this.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 3.2.3 Reducing the Width of the Nanostructures. An additional step, requiring an additional lithographic process with SPL and Ga0.51In0.49P wet etching, was performed to obtain narrower nanostructures from slab waveguides that were 10 µm wide. Narrower waveguides, i.e. around 2 µm wide, have fewer modes propagating inside and so they can guarantee a higher optical coupling and efficiency. However, due to the difficulty of distributing homoge- neously the photoresist to protect the waveguides, this process was not investigated further. We are confident that wider waveguides or more regular geometries can go through this process successfully.

3.3 Summary

In this chapter, the fabrication processes concerning GaP and Ga0.51In0.49P have been investigated. The presented processes can open the door to further use of the materials for nonlinear optical applications since they allowed us to fabricate waveguides with good optical quality. However, the transferring of Ga0.51In0.49P waveguides onto glass and the dry etching of vertical nanostructures in GaP are still a challenge, due to the potential applications of the nanowaveguides, not only for nonlinear optical applications. In that sense, further technological progress is required.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7

Chapter 4

Nonlinear Optical Properties of Gallium Phosphide Nanostructures.

The nanostructures fabricated in gallium phosphide (GaP) presented in Chapter 3 are realised to fulfil the requirements to achieve an efficient second-harmonic generation (SHG). To satisfy the modal phase matching (MPM) condition an excel- lent overlap of the modes interacting (Eq. (2.29)) and a wavevector mismatch (Eq. (2.27)) close to zero are indeed required. Before the fabrication, adequate simulations to estimate the overlap and the wavevector mismatch were performed. In this chapter, more details about the simulations, the mode analysis, and an ap- propriate comparison with the results from the optical measurements are discussed. The mode analysis is presented for both the nanowaveguides in Paper 1 and Pa- per 2, however it is fundamental do distinguish among the different goals. In the first paper, the main goal is evaluating the quality of the fabricated nanostructures as well as the relation between the surface nonlinearity and the dimensions of the nanowaveguides. The latter paper, instead, focuses more on obtaining nanowaveguides that satisfies the phase-matching (PM) condition for several wavelengths.

4.1 Mode Analysis and Phase Matching for Second-Harmonic Generation: Simulations.

Simulations are fundamental tools in the workflow. Through simulations one is able to study the modes that can be generated inside a hypothetical nanostructure before fabricating it, considering a specific wavelength. The only requirement is to know the refractive index of the material, in this case GaP [6]. Once the geometry and the orientation of the waveguide with respect to the crystallographic axes are set, by using a software it is possible to obtain the modes profile and the respective effective refractive index for both the second-harmonic generated and the pump

31 waves. Those two are the main ingredients to evaluate if a certain geometry and wavelength satisfy the conditions to obtain an eﬃcient SHG. Before the simulation, it is necessary to decide several parameters:

1. The wavelength for the pump (and because of the energy conservation principle, the wavelength for the SHG is set as well);

2. The waveguide geometry (slab waveguide, cylindrical waveguide, array or single, etc.) and the material for the core and the cladding of the waveguide);

3. The orientation respect to the crystallographic axes, i.e. if the waveguides are parallel to the crystallographic axes or if they have a certain angle to them. In principle, this can be done also in a second stage, since the materials used in this thesis are isotropic, so the orientation plays an important role for the nonlinear processes but not when considering the eﬀective refractive index.

4. The number of dimensions needed in the simulation in the frequency domain, e.g. 2-dimensions or 1-dimension. Slab waveguides (SWs) and arrays of SWs can be simulated in 1-dimension (considering one of the dimensions as infi- nite) by using the transfer matrix method. In this case, it is sufficient to use a numerical simulator like MATLAB or Wolfram Mathematica. If the geometry has two finite dimensions, not including the direction of propagation of the wave (e.g. cylindrical waveguides, square waveguides, etc.), then it is necessary to use a 2-dimensional simulation. To perform such a simulation, one way is to use the so-called finite element method, whose principle is implemented in COMSOL Multiphysics, to mention an example.

From the simulations, the mode profiles and the effective refractive indices of a waveguide are obtained for a specific wavelength. At this point, one can evaluate the PM condition (Eq. (2.27)) and the coupling efficiency between the electric fields of the pump and the SH-generated wave (Eq. (2.29)). Moreover, it is important to consider the orientation of the waveguide with respect to the crystallographic axes, since one needs to apply the right polarization to the pump that has to be in fundamental mode, HE11 for cylindrical pillars and HE11-like when considering arrays, or TM0 and TE0 for SWs. By applying the Eqs. (2.5)–(2.7) in case of nanostructures parallel to the crystallographic axes or Eqs. (2.30)–(2.31) in case of structures oriented 45o with respect to the crystallographic axes, it is possible to define an equivalent nonlinear polarization. The next step is to evaluate the coupling efficiency of the electric field resulting from the multiplication of the nor- (2) malized fields used for the nonlinear polarization, for example Px ∝ ExEz and the normalized electric field for each mode obtained for the SHG wavelength, in the case of the example Ex. This gives a numerical value of the field profile coupling, i.e. one can identify the modes that are most efficiently excited by the pump. Fur- thermore, knowing the electric field profile of the SHG, the excited mode and the corresponding propagation constant for each mode, it is possible to calculate the

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 4.1: Orientation of the array of SWs with respect to the plane xy and the crystallographic axes [a1, a2]. (1) The array of SWs is aligned along the crystallographic axes. (2) The array of SWs is 45o rotated with respect to the crystallographic axes.

field profile at the end of the nanowaveguide [57]. Repeating the process for a range of wavelengths and geometries allows one to verify the bandwidth of the conversion and the flexibility in the fabrication, respectively. However, the modes interacting in the process are different depending on the orientation of the nanostructures with respect to the crystallographic axis and the shape of the waveguides. It is worth mentioning that all the simulations and the following analysis are made observing two hypotheses: the first one is that the system is in a non-depleted pump approximation regime, the second is that the nonlinear coefficient d, which is wavelength dependent [9], is kept constant.

4.1.1 Mode Analysis for Arrays of Slab Waveguides. In the case of an array of SWs, there are two diﬀerent analyses that were done, the ﬁrst one, concerning the nanostructures in Paper 1 and then a second case, concerning the array of SWs Paper 2.

1. The array of SWs are aligned along the crystallographic axis (Fig. 4.1(1)): different cases depending on the orientation of the polarization of the pump beam are possible. When the pump is horizontally polarized , i.e. oriented along the x-axis, the excited mode is the TM0 (Fig. 4.2(b-c)) mode that can excite the TM1 (Fig. 4.3 (b-c)) and the TE1 mode (Fig. 4.3(a)), corresponding to the surface and the bulk nonlinearity respectively. When the pump is rotated 45o to the crystallographic axes, it propagates through a superposition of TM0 (Fig. 4.2 (b-c)) and TE0 (Fig. 4.2 (a)) modes and excites the superposition of TM1 and TE1. However, it is very complicated to satisfy the PM conditions, since the different TE and TM modes present different effective

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 4.2: Electric ﬁeld proﬁles of the mode TE0 and TM0 in a GaP array of 5 SWs for the wavelength 1140 nm (pump). (a) TE0 mode, Ey (b) TM0 mode, Ex (c) TM0 mode, Ez. Each slab is 205 nm wide and the cladding between them is 150 nm wide.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 4.3: Electric ﬁeld proﬁles of the mode TE1 and TM1 in a GaP array of 5 SWs for the wavelength 570 nm (SHG). (a) TE1 mode, Ey (b) TM1 mode, Ex (c) TM1 mode, Ez. Each slab is 205 nm wide and the cladding between them is 150 nm wide.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 refractive indices due to the modal dispersion. However, in the configuration of Fig. 4.1(1), it is possible to distinguish the surface contribution to the SHG from the contribution of the bulk (see Section 4.3). 2. When the array of SWs are 45o rotated with respect the crystallographic axes (Fig. 4.1(2)), then only TM0 (Fig. 4.2 (b-c)) for the pump and TM1 (Fig. 4.3 (b-c)) for the SHG play a role in the process. In this case the PM condition can be satisfied as presented in Fig. 4.4. However, this configuration doesn’t allow monitoring the surface component of the SHG light. For completeness, it is worth mentioning that depending on the length of the SWs, there are two possible outcomes. If the SWs are short (e.g. 750 nm, Fig. 4.4(a)) the coupling efficiency among the modes is the most important parameter and the bandwidth where it is possible to have efficient generation is larger than when the SWs are longer (e.g. 10 µm, Fig. 4.4 (b)). In the latter case, the effect of the PM is dominating and, therefore, the bandwidth of the generation is limited. However, it is important to remember that the length used for the generation is a fundamental parameter for the evaluation of the conversion efficiency, as reported in Eq. (2.32) it depends on the length L2 of the waveguide [56].

Figure 4.4: (a) Eﬀective refractive index of the interacting modes in SHG process o when the SWs are rotated 45 with respect to the crystallographic axis: TM0 for the pump wavelength (black line), and TM1 for the SHG wavelength (red line). Normalized SHG intensity (Ix) for an array of 5 SWs with the length 0.750 µm. (b) Eﬀective refractive index of the interacting modes in SHG process: TM0 for the pump wavelength (black line), and TM1 for the SHG wavelength (red line). Normalized SHG intensity (Ix) for an array of 5 SWs with the length 10 µm.

4.1.2 Mode Analysis for Arrays of Cylindrical Nanowaveguides. For what concern a single nanopillar (NP), a detailed analysis was made by Sa- natinia, R. et al. in [35]. Here, the attention will focus on an array of NPs in a square lattice of 5 by 5 NPs. The pillar diameter is 220 nm and the pitch of the

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 array is 300 nm. The calculated electric field profiles of Ex and Ez are shown in Fig. 4.5. These corresponds to the fundamental electric field-components for the

Figure 4.5: Electric field profiles of the mode HE11 in a GaP array of 5 by 5 NPs for the wavelength 1140 nm (pump). (a) Ex (b) Ez. pump mode. From this guided mode (HE11 at λ = 1140 nm), two modes of the calculated SHG wavelength that have the largest contribution are HE21 and TM01, presented in Fig. 4.6. While other modes are present, HE21 and TM01 are the ones that best satisfy the PM condition. In Fig. 4.6(d), the obtained effective refractive indices of the three modes are shown: the fundamental mode HE11 of the pump and two second-order modes for SHG wavelength, HE21 and TM01. Also in this case, it is possible to observe a stronger effect of the coupling efficiency then the PM, due to the length of the NPs, by observing the normalized intensity of the SHG (Fig. 4.6(d)). As previously mentioned in Eq. (2.31), for the coordinate system 45o rotated with respect to the crystallographic axes of the crystal, the components of the second-order nonlinear polarization density P(2) can be written:

(2) (−iβ0z) Px = 2ε0dExEze , (4.1)

(2) 2 (−iβ0z) Pz = ε0dExe , (4.2) where β0 is the propagation constant of the guided mode for the fundamental frequency and Ex, Ey, Ez are the corresponding electric field components. For the NPs presented, the strong confinement of the guided pump in the mode HE11 can lead to a comparable amplitudes of the transverse and the longitudi- (2) nal components of the electric filed. In that case, the generated Px has the largest value and can have good overlap with the second-order mode at the second- (2) (2) harmonic frequency. The symmetry of Px and Pz distribution in the xy-plane (2) allows both of them to excite the same TM01 mode and, additionally, Px can

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 4.6: Electric field profiles of the modes excited in the SHG for the mode HE21 and TM01 in a GaP array of 5 by 5 NPs for the wavelength 570 nm. (a) Ex (b) Ez for the mode TM01. (c) Ex for the mode HE21. (d) Effective refractive index of the interacting modes in SHG process: HE11 for the pump wavelength (black line), HE21 and TM01 for the SHG wavelength (red and blue line, respectively). Normalized SHG intensity (Ix) for an array of 5 by 5 GaP NPs with length of 0.750 µm.

excite HE21 mode. When the PM condition is respected, the three processes can take place simultaneously, enhancing the conversion eﬃciency. The amplitude of the mode excited by the polarization P can be calculated in the same way as presented in [57]. By applying the non-depleted pump approximation, the amplitude of the TM01 and HE21 modes (A1 and A2, respectively) can be

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 calculated by using the following equations:

dA ZZ 1 (2) ∗ (2) ∗ i(β1−2β0)z = −i2ω0 dxdy(P E + P E )e , (4.3) dz x x,1 z z,1 dA ZZ 2 (2) ∗ i(β2−2β0)z = −i2ω0 dxdy(P E )e , (4.4) dz x x,2

where the index 1 and 2 of the electric field profiles refer to the normalized TM01 and HE21 modes, respectively, with β1 and β2 as corresponding propagation constants. (2) (2) Substituting Px and Pz from Eqs. (4.1)–(4.2) into Eqs. (4.3)–(4.4), the electric field profiles at the end of the nanowaveguide array can be expressed as:

(2ω) (−iβ1L) (−iβ2L) Ex = A1(L)Ex,1e + A2(L)Ex,2e , (4.5)

(2ω) (−iβ1L) (−iβ2L) Ey = A1(L)Ey,1e + A2(L)Ey,2e , (4.6)

where L is the length of the nanowaveguide. As one can see, the SHG process depends on the field overlap defined by the integrals in Eqs. (4.3)–(4.4) as well as (2) on the relative phase between individual fields. The contribution of Pz to the field overlap with TM01 mode results in an enhancement of the SHG process. On the other hand, the phase-mismatch caused by the difference between β1 and 2β0 as well as between β2 and 2β0 reduces the total SHG intensity. Furthermore, the polarization of SHG light is determined by the excitation and the phase difference between the modes TM01(β1) and HE21(β2).

4.1.3 Modal Phase Matching Vs. Quasi-Phase Matching.

In Section 2.2, several methods of obtaining PM were presented. Among those, the more interesting for III-V semiconductors are the MPM and quasi-phase matching (QPM). For completeness, it is necessary to compare MPM with the QPM in the conﬁguration presented above, i.e. with the waveguides rotated 45o with respect to the crystallographic axes. For simplicity, we consider a single-slab waveguide with air cladding in the same coordinate system (rotated 45o) as in Fig. 4.7(b).

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 4.7: (a) Eﬃciency of SHG process for CW pump at the wavelength of 1.2 µm in a single-slab waveguide with a core thickness of 180 nm. The red line represents the eﬃciency obtained through modal phase matching, while the green line represents the implementation of the quasi-phase matching. (b) The orientation of the waveguide with respect to the crystallographic axes in the case of MPM. (c) The orientation of the waveguide with respect to the crystallographic axes in the case of QPM.

For the MPM case, the waveguide is in the (11¯0) plane with the light propagation in the z-direction (Fig. 4.7(b)). Then, the pump guided in the TM0 mode (ω) (ω) (2ω) (2ω) (Ex , Ez ) excites the SHG light in the TM1 mode (Ex , Ez ), so that the polarization of the generated light is the same as for the pump. In the QPM method, on the other hand, the slab waveguide is in the (001) plane (Fig. 4.7(c)), due to the fabrication process consisting mainly in the growth of the different do- mains [39,98,99]. Here, the light propagation is in the y-direction. Then, the pump (ω) (2ω) guided in the TE0 mode (Ex ) excites the SHG light in the TM0 mode (Ey , (2ω) Ez ). The momentum mismatch between the two interacting modes is compen- sated by the Bloch wave vector KG = 2π/Λ. For the QPM case, we assumed that the waveguide is fabricated with the orientation-patterned growth of GaP and, therefore, Λ=0.4 µm corresponds to the period of alternating sign of the nonlinear coefficient. The simulation results of the SHG light intensity for CW pump with the wavelength 1.2 µm and the slab width of 180 nm are presented in Fig. 4.7(a). The higher efficiency obtained by the MPM method than by QPM is due to both the use (2) of modal dispersion and the longitudinal nonlinear polarization density Pz , which make the total field overlap value in Eq. (4.3) similar to the QPM case. Therefore, MPM is preferable for second-order nonlinear processes in nanowaveguides with a significant longitudinal electric field component, like the ones we presented here.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 4.2 Wide Bandwidth Second-Harmonic Generation.

The results of the simulations for the arrays of SWs and NPs allowed us to conclude that considering the geometry and the shape, which inﬂuence the dispersion inside the material, the range of wavelengths under investigation would allow the generation of SHG light over a wide spectral range. This was tested experimentally

Figure 4.8: Setup used to measure SHG in the arrays of GaP NPs and SWs. in Paper 2. The nanostructures, both NPs and the SWs have been fabricated 45o rotated with respect to the crystallographic axis (see Fig. 3.2), following the procedure presented in Section 3.1.2, and characterized by using the setup presented in Fig. 4.8. The setup consisted in a Ti:Sapphire pulsed laser (150 fs, 82 MHz repetition rate) and an optical parametric oscillator (OPO) tunable from 1090 nm to 1230 nm. In addition, a polarizer in the collection branch allowed us to distinguish the polarization of the SHG light, parallel or orthogonal to the H-polarized pump, identified by H-polarization and V-polarization respectively. Neglecting here the details of the characterization, which can be found in the paper, there is an interesting consideration that is worth highlighting. The conversion efficiency (50 W−1cm−2) is comparable to the one of low-loss dielectric nanoantennas based on GaP [89], but in contrast to nanoantennas, the conversion efficiency is limited by the length (Eq. (2.32)). This was possible considering that the longitudinal (2) component of the nonlinear polarization (Pz - Eq. (2.31)) contributes to enhance the SHG. In Fig. 4.9 the power of the measured SHG light is shown for the NPs and SWs arrays. However, the only conclusion that can be deduced from the results in Fig. 4.9 is that the nanowaveguides satisfy the PM among the two second-order modes. This can be seen considering the large extinction ratio among the two polarization. In Fig. 4.9(a), the strong H-polarization component and a lower but not negligible V-polarization component (due to the symmetry of TM01, Fig. 4.6(b)) is

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 visible. A diﬀerent behaviour, is noticeable in Fig. 4.9(b) where the only TM1 mode is interacting in the waveguide, making the V-polarization contribution negligible. The presence of the mode TM1 is visible for diﬀerent wavelengths in Fig. 4.10, where the SHG was collected by a charge-coupled device (CCD) camera.

Figure 4.9: Power of the SHG light measured in (a) an array of 5 by 5 nanopillars and in (b) an array of 5 slab waveguides. H- and V- polarization refers to the electric ﬁeld polarization parallel or orthogonal to the pump, respectively. The solid line represents the trend of the experimental data. Left axis: measured average power. Right axis: corresponding peak power.

In Fig. 4.11(c), the Fourier’s decomposition of the intensity that results from the superposition for the modes TM01 and HE21 is shown. From this picture, one can see that the result of the simulation corresponds to the SHG light from the NPs array (Fig. 4.11(a-b)), conﬁrming that TM01 and HE21 are the modes that best satisfy the PM condition. Additionally, it is possible to observe that the Fourier’s decomposition in Fig. 4.11(c) corresponds to the one of a single pillar [35]. From the simulation of the decomposition, one can observe higher harmonics generated (Fig. 4.11(c), the grey-scale part). However, these harmonics are not generated since this would be possible only by violating the principle that NA ≤ 1 to exist, where NA stands for numerical aperture. The area that exists is the colored one in the ﬁgure.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 4.10: Second-harmonic generated light from an array of SWs imaged by a CCD camera for (a) λp = 1140 nm and (b) λp = 1150 nm. (c) λp = 1175 nm.

Figure 4.11: Second-harmonic generated light from an array of NPs imaged by a CCD camera for (a) λp = 1140 nm and (b) λp = 1150 nm. (c) Fourier decomposition of the intensity that results from the superposition of the modes HE21 and TM01.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 4.3 Surface Nonlinearity as a Tool to Evaluate the Quality and the Width of the Nanowaveguides.

In Paper 1 the characterization through SHG was made to evaluate the quality of the nanowaveguides as well as the ideal dimensions to enhance the bulk SHG by exciting the surface SHG. However, by using focused ion beam (FIB) (Section 3.1.1) for the fabrication, the nanowaveguide are visibly full of impurities and redeposited material (see Fig. 3.1a)). To summarize what is presented in Paper 1, while a strong surface SHG was excited as soon as the arrays of SWs were illuminated with a laser (in this case Ti:sapphire, 100 fs pulses, 82 MHz repetition rate, λ = 840 nm, see Fig. 4.12(a)), the bulk SHG could not be detected due to the presence of gallium (Ga) droplets on the surface. These have strong absorption at 420 nm [6], and limits the transmission of the SHG light. After some time of exposure to a laser beam, the Ga transformed to gallium oxide (Ga2O3) [100–102], becoming transparent at 420 nm and allowing the bulk SHG to go trough the surface and be detected (Fig. 4.12(b), in black), until a damage point. At the same time, the higher refractive index of Ga2O3 reduced the electric ﬁeld gradient and modiﬁed the material discontinuity at the interface, reducing the surface contribution (Fig. 4.12(b), in red). In order

Figure 4.12: (a) Optical setup used to measure SHG in the arrays of GaP SWs fabricated by FIB. (b) SHG intensity for pulsed laser exposure with average power of 50 mW. The black line corresponds to the bulk contribution to the SHG and the red line corresponds to the surface contribution to the SHG. The measurements are made for a FIB current of 50 pA. to investigate the inﬂuence of the laser exposure presented in Fig. 4.12(b), polarization measurements were performed at a laser average power Pav = 50 mW and

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 are presented in Fig. 4.13. Two configurations are taken into consideration, thanks to the polarizer and the half-wave plate included in the setup (Fig. 4.12(a)), which allowed to record in a parallel configuration, with φSHG = φPUMP, and orthogo- o nal configuration, φSHG = φPUMP + 90 . The polarization plots before the laser o o exposure (b) and after (c) 45 min of exposure show at φSHG = 0 , 180 the surface contribution to the SHG light (in the parallel configuration, in red), which is o o reduced to about 20 % of the initial value, where at φSHG = 90 , 270 the bulk contribution to the SHG light (in the orthogonal configuration, in blue) increased by > 200%. Additionally, the change of the overall shape of the bulk contribution is visible: this can be explained by a modification in the confinement of the light inside the structure. This is attributed to the reduction of the effective waveguide width due to the laser-induced local heating effects, resulting in surface oxidation and material redistribution. Furthermore, the increase of the bulk contribution in- dicates a lower absorption, where the decreasing of the surface contribution is due to the surface oxidization. Further details about how the dimensions of the slabs change the surface and bulk contribution can be found in the Paper 1. It is interesting to point out that the SHG intensity due to the bulk nonlinearity grows with the reduction of the width of the slabs. However, this is not the case for the SHG intensity due to the surface nonlinearity, which is maximum for width of ≈ 210 nm. When the width gets smaller, ≈ 150 nm, the dominant contribution to the SHG is the longitudinal (2) component (Pz ).

Figure 4.13: Polarization plots of SHG light in an array of SWs (fabricated at 50 pA FIB current) of (280 ± 10) nm of width. In red, parallel conﬁguration, in blue, orthogonal conﬁguration. Laser average power Pav = 50 mW. The angle represented in the axis is φSHG. (a) Orientation of the crystallographic axes. (b) before and (c) after the 45 min laser exposure presented in Fig. 4.12(b).

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 4.4 Summary.

This chapter presented an overview of the work done on GaP. The material has satisﬁes many of the requirements for the generation of a strong SHG: high second- order nonlinear coeﬃcient and a large transparency window. However, the material we used has also strong limitations as soon as it is necessary to have longer waveguides, required for other applications, including spontaneous parametric down- conversion (SPDC). This is mainly due to the current lack of fabrication techniques. While more recently, various works show the possibility of growing thin layers of good optical quality GaP [24,88,103]. The top-down approach on the bulk material is not that developed. One interesting option that is well known for silicon, and it is used to fabricate structure with large aspect ratio, is metal-assisted chemical etching. However, this is still work in progress for GaP [104].

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Chapter 5

Nonlinear Applications of Gallium Indium Phosphide Nanowaveguides.

The ideas presented in this chapter have been inspired by the work of A. De Rossi and V. Berger [105]. This work introduces the idea of two counter-propagating photons generated through spontaneous parametric down-conversion (SPDC) by a pump beam which is propagating with an angle with respect to the waveguide. When the waveguide has a thickness equal to an odd number of half pump wavelength the momentum conservation is required only in the plane of the waveguide. In [105], the phase-matching (PM) is satisfied by kp sin θ − k+ + k− = 0, where the positive (negative) subscript designates the down-converted photon which momentum gives a positive (negative) scalar product with the pump photon momentum (kp) and θ is the angle between kp and the normal to the surface of the waveguide. By modifying the angle θ, it is possible to generate two orthogonal phase-matched photons at different wavelengths. However, to obtain entangled photons, the configuration presented in [105] requires the use of two symmetric pump beams. The concept of entanglement is used to describe the states of composite systems that can not be separated into a subsystem, or product states. The entangled states have properties associated with non-local information, a concept that can not be described by using classical physics and therefore it is often difficult to un- derstand1. Although it was originally considered as incompleteness of quantum mechanics describing nature, as remarked by Einstein, Podolsky, and Rosen [106], it became clear that the non-local character of the quantum theory and, therefore, the entanglement, is not a theoretical artifact of quantum theory but rather an experimental fact [107]. Besides the question concerning fundamental properties of physics, there has recently been a growing interest also in applications of entanglement [54, 79, 82, 108].

47 In this chapter, the focus is on several aspects on how to use the nanowaveguides presented in Section 3.2. Most of chapter is about a theoretical evaluation of SPDC by facing two aspects:

• the behaviour of an ideal nanowaveguide when diﬀerent types of mode matching are considered (Paper 4);

• what to expect from the nanowaveguides fabricated, in particular, the PM condition on real samples.

However, the experimental characterization of the nanowaveguide for SPDC is only at a preliminary stage. The main issue that was faced is due to the strong photoluminescence (PL) of gallium indium phosphide (Ga0.51In0.49P). This is a problem that can be avoided if the energy of the pump photons used in the process is smaller than the bandgap, but this also means working with a lower nonlinearity and generation of photons in outside the eﬃcient range of detection of a silicon (Si) single-photon avalanche diode (SPAD). This is why a part of this chapter is about one method to reduce the PL so that it is possible to excite the material close to the bandgap and work with the largest nonlinearity. This topic is investigated in Paper 3.

5.1 Simulations and Analysis of Spontaneous Parametric-Down Conversion in Nanowaveguides.

When a configuration like the one described by [105] is considered as in Fig. 5.1(a), with the nanowaveguide of materials with 43¯ m crystal symmetry as Ga0.51In0.49P parallel to the crystallographic axes, it is possible to generate two photons with orthogonal polarization counter-propagating along the waveguide. The two orthogonal solutions are indicated by light blue and green in Fig. 5.1(a). However, if we introduce the cavity-effect for the propagation and boundary conditions of the pump wave inside the waveguide, this open to the possibility of exploring different thicknesses of the waveguide than the ones described previously. In Fig. 5.1(b) the momentum diagram for the respective photons in the reciprocal space is presented. The profile of deff is a sinc function due to the waveguide rectangular cross-section. However, the pump distribution, as well as the guided modes of the signal and the idler, contributes to the final conversion efficiency. λ Even if the waveguide geometry prevents efficient conversion, since d = p corre- np 2π sponds to kp = d , the profiles of the guided modes and the pump distribution inside the waveguide can constructively contribute to the photon-pairs generation. In Fig. 5.1(b) is also included one of the solution of Fig. 5.1(a), where the signal photon propagates along z and the idler photon propagates along −z and where,

1A popular description of entanglement, made by Frank Wilczek - a Nobel laureate - can be found on: quantamagazine.org from April 2016.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 5.1: Schemes (a) for spontaneous parametric down-conversion process in a nonlinear waveguide with 43¯ m crystal symmetry (crystal axes parallel to xyz) for counter-propagating signal and idler. Two orthogonal solutions for generating photon-pairs are indicated by light blue and green. The pump polarization is in brown. The direction of propagation of the diﬀerent waves is shown by dotted lines. (b) Momentum conservation diagram presented in the reciprocal space, when the idler is propagating along -z and the signal is propagating along +z. in order to satisfy the PM condition for SPDC process, the momentum of the signal and the idler photons have to cancel each other. For the pump photons, the momentum is absorbed by the nonlinear interaction. Applying the non-depleted pump approximation and assuming counter-propagating signal and idler ﬁelds lead to two sets of equations if the coordinate system in Fig. 5.1(a) and the nonlinear polarization tensor Eq. (2.3) are considered:

dEx,s ∗ i∆kz = iκEz,pE e , dz y,i dEy,i ∗ i∆kz = −iκEz,pE e , (5.1) dz x,s

dEx,s ∗ i∆kz = −iκEz,pE e , dz y,i dEy,i ∗ i∆kz = iκEz,pE e , (5.2) dz x,s

q µ0 ωiωsωp with κ = d and ∆k = ks − ki. Since the PM condition is satisfied, ε0 ninsnp ∆k = 0. Furthermore, the Eqs. (5.1)–(5.2), are similar to the ones used for a mirrorless optical parametric oscillators [109,110], where also in that case photons counter-propagate. As one can see from Eqs. (5.1)–(5.2), an efficient light conversion requires a good overlap among the interacting fields. Therefore, we have to

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 consider the profiles for the guided modes of the signal and the idler as well as the pump field distribution inside the waveguide. The complex value of the overlap ηp,i,s contributes to the phase of the generated light and can be written as: Z ∗ ηp,i,s = Ep EiEsdx, (5.3) core where Es and Ei are the normalized field profiles for the guided fundamental modes, and Ep is the electric field of the pump with amplitude normalized to the free space. To simplify the solution, it is possible to use the overlap η among all the three interacting fields defined as:

q ∗ η = ηp,i,s · ηp,i,s. (5.4)

Since the refractive index of the nanowaveguide core is larger than the cladding, the waveguide can create an optical cavity for the pump wavelength. In this case, the enhancement of the pump field, due to the cavity resonance effect of the waveguide will contribute to the overlap η, where η2 is proportional to the efficiency of generation of the photon-pairs. This gives the solutions of Eqs. (5.1)–(5.2), which are respectively the normalized fields:

Ex,s = iA0 sin(κEz,pηz), ∗ Ey,i = A0 cos(κEz,pηz), (5.5)

∗ Ex,s = A0 cos(κEz,pηz), Ey,i = iA0 sin(κEz,pηz), (5.6)

Eqs. (5.1)–(5.2) describe an optical parametric oscillator with the gain threshold condition satisﬁed for the pump: π Ez,p = (5.7) 2κLη where L is the length of the waveguide. However, below the threshold, the classical description is not valid any longer. According to the quantum mechanical formalism, the state can be described as:

1 Ψout = √ |XsiF |YiiB − |YiiF |XsiB , (5.8) 2 where the subscripts s and i represent the signal and the idler photons, the subscripts F and B the direction of propagation, that is forward (along +z) and backward (along −z) and X and Y represent the polarization of the photons. It is worth mentioning that to each photon pair generated is associated a speciﬁc wavelength

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 5.2: Calculated dispersion inside a waveguide made of Ga0.51In0.49P in a air cladding for diﬀerent widths of the waveguide for λp = 700 nm.

(λTE0 and λTM0 respectively), which satisfies the PM condition, because of the polarization dispersion in the waveguide (Fig. 5.2). Since the crystallographic axes of the crystal are parallel to xyz-coordinates (as described above) the excited counter-propagating modes have orthogonal polarization: Eq. (5.4) includes the field profiles of the TE0 and the TM0 modes. In Fig. 5.3 is shown the overlap η as from Eq. (5.4) calculated for considering the field profiles of the signal and the idler as TE0 and TM0 for a waveguide made of Ga0.51In0.49P in a air cladding. In Fig. 5.3, the overlap of the field η is larger for a width that is equal to the wavelength of the pump inside the core, suggesting that due to the symmetry of the modes, the optimum waveguide width is equal to the wavelength of the pump inside the core. However, in this configuration, it is impossible to have photons with degenerate wavelengths. As shown in the in graph presented in Fig. 5.2, for larger waveguides or very small waveguides, the emission wavelength of the photon-pairs gets asymp- totically closer but can not be the same. To get polarization-entangled photon-pair generation, the signal and the idler emitted at the end of the waveguide need to be indistinguishable. As suggested in [105], this can be solved by using two pump beams.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 5.3: Overlap η as in Eq. (5.4) among the modes TE0 and TM0 and the pump field profile for a waveguide made of Ga0.51In0.49P in a air cladding for different widths of the waveguide for λp = 700 nm.

Nonetheless, an alternative solution can be obtained when the crystallographic axes of the nonlinear crystal are rotated 45o around the z-axis (Fig. 5.4). Then, the

Figure 5.4: Schemes of a slab waveguide with 43¯ m crystal symmetry with the crystal axis rotated 45o with respect to xyz. dispersion problem can be avoided by generation of down-converted photons with the same polarization and the same wavelength. After this rotation deﬀ can be

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 written as: 0 0 0 0 d 0 deﬀ = 0 0 0 −d d 0 . (5.9) d −d 0 0 0 d Eqs. (5.1)–(5.2) can then be written as:

dEx,s ∗ i∆kz = iκEz,pE e , dz x,s dEx,i ∗ i∆kz = −iκEz,pE e , (5.10) dz x,i

dEy,s ∗ i∆kz = −iκEz,pE e , dz y,i dEy,i ∗ i∆kz = iκEz,pE e . (5.11) dz y,s Now, the signal and the idler have the same polarization, which in turn leads to a degenerated wavelength of emitted photons. Since in both polarizations the generated photons have the same energy, their superposition represents a polarization- entangled state: 1 Ψout = √ |XsiF |XiiB − |YsiF |YiiB , (5.12) 2 where the subscripts s and i represent the signal and the idler photons, and X and Y represent the polarization of the photons. The Eq. (5.12) can also be calculated from the photon state generated in the coordinates of the crystallographic axes (Fig. 5.4). Due to the crystal symmetry, one can write: 0 1 Ψout = √ |DsiF |AiiB + |AsiF |DiiB , (5.13) 2

It can be seen that this state in the X,Y coordinates is equal to the one presented in Eq. (5.13). In Fig. 5.5, the overlap η as defined in Eq. (5.4) is presented by considering both idler and signal propagating in a waveguide in the same mode but along opposite directions. The polarization-entangled photons can be generated when η of the two cases, i.e. TE0F-TE0B and TM0F-TM0B, are equal as shown in Fig. 5.5: the TE0F-TE0F overlap is generally stronger and it is better to consider when the generation involves only a pair of photons, since the generation can be more efficient. Furthermore, it is important to remember that the SPDC process with the signal and the idler generated at degenerated wavelength with the same polarization can be used to generate a squeezed-state. By placing a mirror at one end of the waveguide as in Fig. 5.6, the phase relation between forward and backward propagating modes can be defined by the mirror position, allowing the idler and signal to propagate in the same direction. The waveguide for the generation of

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Figure 5.5: Overlap η as in Eq. (5.4) among the modes for a waveguide made of GaInP in a air cladding for diﬀerent width of the waveguide for λp = 700 nm.

Figure 5.6: Nanowaveguide geometry including a mirror at the end of the waveguide.

squeezed states can be evaluated by considering the time dependent Hamiltonian of the electromagnetic ﬁeld [111]:

Z H(t) = − E(r, t)P(r, t)d3r (5.14)

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 with P(r, t) from Eq. (2.1). The electric field operators for the pump, the signal, and the idler can be written as: r ~ωp iky † −iky Ez,p = i (ap(t)e − a (t)e ), (5.15) 2V ε p r ~ω ikz † −ikz Ex,s = i (a(t)e − a (t)e ), (5.16) 2V ε r ω ~ −ikz i(2kL+φm) † ikz −i(2kL+φm) Ex,i = i (a(t)e e − a (t)e e ), (5.17) 2V ε where φm is the phase due to the reflection from the mirror, V , is the quantization volume, ε is the dielectric constant, L is the length of the waveguide. By applying energy and momentum conservation (ωp = ωs + ωi and ks = ki), it is possible to obtain the interaction Hamiltonian: iκ η ~ †2 −2ikL−iφm † 2 2ikL+iφm H(t) = (apa e − a a e ) (5.18) 2 P with η as in Eq. (5.4). We solve the Heisenberg equations of motion (as described in [111]) only for a and a† since the pump is in a non-depleted pump approximation and strong pump field, that can be used as a classical field (εp), which has the units of the square root of the mean photon number. So, the equations of motion are: da 1 † −2ikL−iφm = [a, H] = κηapa e , (5.19) dt i~ da† 1 † 2ikL+iφm = [a ,H] = κηapae . (5.20) dt i~

2ikL+iφm Assuming that κηape is a positive and real number, then this give us as a condition that − 2ikL − iφm = −2iφ0. (5.21) Finally, it is possible to obtain:

iφ0 −iφ0 † a(t) = e a(0) cosh(εpηκt) + e a (0) sinh(εpηκt), (5.22) where φ0 = mπ − κL − φm/2 is a phase-constant of the generated light. This shows that the light produced by using this SPDC process is vacuum squeezed.

5.2 Phase-Matching Condition of the Fabricated Nanowaveguides.

As outlined above, it is possible to generate a pair of twin photons by using a pump beam incident at the top of a thin waveguide. This section will investigate what to expect when a nanowaveguide with a certain thickness is in that conﬁguration. The simulations diﬀer from the ones presented in chapter 4 is several ways. First, a

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 range of wavelength for the generation is established. The limitations are given by the possible applications, as well as the possibility of measuring with the detectors available in our laboratory (Si-SPAD). The calculation of the effective refractive index was made by using the refractive indices for Ga0.51In0.49P presented in [50]. Finally, the width of the waveguide was inserted in the algorithm (based on the transfer matrix method) calculating the effective refractive index of the modes TM0 and TE0 for several wavelengths. At this point, it was possible to evaluate which pair of wavelength and effective refractive index satisfied the PM condition. In

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Figure 5.7: PM condition as a function of λPUMP for an asymmetric slab waveguide (SW) 170 nm thick, where the cladding were air (top) and glass (bottom). (a) λPUMP = 405 nm; (b) λPUMP = 532 nm; (c) λPUMP = 658 nm; (d) λPUMP = 685 nm;

Fig. 5.7, some examples are presented by showing the wavevector as a function of the frequency and Table 5.1 summarize the extrapolated wavelengths values, so the wavelengths which satisfy energy conservation and for which the eﬀective refractive

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 λPUMP [µm] λs [µm] λi [µm] TM TE 0.405 0.76 0.87 0.532 0.98 1.16 0.658 1.18 1.49 0.685 1.20 1.60

Table 5.1: Extrapolated wavelengths, which satisfies energy conservation and whose effective refractive index satisfies the PM for several λPUMP for an asymmetric SW 170 nm thick, where the cladding were air (top) and glass (bottom). indices satisfy the momentum conservation. Energy and momentum conservation can be satisfied thanks to modal dispersion since the idler and the signal propagate in two different modes, as well due to the dispersion of the refractive index, which influences the effective refractive index for the mode at a certain wavelength.

5.3 Reducing the Photoluminescence by Laser Irradiation.

When a direct bandgap semiconductor is excited with a light source that provides photons with an energy larger than the bandgap energy, then it is possible to observe photoluminescence (PL). The process consists in the absorption of excitation photons, followed by the formation of electron-hole pairs, that is an electron from the valence band jumps to the conduction band leaving a hole. After the energy and momentum relax toward the band gap minimum, the electrons recombine with holes and this causes the emission of photons. The photon emitted upon recom- bination corresponds to the energy difference between the valence and conduction bands. Thus, it is lower in energy than the excitation photon. PL can be used to find impurities and defects in several semiconductors, and to determine their band-gaps. The Ga0.51In0.49P grown to match the lattice constant of the gallium arsenide (GaAs), such as the one presented in Chapter 3, shows a very strong PL due to its direct bandgap [51]. This is a very strong limitation that is necessary to overcome when one wants to use Ga0.51In0.49P in the region where it has the strongest nonlinearity, i.e. close to the bandgap. In Paper 3, a method to reduce the PL in Ga0.51In0.49P nanostructures was introduced. The method consists in stimulating the PL of the nanostructure through a continuous wave (CW) laser with λ = 405 nm. Different powers were tested, however, any of them causes any modification of the optical nonlinear properties of the nanostructure, which is fundamental for using the nanostructure for further experiments. Through this process, the PL can be reduced below -30 dB for a laser power of 8 mW after continuous exposure for 30 min, and further reduced to -34 dB if the exposure is prolonged to about 20 hours. A further analysis was made about the behaviour of the PL depending on the expo-

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 sure time and the exposure power. This allowed to understand the process further. However, without extensive notions of chemistry, it is very hard to identify the chemical transformations happening inside the material. Nevertheless, it was possible to identify two main processes, similar to the ones

Figure 5.8: Normalized photoluminescence vs. 405 nm pump power. The squares and the circle are used to distinguish the measurements made in air and nitrogen atmosphere, respectively. Power is color coded. (a) Normalized PL measured at the initial time t = 1 s. (b) Normalized PL measured at the time t = 300 s of excitation. reported for some chalcogenide glasses [112,113]: photodarkening (PD) and photobleaching (PB). In Fig. 5.8 the normalized PL as a function of the CW laser power is shown. PD is a fast process, that is happening during the ﬁrst few seconds of

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 exposure; in Fig. 5.8(a), the PD regime can be observed and it is possible to set a threshold value at 500 µW, since it is the highest normalized PL value. PB is a slow process that affects the PL in the long run. In Fig. 5.8(b), the system is in the PB regime also if the shown values of the PL are due to the combined effects of PD and PB. From Fig. 5.8, it is worth noting the behaviour of the black dot. This is a sample measured in a nitrogen atmosphere. While in the PD regime, the behaviour of the sample is similar to the other samples measured in air, its behaviour presents a marked difference in the longer run, confirming the presence of an oxidization process as reported in previous literature when discussing PB [112,113]. It is worth remembering that PL is a linear process, so its intensity is proportional to the intensity of the excitation light. For completeness, in Fig. 5.9 the second-harmonic generation (SHG) measured

(a) (b) 90 120 15000 60 90 120 15000 60 150 30 150 10000 30

180 0 180 ¢SHG 0 ¢SHG

210 330 210 330 240 300 240 270 300 270

Figure 5.9: Polarization measurement of the SHG light for a Ga0.51In0.49P nanostructure. The angle represented in the figure is φSHG. Blue and red lines: orthog- o onal (φSHG = φPUMP + 90 , where φPUMP is the angle between pump polarization and and the x-axis of the crystal) and parallel (φSHG = φPUMP) configurations, respectively. (a) Before PL measurements. (b) After 20 h of PL excitation. on the nanostructure is shown. Since the symmetry and the intensity of the SHG before (Fig. 5.9(a)) and after (Fig. 5.9(b)) the long exposure is maintained, it is possible to confirm that the optical nonlinearity remain unchanged. For details about the measurements and the setup, the reader can refer to Paper 3.

5.4 Summary.

This chapter presented an overview of the work done on Ga0.51In0.49P. The material satisﬁes the requirements for the generation of photon-pairs through SPDC, due to the strong nonlinearity but also due to the fabrication. However, concerning the possibility of generating photon-pairs with degenerate wavelength, the orientation

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 requirements are not satisﬁed by the epitaxial wafer that we used. By investigating the possibility as well as adapting the fabrication technique, it may be possible to obtain interesting experimental results. Finally, most of the theoretical work presented in this chapter can be potentially applied to any material with 43¯ m symmetry. Furthermore, additional work on the PL can be very beneﬁcial also for other applications, which require a good comprehension of the process for improving the PL intensities and prevent the photodegradation.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 Chapter 6

Conclusions and Outlook.

Nanostructures for nonlinear optical application in III-V semiconductors have at- tracted significant research interest for their potential application and integration with other platforms. In this thesis, we proposed several methods and materials to fabricate and simulate nanostructures for applications in nonlinear optics, including second-harmonic generation (SHG) and spontaneous parametric down-conversion (SPDC). Among the several new technologies, we adopted modal phase matching (MPM) as the driv- ing factor in satisfying the phase-matching (PM) condition required by an efficient second-order nonlinear process. The work on gallium phosphide (GaP) focused mostly on the fabrication and characterization of arrays of waveguides with cylindrical and slab shapes. The fabricated structures have been characterized by SHG. An interesting result was that due to the additional contribution of the longitudinal component to the nonlinear polarization density the SHG was enhanced. This result can be generalized to any material that has the 43¯ m crystal symmetry. Additionally, the investigation on the behaviour of focused ion beam (FIB) milling on GaP shows that it is possible to use SHG to evaluate the quality of the surface, to monitor the evolution of the transmittance. Moreover, by using the FIB, it is possible to prototype different geometries in the nanoscale which make the tool suitable for many applications, increasing the importance of knowing and being able to control the quality of the nanostructures as well as being able to recover the damage made by the high-energy ions. In this sense, we introduced an innovative post-milling process including laser exposure and hydrofluoric acid (HF) wet etching to improve the quality of the milled nanostructures. However, the short length of the fabricated nanostructures limits the applications to SHG, since longer structures are required for SPDC. The geometry required for this kind of applications suggested us to move onto an in-plane type of geometry. This can be done by making use of the most recent high- quality epitaxially grown GaP on silicon (Si) or dielectric, as well as working on

61 other materials that can be commercially grown as thin films. Preferring the last approach, we used gallium indium phosphide (Ga0.51In0.49P). This material was preferred compared to other possible options, including gallium arsenide (GaAs) for example, due to the affinity to GaP. The latter has the same crystallographic symmetry as well as a strong nonlinear optical coefficient. The most important characteristics are the bandgap close to the visible range and a transparency range starting at 680 nm. This allows the generation of photons trough SPDC in a wavelength range that is measurable with a high efficiency Si-based detector. However, the direct bandgap of the material resulted in a major problem, a strong photoluminescence (PL). We solved the problem by suppressing the PL through the use of a CW laser to stimulate the photodarkening (PD) and photobleaching (PB) of the material. The photodegradation of the material showed in a strong reduction of the PL, but allowed us to use the material for nonlinear applications since it didn’t modify the nonlinear property of the material. The fabrication of mm-long nanowaveguides in Ga0.51In0.49P and the possibility to transfer them onto glass (and virtually onto many different substrates) was an important step towards the realization of a device to generate photon-pairs. We proved in fact that different thicknesses of waveguides and different geometries and sizes can be transferred, making this material very versatile for the generation of photons through SPDC at different wavelengths and polarizations, by optimization of the fabrication process towards one’s needs. While preliminary measurements on the waveguide have been done, a certain cau- tion is required on the interpretation of the data due to the low efficiency of the SPDC process. Therefore they are not reported here. However, the theoretical approach guiding the last experiments represents an important achievement in the evaluation of the full potential of these waveguides.

As one can see browsing in the bibliography at the end of the thesis, most of the works reported about III-V semiconductors in nonlinear optics are from the last decade. This field is in expansion due to the technological achievements of the last few years. Further improving the technology for the epitaxial and bulk crystal growth is an essential step to improve further the quality but also to make these materials commercially appealing, meaning that it would be easier to buy wafers with specific characteristics. The obvious next step for what concern the work presented in this thesis would be measuring the SPDC on the Ga0.51In0.49P waveguides. Additional work to reduce the scattering due to the irregular surfaces is another interesting point to address in the next future. Furthermore, transferring wider than 10-µm-structures looked promising and therefore should be taken into account as a possible development of this work. GaP remains a promising material. Investigating a new technological approach to obtain vertical nanostructures from the bulk material can be of interest. The next step may be as well to try a different material as a mask for the dry etching process.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7 The main characteristics that the new mask needs to have are high selectivity, being easily removable, without damage of the GaP and, possibly, having good compati- bility with the electron-beam lithography (EBL). Although the metal-assisted chemical etching is barely known for III-V semiconductors, the potential of this fully developed technology would be priceless.

TRITA-SCI-FOU 2019:45 • ISBN 978-91-7873-318-7

Appendix A

Rotation of the axis to evaluate the nonlinear components.

In case of a system as the one represented in Fig. A.1, it is possible to evaluate the components by using a simple rotation of the axis. The Eq. (2.4) is written with respect to the coordination system of the crystallographic axis, that coincides with the xyz-system of coordinate when a structure parallel to them is considered. In the thesis, instead, the coordinates xyz-system is the one corresponding to the position of the nanostructures, and for this reason it is important to know how the two systems are connected. By applying the following transformation, i.e. the new axes rotate clockwise of an angle θ around the z-axis:

 (2ω)    (2ω) Px0 cos θ − sin θ 0 Px  (2ω) θ θ  (2ω) Py0  sin cos 0 Py  (A.1) (2ω) 0 0 1 (2ω) Pz0 Pz

Figure A.1: Slab waveguide rotated 45o with respect to the crystallographic axis.

65 (2ω) (2ω) (2ω) Px0 = Px cos θ − Py sin θ, (A.2) (2ω) (2ω) (2ω) Py0 = Py cos θ + Px sin θ, (A.3) (2ω) Pz0 = Pz. (A.4) If we apply the same transformation, Eq. (A.1), to the electric ﬁeld:

Ex0 = Ex cos θ + Ey sin θ, (A.5)

Ey0 = Ey cos θ − Ex sin θ, (A.6)

Ez0 = Ez, (A.7) and we rewrite Eqs. (2.5)–(2.7) using the inverse rotation to in the new coordinate system, we obtain: (2ω) Px = 2dEz(Ey0 cos θ + Ex0 sin θ), (A.8) (2ω) Py = 2dEz(Ex0 cos θ − Ey0 sin θ), (A.9) (2ω) Pz = 2d(Ex0 cos θ − Ey0 sin θ)(Ey0 cos θ + Ex0 sin θ). (A.10) Finally we can insert the results from Eq. (A.10) in to Eq. (A.4), obtaining: (2ω) Px0 = 2dEz0 (Ey0 cos 2θ + Ex0 sin(2θ), (A.11) (2ω) Py0 = 2dEz0 (Ex0 cos 2θ − Ey0 sin 2θ), (A.12) (2ω) Pz0 = 2d(Ey0 cos θ + Ex0 sin θ)(Ex0 cos θ − Ey0 sin θ). (A.13) In the case of θ = 45◦, the Eq. (A.13) simpliﬁes to: (2ω) Px0 = 2dEx0 Ez0 , (A.14) (2ω) Py0 = −2dEy0 Ez0 , (A.15) (2ω) 2 2 Pz0 = d(Ex0 − Ey0 ). (A.16) At this point, it is possible to see two cases: the pump has a horizontal polarization (case a), i.e. Ey0 = 0 , or the pump has a vertical polarization (case b), i.e. Ex0 = 0. In case (a), we obtain: (2ω) Px0 = 2dEx0 Ez0 , (A.17) (2ω) Py0 = 0, (A.18) (2ω) 2 Pz0 = dEx0 , (A.19) while in case (b), (2ω) Px0 = 0, (A.20) (2ω) Py0 = −2dEy0 Ez0 , (A.21) (2ω) 2 Pz0 = −dEy0 . (A.22)

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