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Novel Electromagnetic Scattering Phenomena MASSACHUSETTS INSTITUTE by OFTECHNOLOGY Yi Yang OCT 0 3 2019

B.S., Peking University (2011) LIBRARIES I S.M., Peking University (2014) ARCHIVES Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Electrical Engineering at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2019

Signature redacted Author ...... Department of Electrical Engineering and Computer Science August 30, 2019 Signature redacted C ertified by ...... I Marin Soja-in Professor of Physics and MacArthur Fellow Thesis Supervisor Signature redacted Accepted by...... L 9 sle A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 77 Massachusetts Avenue Cambridge, MA 02139 MITLibraries http://Iibraries.mit.edu/ask

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Novel Electromagnetic Scattering Phenomena by Yi Yang

Submitted to the Department of Electrical Engineering and Computer Science on August 30, 2019, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering

Abstract

Scattering of electromagnetic waves is fundamentally related to the inhomogeneity of a system. This thesis focuses on several theoretical and experimental findings of electromagnetic scattering under contemporary context. These results vary from scattering off real structures and off synthetic gauge fields. The source of scattering also varies from near-field to far-field excitations. First, we present a general framework for nanoscale electromagnetism with experimental verifications based on far-field plasmonic scattering. We also theoretically propose two schemes featured by thin metallic films and hybrid plasmonic dielectric nanoresonantors, respectively, aiming at achieving high radiative efficiency in plasmonics. Second, treating free electrons as a near-field scattering excitation, we derive a universal upper limit to the spontaneous free electron radiation and energy loss, verified by measurements on the Smith-Purcell radiation. Such an upper limit allows us to identify a new regime of radiation operation where slow electrons are more efficient than fast ones. The limit also exhibits a emission probability divergence, which we show can be physically approached -by coupling free electrons to photonic bound states in the continuum. Finally, we will discuss the scattering of optical waves off synthetic magnetic fields. Specifically, we will describe a synthesis non-Abelian (non-commutative) gauge fields in real space, enabled time-reversal symmetry breaking with distinct manners. These synthetic non-Abelian gauge fields enables us to observe the non-Abelian Aharonov-Bohm effect with classical waves and classical fluxes, relevant for classical and quantum topological phenomena.

Thesis Supervisor: Marin Soljaid6 Title: Professor of Physics and MacArthur Fellow

In memory of my mother. 6 Acknowledgments

The past five years at MIT has been a transformational journey. It transformed me from a student who was curious about science into a serious researcher. It gave me valuable chances to interact with talented minds from all over the world. About life, it gave me heavy lessons on birth and death and made me become a stronger person. I would like to sincerely thank my advisor Prof. Marin Soljaeid for his guidance on my research and career development, as well as for his care on me and my family. It is an honor for me to have the chance to work with Marin. I also thank Prof. John D. Joannopoulos for his advice, encouragement, and inspiring discussions. I thank Prof. Steven G. Johnson and Prof. Karl K. Berggren for their great support and help on my projects. I thank Prof. Chao Peng for his visit and collaboration. I thank Prof. Qing Hu for counselling my graduate study.

I sincerely thank Frederick Sangyeon Cho, Thomas Christensen, Chia Wei Hsu, Ido Kaminer, Steve E. Kooi, Xiao Lin, Josue Lopez, Aviram Massuda, Owen D. Miller, Nick Rivera, Charles Roques-Carmes, Yichen Shen, Scott Skirlo, Jamison

Sloan, Xuefan Yin, Bo Zhen, Di Zhu, and the entire JDJ group for our friendship and fruitful discussions.

I cannot make it through without the support of my family and friends. I thank my parents Shouzhi Yang and Changzhen Xie for their deepest love. I thank my wife Yifan for her company, support, and the greatest gift-our son Aspen, who is always my source of joy. I thank Di Zhu, as well as his family, for their daily helping hands and the leisure time we spent together.

7 Citations to Previously Published Work

A portion of Chapters 2 has been published in the following papers:

"Optically thin metallic films for high-radiative-efficiency plasmonics." Yi Yang, Bo Zhen, Chia Wei Hsu, Owen D. Miller, John D. Joannopoulos, and Marin Sol- jaeid.Nano letters 16, 4110, (2016). "Low-loss plasmonic dielectric nanoresonators." Yi Yang, Owen D. Miller, Thomas

Christensen, John D. Joannopoulos, and Marin Sojaei6. Nano letters 17, 3238, (2017).

A portion of the following paper appears in Chapter 3: "A General Theoretical and Experimental Framework for Nanoscale Electromag- netism." Yi Yang, Di Zhu, Wei Yan, Akshay Agarwal, Mengjie Zheng, John D. Joan- nopoulos, Philippe Lalanne, Thomas Christensen, Karl K. Berggren, and Marin Sol- jaei6. arXiv preprint arXiv:1901.03988 (2019). To appear in Nature.

A portion of the following paper appears in Chapter 4: "Maximal spontaneous photon emission and energy loss from free electrons." Yi

Yang, Aviram Massuda, Charles Roques-Carmes, Steven E. Kooi, Thomas Christensen, Steven G. Johnson, John D. Joannopoulos, Owen D. Miller, Ido Kaminer, and Marin Sojaeid. Nature Physics 14, 894, (2018).

A portion of the following paper appears in Chapter 5:

"Synthesis and Observation of Non-Abelian Gauge Fields in Real Space." Yi Yang, Chao Peng, Di Zhu, Hrvoje Buljan, John D. Joannopoulos, Bo Zhen, and Marin Soljaid. arXiv preprint arXiv:1906.03369 (2019). To appear in Science.

8 Contents

1 Overview 27

2 High Radiative-efficiency Nanoresonators 31 2.1 Introduction ...... 31

2.2 Far-field scattering ...... 33

2.2.1 Enhanced quality factors with thin metallic films ...... 33

2.2.2 Scattering cross-section upper limit of hybrid nanoresonators . 39

2.3 Near-field spontaneous emission ...... 42

2.4 Widely tunable quality factors ...... 45

2.5 Robustness to nonclassical corrections ...... 47

2.6 D iscussion ...... 48

3 General Framework for Nanoscale Electromagnetism 51

3.1 Introduction ...... 51

3.2 M esoscopic theory ...... 53

3.3 Numerical implementation ...... 55

3.4 Quasi-normal-mode perturbation theory ...... 57

3.4.1 Perturbation theory framework ...... 57

3.4.2 Perturbation result in the cylindrical coordinates ...... 58

3.4.3 Perturbation strength comparison ...... 59

3.4.4 Structural dependence of the spectral shift ...... 60

3.5 Experim ental setup ...... 61 3.6 Au-Au results: dispersion measurement of surface response functions 64

9 3.7 Si-Au results: robustness to detrimental nonclassical corrections .. . 66

3.8 Al-Au results: partial cancellation of nonclassical corrections between spill-in/out m aterials ...... 69

3.9 Materials and methods ...... 70

3.9.1 Ellipsom etry ...... 70

3.9.2 Impact of surface roughness on optical response ...... 72

3.9.3 Data analysis ...... 75

3.9.4 Oxide layers of Si and Al nanodisks ...... 76

3.9.5 Index dependence of measured surface response functions . . . 77

3.10 D iscussion ...... 79

4 Fundamental Limits to Spontaneous Free Electron Radiation and Energy Loss 81

4.1 Introduction ...... 81

4.2 Theoretical framework ...... 82

4.2.1 Three-dimensional general upper limits .. . . 82

4.2.2 Smith-Purcell radiation upper limit in three dimensions for rectangular gratings ...... 87

4.2.3 Limit asymptotics ...... 89

4.2.4 Penetrating electron trajectories ...... 91

4.2.5 Upper limit in two dimensions ...... 94

4.3 Key predictions ...... 97

4.3.1 Slow-electron-efficient regime ...... 97

4.3.2 Enhanced radiation by coupling electrons with photonic bound states in the continuum ...... 97

4.4 Experimental verification ...... 100

4.4.1 Measurement in the visible wavelengths .. . . 100

4.4.2 Measurement in the infrared wavelengths .. . 102

4.4.3 Experimental methods and data analysis .. . 104

4.5 Discussion ...... 107

10 5 Synthesis and Observation of Non-Abelian Gauge Fields 111 5.1 Introduction ...... 111 5.2 Non-Abelian gauge fields and state evolution ...... 114 5.3 Synthesis of non-Abelian gauge fields ...... 115 5.4 Observation of Non-Abelian Aharonov-Bohm effect ...... 118 5.5 Materials and Methods ...... 122 5.5.1 Experimental setup ...... 122 5.5.2 Magnetic field characterization ...... 124 5.5.3 Rotation angle characterization ...... 124 5.5.4 Reciprocal dynamic phase characterization ...... 125 5.5.5 Breaking reciprocity and time-reversal symmetry ...... 126 5.5.6 Faraday rotation ...... 126 5.5.7 Dynamic modulation ...... 127 5.5.8 non-Abelian Aharonov-Bohm effect ...... 127 5.5.9 Criteria for non-Abelian gauge fields ...... 128 5.5.10 Multifrequency modulation scheme ...... 133 5.5.11 Contrast function ...... 135 5.6 D iscussion ...... 138

6 Conclusion and Outlook 139

11 12 List of Figures

2-1 (a) Structure: a torus sitting on top of ametallic multifilm. The major and minor (cross-section) radii are denoted by R = 36 nm and r = 14 nm, respectively. The thicknesses of the upper and lower amorphous TiO2 layers are fixed at 5 nn and 20 nm respectively. The thickness of the middle epitaxial silver layer is denoted by t. (b) E, profiles of

two eigenmodes when t = 3.4 nm in x - z (left) and x - y (right)

planes. Upper: gap plasnmon resonance, Lower: torus (Mie) plasmon resonance. Scattering and extinction cross-sections of the torus on a

(c) thick metal film (t = 30 nn) and (d) thin metal film (t = 3.4 nm). respectively. The radiative efficiency r; increases significantly when metal thickness is reduced...... 34

2-2 (a) The scattering cross section o of torus plasnion resonance de-

creases as the silver film thickness t increases. (b)) FOMsca - Qtot/ (1- r) ~Qsshows that our structure can exceed the quasistatic limits for the Palik silver used in the nanoparticle. When the silver film is optically thin (t = 3 ~ 10 nm), a plateau of FOMsca -40 exceeding quasistatic limit of the Palik silver is achieved for resonant wavelengths at 600~800 nm, as denoted by the dashed green lines. The blue dots are calculated via Eq. 2.3 from the time-domain scattering simulation. The blue line is calculated via Eq. 2.7 from the frequency-domain eigenmode simulation. (c) Angular dependence of the scattering cross section of the torus plasmon resonance with t = 3.4 nm under the excitation of TE and TM polarizations...... 37

13 2-3 Dielectric-metal resonances offer strong scattering accompanied by modest absorption, at combined rates that cannot be achieved by all-metal or all-dielectric structures. Top: Scatter-

ing and absorption cross sections of nanoparticles under varying material and environment composition: (a) Si cylinder in free-space; (b-c) Si and Ag cylinders, respectively, above a semi-infinite Ag substrate with gap thickness g =2nm. Geometrical parameters (insets) are chosen to align their resonant wavelengths at 700 im. The three structures are all illuminated by normally-incident plane waves. In (b-c), the absorption includes the dissipation in both the particle and the substrate. (d) The dielectric-metal structure shows the highest per-volume scattering cross-section, because it simultaneously achieves large scattering cross-section osca, high radiative efficiency t, and a small particle volume V. (e) In the visible regime, the scattering capabilities of metal-metal geometries (Ag-Ag and Au-Au bounds), free-space metallic (Ag bound), and free-space dielectric (Si free-space) scatterers all

fall short when compared with the dielectric-metal (Si-Ag) scatterer, which also approaches its own upper bound, per Eq. (2.8)...... 41

14 2-4 High-Purcell, high-efficiency, high-directionality spontaneous emission enhancement with the hybrid resonances. (a) Struc-

ture and its (1, 1) modal profile for photon emission. An r 80 nm, h= 100nm silicon cylinder above semi-infinite Ag with ag 2 nm gap. A z-oriented dipole (red arrow) is located in the middle of the gap and at x = 67nm. (b) Enhancement decomposition reveals strong and efficient photon emission. A high quantum efficiency >90% and photon efficiency >75% are achieved using the (1,1) mode. (c) Far-field photon radiation pattern of the (1, 1) and (1, 2) mode. Highly directional photon emission is achieved using the (1, 2) mode. (d) Structure and its (1, 1) resonance profile for plasmon emission. A finite-thickness (t = 5 nm) metallic film is considered; all other parameters mirror those in (a). (e) Enhancement decomposition reveals strong and efficient plasmon launching. The (1,1) mode achieves a total radiative efficiency >90% and a plasmon efficiency >75%. (f) Directional plas-

mion propagation with the (1, 2) and (1, 3) mode...... 43

2-5 Low- and high-order (whispering-gallery-like) hybrid resonances offer a large continuous design space for plasmonic quality factors. (a-b) Field profiles of the plasmonic-like [P(1,6)] and

dielectric-like [D(1,5)} resonances in the (a) r-z and (b) x-y planes. E, are evaluated in the middle of the gap (particle) for the plasmonic-like

(dielectric-like) resonance. (c) Total (blue), radiative (red), and absorptive (green) quality factors of the hybrid resonances. Inset: struc-

ture and dipole excitation for quality-factor extraction...... 46

15 2-6 Hybrid resonances show increased robustness to the detrimental effects of quantum corrections than their metal-metal counterparts. The (1,1) resonances of Ag or Si nanocylinders above

a semi-infinite Ag film, separated by a finite gap [inset (i)]. The radius (height) of the Si cylinder is 50 nm (40 nm). The Ag cylinder is of identical height but of variable radius, 24-34 nm, to spectrally align the distinct structures' (nonlocal) resonance wavelength. An effective nonlocal model [1681 reveals that (a) spectral blueshifting, (b) linewidth broadening, and (c) field enhancement (at gap center) reduction, relative to classical (local) predictions, are greatly mitigated in the hybrid resonators relative to metal-metal resonators. (d) Accounting for nonlocal response, hybrid resonances exhibit higher field enhancement than the metal-metal resonance for gap sizes 5nm (crossover in green marker). Inset (ii), the induced current distribution, |JzI, of the metal-metal resonance (gap, g = 4 ni)...... 47

16 3-1 Theoretical framework, experimental structure, and measured resonance frequencies versus theory. a. Classical and b. mesocopic electromagnetic BCs. c. Equilibrium and induced densities, n(r) and p(r, w) (not to scale), at a jellium-vacuum interface (Wigner-Seitz

radius, r, = 3.93; hw = 1 eV) computed from (TD)DFT: di is the centroid of induced charge. d. Nonclassical corrections can be for-

mulated as self-consistent surface polarizations, representing effective surface dipole density r(r) and current density K(r). e. Schematic of the experimental structure: filn-coupled Au nanodisks on an Au-Ti-Si substrate, separated by a nanoscale AlO gap g (Si and Al nanodisks have also been studied). f. The nonclassical correction wl due to the d-parameters can be obtained from Eq. (3.2): the contribution from

d1 is proportional to the surface dipole density 7r(r), here shown for the (1,1) gap plasmon of a film-coupled Au nanodisk (D = 63nm, 9g= 4un). g. Observation of large nonclassical corrections(aspectral

shift > 400 nm) in film-coupled Au nanodisks (D = 63 ni). Measured

resonance frequencies of the (1, 1) plasmon blueshift (circles) relative to the classical prediction (dashed line) and quantitatively agree with our nonclassical calculations [solid line and intensity map (scattering efficiency o'sca/A where A = -rD2/4)]...... 54

3-2 Nonclassical perturbation strength comparison of various Au structures. a. Au disk (diameter, 70.4nm; height, 31nm) on Au substrate. b. Au sphere (diameter, 70nm) on Au substrate. c. Au sphere in vacuum; its di and d perturbation strength are the same in

magnitude, but of opposite signs...... 61

17 3-3 Schematic of measurement setup and micrographs of fabricated nanostructures. a. Tabletop dark-field scattering setup. It has a tunable magnification, and can record the dark-field image and measure the scattering spectrum. b. Dark-field micrograph of a Au nanodisk array (scale bar, 2 im). c. SEM image of a single Au nanodisk (scale bar, 40 nm). d. Cross-sectional TEM image of an AlO, gap

(scale bar, 10 nm )...... 61

3-4 Systematic measurement of the complex surface-response function di(w) of the Au-AlO, interface. a. Nonclassical perturbation strengths. calculated from QNM-based perturbation theory, Eq. (3.3),

in a film-coupled Au nanodisk (inset, D = 70.4nm); A"-AO- is dominant. b-c. Measured (markers) dispersion of Red"-^1°ix (b) and limd l (c) and their linear fits (lines). Gap sizes are distinguished

by color and diameters (D1 ~ 82.9nm, D 2 ~ 70.4 nm, and D 3 63.0 nm) decrement rightward. d. Measured thickness-dependent r- fractive indices of bare AlO, films grown on Au. e-j. Scattering efficiency (e,g,i) across distinct diameters and gap sizes and the extracted complex (1,1) resonance eigenfrequencies (markers; f,h,j). While classical predictions (brown, dashed lines) deviate significantly from observations, our nonclassical calculations (black, solid lines;), employing

the aforementioned linear dAu-AIO (W) fit, are in quantitative agreement across all diameters. Shadings indicate fit-derived confidence intervals for our calculations; 2o-~ 95% for Re dAu-Alo and Re W (b,f,h,j) and lo - ~68% for Imnd^"~A-Or and in c (c.f,h.j)...... 64

18 3-5 Robustness to nonclassical corrections. a. The nonclassical perturbation strength is one order of magnitude smaller in the hybrid Si-Au system than in its Au-Au counterpart. Si and Au nanodisk diameters are chosen to ensure spectral alignment of the (1, 1) resonance at every gap size (spanning D E [80,160] nm and D E [15, 40] nim, respect-

ively). b-d. Observation of robust optical response in Si-Au setup with the detrimental quantum corrections mitigated. The nonclassical calculation for the Si-Au setup assumes dA~1°. = -0.5 + 0.3inm, a constant extrapolation tohigher frequencies from Fig. 3-4b-c. In d, measured and calculated spectra are normalized separately. Calculated

spectra incorporate inhonogeneous broadening (~ 6%) due to disk-size inihom ogeneity (3.9.3) ...... 67

3-6 Additional measurement showing robustness to nonclassical

corrections from Si nanodisks with gap sizes of 1.1 nm, 1.8 nm, and 2.7nm. In Fig. 3-5d, we show the measured spectra and optical response robustness for the thinnest gap 1.1nm. Here in Fig. 3-6 we include additional measurement and comparisons with nonclassical and classical simulations to further demonstrate the robustness. Again, we

observe minor nonclassical conrrections ...... 68

3-7 Cancellation of nonclassical corrections in Au-film-coupled Al nanodisks (structure and disk-bottom oxide modeling shown in b- c insets). Complex resonant frequencies obtained from per-volume scattering cross-sections (a, e and i) are in agreement with nonclassical calculations (solid black lines in c-d, g-h, and k-1), indicating the

cancellation between the nonclassical corrections from the spill-in (Au)

and spill-out (Al) materials (b, f, and k)...... 69

3-8 Ellipsometric measurement of bulk permittivities. a. Au, b. Al, and c. Si...... 71

19 3-9 Ellipsometric measurement of A1O, thickness and refractive indices. a. Measured thicknesses as a function of deposition cycles (dots) and the linear fitting (line and inset equation) with 95% confidence interval (shading). b,c. Measured Cauchy coefficients A and B (dots) and the exponential fitting (y = ae-bx - c) with 95% confidence interval...... 72

3-10 Analysis of the surface roughness dependence on the optical responses of ALD-fabricated gap plasmonic nanoresonators. a,b. Schematic illustration of the roughness-free (a) and conformal

surface roughness model (b). The roughness of the evaporated Au

substrate is modeled by spatial sinusoidal variations. The gap size,

sinuosodial periodicity and peak-to-peak amplitude are denoted by g, K, arid pp, respectively. The position of the center of the nanoresonator with regard to the spatial variation introduces another degree of freedom, the initial 'phase' factor 0. c. Roughness-induced shifts of the resonant frequency and width are negligible given the smoothness level of our fabrication process. The red box denotes the minor resonant frequency shift range (<1.2%) for all possible random structural

variations (i.e., 0 E --7r])...... 73

3-11 Data analysis. a. Schematic for the extraction of the complex resonant frequency from the measurement. The convolution of the Gaussian size distribution and the single-particle Lorentzian resonance spectra yields the measured Voigt profile. b,c. Fitting the measured scattering of Au nanodisks with a Voigt profile and noise background [see Eq. (3.19)]. Two extreme cases are shown-smallest disk diameter on thinnest gap (b) and largest disk on thickest gap (c) d. Resonance broadening (blue) on the simulated nonclassical single-particle scattering spectrum (red) due to inhomogeneity (green) of the nanodisk

array...... 74

3-12 Native oxide bottom layer of Al and Si nanodisks...... 76

20 3-13 Frequency and index dependence of measured dAiu-Al° . Fei- belnan d-parameters generally depend on both frequency and interface

composition. Since the cladding response (nAio,) varies with gap-size (Sec. 3.9.1), the measured d^"-°IO inherits this dependence, leading to

an approximate overall ((A, nA1o,)-dependence (here, JD = Re C). Our

measurements of a. Re d^".AIO and b. Im d"u-AIO reveal the surface-

response dispersion along the thin (G', nAiO,)-band sampled by our 18 (g, D)-combinations. For Re d^-Ao the perturbation centers are the

classical eigenfrequencies, i.e. ( 1Ao,( )); for Im du-AO the perturbation centers are chosen as the measured eigenfrequencies, i.e.

( noAO, (c,)), cf. the largeness of the nonclassical c, -correction (see Sec. 3.2)...... 78

4-1 Theoretical framework and predictions. (a) The interaction between a free electron and an obstacle defined by a susceptibility tensor X(r, w) within a volume V. located at a distance d, generates electron energy loss into radiation and absorption. (b) 11/Imy constrains the max- imun material response to the optical excitations of free electrons over different spectral ranges for representative materials (from Ref. [175j)

At the X-ray and EUV regime, Si is optimal near the technologically relevant 13.5 n (dashed circle). Contrary to the inage charge intuition for the optical excitations of electrons, low-loss dielectrics (such as Si in the visible and infrared regimes) can be superior to metals. (c) Shape-independent upper limit showing superiority of slow or fast electrons at small or large separations; the material X only affects the overall scaling. (d-e) Numerical simulations (circles) compared to ana-

lytical upper limits [lines; Eq. (4.10a) for (d) and Eq. (4.18) for (e), respectively] for the radiation (blue) and energy loss (red) of electrons (d) penetrating the center of an annular bowtie antenna and (e) passing

above a grating...... 83

21 4-2 Optimal electron velocities for maximal Smith-Purcell radiation. (a) Behavior of g (B, kd), Eq. (4.18), whose maxima indicate separation-dependent optimal electron velocities. Here g is normalized between 0 and 1 for each separation. The limit yields sharply- contrasting predictions: slow electrons are optimal in the near field (kd « 1) and fast electrons are optimal in the far field (kd > 1). (b- c) Energy loss (red) and radiation (blue) rates [circles: full-wave simulations; lines: grating limit, Eq. (4.18); shadings: shape-independent limit, Eq. (4.10)] at two representative near/far-field separation distances [white dashed slices in (a)]...... 96

4-3 Strong enhancement of Smith-Purcell radiation via high-Q resonances near a photonic bound state in the continuum (BIC). (a) Schematic drawing of a silicon-on-insulator grating (one-

dimensional photonic crystal slab: periodic in x and infinite in y). (b) Calculated TE band structure (solid black lines) in the I-X direction. The area shaded in light and, dark yellow indicates the light cone of air and silica, respectively. The electron lines (blue for velocity v, and green for u/2) can phase match with either the guidedmodes (circles) or high-Q resonances near a BIC (red square). (c) Upper: Incident field of electrons. Lower: resonant quality factors (left) and eigenmode profile (right) near a BIC. (d) Strongly enhanced Smith-Purcell radiation near the BIC. (e) Vertical slices of (d). (f) The limit (shaded area) comparing with the horizontal slice of (d), with material loss considered. Strong enhancement happens at electron velocities # = a/mA (m = 1, 2, 3 ... )...... 98

22 4-4 Experimental probing of the upper limit. (a) Experimental

setup. OBJ, objective (NA = 0.3); BS, beam splitter; SP, spectrometer; CAM, camera. (b-c) SEM images of the structure in (b) top view and (c) cross-sectional view. (d) Quantitative measurement of Smith-Purcell radiation (inset: camera image of the radiation). Solid lines mark the theoretical radiation wavelengths at the normal angle [Eq. (4.1)]. The envelope (peak outline) of the measured spectra (dots) follows the theoretical upper limit (shaded to account for fabrication tolerance; calculated at each wavelength with the corresponding electron velocity for surface-normal radiation)...... 101

4-5 Smith-Purcell radiation observed in the near-infrared regime and the

comparison with the upper limit theory...... 103

4-6 (a) Measured current of the near-infrared experiment. (b) Electron structure separations d obtained from the model (dots; see 4.4.3) and their polynomial fitting (curve) for calculating theoretical upper limits. 103

4-7 Experimental setup of the calibration measurement...... 104

4-8 Fraction of the generated photons into the substrate for different accel- erating energies at normal emission angle (A = a/3) for the first-order Smith-Purcell radiation...... 105

4-9 (a) Measured current of the experiment. (b) Schematic of the model to evaluate the interaction length of the electron beam with the structure.

(c) Electron structure separations d obtained from the model (dots) and their polynomial fitting (curve; the 20 kV outlier data point dropped from fitting) for calculating theoretical upper limits...... 106

23 5-1 Comparison between SU(2) Abelian and non-Abelian gauge fields in real space and in Hilbert space. a-d. Along a closed loop inside an Abelian gauge field A oc o (a) or uy (c), the state evolves by rotating around the z (b) or y (d) axis of the Poincar6 sphere. Within each case (a-b or c-d), the state evolution are always commutative. e-f. In non-Abelian gauge fields, the evolution operators for different loops are no longer commutative, which leads to different final states,

s( and sO, for the same initial state si. The non-commutativity can be tested by an Aharonov--Bohm interference of the two final states. 113

5-2 Synthesis of non-Abelian gauge fields. a. Non-Abelian gauge fields for photons. Temporal modulation and the Faraday effect, which break T-symmetry in two orthogonal bases of the Hilbert space, are used tosynthesizeoando-gauge fields, respectively. b. Pseudospin-

dependent non-reciprocal phase shifts are created through sawtooth

phase modulations, which corresponds a synthetic gauge field along o. c. Non-reciprocal rotation of the pseudospin is achieved via the Faraday effect in a terbium gallium garnet crystal, which corresponds to a synthetic gauge field along cy. d. Experimental setup. The interference between different final pseudospin states.-originated

from reversed ordering of the gauge structures (CW and CCW, e) - is read out through a Sagnac interferometer, which gives rise to the non-Abelian Aharonov-Bohm effect. PBS/C: polarization beam splitter/combiner; PM: phase modulator; AWG: arbitrary waveform gen-

erator; COL: collimator; TGG: Terbium Gallium Garnet; PD: photodetector...... 116

24 5-3 Non-Abelian Aharonov-Bohm interference. a. Contrast function p on the Poincare sphere, featured by a fixed zero'pole pair on the equator, and a tunable zero//pole pair (which indicates the consequence of gauge fluxes). The two pairs of zeros and poles are always antipodal. b-c. Location (latitude and longitude) of the tunable pole on the Poin- cars sphere as a function of the gauge fluxes (0, ).Abelian gauge fields correspond to on-equator poles (red dashed lines); non-Abelian gauge fields correspond to off-equator poles-both of which are experiment-

ally demonstrated. d. Wilson loops W on the synthetic torus (0,#). |W| = 2 (red dashed lines) is a necessary but insufficient condition for non-Abelian guage fields (cf. b) . e-f. Examples of predicted and observed contrast functions p for Abelian (Q, U, and V) and non-Abelian (X and Y) gauge fields...... 119

5-4 Tunability of the non-Abelian gauge fields. Predicted (a) and

measured (b) contrast function p for a fixed incident pseudospin state

( 3) ~ (-51 - 12. The gauge fields 0c- and Oo are continuously tuned by respectively varying the modulation frequencies in the arbitrary waveform generators and the voltages applied to the solenoid...... 1 2 1

5-5 Characterization of pulsed magnetic fields of the solenoid system. a. Pulsed Magnetic field at the center of the solenoid as a function of charged DC voltage. The duration of the magnetic field 10 is. b. Synchronization signal for triggering and the associated

measurement time window (shaded area) Tscope= 0.5 ms...... 124

5-6 Characterization of the rotation angle of the crystal in solenoid. a. Characterization setup. b. Measured rotation angle as a

function of driving DC voltage...... 125

5-7 Temporal fluctuation of the calibrated dynamic birefringent

phase factor...... 125

25 5-8 Time-reversed Berry holonomies in Abelian and non-Abelian

Aharonov-Bohm experiments. The two Berry holonomies W1 andW2 share the same physical loop, but are path-ordered reversely. 129

5-9 Venn diagram illustration of the relation among criteria for U(N) Abelian and non-Abelian gauge fields. The loose definition

[Am, A,] z 0 (orange color) and a nontrivial Wilson loop |WI N (blue color) are both necessary but insufficient conditions for genuine non-Abelian gauge fields, which is defined by the non-commutability

between two Berry holonomies W 1 andW2 (green color). Note that

[W 1, W 2 ] z 0 is not the intersection set of [Am, A,] 4 0 and IW| N. 133

26 Chapter 1

Overview

Scattering of electromagnetic waves is fundamentally related to the inhomogeneity of a system. The inohomogeneity, i.e. the obstacle, can vary between microscopic to macroscopic scales; it can be an electron, a nanoantenna, or even a uniform slab. When an external electromagnetic excitation acts upon the obstacle, it creates secondary oscillation (besides the original incident waves) in the obstacle. The extra, secondary electromagnetic oscillation also re-radiates. These secondary oscillation

and re-radiation are known as electromagnetic scattering. Aside from scattering, the secondary oscillation also transforms into thermal energy, which is known as absorption. Taken together, the summation of scattering and absorption are called extinction.

Perhaps the most famous electromagnetic scattering theory is the Mie theory that was developed over a hundred years ago. In the Mie theory, the obstacles are limited to spherical particles. Beyond the Mie theory, the development of electromagnetic computational techniques has enabled rigorous numerical solutions to various scattering problems, including both the nonretarded and retarded regimes. Moreover, thanks to nanotechnology and the emergence of plasmonics and topological photonics, there have been continuous advances in electromagnetic scattering over the past two decades.

This thesis aims to study, both theoretically and experimental several novel aspects of electromagnetic scattering under contemporary context. In chapter 2, we will

27 investigate and design resonating nanostructures (both photonic and plasmonic) to achieve high radiative-efficiency 7 [defined as 7 = sca/Oext = Osca/(Usca+ Jabs), where

Osca, aabs, and oext arescattering, absorption, and extinction cross-sections, respectively]. The motivation to achieve high-radiative-efficiency nanoresonators is because material losses in metals are a central bottleneck in plasmonics for many applications. To improve radiative efficiency, we will theoretically explore two schemes. The first scheme employs optically-thin metallic films. We show that the quality factors of resonances can be improved by placing plasmonic nanoresonators in proximity to a high-quality thin metallic films. In the near-field spontaneous emission, a thin metallic substrate, of high quality or not, greatly improves the field overlap between the emitter environment and propagating surface plasmons, enabling high-Purcell (total enhancement >104), high-quantum-yield (>50%) spontaneous emission, even as the gap size vanishes (3-5 nm). The second scheme is featured by hybrid plasmonic dielectric nanoresonators. We show that metal losses can be successfully mitigated with dielectric particles on metallic films, giving rise to hybrid dielectric-metal resonances. In the far field, they yield strong and efficient scattering, beyond even the theoretical limits of all-metal and all-dielectric structures. In the near field, they offer high Pur- cell factor (>5000), high quantum efficiency (>90%), and highly directional emission at visible and infrared wavelengths. Their quality factors can be readily tailored from plasmonic-like (~10) to dielectric-like (~ 103), with wide control over the individual resonant coupling to photon, plasmon, and dissipative channels.

In Chapter 3, we will go beyond the classical electromagnetic theory and describe a general theoretical and experimental framework for electromagnetism at the nanoscale, where nonclassical effects becomes non-negligible. Local, bulk response functions, e.g. permittivity, and the macroscopic Maxwell equations completely spe- cify the classical electromagnetic problem, which features only wavelength and geometric scales. The above neglect of intrinsic electronic length scales leads to an eventual breakdown in the nanoscopic limit. In the nanoscopic framework, we employ the surface response function known as Feibelman d-parameters to reintroduce the missing electronic length scales. This mesoscopic approach naturally incorpor-

28 ates nonlocality, spill-in/out, and surface enabled Landau damping effects. Exper- imentally, we establish a procedure to measure these complex, dispersive surface

response functions, enabled by quasi-normal-mode perturbation theory and observations of pronounced nonclassical effects-spectral shifts in excess of 30% and the breakdown of Kreibig-like broadening-in a quintessential multiscale architecture: film-coupled nanoresonators, with feature-sizes comparable to both wavelength and electronic length scales. Moreover, we observe robustness to nonclassical corrections

in hybrid plasmonic dielectric nanoresonators (predicted in Chapter 2). In Chapter 4, we will switch gear from far-field excitation with plane waves to near-field excitation with free electrons. We will derive a universal upper limit to the spontaneous free electron radiation and energy loss. Such an upper limit allows us to make two predictions. One is a new regime of radiation operationATat subwavelength separations, slower (non-relativistic) electrons can achieve stronger radiation than fast (relativistic) electrons. The other is a divergence of the emission probability in the limit of lossless materials. We further reveal that such divergences can be approached by coupling free electrons to photonic bound states in the continuum.

In Chapter 5, we will discuss the scattering of photons and their associated synthetic non-Abelian magnetic fluxes. This is in contrast to all previous chapters

where photons scatters off real structures. Based on an optical mode degeneracy, we break time-reversal symmetry in different manners-via temporal modulation and the Faraday effect-to synthesize tunable non-Abelian gauge fields. Via a real-space closed loop configuration, we observe the non-Abelian Aharonov-Bohm effect with classical waves and classical fluxes. The Sagnac interference of two final states, obtained by reversely-ordered path integrals, demonstrates the non-commutativity of the gauge fields.

In Chapter 6, we will conclude these scattering phenomena and discuss future research directions.

29 30 Chapter 2

High Radiative-efficiency Nanoresonators

2.1 Introduction

The material composition of an optical nanoresonator dictates sharply contrasting properties: metallic nanoparticles [172, 78, 28, 228, 119, 37] support highly subwavelength plasmons with large field strengths, but which suffer from intrinsic material losses [236, 211, 160, 162, 1611, whereas dielectric nanoparticles [130, 73, 134, 1111 support exquisite low-loss versatility, but only moderate confinement as their sizes must generally be wavelength-scale or larger.

Mitigating loss is a pivotal goal [120, 221, 33, 169] in plasmonics. When nanoparticles interact with plane waves, their cross-sections are typically dominated by dissipative absorption. In the near field, large spontaneous-emission enhancements (Purcell factors) have been demonstrated [189, 127, 197, 10, 591 through mode-volume squeezing, but it has been typically accompanied by sub-50% quantum efficiencies at visible frequencies. A subsequent question that emerges is whether dielectric-like near-unity efficiency and large plasmonic confinement can be simultaneously achieved. Previously proposed hybrid structures [55, 1951 with separate dielectric (director) and metal (feed) functionality exhibit better radiative efficiency, but at the cost of lower enhancements. This tradeoff suggests the notion that strong and efficient plasmonic

31 antennas are only possible at infrared frequencies [1201, where they behave akin to perfect conductors and "plasmonic" effects are minor. Quantum corrections in plasmonics [67, 261, 234, 44], e.g. due to electron tunneling [62, 202, 2061 and nonlocality {76, 46, 168], further limit the ultimate enhancement of plasmonic resonators.

The difficulty of achieving low-loss plasmons has led to the perception that high confinement is simply incompatible with low loss, as large fields near/in a metal surface may necessarily generate significant dissipation. This intuition has led to the burgeoning field of alternative plasmonic materials [238, 169, 121], whereby highly doped semiconductors or polar dielectrics ideally exhibit negative real permittivities with small imaginary (lossy) parts. There has been a complementary effort in all-dielectric nanoparticles[130, 73, 134] and metamaterials [134, 111], but subwavelength resonances fundamentally require metallic components with negative permittivities [120, 236, 13]. Material engineering has also been proposed in the form of band engineering [123] and gain offsets [258]. The perceived confinement-loss tradeoff is rigorously correct for quasistatic plasmonic resonators [236], in which the desired resonant frequency directly sets the fraction of the field intensity that must reside within the lossy metal [236, 183, 122].

In this chapter, we propose and theoretically demonstrate two schemes to achieve higher radiative efficiencies in plasmonics. The first scheme employs optically thin metallic films [254]. We propose and theoretically investigate optically thin metallic films as an ideal platform forhigh-radiative-efficiency plasmonics. For far-field scattering, adding a thin high-quality metallic substrate enables a higher quality factor while maintaining the localization and tunability that the nanoparticle provides. For near-field spontaneous emission, a thin metallic substrate, of high quality or not, greatly improves the field overlap between the emitter environment and propagating surface plasmons, enabling high-Purcell (total enhancement > 104), high-quantum- yield (>50%) spontaneous emission, even as the gap size vanishes (3-5 nm). The enhancement has almost spatially independent efficiency and does not suffer from quenching effects that commonly exist in previous structures.

The second scheme is a combined approach-dielectric nanoparticles on metallic

32 films [253]-can exhibit a unique combination of strong fields and high confinement alongside small dissipative losses. We show the utility of such hybrid plasmonic dielectric resonators for (i) far-field excitations, where subwavelength silicon-on-silver

nanoparticles can scatter more efficiently than is even theoretically possible for any all-metal or all-dielectric approach, and (ii) near-field excitations, where highly directional spontaneous emission enhancements >5000 are possible with quantum efficiencies >90% and even approaching unity. Moreover, the dielectric composition of the nanoparticle, when placed atop a metallic supporting film, should mitigate much of the quantum- and surface-induced nonlocal damping that occurs at nanometer scales, an effect we confirm quantitatively with a hydrodynamic susceptibility model. Furthermore, as our approach does not rely on nanostructured metallic components, it strongly constrains parasitic dissipation arising from fabrication imperfections. More broadly, simple geometrical variations provide wide control over the individual resonant-coupling rates to photon, plasmon, and dissipative degrees of freedom, opening a pathway to low-loss, high-efficiency plasmonics.

2.2 Far-field scattering

In this section, we investigate the scattering properties of high-efficiency plasmonic nanoresonators with focuses on improved quality factors and amplitudes of scattering cross-sections.

2.2.1 Enhanced quality factors with thin metallic films

Enhanced quality factors is important in plasmonic applications such as biomedical sensing and transparent displays. We show that in plasmonic optical scattering, the quasistatic Q of a deep subwavelength nanoparticles can be exceeded with the help of an optically thin high-quality metal film, while maintaining considerably high radiative efficiencies 77, which is also known as the scattering quantum yield [137] or the albedo [170] in scattering problems. For a subwavelength scattering process, based on temporal coupled-mode theory[97, 193], the radiative efficiency q and the

33 A: Gap plasmon (a) (b)

k y

Pahik AgR'I L $t B:Torus plasmon

10 5 5

0 0 500 1000 1500 2000 500 1000 1500 2000 Wavelength [nm] Wavelength [nm]

Figure 2-1: (a) Structure: a torus sitting on top of a metallic multifilm. The major and minor (cross-section) radii are denoted by R = 36 nm and r = 14 nm, respectively.

The thicknesses of the upper and lower amorphous TiO2 layers are fixed at 5 nm and 20 nm respectively. The thickness of the middle epitaxial silver layer is denoted by t. (b) Ez profiles of two eigenmodes when t'= 3.4 nm in x - z (left) and x - y (right) planes. Upper: gap plasmon resonance; Lower: torus (Mie) plasmon resonance. Scattering and extinction cross-sections of the torus on a (c) thick metal film (t = 30 nm) and (d) thin metal film (t = 3.4 nm), respectively. The radiative efficiencyr increases significantly when metal thickness is reduced.

34 total quality factor Qtot for a single resonance are given by

= rad_= Usca , (2.1) 'Ytot Oext on resonance Qtot WO/ 27)tot, (2.2)

where wo is the resonant frequency, ytot = rad + 7abs is the total decay rate, and

Oext = Usca+abs is the extinction cross-section. As 7abs is mostly dictated by material absorption [236, 183], to get a high q, one has to increase 7rad. This in turn spoils the quality factor (Eq. 2.2), which reveals the trade-off between 7 and Qtot, as we described previously. Because simultaneously achieving a high Q and a high 77 is important for many applications, like biomedical sensing[137, 60, 16, 198] and transparent displays[103, 102, 199], we define the figure of merit (FOM) for scattering as

FOMsca = Q to t (2.3) 1 - 77

It follows that this FOM reduces to the quasistatic quality factor Qq, [236]

Wdw' FOMsca = WO/ 2 7abs = Qabs , Qqs-- &-. which only depends on the material property of the nanoparticle. Here, e' and E" are real and imaginary parts of the complex permittivity. For subwavelength metallic nanoparticles (dimension < A), their plasmon properties are typically dominated by quasistatic considerations[236] and thus the approximation Qabs f Qqs holds, which also indicates that the material loss inside the metallic nanoparticle cannot be further reduced. Therefore, our strategy is to squeeze parts of the resonant mode into a high-quality metallic film[243, 156] with much lower loss, while maintaining efficient radiation rates.

Below, as an example, we investigate a silver torus [153, 57, 222, 182] scatterer, sitting on top of a TiO 2-Ag-TiO 2 multifilm, whose structural geometry is shown in Fig. 2-1(a). The permittivities of the silver film and the torus are obtained from

35 Wu[243] and Palik[175], respectively; the former has substantially lower loss since it is assumed to be made epitaxially. The permittivity of amorphous TiO 2 (refractive index ~ 2.5 in the visible and near-infrared spectra) is from Kim [1261. The material

absorption in TiO 2 is negligible compared with the absorption in silver, as Im(eTio 2 ) is several orders of magnitude lower than thatof Im(EAg)within the wavelength range of interest. Thus the absorption in TiO 2 is not considered in the calculation. The ambient index of refraction is 1.38 (near the refractive index of water, tissue fluids, and various polymers). If the structure is probed with normally incident plane waves, only the m = 1 (m is the azimuthal index of the modes since the structure is axially- symmetric) modes of the structure can be excited [32]. Fig. 2-1(b) shows the mode profiles of the two m = 1 resonances in this structure. Resonance A is a gap plasmon resonance [166] whose field is mostly confined in the upper TiO 2 layer. Resonance B corresponds to the torus (Mie) plasmon resonance [248], given that it maintains a nodal line [green dashed line in Fig. 2-1(b)] along z = r (r is the minor radius of the torus), which is a feature of the torus resonance in free space [13, 7, 222,182. Fig. 2-1(c) and (d) compare o-sca and o-ext of the torus when the silver layer in the multifilm is optically thick (t = 30 nm) or thin (t = 3.4 nm). For both resonances, the radiative efficiency in the thin-film case is much higher than that in the thick- film case. Moreover, when the torus moves away from the multifilm, the response of resonances is very different for the thin film case from that for the thick film, as shown in Fig. Si. We now focus on the Mie resonance B for high-Q scattering as most of its entire radiation (photon and plasmon combined) goes into the far field (photon). We will return to the gap plasmon resonance A later for enhanced emission applications.

By changing t from 0 nm to 50nm while keeping other parameters unchanged

(t = 0 nm corresponds to a single 25-nm TiO2 layer), we are able to track the torus plasmon resonance B and evaluate its FOMsa, as shown in Fig. 2-2. As t increases, the resonance blueshifts, along with a reduced linewidth [Fig. 2-2(a)]. In Fig. 2-2(b), we compare the FOMscain our structure to the quasistatic limit Qqsfor different materials in the system: the Palik silver[175] that is used for the torus and the epitaxial silver that is used for the substrate[243 (FOMscaand Qqs are directly comparable, see Eqs.

36 2 R (a) 40 a /r 10 20

Q 10

E i~ 3

(b)0

200 10

500 600 700 800 900 CYM20 (c)wavelength [nm] am/R 2 0

5 -0 700 750 800 850 CW lavelength[n m]

Figure 2-2: (a) The scattering cross section ca ,of torus plasmon resonance decreases as the silver film thickness tincreases. (b) FOMsca =Qtot/(1 - r7) ~Qqs shows that our structure can exceed the quasistatic limits for the Palik silver used in the nanoparticle. When the silver film is optically thin (t = 3 ~- 10 nm), aplateau of FOMsca'-~-40exceeding quasistaticlimit ofthe Palik silver is achieved for resonant wavelengthsat 600~80nm, asdenoted by the dashed green lines. The blue dots are calculated via Eq. 2.3 from the time-domain scattering simulation. The blue line is calculated via Eq. 2.7from thefrequency-domain eigenmode simulation. (c) Angular dependence of the scattering cross section of the torus plasmon resonance with t= 3.4 nmunder theexcitation of TEand TMpolarizations.

37 2.3 and 2.4). There exists a plateau of higher FOMsca at t = 3 ~ 10 nm. At these thicknesses, the multifilm still has very high transmission > 80% (Fig. S2). The

FOMscaof the torus plasmon resonance exceeds and becomes twice as high as the Qqs of the torus material (Palik [175]). When the silver layer is either too thin (< 3 nm) or too thick (> 20 nm), the FOMsca drops considerably and FOMsca Qs(Palik), the quasistatic quality factor of the torus material. Fig. 2-2(c) shows that the high FOMscacan be maintained for both polarizations over a wide range of incident angles.

The increased quality factor is the result of effective mode squeezing that only occurs in thin silver films - an effect we qualitatively demonstrate in Fig. S3 of the Supporting Information. The mode squeezing mechanism can be quantitatively demonstrated by calculating the energy density integral of the eigenmode. The energy density u in lossy media is generally defined as u = co (e'+ 2wc"/y)IE1 2 /2 [1941, where e' and c" are real and imaginary parts of permittivity respectively, and y is the damping of the metal. We adopt y = 1.4 x 10" rad/s for the Palik silver and

-y = 3.14 x 1013 rad/s for the epitaxial silver to best match the tabulated data. Since the metallic objects (Palik silver torus and epitaxial silver film) dominate the absorption loss in this system, we define the energy concentration coefficients in the torus and the film as

Ctor ftorus udV , (2.5) rus f udV + film udV fIM udV Cfilm ftorus udV + ffilm udV (2.6)

Thus, the Qas of the system can be estimated as

1 Ctorus Cfim (2.7) Qqs Qqs(Palik) Qqs(Epitaxy)

As shown in Fig. 2-2(b), the Qqs of the system, calculated from the scattering (blue dots) and eigenmode (blue curve) simulations respectively, match each other well. Our calculation shows the high energy concentration in the film only happens when the film is optically thin (see Fig. S4). Near the maximum of the Qs (wavelength

38 -700 nm, silver film thickness -7 nm), the energy concentrated in the film is three

times higher than that in the torus (cm - 3torus). We also note that the Qs curves of the two materials are quite flat within the wavelength of interest. Thus, it is the effective mode squeezing into a high-quality film, rather than the dispersion of an individual material, that contributes to the improved quality factor of the system.

The aforementioned enhanced Q is different from the linewidth narrowing that is

based on the interference between multiple resonances [214]. For coupled resonances, as the trace of the full Hamiltonian is conserved, the linewidth reduction of one resonance necessarily implies the broadening of the others'. This coupling also typically renders the spectrum Fano-like with dark states in the middle of the spectrum [101]. In contrast, here the linewidth reduction is realized via effectively squeezing a single Mie plasmon mode into an optically-thin metallic film. Scattering spectrum is kept

single-Lorentzian, which is favorable for many applications [60, 16, 198, 103, 199] as it maintains a high resolution and SNR. Moreover, as the resonance for scattering uses the Mie plasmon and the ambient environment is the perturbed free space, most of

the reradiated energy goes into the far field with weak plasmon excitation (see supporting information). We also note that optically thin metallic films are not restricted

to high-Q applications shown above. Applications based on broadband strong scattering (like solar cells requiring longer optical path) can also be implemented on this platform, utilizing its high radiative efficiency.

2.2.2 Scattering cross-section upper limit of hybrid nanores-

onators

Metallic nanoparticles generally scatter more strongly than all-dielectric nanoparticles.

Yet this large scattering strength-as measured, e.g., by the optical cross-section per unit particle volume-is typically accompanied by significant absorption. Thus for many applications where absorption is undesirable (such as photovoltaics [20]), the critical figure of merit is scattering strength accompanied by high radiative efficiency.

Here we leverage recently developed optical-response bounds to show that low-loss

39 dielectric nanoparticles on metallic films can achieve subwavelength scattering with large radiative efficiency, surpassing all-metal and all-dielectric scatterers and approaching fundamental limits.

There has been significant interest in finding general upper bounds to optical response [91, 106], and recently we developed new such bounds [160, 162, 161]. Passivity, which requires non-negative absorbed and scattered powers, imposes limits to the currents that can be excited in an absorptive scatterer, leading to bounds that are independent of shape, which account for material loss (oc ImX, for material susceptibility x), and which can incorporate radiative-efficiency constraints. The bounds demonstrate [162] that high radiative efficiency, defined as77= Usca/ (0sca + abs) -- sca/0ext

(where osca, cabs, and Uext are the scattering, absorption, and extinction cross sections, respectively), necessarily reduces the largest cross-section per volume that can be achieved. A natural figure of merit (FOMsca) emerges: o-sca/V x 1/[n(1- q) (equivalently, Uext/0abs x oext/V), which rewards high scattering cross-section (sca/V) as well as high radiative efficiency (rI» 0.5). The FOMsca is subject to the bound [1621

-Osca/V W |XGo)|2 L n FOMsca -<; ,( (2.8) 77 (1 - 77) c Im X(w) 10 which depends only on the frequency w, the material composition, and the incident field properties. IiJc/Io is the ratio of the incident-field intensity Iinc (including e.g., reflection from a planar film in the absence of the nanoparticle) integrated over particle volume to the intensity of theplane wave. Perfect radiative efficiency (77= 1) is unachievable for lossy scatterers, such that Eq. (2.8) cannot diverge. Equation (2.8) clearly shows that low-loss materials offer the possibility for strong and high-efficiency scattering, but all-dielectric structures cannot reach their bounds (in most parameter regimes) for lack of subwavelength resonances. On the other hand, by equipping dielectric nanoparticles with a subwavelength resonant mechanism, achieved by coupling to a metallic substrate, these high limits may actually be approached.

We compare scattering by three types of resonators-(i) a free-space, all-dielectric resonator, (ii) a hybrid dielectric-on-metal resonator, and (iii) a metal-on-metal resonator-

40 Free-space Si Si-Ag Ag-Ag 2 2 15o- 0.5 Ag

i Scatt e0 1p 10 bsorption 0.5 e 01 Absor AbS~ption 600 700 800 600 700 800 600 700 800 Wavelength (nm) Wavelength (nm) Wavelength (nm)

(d) 0.4 -(b)Si-Ag E 0.2 . c) A-Ag- ,(a) space Si

0 600 650 700 750 800 Wavelength (nm) (e) h-60nm 16 Analytical Computational

h=40nm

Si-Ag 8

Ag-Agg

0 400 500 600 700 800 Wavelength (nm)

Figure 2-3: Dielectric-metal resonances offer strong scattering accompanied by modest absorption, at combined rates that cannot beachieved by all-metal orall-dielectric structures.Top: Scatteringand absorptioncross sections of nanoparticles under varying material and environment composition: (a) Si cylinder in free-space; (b-c) Si and Ag cylinders, respectively, above asemi-infinite Ag substrate with gap thicknessg9=2nm. Geometrical parameters (insets) are chosen to align their resonant wavelengths at 700 nm. The three structures are all illuminated by normally-incident plane waves. In(b-c), the absorption includes the dissipation in both the particle and thesubstrate. (d) Thedielectric-metal structure shows the highest per-volume scattering cross-section, because it simultaneously achieves large scattering cross-section osca, high radiative efficiencyi'r, and asmall particle volume V. (e) In the visible regime, thescattering capabilities ofmetal-metal geometries (Ag-AgandAu-Aubounds),free-space metallic(Ag bound),andfree-spacedielec- tric (Si free-space) scatterers all fall short when compared with thedielectric-metal (Si-Ag) scatterer,whichalso approachesitsown upperbound, per Eq.a(2.8).

41 at 700 nm wavelength. For each resonator, the dielectric is Si. The free-space dielectric resonator [Fig. 2-3(a)] is designed to achieve super-scattering [192], with j.~ 96%, via aligned electric- and magnetic-dipole moments. The hybrid silicon-on-silver resonator [Fig. 2-3(b)] is optimized to have a similar scattering cross-section, which is achieved in roughly one-fifth of the volume and with 7 ~ 93%. Finally, the radius of the Ag-on- Ag resonator [Fig. 2-3(c)] is optimized by radius [cylinder height and gap size same as Fig. 2-3(b) for constant I1]; notably, it only achieves only ~ 17% radiative efficiency. Figure 2-3(d) compares the scattering strengths of the three architectures, measured by o-sca/V, clearly showing the dielectric-metal structure's advantage, which remains compelling across visible frequencies [Fig. 2-3(e)]. Fig. 2-3(e) compares FOMsca of different structures and includes corresponding bounds (shaded regions) based on the cylinder height due to the oscillatory incident fields in the presence of the reflective film. Different from Fig. 2-3(a-d), all cross-sections in Fig. 2-3(e) (except the dashed line) isolate the contributions of the nanoparticles themselves, and the substrate is incorporated in the incident-field definition [161]. At longer wavelengths, the scattering strength of the Si cylinder (blue solid line) approaches its bound, the highest among all bounds. Including film absorption and scattering in the dielectric-metal structure (blue dashed line), the hybrid resonance retains large FOMsca, still outper- forming all-metal and all-dielectric resonators. In the following section, we translate this large-response, high-radiative-efficiency capability from the far field to the near field.

2.3 Near-field spontaneous emission

Plasmonic losses are particularly acute in the near field, for sources in close proximity to the resonator, as the source readily accesses lossy channels that dissipate energy before it can escape into a propagating far-field photons or guided plasmons.

In contrast, with negligible local dissipation, dielectric-metal resonances can provide high-Purcell, high-efficiency, and high-directionality spontaneous emission enhancements. A Purcell factor >5000 with quantum efficiency (including both photon and

42 Semi-infinite Ag substrate (t = o) (a) m-),n IE/J (b) 100 s]k 1 0

00 EW 2 0 x(nm) 01 (C) Photon radiation pattern '90% (I) Qii)%

((nm)=0 2)

0 is 0 90 iso o C 500 60) 700 M0O 900 Wavelength (rnt)

Finite Ag film (t=5 nm) ,10m) (d) (e) 3)(Z2)(1,3) (1,2) ( jEAEJ 100x 1>10k 160G 0o -200 0 ou x (nm)

E 1m (f) Plasmon propagation pattern V io 600 0 00 WaeeChnn

20 y 220 (m)=0,2) L (tm )=(3

Figure 2-4: High-Purcell, high-efficiency, high-directionality spontaneous emission enhancement with the hybrid resonances. (a) Structure and its (1, 1) modal profile for photon emission. An r = 80 nm, h = 100 n msilicon cylinder above semi-infinite Ag with a g = 2 nm gap. A z-oriented dipole (red arrow) is located in the middle of the gap and atx = 67nm. (b) Enhancement decomposition reveals strong and efficient photon emission. A high quantum efficiency >90% and photon efficiency >75% are achieved using the (1,1) mode. (c) Far-field photon radiation pattern of the (1, 1) and (1, 2) mode. Highly directional photon emission is achieved using the (1, 2) mode. (d) Structure and its (1, 1) resonance profile for plasmon emission. A finite-thickness (t = 5 nm) metallic film is considered; all other parameters mirror those in (a). (e) Enhancement decomposition reveals strong and efficient plasmon launching. The (1, 1) mode achieves a total radiative efficiency >90% and a plasmon efficiency >75%. (f) Directional plasmon propagation with the (1, 2) and (1, 3) mode.

43 plasmon emission) >90% can be achieved in the optical regime. Whereas some previous work (e.g., Ref. [10]) has not distinguished between emission into guided plasmons and emission into radiating photons, we separate each contribution and show that a simple geometrical reconfiguration (increasing/reducing the metal-film thickness) can swing the emission rate from plasmon-dominant (> 75%) to photon-dominant

(> 75%) or vice versa. Directional photon and plasmon emission can also be realized via high-order resonances.

We first demonstrate photon emission enhancement with a silicon cylinder on a semi-infinite Ag substrate, separated by a 2nm gap [Fig. 2-4(a)]. Planar dispersion analysis suggests that this geometry should provide similar Purcell enhancement, and much higher quantum efficiency, as compared to a 5nm-gap-size metal-metal structure. We decompose [252] the enhanced emission from a z-oriented dipole into far-field photon, guided plasmon, and local dissipative channels and obtain corresponding efficiencies [Fig. 2-4(b)]. The (1, 1) and (1, 2) modes achieve Purcell factors

(total enhancement) >5000 and >104, respectively. As importantly, the (1, 1) mode exhibits >90% quantum efficiency and >75% photon efficiency. Similar efficiencies are achieved for emitters located throughout the gap region (not shown; adopting the approach in [254]). In the far field [Fig. 2-4(c)], the (1, 1) mode exhibits wide-angle emission, while the (1, 2) mode enables highly directional photon emission, without the Yagi-Uda configuration [55, 49] or a periodic lattice [148].

Even higher quantum efficiencies, with similar enhancements, are possible with alternative low-loss dielectric materials (on Ag). AlSb [263] nanoparticles offer close-to- unity efficiencies below their 2.2 eV direct bandgap. Ge nanoparticles exhibit Purcell factors of 2 x 104 with high radiative (; 95%) and photon (~ 85%) efficiencies at the technologically relevant 1.55 pm wavelength. Relative to a previously proposed [189] infrared antenna with similar efficiency, this Purcell factor is 10 times higher.

We further demonstrate plasmon generation [75] with high efficiency by using an optically thin (t = 5nm) metal layer [Fig. 2-4(d)]. The thin metal improves the modal overlap between the gap and propagating plasmons [254]. The Purcell factors exceed 104 for all the modes in Fig. 2-4(e). Similar to the thick-metal case, high total

44 quantum efficiencies are achieved, with that of the (1, 1) mode still >90%. Contrary to the thick-metal case, photon emission is suppressed while plasmon emission is strongly boosted: the plasmon efficiency exceeds 60% for each of the (1, 1), (1, 2) and (1, 3) modes. The guided-plasmon propagation pattern [Fig. 2-4(f)] reveals highly directional plasmon launching.

2.4 Widely tunable quality factors

The quasistatic properties of metals [236] limit the quality factors of conventional plasmonic resonances (typically <100 in the optical regime), imposing severe restrictions on many plasmonic applications. In contrast, dielectric-metal resonances provide control over the individual absorptive- and radiative-loss rates, providing options along the entire continuum between the all-metal and all-dielectric extremes.

Using approximately lossless dielectrics, such as TiO 2 at visible frequencies, plasmonic modes with extraordinarily high quality factors can be designed (Fig. 2-5). As evidenced by their field patterns [Fig. 2-5(a-b)], the modes of the dielectric-metal resonator partition into dielectric-like and plasmonic-like resonances-both of which display strong field confinement within the gap. Figure 5(c) shows the total, radiative, and absorptive quality factors (ot, Qrad, and Qabs) of the resonances. The dielectric-like modes generally have higher Qabs than the plasmonic-like modes because of their larger field intensity in the interior of the dielectric [Fig. 2-5(a)]. Unlike conventional plasmonic modes, for which Qtot is mainly limited by material loss, here

QtOt is primarily limited by radiation loss, which can be readily tailored via the nanoparticle geometry and size. The Qtt of these resonances ranges widely from ~10 to ~103, offering a wide, continuous design space for narrow- or broad-band plasmonic applications.

45 (a) P(1,6)

max max

1P |E Er

r 62eecti 366k D (14) D15 i (C)

0) 200 .32 3441s Qtot Qrad Qabs

100

450 500 550 600 650 700 Wavelength (nm)

Figure 2-5: Low- and high-order (whispering-gallery-like) hybrid resonances offer alarge continuous design space for plasmonic quality factors. (a- b) Field profiles ofthe plasmonic-like [P(1,6)] and dieectic-like [D(1,5)] resonances in the (a) r-zand (b) x-yplanes. Eare evaluated inthe middle ofthe gap(particle) for the plasmonic-like (dielectric-like) resonance. (c) Total (blue), radiative (red), and absorptive (green) quality factors of the hybrid resonances. Inset: structure and dipole excitation for quality-factor extraction.

46 2.5 Robustness to nonclassical corrections

Quantum phenomena beyond the classical description set the ultimate limitations on the achievable response in plasmonic nanostructures. Chief among these phenomena are nonlocality, spill-out, and surface-enabled damping [67]. In Ag, their joint impacts are well-described by a nonlocal, effective model-GNOR [168], a convective-diffusive hydrodynamic model-causing spectral blueshifting and broadening in structures with nanoscale features. In comparison, analogous quantum corrections in dielectrics are negligible due to the absence of free carriers. We show here that the dielectric- metal resonances display increased robustness to these detrimental quantum corrections compared to their metal-metal counterparts; taking field enhancement as a measure, the former is even superior for gaps < 5nm.

(a) 140 00 M (d 600 Ag local (a) 1 z (d) CM Si local 120

4500 > Nonlocal 100 r 400 .a".() g E

800 0.5 Ag 0 60 cY 1 . (b) 0 . CosoeCross-over

0.5 - Ag 20

0 . . 0 --- Nonlocal 2 4 6 8 10 2 4 6 8 10 Gap size g (nm) Gap size g (nm)

Figure 2-6: Hybrid resonances show increased robustness to the detrimental effects of quantum corrections than their metal-metal counterparts. The (1,1) resonances of Ag or Si nanocylinders above a semi-infinite Ag film, separated by a finite gap [inset (i)]. The radius (height) of the Si cylinder is 50 nm (40 nm). The Ag cylinder is of identical height but of variable radius, 24-34 nm, to spectrally align the distinct structures' (nonlocal) resonance wavelength. An effective nonlocal model [168] reveals that (a) spectral blueshifting, (b) linewidth broadening, and (c) field enhancement (at gap center) reduction, relative to classical (local) predictions, are greatly mitigated in the hybrid resonators relative to metal-metal resonators. (d) Accounting for nonlocal response, hybrid resonances exhibit higher field enhancement than the metal-metal resonance for gap sizes <, 5nm (crossover in green marker). Inset (ii), the induced current distribution, |Jz|, of the metal-metal resonance (gap, g = 4 nm).

47 Figure 2-6 examines these quantum corrections for 2-10 nm gap sizes, where inter- surface electron tunneling is absent [261]. For both dielectric-metal and metal-metal structures (with equal nonlocal resonant frequencies), the resonant wavelength, quality factor, and field enhancement of the (1,1) resonance are shown [Fig. 2-6(a-c)] as functions of gap size. Relative to local, classical predictions, both configurations exhibit blueshifted resonant wavelengths and reductions in quality factor and field enhancement-all of which increase as the gap size decreases. Crucially, the metal- metal system suffers more severe reductions than its counterpart. This observation can be attributed to two cooperating effects: first, in light of the plasmon-Bessel framework [253], the planar multilayer equivalent approximately dictates the gap- dependent impact of quantum corrections. Accordingly, since the surface plasmon of the planar metal-dielectric-metal system suffers increased impact of quantum corrections compared to the planar dielectric-metal system (by a factor 1+ e-k [44), the metal-metal nanoparticle's performance is similarly reduced. Second, the metal nanoparticle's edges host sharply varying current densities [Fig. 2-6(d), inset (ii)] and consequently incur large nonlocal corrections in these regions.

Strikingly, the relative robustness of the hybrid resonances to quantum corrections enables them to demonstrate larger absolute field enhancements, for equal gap sizes < 5 nm [Fig. 2-6(d)], than the high-intensity, pure-plasmonic metal-metal resonators.

The enhancement in the latter system deteriorates drastically at these gap sizes, due to the above-noted distinguishing aspects. The comparative robustness of the hybrid resonances suggests a pathway to stronger light-matter interactions in extreme nanoscale gaps [42].

2.6 Discussion

We have shown the possibility for low-loss plasmonics by with optically-thin metallic films and by coupling low-loss dielectric nanoparticles with high-confinement metallic substrates.

The high-quality thin metal films can help improve the quality factors of the

48 Mie resonances of metallic nanoparticles. Besides, the hybrid dielectric-metal resonances exhibit strong and efficient scattering and near-field emission enhancements, large quality factors, and nonlocal robustness beyond those of conventional plasmonic nanostructures.

The approach to high efficiency presented here can work in tandem with future material improvements. Just as we have shown that re-architecting common materials can improve their plasmonic response, new, low-loss materials should be integrated into these hybrid geometries rather than conventional all-metal structures. Graphene sheets behave optically very much like ultrathin metallic films, and thus our approach extends to dielectric-on-graphene architectures for efficient graphene plasmon confinement.

Looking forward, the dielectric-metal approach prompts two directions for new ex- ploration. First, the strong emission enhancement of the dielectric-metal resonances rely on the high index contrast between the dielectric scatterer and the dielectric spacer (comprising the gap). When the index contrast is reduced, the high efficiencies can be maintained though at the expense of reduced optical confinement. Thus con- tinued development of very-low-index (n 1) materials, such as low-indexSiO 2 (245], aerogels [219], and low-index polymers [89], would further increase enhancements and improve efficiencies. Second, quantum effects in dielectric and dielectric-metal structures at few-nanometer length scales are of increasing interest, and should be explored further with alternative (e.g., time-dependent density functional theory) electronic and optical models. The prospect of dielectric-metal structures that are robust to deleterious nonlocal effects is especially enticing for the growing field of quantum plasmonics [71].

49 50 Chapter 3

General Framework for Nanoscale Electromagnetism

3.1 Introduction

The macroscopic electromagnetic boundary conditions (BCs)-the continuity of the

tangential E- and B-fields and the normal D- and H-fields (Fig. 3-la) across interfaces- have been well-established for over a century [155]. They have proven extremely successful at macroscopic length scales, across all branches of photonics. Even state- of-the-art nanoplasmonic studies [208, 94, 171, 24, 42, 11, 119, 39, 99], exemplars of

extremely interface-localized fields, rely on their validity. This classical description, however, neglects the intrinsic electronic length scale associated with interfaces. This omission leads to significant discrepancies between classical predictions and experimental observations in systems with deeply nanoscale feature-sizes, typically evident below ~10 -20 nm [131, 48, 87, 25, 47, 224, 205, 186, 1081. The onset has a mesoscopic character: it lies between the domains of granular microscopic (atomic-scale)

and continuous macroscopic (wavelength-scale) frameworks. This scale-delimited, mesoscopic borderland has been approached from above by phenomenological accounts of individual nonclassical effects-chiefly spill-out [52, 260, 212] and nonlocality [30, 51, 185, 70, 151, 167, 124]-and from below, using explicit time-dependent density functional theory (TDDFT) [264, 226, 216, 223, 233]. The former approaches

51 are uncontrollable and disregard quantities comparable to those they include; the latter is severely constrained by computational demands. A practical, general, and unified framework remains absent.

We briefly review the key nonclassical mechanisms that impact plasmonic response at nanoscopic scales [68]. First, equilibrium charge carriers spill out beyond the ionic interface [136], blurring the classically-assumed step-wise transition between material properties; and second, dielectric response is nonlocal [181, 81], i.e. the D- and E-fields are related by a nonlocal response function (r, r'; ) rather than the local response function e(r, w)6(r -r') implicitly assumed in classical treatments (additionally, tunnelling [61, 201, 14, 220, 250, 260] and size-quantization effects [226, 227, 96], ignored in this work, are non-negligible at feature-sizes below ~1nm). Individual aspects and consequences of these omissions have been studied extensively-e.g. hydrodynamic nonlocality [167, 185, 70, 151], local-response spill-out [52, 225, 212], and surface-enhanced Landau damping [124]. However, to attain meaningful, quantitative comparisons of experiments and theory, a unified, general framework that incorporates these mechanisms on equal footing is required. In principle, TDDFT [152] provides such a framework, but its range of applicability is limited to highly symmetric or sub-nanometric systems [264, 226, 216, 223, 233] due to prohibitive computational scaling. Many promising electromagnetic systems, particularly plasmonic systems with multiscale features, are thus simultaneously incompletely described by macroscopic, classical electromagnetism and inaccessible to microscopic, quantum- mechanical frameworks like TDDFT.

Here, we introduce and experimentally demonstrate a general theoretical and experimental framework-amenable to both analytical and numerical calculations and applicable to multiscale problems-that reintroduces the missing length scales. Our framework should be generally applicable for modelling and understanding of any nanoscale (i.e. all relevant length scales >1nm) electromagnetic phenomena.

52 3.2 Mesoscopic theory

We achieve the reintroduction of electronic length scales by amending the classical

BCs with a set of mesoscopic complex surface response functions, known as the Fei- belman di- and dii-parameters (Fig. 3-1b) [68, 142]: these parameters play a role

analogous to the local bulk permittivity, but for interfaces between two materials. d

and d1 are the missing electronic length scales-respectively equal to the frequency- dependent centroids of induced charge and normal derivative of tangential current at an equivalent planar interface (Fig. 3-1c). They enable a leading-order-accurate

incorporation of nonlocality, spill-out, and surface-enabled Landau damping.

These d-parameters drive an effective nonclassical surface polarization P, = Ir + iw- 1K (Fig. 3-1d), with di contributing an out-of-plane surface dipole density7r

dieoE[E 1jfn and d 1 an in-plane surface current density K = iWdiijDii]. Here, [f] f+ - f- denotes the discontinuity of a field f across an interface 8 with outward normal f; similarly, fi - n -f and fl = (I - fnfT)f denote the (scalar) perpendicular and (vectorial) parallel components of f relative to &Q. These surface terms can be equivalently incorporated as a set of mesoscopic BCs for the conventional macroscopic Maxwell equations (also shown in Fig. 3-1b):

D_] = -i-Vi- K = djiVi - [DIi], (3.1a) B1] = 0, (3.1b)

Eli] = -- Vi7r= -dV [Ei, (3.1c)

Hii] = K x ii = iwd1i[D1j x fi. (3.1d)

These mesoscopic BCs are a two-fold generalization from opposite directions. First, they generalize the usual macroscopic electromagnetic BCs-[Di]1= [B1] = 0 and

[Eli] = [Hj = 0-to which they reduce in the limit di = dii = 0. Second, they represent a profound conceptual and practical generalization of the Feibelman d- parameters' applicability-elevated from their original purview of planar [681 and spherical [17] interfaces, and beyond recent quasistatic considerations [43], to a fully

53 general electrodynamic framework.

netic______B__s.___c._ Eequninumd r) a ellsi u ir (inded r= d, 1200 H ------

structe : o Positi o n a, s

correction(; .) dueo the m ca beq thct

vaiguref-largeocicalfcrewtosk(aespetraintsind imeued resonance De=u6nm). Measur reona frequenciescal andb.mesocopic electromagnetic BCs.c. Equilibrium andinduceddensities, n(r) andp(r,w)(nottoscale), at a jellium-vacuum interface (Wigner-Seitz radius, r,,= 3.93; hw = 1 eV) computed from (TD)DFT:d is the centroid ofinduced charge. d. Nonclassical corrections can beformulated asself-consistent surfacepolarizations,representing effectives ur- facedipole density7r(r)and currentdensity K(r). e. Schematicofthe experimental structure:film-coupledAu nanodiskson an Au-Ti-Si substrate, separated bya nano- scaleAlO gaplg(Si andAl nanodisksnhavecalso beenstudied). The nonclassical correction duetothed-parameterscan be obtained from Eq.(3.2): thecontri- bution from djis proportional to the surface dipole densityir(r), here shown for the (1, 1) gap plasmon of afilm-coupled Au nanodisk (D =63 nm, g =4 nm). g. Obser-

vation of large nonclassical corrections (a spectral shift >400 nm) in film-coupled Au nanodisks (D = 63 nm). Measured resonance frequencies of the (1, 1) plasmon blueshift (circles) relative to the classical prediction (dashed line) and quantitatively agree with our nonclassical calculations[solid lineand intensitymap(scatteringefficiency oscaAwhereA= 7lDe/4)n.

To extract the surface response functions from observables, we develop aperturbation- theoretical description of the nonclas'sical spectral shift under the QNMframework 12511 with retardation explicitly incorporated: the true eigenfrequency' (D=-M+'()+. (eigenindex implicit) exhibits afirst-order nonclassical correctionCZ'A)()to its classical value (()(Sec. 3.4)

- ~O)K~di± (3.2)

T with mode-, shape-, and scale-dependent nonclassical perturbation strengths (units of inverse length)

2 0 2 i JKGiE- £65]f)J drand j JO) -[b )] d r. (3.3) V1)* DI1 33

54 Here, r runs over all material interfaces such that UJ, 8 7 = o, i.e. T E {Au-AlO , Au-air} for our setup, while b() and E(o) denote the D- and E-fields of the (suitably normalized) classical QNM under consideration. Conceptually, Eq. (3.3) states that the nonclassical perturbation strength is proportional to the difference between classical surface energy densities evaluated on either side of the interface.

3.3 Numerical implementation

Here we consider a general electromagnetic scattering problem (as in our experiment) to illustrate the numerical implementation in COMSOL Multiphysics. Other electromagnetic problems can be treated in a similar manner.

The scattering formulation of Maxwell's equations yields

V x --iV x Esca - E(r, w)k 2 Esca = AE(r, w)k2Einc, (3.4)

where k = w/c, Einc is the incident field, Esca is the scattered field, and Ae(r) = E(r)-

Ebg(r) is the permittivity contrast between the scattering object and its background

(vanishes outside the scatterer). Eq. (3.4) and Eqs. (3.1) constitute the full ingredients for numerically solving

our scattering problem. Specifically, in our gap plasmon nanoresonator structures, the incident field Einc in Eq. (3.4) is defined as the summation of the incident plane waves and their reflected fields from the substrate. Using the TM polarization for example,

Einc, TM -EO [(,b cos 0 + z sinO)eikzz+ikpp +rTM(-p cos + z sin )e-ikz+ikPp], (3.5) where 0 is the incident angle and rTM is the (Feibelman-corrected) TM reflection coefficient. Notably, the nonclassical correction to rTM is small - kod, 11 . We ex- ploit the structure's rotational symmetry by decomposing the incident plane waves in cylindrical harmonics [109) which allows us to calculate the scattering response for each azimuthal index m separately. This reduces the dimensionality of the compu-

55 tational problem from three to two, allowing significant reductions in computational time and memory requirements. As our focus is on the lowest order (1, 1) (denotes the radial and azimuthal index number respectively [2531) resonance, we restrict our considerations to its m = 1 channel.

Implementation wise, the master equation Eq. (3.4) is incorporated via weak- form integrals. For the nonclassical BCs [Eqs. (3.1)], only two of them [Eqs. (3.1c) and (3.1d)] are needed to uniquely and completely define the computation. In principle, the boundary conditions Eqs. (3.1) can be straightforwardly incorporated by point-wise or weak constraints. However, numerical instability may arise due to the derivative form of Eq. (3.1c).

As an alternative, we develop a numerical stable approach to implement those nonclassical BCs. The d 1 contribution [Eq. (3.1d)] can be incorporated via a surface current term K(r) = idjw[Djj. Specifically for the di contribution [Eq. (3.1c)], we describe below the auxiliary potential method employing its integral form for better numerical stability.

For the scattered field, Eq. (3.1c) can be rewritten as

Esca= Eca+V@, (3.6) where Eca is the continuous classical scattered field, the scalar auxiliary potential ? defined on the boundary 8O is given by

|)' = -d 1 [[E]. (3.7)

The potential needs to change sign depending on whether it is defined on the "+" or

"-" side of the boundary. Plugging Eq. (3.6) into Eq. (3.4) yields

1 2 2 2 V x p-I V x Esca - e(r, w)k Esca = Ae(r, w)k Eic + E(r, w)k Vb. (3.8) where the nonclassical contribution (last term on the right-hand size) is implemented via weak-form integrals.

56 3.4 Quasi-normal-mode perturbation theory

3.4.1 Perturbation theory framework

Classically, the electromagnetic resonant eigenfrequency and eigenfield E of a general geometry satisfies the following master equation [(r,C)-dependence implicit]

-2 V x p-1V x E e E= 0. (3.9) C2

For open resonators with radiative and potentially absorptive losses, C is complex- valued and the outgoing-wave boundary condition is imposed. Hereafter we denote the classical non-perturbed frequencies and fields as c(o) and E(0 ) (throughout, the tilde notation indicates the QNM eigenfrequencies and eigenfields).

Under the QNM framework, eigenfields are normalized under the convention that

EE - E- . d N 3r =1 (3.10) where R3 in practice includes both physical and perfectly-matched-layer domains [135].

Next we "turn on" the nonclassical perturbation, i.e. the surface polarization

PS(di, d; E) = -r + iw--K. Evidently, the nonclassical surface polarization is a functional of d-parameters and the nonlassical eigenfield E. The nonclassical master equation naturally emerges as

-2 -2 V X -V X N - E -W P (d±, d 1; t)6(r - oQ) = 0. (3.11)

Within first-order perturbation, - +.0(1) + 0[Q - (o))2]. The first-order correction, C(1), is given by an inner product between the unperturbed eigenfield and the perturbing polarization [251]. Specifically, the nonclassical surface correction can

57 be calculated by a surface integral runs over all interfaces

= -°) E(0) p(0) d2r (3.12) Jan with = P_(d_, di; E(°)). In contrast to conventional perturbation theory, an unconjugated inner product is used between the ground-state and perturbing components (and in the normalization condition), reflecting the non-Hermitian nature of the QNM eigenproblem [135].

The nonclassical surface polarization P, requires further specifications-in terms of which medium it placed-to render Eq. (3.11) consistent with the generalized boundary conditions, Eqs. (3.1). To illustrate such necessity, we observe that, if Ps is simply chosen to place in metal or dielectric sides of metal-dielectric interfaces, the generalized boundary condition of El shall be modified to [E]j = -dVj[EI]I/eb, where the extra termebrepresents the permittivity of background media in which P, is embedded. This modification comes from the fact that, when defining the surface polarization 7r and current K in relation with di and d, respectively. We choose vacuum as background, in which7r and K radiate. Respecting this prerequisite, we, thus, introduce an infinitely thin vacuum layer separating the metal and the dielectric, and P, is placed on this vacuum layer; in this way, it could be examined that the generalized boundary condition of Eq. (3.1c) is restored.

3.4.2 Perturbation result in the cylindrical coordinates

Next, we describe some unique considerations necessary to facilitate the perturbation calculations in cylindrical coordinates. Given the axial symmetry of the experimental nanodisk structure, the resonance modes' E-field assume the form Em(r, C)=

Em(p, z)eimqe"1t, expressed in cylindrical coordinates (p, #, z) and with m = 0, 1, 2 (the modes are also index by a radial quantization number n; it is suppressed here for clarity). For nonzero m, Eq. (3.10) suggests a vanishing normalization integral. For these

58 azimuthally varying modes, the normalization condition is revised to

/ l a(O)O Nm -i-m d3 r = 1, (3.13) where E-m = [tm - i, tm -Nm - 2 and N-m [-m ,m• ,4-m-]: this produces a

The perturbation strengths ri" are similarly revised:

2 2 K f)-j _mVN m] d r, j 7- E I-[D ] d r. (3.14)

These revisions are necessary due to the degeneracy of m = Im (# 0) modes in the axially symmetric structure: the perturbation expressions in Eqs. (3.13)-(3.14) are derived under the framework of non-degenerate perturbation theory, and so are not directly applicable to the degenerate case. In this doubly-degenerate case, non- degenerate perturbation theory produces a 2x2 matrix-form: the diagonal terms vanish and the off-diagonal terms [i.e. Eqs. (3.13)-(3.14)] remain.

3.4.3 Perturbation strength comparison

Based on Eqs. (3.12) and (3.2), the nonclassical corrections can be reformulated as [shown in Eq. (3.2)]

j) =sd(O) i S+d , (3.15)

where T E {Au-AlO , Au-air} for our Au-Au setup. The summation gives rise to four discrete corrections on the Au-air and Au-AlOx interfaces in our structure.

Figure 3-2a (same as Fig. 3-4a) shows the magnitude of the perturbation prefactors |, I(o E {-L,}) as a function of gap size for the film-coupled Au nanodisk. For our experimentally considered gap sizes, we find that ' are negligible compared to

,u-AO1 [about one and two order(s) of magnitude smaller for big (8nm)and small ( 1nm) gaps, respectively]. The sharp difference in perturbation strengths is

59 a consequence of the highly confined electric field (in the gap) being mostly in the surface normal direction.

For the perturbation strengths of the two di corrections, inAu--A1lO| is about one magnitude larger than |K u-air for smaller gaps (4nm), and remains larger but comparable for bigger gaps (6nm). On the other hand, due to screening from the dielectric cladding, the magnitude of the d parameters of a Au-dielectric interface is larger than that of a Au-vaccum(air) interface. We note that a recent work [1131 reports a similar conclusion for Ag in its density-function-theory calculation.

Based on the aforementioned complementary arguments for the relative magnitude of i and d" respectively, we approximate Eq. (3.15) by

(l) ~ D(O), Au-AlO Au-AlO.(1

Applying Eq. (3.16) to the experimental spectral shift and broadening, we are able to extract dAu-AlOx explicitly from the measurements, as shown in Fig. 3-4a,b.

In our analysis, the perturbing term is eigenvalue-dependent (i.e. PS depends on c) and therefore the entire perturbation becomes dispersive/nonlinear. Therefore, the perturbation center should be judiciously chosen. For Re dAu-AOx (which is quite dispersionless; see Fig. 3-4a), the classical frequency is treated as the perturbation center (i.e. a simple pole approximation). For Im diu-AlO, the nonclassical frequency is treated as the perturbation center to count for its strong frequency-dispersion (see Fig. 3-4b). In other words, the spectral shift (Re d contribution) and broadening (Im d contribution) are 'turned on' successively (not simultaneously) in order to reduce the second-order O[(J- CO(O)) 2 ] error in the perturbation analysis.

3.4.4 Structural dependence of the spectral shift

In our experiment we observe large nonclassical corrections in the film-coupled Au disk structures (Fig. 3-4). The observed nonclassical corrections are much larger than those in standalone nanospheres [25] or film-coupled nanospheres [45] for similar separations (or particle sizes). Such contrast highlights the structural dependence

60 aC 10,a<' OlD Sg

10-2

2 4 6 8 2 4 6 8 2 4 6 8 gap size g (nm) gap size g (nm) diameter D (nm)

Figure 3-2: Nonclassical perturbation strength comparison of various Au structures. a. Au disk (diameter, 70.4nm; height, 31nm) on Au substrate. b. Au sphere (diameter, 70nm) on Au substrate. c. Au sphere in vacuum; its di and d perturbation strength are the same in magnitude, but of opposite signs.

azom lers Rip mrrw cam b

C d

Ianw dw-m

Figure 3-3: Schematic of measurement setup and micrographs of fabricated nanostructures. a. Tabletop dark-field scattering setup. It has a tunable magnification, and can record the dark-field image and measure the scattering spectrum. b. Dark-field micrograph of a Au nanodisk array (scale bar, 2 Pm). c. SEM image of a single Au nanodisk (scale bar, 40 nm). d. Cross-sectional TEM image of an AlO, gap (scale bar, 10 nm). of the nonclassical corrections, as confirmed by our perturbation analysis in Fig. 3-

2. We find that a film-coupled nanosphere in the retarded regime and a standalone nanosphere in the nonretarded regime exhibit similar maximal perturbation strengths (Fig. 3-2b,c). In contrast, the perturbation strength of our experimental Au-Au nanodisk structure is about one order of magnitude larger (Fig. 3-2a).

3.5 Experimental setup

Experimentally, we establish a systematic approach to measure the d-parameter dispersion of a general two-material interface, and illustrate it using an Au-AlO. interface. While the dparameters of simple metals can be accurately computed within

61 jellium time-dependent density functional theory (TDDFT) [68, 140], d-parameters of noble metals, such as Au, require TDDFT beyond the jellium-approximation due to non-negligible screening from lower-lying orbitals [141, 69, 43]. We show that d- parameters can instead be measured experimentally: by developing and exploiting a quasi-normal-mode (QNM)-based [135] perturbation expression, we translate these mesoscopic quantities directly into observables-spectral shifting and broadening- and measure them in designed plasmonic systems that exhibit pronounced nonclassical corrections. Our experimental testbed enables a direct procedure for the extraction of d-parameters from standard dark-field measurements, in a manner analogous to ellipsometric measurements of the local bulk permittivity. Moreover, by investigating a complementary hybrid plasmonic setup, we discover and experimentally demonstrate design principles for structures that are classically robust-i.e. exhibit minimal nonclassical corrections-even under nanoscopic conditions.

The extensive interest in film-coupled nanoresonators [166, 11, 42, 24, 63, 23], which combine wavelength-scale resonators with a nanometric gap that approaches the intrinsic electronic length scale, is a particularly pertinent example that underscores the need for multiscale electrodynamic tools that incorporate nonclassical effects. We designed and fabricated film-coupled nanodisks (Figs. 3-le and 3-3b-d) of various materials, to verify our framework and directly measure the d-parameters: specifically, an optically-thick Au film (atop a Si substrate) is separated from litho- graphically defined Au, Si, or Al nanodisks (diameter, D) by a nanoscale AlO. spacer, deposited by atomic layer deposition, demarcating a film-nanodisk gap of thickness g. Such film-coupled nanodisks support localized gap plasmon resonances [22], which are (m, n) integer-indexable according to their field variations in the azimuthal and radial directions, respectively [253]. The fundamental mode (1, 1) is optically access- ible in the far field and exhibits highly confined electromagnetic fields within the gap, suggesting potentially large nonclassical corrections.

We built a table-top optical setup for the dark-field (DF) scattering measurement, as shown in Fig. 3-3a. Collimated broadband visible and infrared lamp (Oshio Halo- gen EKE and Kahoku OSL2BIR) illumination (path shown in yellow) is used as the

62 source for the setup. To create dark field illumination, a beam block with diameter

2cm blocks the center of the beam. An objective lens (Nikon, TU Plan ELWD, 100x magnification, 0.8 NA) focuses lamp light onto the mounted sample. Light scattered from the sample (path shown in gray) is collected with the same objective and goes through another magnification element (Optem Zoom 70XL, 7:1 zoom) with tunable focus and magnification. The system thus provides a total of 700x700 maximal magnification, at which the scattering signal of an ensemble of < 100 nanoparticles is collected. With the flip mirror, the scattered light is subsequently directed into either a CMOS camera (AmScope MU1000) for imaging or a spectrometer (Prin- ceton Instrument Action SP-2360-2300i). Measured counts were normalized to the reflection of a Ag mirror placed at the sample position to obtain the scattering cross- section.

Optical spectra were recorded at full zoom, capturing the scattered light from an ensemble of <100 nanodisks. Mutual coupling between nanodisks in the array is negligible, which is ensured by a lattice periodicity of 2 pm, corresponding to an in-plane filling factor of less than 1%. This allows an isolated-particle treatment. The size distribution of the nanodisks was characterized systematically to adjust for the impact of inhomogeneous broadening in the measured scattering spectrum from the ensemble (see Fig. 3-3c), We measured the AlO, gap size g using a variable-angle UV-VIS ellipsometer and confirmed the results through cross-sectional transmission electron microscopy (TEM; see Fig. 3-3d), finding good agreement with nominal ALD cycle expectations. The Au substrate's surface roughness was measured to be ~ 0.6nm (RMS) using atomic force microscopy (AFM) and was taken as the gap size uncertainty. Due to the conformal nature of the ALD [771, such roughness should have negligible influence on the scattering spectra, as we verified by numerical simulations

(Sec. 3.9.2). These detailed characterizations eliminate the main sources of geometric uncertainty in the mapping between calculated j(o) and measured C, facilitating an accurate evaluation of the nonclassical shift C - C(o).

63 3.6 Au-Au results: dispersion measurement of surface response functions

aD b o I C M4 d -nis-A + +Mesured Fit

Id0' 13

0 40 2 468 09 12 1. 1. 1.2 1.5 18 1. 1 2.0 gap Sieg (n-) Frequency (eV) Frequency (eV) Frequency (eV)

D=82.9 nm Da=70.4nm D 63.0 nrm 9 I IA 1-1 3 IeA -d ~ Nonasscw ZE Caeaical 0A + Measured S104

100- 100 60E

A 2 131.72.1 2 4 6 8 1.3 1.7 2.1 2 4 6 8 1.3 1.7 2.1 2 4 6 8 Frequency(eV) Gap size 9(im) Frequency (eV) Gap size 9 (nm) Frequency (eV) Gap size g (nm)

Figure 3-4: Systematic measurement of the complex surface-response function d-(w) of the Au-AO, interface. a. Nonclassical perturbation strengths, calculated from QNM-based perturbation theory, Eq. (3.3), in a film-coupled Au nanodisk (inset, D = 70.4nm); A-A1O- is dominant. b-c. Measured (markers) dispersion of Re dA = o(b) and Imd Au-AIO (c) and their linear fits (lines). Gap sizes are distinguished by color and diameters (D1 82.9rnm, D2 70.4nm, and D3 s 63.0nm) decrement rightward. d. Measured thickness-dependent refractive indices of bare AlO., films grown on Au. e-j. Scattering efficiency (e,g,i) across distinct diameters and gap sizes and the extracted complex (1,1) resonance eigenfrequencies (markers; f,h,j). While classical predictions (brown, dashed lines) deviate significantly from observations, our nonclassical calculations (black, solid lines;), employing the aforementioned linear dAu-AO- () fit, are in quantitative agreement across all diameters. Shadings indicate fit-derived confidence intervals for our calculations; 2o - 95% for Red'"~AlO and Re C(b,f,h,j) and la - 68% for Im dAuO and Im (c,f,hj).

Figure 3-4a shows the magnitudes of the nonclassical perturbation strengths in a film-coupled Au nanodisk: uO u-air7 'uAlOl , and u-r*. Evidently, Kj far exceeds x, for all gap sizes of interest, rendering the impact of dr negligible. Similarly,

64 the impact of duairis negligible relative to dAu-AlOx since 1 Au-AO ,» Au-air for small g; and more generally since d"Al~ > dAu-air due to screening from AlO [113]. Jointly, this justifies the approximation

/(1) ~ (O),Au-A] Au-Al d1" (3.17)

Inversion of Eq. (3.17) enables the direct experimental inference of du-AlOgiven measured C and calculated (O) (since, to first order, (1) _ C - CDo)). We note that I Re I Im I(by1-2 orders of magnitude) for the considered gap-sizes: consequently, Re dr contributes to spectral shifting and Imd 11 to broadening.

Ensemble scattering spectra of 18 Au nanodisk (height, 31nm) arrays were measured by dark-field scattering microscopy (Fig. 3-3 and Methods), spanning three diameters and six gaps sizes (Figs. 3-4e-j). Associated complex eigenfrequencies C were subsequently extracted by Voigt profile deconvolution (Sec. 3.9.3), using the measured particle size distribution. For the AlO2 spacer, we observed ellipsometrically-and include in our calculations-a thickness-dependent refractive index nAlO, (Fig. 3-4d and Sec. 3.9.1), a commonly-observed effect in ultrathin ALD-grown AlO, layers [90] and other ALD-grown materials [21].

Figures 3-4bc show the complex surface-response function dAu-AlOx(w), extracted via Eq. (3.17). Within the considered spectral range, Red u-AlO- (Fig. 3-4b) reveal a nearly dispersionless surface response of comparatively large magnitude, from -0.5nm to -0.4nm. In contrast, ImdAu-AlO (Fig. 3-4c) is strongly dispersive, increasing from <0.1nm in the near-infrared to ~~0.3nm in the visible. The thickness dependence of nAlO, imparts an attendant, implicit dependence to the inferred d Au-AlO- (w) (Sec. 3.9.5). As a result, the frequency-fits in Figs. 3-4bc convey a composite dependence along the (w, nAlO.)-space (Fig. 3-4d, circles) sampled by our data.

While the negative sign of RedAu-AlO--and the associated blueshift of Re(CO - CD(O)) (Figs. 3-4fhj, top panel)-agrees with earlier observations in Au [48, 471 and Ag [224, 205, 186] nanoparticles, the spectral shift is significantly larger. There are two reasons: first, the nonclassical perturbation strength ru-AIOz is much larger

65 than in e.g. standalone nanospheres or film-coupled nanospheres, due to strong field- confinement beneath the entire nanodisk footprint (Sec. 3.4); and second, screening from the AlO2 cladding expels induced charge into Au, thereby enhancing d Au-AlOx relative to the unscreened interface, i.e. relative to du-air 113.

Nonclassical broadening due to surface-enhanced Landau damping, i.e. Im(o - o( 0 )), is similarly enhanced for the same reasons (Figs. 3-4fhj, lower panels). Clas- sically, the linewidth reduces near-monotonically with gap-size, primarily due to increased light confinement (reduced radiative coupling). Instead, we observed-and predict, nonclassically-a near-constant broadening that is reduced slightly for very small gaps. The near-constant broadening results from an interplay [Eq. (3.17)] among the strong (classical) gap-dependence of Re a, the increase of nonclassical perturbation strength (Fig. 3-4a) at smaller g, and the decrease of Im dAu-AlOx towards the infrared (Fig. 3-4c). Strikingly, the smallest gap does not produce the strongest nonclassical broadening (i.e. ImcY(), in contrast to the natural expectation of monotonically increasing ImCoM with decreasing g. Instead, Imca() is minimal there-a consequence of the near-vanishing magnitude of the strongly dispersive ImdAu-AlO

(Fig. 3-4c). This behavior demonstrates the apparent breakdown of the empirical understanding of nonclassical broadening in nanostructures, known as Kreibig damping 1131], which holds that ImcD ( 1/g.

3.7 Si-Au results: robustness to detrimental nonclassical corrections

The observation of large nonclassical corrections in our coupled Au-Au setup frames a natural question: can nonclassical effects-which are often detrimental-be efficiently mitigated even in nanoscopic settings? To answer by example, we consider a hybrid dielectric-metal design, replacing Au nanodisks with Si. Such hybrid configurations have been predicted to yield higher radiative efficiency with comparable overall plasmonic response [2531 and have two key advantages for mitigating nonclassical effects:

66 ow factor WI.~U ___ _D =104.4nm a 2.2 D 3 W as. b C 2-3. 2 I+ 2.1 Is 1.9

2.2 IS Gap size g(nm) Frequency (eV) Gap size g(nm)

d g=1.1nm

.5 1 1A

1.8 2 22 1.8 2 2.2 1.8 2 22 Frequency (eV)

Figure 3-5: Robustness to nonclassical corrections. a.The nonclassical perturbation strength is one order of magnitude smaller in the hybrid Si-Au system than in its Au-Au counterpart. Si and Aunanodisk diameters are chosen to ensure spectral alignment of the (1, 1) resonance at every gap size (spanning D E[80,160unmand D E[15, 40]un,respectively). b-d. Observation of robust optical response in Si-Au setup with the detrimental quantum corrections mitigated. Thenonclassical calculation for the Si-Au setup assumes di~I== -0.5+0.3inm, aconstant extrapolation tohigherfrequenciesfromFig.3-4b-c. Ind,measuredand calculatedspectraare normalized separately. Calculated spectra incorporate inhomogeneous broadening (~6%) duetodisk-sizeinhomogeneity(3.9.3)

67 -g=1.1nm- -- g=1.8nrn--- g 27nm-

114

z 104

C cC S1.4

1.8 2 2.2 1.8 2 2.2 1.8 2 2.2 Frequency (eV)

Figure 3-6: Additional measurement showing robustness to nonclassical corrections from Si nanodisks with gap sizes of 1.1nm, 1.8 nm, and 2.7 nm. In Fig. 3-5d, we show the measured spectra and optical response robustness for the thinnest gap 1.1nm. Here in Fig. 3-6 we include additional measurement and comparisons with nonclassical and classical simulations to further demonstrate the robustness. Again, we observe minor nonclassical conrrections.

first, undoped Si is effectively a purely classical material, i.e. ds0Alo ~ 0, under the jellium approximation as it lacks free electrons; and second, high-index nanoresonators reduce field intensity at the metal interface while maintaining confinement in the gap region. This hybridization can be exploited to reduce the nonclassical perturbation strengthK Au-A10 by an, order of magnitude relative to that in the Au-Au design as shown in Fig. 3-5a. Our measurements confirm this prediction: for D ~ 104.4nm Si nanodisks, we observe a high-quality scattering spectra with a symmetric, single- resonance feature for all gap sizes (Fig. 3-5b). The measured resonance frequencies (Fig. 3-5c) show only minor deviations from classical predictions, in both real and imaginary parts. While the inclusion of nonclassical effects improves the experimental agreement, the overall shift remains small, comparable to the uncertainties owing to the intrinsic oxide thickness beneath the Si nanodisk (Sec. 3.9.4). Considering a range of nanodisk diameters (Fig. 3-5d), we reach an identical conclusion, even for the smallest considered gap (~1.1nm): classical scattering spectra agree well with

68 measurements, and nonclassical corrections are minor relative to those in the Au-Au system. We found similar robustness across several additional gap sizes and diameters (Fig. 3-6).

3.8 Al-Au results: partial cancellation of nonclas-

sical corrections between spill-in/out materials

D =83.5 nr O=62.7

a "" b * f

2 2.2

;2 6 k

t~. C Og

6Aen d5k 30

30 d1

.. 100 -...... - 100 1.7 1.2 2 4 6 8 1.7 2.2 2 4 6 8 1.7 2.2 2 4 6 8 Frequency (eV) gap sizeg9(-m) Frequency(.W gap size 9 (m) Frequency (eV) gap size g(nm)

Figure 3-7: Cancellation of nonclassical corrections in Au-film-coupled Al nanodisks (structure and disk-bottom oxide modeling shown in b-c insets). Com- plex resonant frequencies obtained from per-volume scattering cross-sections (a, e and i) are in agreement with nonclassical calculations (solid black lines in c-d, g-h, and k-1), indicating the cancellation between the nonclassical corrections from the spill-in (Au) add spill-out (Al) materials (b, f, and k).

Here in Fig. 3-7a-1, we demonstrate the cancellation of nonclassical corrections from different materials of spill-in or spill-out induced charge density (i.e. opposite signs of Re d_±). The spill-in material is chosen as Au, whose surface response function is taken from our measurement (Fig. 3-4) .The spill-out material is chosen as Al and its surface response function is obtained from TDDFT calculation of lossless homogeneous electron gas of Wigner-Seitz radius r, = 2. The scattering spectra of 18 Al nanodisk arrays were collected (Fig 3-7a,e,i), spanning three diameters and six gaps sizes. Following the same data analysis approach

69 (see Sec. 3.9.3), we obtain the the measured complex resonant frequencies. As shown in Fig. 3-5b,f,k we theoretically predict that the Au substrate and the Al nanodisk introduce blueshift and redshift nonclassical corrections respectively, and partially cancel with each other. The total nonclassical shift is still blueshift due to the more pronounced correction from Au, and agrees with our measurement (Fig. 3-5c,g,k).

Despite the cancellation in Re C, passivity requires that the spectral broadening is still cumulative for ImC (Fig. 3-5d,h,l). In these theoretical calculations, we include an exponential bottom-disk oxide model with a single free parameter r for both classical and nonclassical considerations (see Fig. 3-7b,c insets and Sec. 3.9.4). The same model is applied to the results of three discrete Al disk diameters, which yields similar agreement between experiment and theory. We also note that the prediction of nonclassical correction cancellation is unaffected by the choice of bottom-oxide layer model.

3.9 Materials and methods

3.9.1 Ellipsometry

The bulk permittivities of the evaporated Au, Al, and Si used in the experiments were measured using a variable-angle spectroscopic ellipsometer (J.A. Woolam Co., Inc., benchtop XLS-100 UV-VIS continuous spectroscopic ellipsometer and WVASE32 software).

The bare Au sample for ellipsometry measurement was prepared by evaporating 10 nm Ti followed by 60 nm Au (nominal thicknesses) on a 4-inch Si wafer. The wafer was cleaved into smaller pieces as the base substrates for other samples. We then evaporated 40 nm Si and 40 nm Al on the Au-coated substrates for measuring the permittivities of Si and Al, respectively.

Spectroscopic scans were performed at the 70 relative to the surface normal of the samples. We verified that the measured permittivities were consistent across various incident angles (60-80°). Ellipsometry data were analyzed using multi-layer

70 models and the surface roughness (measured by AFM) was taken into account via the Bruggeman effective medium theory [74]. The measured permittivities for the three materials axe shown in Fig. 3-8 and are used for our subsequent measurements and theoretical calculations. Specifically for the permittivities of Au, our results agree well with the Johnson-Christy data [1141 in the visible regime. Therefore, we adopted the Johnson-Christy data for energies below 1.24eV (equivalently, for wavelengths above

1000 nm, corresponding to the cut-off wavelength of our ellipsometer). For Si and Al evaporated films, an extra native oxide layer was taken into account to measure their native oxide thickness in ambient conditions. We found the native oxide thickness of the Si film ~1.5nm as of measurement. The native oxide thickness of the Al film is ~ 4 nm, consistent with previous studies (e.g. [361).

a o ' . b -20 C

Measured -40

-6 -50 o 13

-60 K Johnson-Chrisy 0 Au Al Si -100

10 40 10

-5. 20 5

01 10

0' 0 1 2 3 1.5 2 2.5 3 2 2.2 2.4 2.6 Energy(eV) Energy(eV) Energy (eV)

Figure 3-8: Ellipsometric measurement of bulk permittivities. a. Au, b. Al, and c. Si.

Ellipsometry was also used to measure the thicknesses and permittivities of the ALD-deposited AlO, gap on the Au substrate (permittivities experimentally determined; see Fig. 3-8a). The ellipsometric data were collected with the same approach as described for bulk permittivities. The thicknesses g and the AlO, refractive indices

(modelled by the Cauchy dispersion: n(A) = A + B/A2 , where A is in pm) are fitted to achieve minimal root-mean-square errors.

The measured gap size as a function of deposition cycle is shown in Fig. 3-9a.

71 a 100 b 1.6 C 0.1

1.5 C 0.05

50 -1.4 0

1.3 -0.05 Sg = / 097N + 6.49 (A) 3 0 1.2 ' ' '-0.1 20 40 60 80 2 4 6 8 2 4 6 8 cycle # gap size (nm) gap size (nm)

Figure 3-9: Ellipsometric measurement of AlO, thickness and refractive indices. a. Measured thicknesses as a function of deposition cycles (dots) and the linear fitting (line and inset equation) with 95% confidence interval (shading). b,c. Measured Cauchy coefficients A and B (dots) and the exponential fitting (y= ae-± +c) with 95% confidence interval.

The linear fitting yields a deposition rate of 0.97 A, consistent with the ~ 1 A/cycle rate reported in literature [77, 45, 90]. The 6.49A offset may be due to the extra preconditioning purging cycles in the ALD process.

The Cauchy model parameters A and B, as well as their exponential fittings are shown in Figs. 3-9b and 3-9c. We find that the dispersion-less part (coefficient A) of the refractive index decreases for smaller gap sizes and approaches the bulk index for larger gaps. This gap-dependent refractive index of ultrathin AlOx ALD layers has been observed elsewhere [90] and may be explained by the substrate lattice mismatch, interfacial contaminants, or the saturation of the phase transition layer as deposition cycles increase. Similar thickness-dependent permittivity was also reported in other ALD-grown thin-filrps [21]. The dispersive part (coefficient B) of the refractive index slightly increases for smaller gaps (Fig. 3-9c). The AlOx refractive indices for different gap sizes are shown in Fig. 3-4d.

3.9.2 Impact of surface roughness on optical response

In this section we numerically show that the surface roughness introduces negligible error to the resonant eigenvalue on the complex plane, since the ALD process [771 is highly conformal.

Fig. 3-10 illustrates the cross-sectional view (p, z) of the Au nanoresonator on the

72 Au substrate without (Fig. 3-10a) and with (Fig. 3-10b) surface roughness taken into account, respectively. The roughness of the evaporated Au substrate is modeled by spatial sinusoidal variations. Since the fabrication process is conformal, the variation is inherited by the subsequent ALD gap layer and the nanoresonator. The periodicity

, and the peak-to-peak value 6pp = 2v-/Zrms of the spatial variation are taken from the experimental (AFM) measurement of the grain size ~ 50 nm and the surface roughness

rms ~~ 0.6 nm.

Fig. 3-10c shows the negligible influence of measured surface roughness on the eigenfrequency of the nanoresonator. The resonant frequency drift is <1.2% for all possible random configurations (parametrized by the variable 0) of our structure due to surface roughness. We also verified that this conclusion also holds for Al and Si nanoresonators.

a b C

Au V Au f.05 T~ 6P Au 095

-1 0 -1 O/r

Figure 3-10: Analysis of the surface roughness dependence on the optical responses of ALD-fabricated gap plasmonic nanbresonators. a,b. Schematic illustration of the roughness-free (a) and conformal surface roughness model (b). The roughness of the evaporated Au substrate is modeled by spatial sinusoidal variations. The gap size, sinuosodial periodicity and peak-to-peak amplitude are denoted by g, ., and opp, respectively. The position of the center of the nanoresonator with regard to the spatial variation introduces another degree of freedom, the initial 'phase' factor 0. c. Roughness-induced shifts of the resonant frequency and width are negligible given the smoothness level of our fabrication process. The red box denotes the minor resonant frequency shift range (<1.2%) for all possible random structural variations (i.e., 0 E [-7r, 7]).

73 a Measurement Gaussian Lorentzian Voigtprofile distribution resonance

* V 21mw

ia)n LU renquency= iaic esize requency

g=1.1 nm g =8.6 nm g=1.1 nm b D=63.0 nm C D=82.9nm d D = 24 nn 0.15 Au 1 .5 verall SAu Au Si Measured Voigt 0.1

back - 0.05 , 0 .5 rounec Lorentzian E

Voigi Gaussian ' 0 1.2 1.4 1.6 1.8 1.2 1 4 1.6 1.8 2 1.8 1.9 2.0 2.1 2.2 2.3 Frequency (eV) Frequency (eV) Frequency (eV)

Figure 3-11: Data analysis. a. Schematic for the extraction of the complex resonant frequency from the measurement. The convolution of the Gaussian size distribution and the single-particle Lorentzian resonance spectra yields the measured Voigt profile. b,c. Fitting the measured scattering of Au nanodisks with a Voigt profile and noise background [see Eq. (3.19)]. Two extreme cases are shown-smallest disk diameter on thinnest gap (b) and largest disk on thickest gap (c) d. Resonance broadening (blue) on the simulated nonclassical single-particle scattering spectrum (red) due to inhomogeneity (green) of the nanodisk array.

74 3.9.3 Data analysis

Complex resonant frequency extraction from experiment

Our measurement system captures the scattering from < 100 nanoparticles. There- fore, to obtain the complex resonant frequency of a single nanoresonator, we need to take into account another broadening mechanism, the inhomogeneity of the particle sizes. As illustrated in Fig. 3-11a, the measured spectra are Voigt profiles [18], i.e. the convolution of the co-centered Lorentzian profile of the resonance and the Gaussian distribution of the particle size

V(w; a,') j G(w'; o-,)L(w - w'; 7) dw', (3.18) where V, G, and L are Voigt, Gaussian, and Lorentzian profiles, respectively. cx is the Gaussian standard deviation and y = ImcD is the Lorentzian half-linewidth.

We extract the complex resonant frequency C in the following manner. The measured spectra S(w) is treated as the summation of the Voigt profile V(w; o,,-) and the noise background N(w):

S(w) = V(w; o., y) + N(w). (3.19)

Here the noise background N(w) is modeled by polynomials with order ; 2. The spectral Gaussian standard deviation a, is measured experimentally. We first obtain the particle diameter standard deviation UD with particle size statistics. Then, oa,

d D, Where the slope - is obtained from the resonant frequencies ReC 'of the particles atop the gap of same thickness but with different diameters.

The real part of the resonant frequency ReCo can be obtained straightforwardly- the center frequency of the Voigt profile, since the Lorentzian and Gaussian profiles are co-centered. The imaginary part Im is obtained from Eqs. (3.18) and (3.19) via fitting to minimize root-mean-square error. Figures 3-11b and 3-11c shows two examples of the fitting process.

75 Incorporation of inhomogeneous broadening in theoretical calculations for Si nanodisks

In Fig. 3-5d, in order to directly compare with measured spectra, the theoretical calculations (both nonclassical and classical) of the scattering cross-sections of the Si disks also incorporates the broadening due to disk size inhomogeneity. Such incorporation can be understood as the inverse process of that described in Sec. 3.9.3. Figure 3-11d shows an example of the consequence of broadening due to nanodisk inhomogeneity, which contributes to an extra 6% broadening on the linewidth of the Si nanodisks.

3.9.4 Oxide layers of Si and Al nanodisks

native oxide thickness

Si/Al SiO/AIO, native oxide

bottom oxide thickness

ALD AO: Ayu

Figure 3-12: Native oxide bottom layer of Al and Si nanodisks.

Unlike Au, both Al and Si oxidize in ambient conditions, adding extra uncertainty to the structural parameters of the resonators (see Fig. 3-12). For our Al and Si nanodisks, the top and side surfaces are fully exposed to air; therefore, the native oxide thickness 6 (-r E {SiO2, AlO,}) should follow that in our ellipsometry measurement

(see 3.9.1). On the other hand, the bottom surface of the nanodisk does not expose to air (oxygen) directly; however, it may still form an oxide layer (thickness denoted by

76 Ag,, T E {SiOx, AlOx}) by reacting with diffused oxygen since the ALD AlOx layers are only few nanometer thick. Practically, we did not have a good in-situ method to directly measure the thickness of the bottom oxide.

For Si, 6 siox a 1.5nm as of measurement (see 3.9.1). We account for the uncertainty of the disk bottom oxide thickness by Agsiox E (0, 5sioxj, since the bottom oxide should be no thicker than the native oxide (which is fully exposed to oxygen). Such structural uncertainty is taken into account by the uncertainty of the theoretical calculations of the resonant eigenfrequencies in both classical and nonclassical considerations (see Fig. 3-5c).

For Al, the measured native oxide6 AO, 4nm (see 3.9.1). For such wide thickness, the applicability of the uncertainty analysis (as in the Si case) becomes limited. As an alternative, we adopt single-parameter fitting for the bottom native oxide layer with the model A9AlOx =Aloxe K9/6A1o=, where g is the thickness of the ALD AlO2 spacer. In this model, r is the only free parameter. We choose this model because of its reasonable asymptotics-lim+ A9AlOx = 0 and limg+o A9AlOx = 6 AIOx. The first asymptotic means that sufficiently thick ALD AlO can already passivate the bottom surface of the Al disk, and the extra bottom oxide should vanish. The second asymptotic means that without the passivation from the ALD AlOx, the bottom surface, like other surfaces of the disk, should develop oxide layer of similar thickness.

We adopt K = 0.6 in the model to compare with our experimental results for Al (see Sec. 3.8).

3.9.5 Index dependence of measured surface response functions

The magnitude of the measured surface response functions depends on frequency and the materials that compose the interface [113]. In our measurements, due to the thickness-dependent refractive index (nAIo,) of the AlO2 spacer (see Sec. 3.9.1 and Fig. 3-4d), the measured dAu-AlOx-parameter (Fig. 3-4b,c) inherits an effective nAIox-dependence in addition to its frequency dependence.

77 A 9,-eVm b a 3.1 A8,-Sm A #W=&4D a U a e dA-M A A N 0 *O-7t4m V D3-63iin A U V A A v 9,-ow sin~y

,jd1(A) -2 _0 IMd (A) 3.1 1.55A .il:1V 1.55. AA - P A s .6.... 64m 1i~

A 43R 431

2.3-

A AI A UA I7e= A .... 1 ... A *- raJ

121 - - i.. I 13 0.8 1.9 1.1 2 Frequency Q"(eV) Frequency W,(eV)

Figure 3-13: Frequency and index dependence of measured dA ". Fei- belman d-parameters generally depend on both frequency and interface composition. Since the cladding response (nAO.) varies with gap-size (Sec. 3.9.1), the measured d u-AlOx inherits this dependence, leading to an approximate overall (CR,nAIO.)-dependence (here, Da = ReCD). Our measurements of a. Red^"-Al1- and b. Imdu-AlO reveal the surface-response dispersion along the thin (a, nAO.)-band sampled by our 18 (g,D)-combinations. For Re duAlO- the perturbation centers are the classical eigenfrequencies, i.e. (a), nAO ( 0)));for Im duAlOtheperturba- tion centers are chosen as the measured eigenfrequencies, i.e. (V,nAO (CUR)), cf- the largeness of the nonclassical l-correction (see Sec. 3.2).

78 In practice, we only sample a narrow slice of the entire (C, nA1O,)-space, given the finite selection of (g, D)-combinations considered (and the associated resonance dispersion with g and D). The resolution limitation of fabrication-joint constraints from electron beam lithography and lift-off processes-restricts us from decreasing the disk diameters. On the other hand, the long-wavelength cut-off (~1500 nm) of the spectrometer restricts us from increasing the disk diameters. The dispersion of dAu-AlO along this slice is shown in Fig. 3-13, illustrating the various "projection"-perspectives one may consider. As noted, the (C, A1O)-space sampled by our measurements resembles a relatively thin band: the narrowness of this band precludes us from simultaneously disentangling both dependencies separately. Therefore, the linear frequency-fits in Figs. 3-4b,c ultimately reflect a composite dependence along the sampled (C, nAlo,) space.

3.10 Discussion

The mesoscopic framework presented here introduces a general approach for incor- porating nonclassical effects in electromagnetic problems by a simple generalization of the associated BCs. Our experiments show how to directly measure the nonclassical surface-response functions-the Feibelman d-parameters-in general and technologically relevant plasmonic platforms. Our findings establish the Feibelman d-parameters as first-class response functions of general utility. This calls for the compilation of databases of d-parameters at interfaces of photonic and plasmonic prominence, analogous and complimentary to the existing databases of local bulk permittivities.

79 80 Chapter 4

Fundamental Limits to Spontaneous Free Electron Radiation and Energy Loss

4.1 Introduction

In the previous chapter, the background fields in the dark-field scattering measurement are plane waves from the far field. In this chapter, we will focus on the scattering problem of a near-field excitation, which are free electrons. We will the electron energy loss and radiation in the spontaneous regime, where the velocity of electrons is assumed constant.

Spontaneous free electron radiation such as Cerenkov [411, Smith-Purcell [213], and transition radiation [80, 85] can be greatly affected by structured optical envir-

onments, as has been demonstrated in a variety of polaritonic [146, 117], photonic- crystal [1501, and metamaterial [4, 79,1451 systems. However, the amount of radiation that can ultimately be extracted from free electrons near an arbitrary material structure has remained elusive.

The Smith-Purcell effect epitomizes the potential of free-electron radiation. Con- sider an electron at velocity # = v/c traversing a structure with periodicity a; it

81 generates far-field radiation at wavelength A and polar angle 0, dictated by [213]

a 1 A-= - Cos 0 (4.1) where m is the integer diffraction order. The absence of a minimum velocity in Eq. (4.1) offers prospects for threshold-free and spectrally tunable light sources, spanning from microwave and Terahertz [231, 129, 56], across visible [132, 249, 116], and towards X-ray [1651 frequencies. In stark contrast to the simple momentum-conservation determination of wavelength and angle, there is no unified yet simple analytical equation for the radiation intensity. Previous theories only offer explicit solutions either under strong assumptions (e.g., assuming perfect conductors or employing effective medium descriptions) or for simple, symmetric geometries [232, 92, 2091. Con- sequently, heavily numerical strategies are often an unavoidable resort [177, 53]. The inherent complexity of the interactions between electrons and photonic media have prevented a more general understanding of how pronounced Smith-Purcell radiation and its siblings can ultimately be for arbitrary structures, and consequently, how to design the maximum enhancement for free-electron light-emitting devices.

4.2 Theoretical framework

4.2.1 Three-dimensional general upper limits

We begin our analysis by considering an electron (charge -e) of constant velocity vk traversing a generic scatterer (plasmonic or dielectric, finite or extended) of arbitrary size and material composition, as in Fig. 4-1(a). The free current density of the electron, J(r,t) =- kev6(y)6(z)(x - vt), generates a frequency-dependent (e-W convention) incident field [54]

Einc(r,w) = KPic [ kinpKo( p p) - pkvK1(rpp)], (4.2) 27rwe

82 b)102 (C) 10 (a = /C election fastele5tron (a) ) favored

10-1 E X10 2

101 avored .2 I visible (0 radiation absorption 10' + rMW EON UV AIS/ IR *Aenergylosb 10' 102 0 1010 102 10- 100 101 Wavelength (nm) kd

(e) (d) 0 10.0 Enegyioss

i 101 0'1 d 10' P=0.2 Au M10n 10-1 4~~~ di~ mmtno

10 4 a 12 14 20 4 a 12 16 20 Separationtdnm) Separationd(nm)

Figure 4-1: Theoretical framework and predictions. (a) The interaction between a free electron and an obstacle defined by a susceptibility tensor x(rw) within a volume V, located at a distance d, generates electron energy loss into radiation and absorption. (b) |X12 /ImX constrains the maximum material response to the optical excitations of free electrons over different spectral ranges for representative materials (from Ref. [175]) . At the X-ray and EUV regime, Si is optimal near the technologically relevant 13.5 nm (dashed circle). Contrary to the image charge intuition for the optical excitations of electrons, low-loss dielectrics (such as Si in the visible and infrared regimes) can be superior to metEls. (c) Shape-independent upper limit showing superiority of slow or fast electrons at small or large separations; the material X only affects the overall scaling. (d-e) Numerical simulations (circles) compared to analytical upper limits [lines; Eq. (4.10a) for (d) and Eq. (4.18) for (e), respectively] for the radiation (blue) and energy loss (red) of electrons (d) penetrating the center of an annular bowtie antenna and (e) passing above a grating.

83 written in cylindrical coordinates (x, p,,); here, K is the modified Bessel function of the second kind131, kv = w/v, and'9 = 2- -= k/7y (k = w/c, free-space wavevector; y = 1/1 - #2, Lorentz factor). Hence, the photon emission and energy loss of free electrons can be treated as a scattering problem: the electromagnetic fields

Finc =(Einc, ZoHinc) (for free-space impedance Zo) are incident upon a photonic medium with material susceptibility X (a 6 x 6 tensor for a general medium), causing both absorption and far-field scattering-i.e., photon emission-that together com- prise electron energy loss [Fig. 4-1(a)].

As recently shown in Refs. [162, 253, 161], for a generic electromagnetic scattering problem, passivity-the condition that polarization currents do no net work- constrains the maximum optical response from a given incident field. Consider three power quantities derived from Finc and the total field F within the scatterer volume V: the total power lost by the electron, PicS = -(1/2) Re fv J*.E dV = t (eow/2) Im fy FucXF dV, the power absorbed by the medium, Pabs = (Cow/2) Im f F XF dV, and their difference, the power radiated to the far field, Prad = Ploss - Pabs. Treating F as an independent variable, the total loss PIO.s is a linear function of F, whereas the fraction that is dissipated is a quadratic function of F. Passivity requires nonnegative radiated power, represented by the inequality Pabs < Pioss, which in this framework is therefore a convex constraint on any response function. Constrained maximization of the energy-loss and photon-emission power quantities, Poss and Pra, directly yields the limits

P(w) < FF cXt(Imx)-xFinc dV, (4.3) where r E {rad, loss} and (- accounts for a variable radiative efficiency r (defined as the ratio of radiative to total energy loss): (1oss = 1 and rad = 7(1 - 77) < 1/4. Hereafter, we consider isotropic and nonmagnetic materials (and thus a scalar susceptibility x), but the generalizations to anisotropic and/or magnetic media are straightforward.

84 Combining Eqs. (4.2) and (4.3) yields

2K T <(w) e 2 22 I x[K (PP)+ k 2(pp)] dV. (4.4) 8how Ir v ImX " where a is the fine-structure constant. Equation (4.9) imposes, without solving Max- well's equations, a maximum rate of photon generation based on the electron velocity

# (through kv and ,), the material composition X(r), and the volume V. We assume the structure is made of a single material

IF,(W) :! hcW2 2 (x K pp+ Pkv K (rpp)] dV. (4.5) 8heow2 2ImxyV

We now simplify the integral

Ac= j[(K g(,yp)+ ) - (pp)] dV. (4.6)

For an arbitrarily-shaped structure, whether isolated or extended, one can always find a circular concentric hollow cylinder (height L, opening azimuthal angle V E [0, 27r], minor radius being the electron structure separation, major radius can be finite or infinite) that encloses it. Therefore, we can evaluate the integral in the cylindrical coordinate

AC< LO p [K (rpp)+ k K1 (p)] dp

L x P (K(x)+k K (x)] dx,

{ P{(Ki(xo)- K(xo)] + k [Ko(xo)K2(xo) - Ki(xo)]},

{{kKo(xo) [K (x) - 2 2 2 Ko(xo)]- k (K (xo)- K2(xo)]} = xok Ko(x)Ki(x) -- x2k2[K -(xo)- ],

85 where xo = r~d. In the derivation above, we use the following relations [3]

xK (x)dx= 2 [K2(x)- Kn_1(x)Kn+1 (x)] , (4.8a)

K_1(x) = K1(x), (4.8b)

K2 (x) - Ko(x) = 2K1 (x)/x, (4.8c) K1(x) > Ko(x). (4.8d)

a general limit on the loss or emission spectral probabilities r,(w)=P,(w)/hw:

T(w) a r x [ K2 k K (rp)] dV, (4.9) 2 Vw Im X

Eq. (4.7) corresponds to a general closed-form shape-independent limit that highlights the pivotal role of the impact parametercd:

2 'T (w) ( X1 LO (/pd)Ko(rpd)K1(,pd)], (4.1Oa) 27c Im X #2

1 In (1/,d) for icd « 1, oc (4.10b) 2ire- 2xd/ 2 for ,rd> 1.

The limits of Eqs. (4.9,4.10) are completely general; they set the maximum photon emission and energy loss of an electron beam coupled to an arbitrary photonic environment in either the nonretarded or retarded regimes, given only the beam properties and material composition. The key factors that determine maximal radiation are identified: intrinsic material loss (represented by ImX), electron velocity #, and impact parameter Kd. The metric |x12/ImX reflects the influence of the material choice, which depends sensitively on the radiation wavelength [Fig. 4-1(b)]. The electron velocity # also appears implicitly in the impact parameter s~d = kd/3y, showing that the relevant length scale is set by the relativistic velocity of the electron. The impact parameter ipd reflects the influence of the Lorentz contraction d/y; a well-known feature of both electron radiation and acceleration [72, 54, 165].

86 The tightness of the limit [Eqs. (4.9,4.10)] is demonstrated by comparison with full-wave numerical calculations (see Methods.) in Figs. 4-1(d-e). Two scenarios are considered: in Fig. 4-1(d), an electron traverses the center of an annular Au bowtie antenna and undergoes antenna-enabled transition radiation (q ~ 0.07%), while, in Fig. 4-1(e), an electron traverses a Au grating, undergoing Smith-Purcell radiation (q ~ 0.9%). In both cases, the numerical results closely trail the upper limit at the considered wavelengths, showing that the limits can be approached or even attained with modest effort.

4.2.2 Smith-Purcell radiation upper limit in three dimensions

for rectangular gratings

Next, we specialize in the canonical Smith-Purcell setup illustrated in Fig. 4-1(e) inset. This setup warrants a particularly close study, given its prominent historical and practical role in free-electron radiation. Aside from the shape-independent limit

[Eq. (4.10)], we can find a sharper limit (in per unit length for periodic structure) specifically for Smith-Purcell radiation using rectangular gratings of filling factor A.

We choose coordinates such that (vt, yo, zo) depicts the trajectory of the charged particle. In the cylindrical coordinate (p, 4, x), the current density can be rewritten as -ev J(r, t) = 6(x - vt)(p)x. (4.11) 27rp

Fourier transform on Eq. (4.11) yields the current density in the frequency domain

J(r, w) = eikvxJ(p)k, (4.12) 27rp

87 whose external electromagnetic field is given by [128]

Einc(r,w)= -(k 2 k+ikV)Ho(1(ipp)eik v, 4wco e [(k2 - k )H(o)(iipp) +ikv dpH l)(ir,,p)]eikx, (4.13) 4wE0

[i2Ko(Kpp)k - rpkK1(Ppp)p eikvx. = 21rwce where Ho is the Hankel function of the first kind with zero order. Here we utilize the relation Ko(z) = "Hol)(iz), where z is a real argument. Insert Eq. (4.13) into Eq. (4.3) yields the general three-dimensional limit shown in Eq. (4.9).

Next we consider Smith-Purcell radiation from rectangular gratings in three dimensions. The volume integral of the evanescent field is given by

222 r/2 o0 p [o4K2(rP)+o222(p E(r)| dV= 2 dx d$ p dp [kK (pp)] v w272 J-,r/2 Jd/costb (4.14) Closed-form integral can be obtained by using the relation

2 p dpK2 (,p) (G, 2d2 sec V0 i12, (4.15) fd/ Cos V)p and

00,2 . (4.16) osV)pdpKi = 4p 2 G1,0 d2 se

Here G is the Meijer G-function [159, 3] defined as a line integral in the complex plane

ai, ... , a _ i j f1 I(b _- s) H ,(1- aj + s) zds bi, .... , bq 2wi L=m+iP(1- + s) Hj_,+ F(aj - s) (4.17) where F is the gamma function.

Plug Eq. (4.15) and Eq. (4.16) into Eq. (4.9) yields

d a& IX|2 Ag (# kd), (4.18a) dx - 27rc Im X

88 where

g(, kd) = g (, kd) +g,(#, kd), (4.18b)

x(#3, kd) = VJ7:;2 :3G d 2 sec2 V, 132 , (4.18c)

2 g,(,3kd) = dObG"3(12G/fr/2 sec @| 30 22 . (4.18d) 4 f-7/2 32 3,0 2y2 00,

Here, k, = w/c#, / , = w/c#, and a = e2/47reohc.

4.2.3 Limit asymptotics

For the asymptotic behavior of the limit, here we consider four scenarios: electrons in the near field (kd- 0), electrons in the far field (kd- oo), extreme nonrealistic electrons (v -+ 0), and extreme relativistic electrons (v- c). In this section we only consider the three-dimensional problem [Eq. (4.9)].

First, we consider near field kd -+ 0. We also assume the electron speed is intermediate so neither 3 -+ 0 (extremely slow) nor - -+ oc (extremely fast), which we will discuss later. In the expression of the general limit [Eq. (4.9)], there are two terms in the integrand where the first term (containing KO) is the contribution from the longitudinal polarization E, and the second term (containing K 1) is the contribution fron the transverse polarization Ep. The hyperbolic Bessel functions K, in these two terms has the same argument np = kp3y, which also approaches zero for p > d. Both Ko(Kp) and K1 (p) diverge when 'pp - 0 but at different divergence rates [3]:

lim Ko(rpp) - - ln(xpp/2) - yo, (4.19a) p-* 0

lim K1 (,p) 1 , (4.19b) p-40 K pp where yo is the Euler-Mascheroni constant. Therefore, Ki(rpp) » Ko(Kpp) when pp -> 0 and Ep has the major contribution to the limit.

89 Second, we consider electron beams in the far field kd -+ oo:

1 /2 lim Ko(,p)~ e-KPP 1 - + O( p2), (4.20a) P) rx + 31 lim K1 (,pp) ~ 2 e-PP1i +8-p + O(r2 p2) . (4.20b) 8rp 1

Therefore, both Er-limit and E,-limit decay exponentially at the same rate and E,-limit remains be higher.

Third, we consider asymptotic behavior of the limit when the electrons are extremely nonrelativistic(#- 0). In this limit, we have limp_>o , = k, -+ oo. Thus in Eq. (4.9)

lim 2 Ko(rp) ~ 2e-"a 1 - 1 + 2( p2)= 0 (4.21a) no-+0oo 2r, p 08 _r.p

lim 4rpkK1(rp) ~ 2e-"PP 1 + 3+ O(K 2p2) =0, (4.21b) p- oo 2pp I 8pp which is consistent with the fact that static charges do not generate radiation. Our computational verification is shown in Fig. 4-2(b) and (c) where both the limit and numerical results approach zero as # -+ 0 for either small or large separations (whether slow or fast electrons are preferred) between the electron beams and the structure.

Last, we consider the limit behavior when the electrons are extremely relativistic, where lim 1 ,= W2v2 _ 2/c2 = 0:

lim r Ko(npp) ~ ,[- ln(Kp/2) - yo] = 0, (4.22a) Kp-+o

lim nkK1(Kpp) ~pkv/rpp = kv/p. (4.22b) Kp--+o

Therefore, in this limit, E, contribution vanishes but E, remains finite. The entire problem becomes equivalent to a plane-wave scattering problem since the incident field is purely transverse.

90 4.2.4 Penetrating electron trajectories

In the main text, we discuss electron trajectories near photonic structures. For penetrating electron trajectories-that is, when the electron trajectory re(t) intersects

X(r) , 0 regions-a subtlety arises: the limit, Eq. (4.3), then apparently diverges even in lossy materials Im x4 0. In specific terms, the norm-squared incident field Einc is non-integrable over the electron trajectory, that is fy dV IEinc(r)12 fv dV1is ln pp + p p-12 diverges if V includes regions where p = 0. Here, we discuss the regularization of this divergence with emphasis on the implications to electron energy loss spectroscopy (EELS).

Though at first sight disconcerting, the divergence is not a surprise: the direct calculation of the EEL spectrum, F(p,j) = e Ref dxEx(p+x,w)ekZ, is also divergent for penetrating trajectories when Im x # 0. For an extended bulk material, of permittivity e = 1+ X, the EEL spectrum (per unit length L) can be evaluated from the momentum-space representation of the total field (to be introduced shortly), yielding [541:

e 2L V 1 q2 + k2 -ek 2- 2 2-* FEELS(W) o- -}ln (4.23) 7ho2 c C k2- ek _

The denominator of the logarithm describes the emergence of Cherenkov losses for v > c/E and is finite-in contrast, the numerator, which describes EEL due to material loss, diverges logarithmically in a momentum cut-off qc. Of course, the divergence is merely an artifact of an idealized description of the system-several physical and practical considerations impose natural momentum cut-offs, e.g.:

Collection angle The collection semi-angle of the microscope's spectrometer W restricts momentum transfer collection to in-plane momenta qp < q,, with hqc=

mev sin W ~mevo. At typical collection semi-angles and acceleration voltages-

say, s = 10 mrad and 100 keV-this sets a cut-off at hqc ~ 2.8 x 103 eV/c, or equivalently, a spatial spread 1/qc ~ 1A.

Nonlocality Nonlocality effectively suppresses the dielectric response to large-momentum

91 plane-wave components, i.e., E(q, w) -+ 1 for q »1/a (lattice constant a). The free-electron response is quenched even earlier, at a threshold set by the Thomas-Fermi momentum.

Electron spread The spread, AR, of the electron's in-plane density imposes a cut- off qc ~ 1/AR.

To summarize; the divergence of the limit for penetrating trajectories is simply the mirror of the divergence of the direct calculation. Accordingly, the divergence's rem- edy is also mirrored: the limit is regularized upon introducing a momentum cut-off in the electron's (incident) field Einc. Denoting this regularized field Einc,qc, we next verify that this field is indeed regular as p -+ 0. Coincidentally, this also outlines the derivation of the conventional, non-regularized field [Eq. (4.2)]. The derivation proceeds as follows: in momentum-frequency space, the electron charge density p(r, t) = -e6(r - vt) equals p(q, w) = -2xe(w - q - v) and is accompanied by a current density J(q, w) = -27rev6(w - q -v). Jointly with Maxwell's

2 2 equations, in the form of the wave-equation (q - ek )Einc = iE'(Jk/c - pq/c), this gives the associated electric field's (q, w)-representation:

2irie kv/c - q/ q Einc(q, w)= ~ 2 / 6 (w - q - v). (4.24) 6o q 2 - ck2

An inverse transform then yields the (r, w)-representation (specializing to v =vk and

3 - Einc,qj(r,w)= 22rie d q2k/c-3 2 q( - v - Co JqI

ie eikx f d2q, (kv/c - kv)k - qpiqp.p Eov jIq~p

ie ike k f°dqp qpJo(qpp) ± d q q J1(qpp) 2eo 2 0 2g + k -k Vg +k k2 '

SLq + xp K(Kpp)/y for qc-+oo Tqe --+ pK1(ri pp) for qc-+oo (4.25) reproducing Eq. (4.2) as qc- oo (we remind that rp kv/-y). Written in terms

92 of the transverse and longitudinal parts introduced in the above, Lc and T,, the regularized version of Eq. (4.3) reads

PT (w) < 2 dV F (L 2 + T2). (4.26) -167r3EOV2J ImX q° q

To demonstrate the limits' finiteness, we require the small-p behavior of L. and T, at finite qc. Since qc is large, much larger than sP, this is straightforward-particularly

for Tc:

Tqc fcd qPJ1(qp) °° qJ1(qpp) °° dq J1(qp) 0 q J0 q +q, +

00 ~ K1(p) - f dq, J1(qp) = -pK1(Kp)Jo(qcp) (4.27)

The small-p behavior then follows from the small-argument asymptotics of the Bessel functions [for x « 1, K1(x) = x-- x (j -EM In }x) + 0(x3 Inx) and x-Jo(x) =

3 X -jX + O( ) withYEM denoting the Euler-Mascheroni constant):

T I 2 12p+ __ )EMp + In 1(PPp for p < q-1 < -'1. (4.28)

Thus, the regularized transverse component Tq, vanishes as p -4 0-for slightly larger p-values, however, Tqc has a global maximum: maxqep Tq~ Tq,(qcp 2.76) 0.42qc (assuming qc » p).

The longitudinal contribution Lq, does not find as neat a closed form expression as Eq. (4.27), though it may still be expressed in terms of known functions:

Lq~A Pjdqp qpjo(qpp) /rP [jc qpJo(qpp) jdC* Oqp) Lc - 0dq 2 + ~ dqp q 2 -K dqp q

{Ko(Kpp)+1 1qcp+EM - 1(qcp 2 F3 [ (12',22qcp (4.29)

where 2 F3 is a generalized hypergeometric function with the asymptotic behavior 1-0[(qcp) 2]. The small-p behavior again follows from the Bessel function asymptotics

93 [Ko(x)= -nj-EM +00(x In x)], such that:

Le ~_ - ,PI for p « qc « 1 . (4.30)

Thus, the longitudinal contribution Lqe tends to a finite, nonzero value oc ln qc/', as p -+ 0; this is also the maximum of Lc.

Equations (4.28) and (4.30) demonstrate that the p = 0 singularity of the incident field is regularized for finite cut-off momenta q. This ensures that both direct calculations and limits similarly yield finite, regularized values, with bulk contributions dependent on the cut-off momentum.

4.2.5 Upper limit in two dimensions

The limits can be derived in both the three-dimensional or the two-dimensional case. For completeness, here we also derive the limit in the two-dimensional case, which correspond to sheet electron beams that are assumed in Fig. 4-3(f).

We consider an electron sheet beam in the (x, z) plane with charge density being one electron per nanometer, i.e., q = 1.6 x 10-19 C/nm [consistent with our unit

2 2 for probability in two dimensions dxd dy' /h (eVlnm )]. Precisely, the probability is invariant of the choice of the transverse (y) length scale, as long as the length scale is in the same unit for both the source current density and the probability. Here the length scale is chosen as nanometer for both of the quantities.

The source current density in the time domain can be written as J(r, t) = qvJ(z - zo)J(x - vt). In the frequency domain, the current density is given by

J(r, w) = qo(z - zo)eikxi, (4.31)

94 The induced fields are

H(r, w) = -eikvx-(z-zo)S (4.32a) 2 E(r, w) = q (ki - isp,)eikx-Kp(z-zo) (4.32b) 2wco

for z > zo and

H(r, w) = qeikx+KP(z-zo)g (4.33a) 2 E(r, w) = - q (kvi + izp)eikx+Kp(z-zo) (4.33b) 2we

for z < zo, where co is the vacuum permittivity, and i, also defined as K = k2 - k2 same as the main text. where k = w/c is the light wavevector.

Insert Eq. (4.33b) into Eq. (4.9), we obtain the limit in two dimensions

dF<(w) |x| 2 q2r (k2 +Ks) K _- dy - Imx 32heow 2 e2Kp 1`zo dS, (4.34) where S is the area defined by the profile of the structure.

As in the main text, we also consider a concrete example: Smith-Purcell radiation from a rectangular grating with filling factor A. Applied the rectangular profile to Eq. (4.34), the radiated photon per frequency per electron per unit area is bounded by d2Fr(W) 1 2 Aq 2 r(kv + K)e2Kpd (4.35) dx dy - ImX 64heoKsw2 where d is the distance between the electron and the grating.

95 0.8

(0,kd) 0.6

0.4 0

0.2

10-2 101 100 10 1 kd

2 = 0.O15, kd =0.09 p--. 1, kd =0.47 102 Energy loss -4104 Energy loss

E 0 10-6 10 O

a. X=400nm 10 0 X=400nm n 10 - 10

*~ 10 10° 0 -* x .0-10 6 nm o r d=30 nm At10 XPhtn A u 0.1012( Photon 10-12 - Photon Au emission (b) (emission (C)

101 10~4 0.01 0.1 1 0.05 0.2 - 0.8 13

Figure 4-2: Optimal electron velocities for maximal Smith-Purcell radiation. (a) Behavior of g (3, kd), Eq. (4.18), whose maxima indicate separation- dependent optimal electron velocities. Here g is normalized between 0 and 1 for each separation. The limit yields sharply-contrasting predictions: slow electrons are optimal in the near field (kd « 1) and fast electrons are optimal in the far field (kd » 1). (b-c) Energy loss (red) and radiation (blue) rates [circles: full-wave simulations; lines: grating limit, Eq. (4.18); shadings: shape-independent limit, Eq. (4.10)] at two representative near/far-field separation distances [white dashed slices in (a)].

96 4.3 Key predictions

4.3.1 Slow-electron-efficient regime

A surprising feature of the limits in Eqs (4.9,4.10) is their prediction for optimal electron velocities. As shown in Fig. 4-1(c), when electrons are in the far field of the structure (Kpd » 1), stronger photon emission and energy loss are achieved by faster electrons-a well-known result. On the contrary, if electrons are in the near field

(nid « 1), slower electrons are optimal. This contrasting behavior is evident in the asymptotics of Eq. (4.1Ob), where the 1/#2 or e-2pd dependence is dominant at short or large separations. Physically, the optimal velocities are determined by the incident- field properties [Eq. (4.2)]: slow electrons generate stronger near field amplitudes although they are more evanescent. There has been a recent interest in using low- energy electrons for Cherenkov [145] and Smith-Purcell [154] radiation; our prediction that they can be optimal at subwavelength interaction distances underscores the substantial technological potential of nonrelativistic free-electron radiation sources.

On the other hand, the grating limit [Eq. (4.18)] exhibits the same asymptotics as Eq. (4.10), thereby reinforcing the optimal-velocity predictions of Fig. 4-1(c). The (0, kd) dependence of g, see Fig. 4-2(a), shows that slow (fast) electrons maximize Smith-Purcell radiation in the small (large) separation regime. We verify the limit predictions by comparison with numerical simulations: At small separations [Figs. 4- 2(b)], radiation and energy loss peak at velocity # :::: 0.15, consistent with the limit maximum; at large separations [Figs. 4-2(c)], both the limit and the numerical results grow monotonically with #.

4.3.2 Enhanced radiation by coupling electrons with photonic

bound states in the continuum

Finally, we turn our attention to an ostensible peculiarity of the limits: Eq. (4.9) evidently diverges for lossless materials (Im X -+ 0), seemingly providing little insight.

On the contrary, this divergence suggests the existence of a mechanism capable of

97 WC P=aA 0~=4M (e)

10 0.63

0.8 TO E M) -C 10 010 0.6 -

0 k a2 0.04 x

0.4 (d) Probability 0.6 (log scale a.u-) max 10 0.65 w=vk 0.2 0.64 81 r4 40 iC 0.63 B 1 0.6 0.65 0.7 0.2 0.4 0.6 r k a/2n X

Figure 4-3: Strong enhancement of Smith-Purcell radiation via high-Q resonances near a photonic bound state in the continuum (BIC). (a) Schematic drawing of a silicon-on-insulator grating (one-dimensional photonic crystal slab: periodic in x and infinite in y). (b) Calculated TE band structure (solid black lines) in the F-X direction. The area shaded in light and dark yellow indicates the light cone of air and silica, respectively. The electron lines (blue for velocity v, and green for v/2) can phase match with either the guided modes (circles) or high-Q resonances near a BIC (red square). (c) Upper: Incident field of electrons. Lower: resonant quality factors (left) and eigenmode profile (right) near a BIC. (d) Strongly enhanced Smith-Purcell radiation near the BIC. (e) Vertical slices of (d). (f) The limit (shaded area) comparing with the horizontal slice of (d), with material loss considered. Strong enhancement happens at electron velocities 3= a/mA (m = 1, 2, 3...).

98 strongly enhancing Smith-Purcell radiation. Indeed, by exploiting high-Q resonances near BICs [104] in photonic crystal slabs, we find that Smith-Purcell radiation can be enhanced by orders of magnitude, when specific frequency, phase, and polarization matching conditions are met.

A one-dimensional silicon (x = 11.25)-on-insulator (SiO 2 , X =1.07) grating interacting with a sheet electron beam illustrates the core conceptual idea most clearly. The transverse electric (TE) (E., Hy, E2) band structure (lowest two bands labeled

TEO and TE1 ), matched polarization for a sheet electron beam [Eq. (S41b)], is depic- ted in Fig. 4-3(b) along the F - X direction. Folded electron wave vectors, kv = w/v, are overlaid for two distinct velocities (blue and green). Strong electron-photon interactions are possible when the electron and photon dispersions intersect: for instance, kv and the TEO band intersect (grey circles) below the air light cone (light yellow shading). However, these intersections are largely impractical: the TEO band is evanescent in the air region, precluding free-space radiation. Still, analogous ideas, employing similar partially guides modes, e.g., spoof plasmons11781, have been explored for generating electron-enabled guided waves [15, 133].

To overcome this deficiency, we theoretically propose a new mechanism for enhanced Smith-Purcell radiation: coupling of electrons with BICs [104]. The latter have the extreme quality factors of guided modes but are, crucially, embedded in the radiation continuum, guaranteeing any resulting Smith-Purcell radiation into the far field. By choosing appropriate velocities#= a/mA (m any integer; A the BIC wavelength) such that the electron line (blue or green) intersects the TE1 mode at the BIC [red square in Fig. 4-3(b)], the strong enhancements of a guided mode can be achieved in tandem with the radiative coupling of a continuum resonance. In Fig. 4-3(c), the incident fields of electrons and the field profile of the BIC indicate their large modal overlaps. The BIC field profile shows complete confinement without radiation, unlike conventional multipolar radiation modes. The Qs of the resonances are also provided near a symmetry-protected BIC [104] at the F point. Figs. 4-3(d) and (e) demonstrate the velocity tunability of BIC-enhanced radiation-as the phase matching approaches the BIC, a divergent radiation rate is achieved.

99 The BIC-enhancement mechanism is entirely accordant with our upper limits. Practically, silicon has nonzero loss across the visible and near infrared wavelengths.

E.g., for a period of a = 676 nm, the optimally enhanced radiation wavelength is -

1050 nm, at which Xsi = 11.25 + 0.001i [88]. For an electron-structure separation of

300 nm, we theoretically show in Fig. 4-3(f) the strong radiation enhancements (> 3 orders of magnitude) attainable by BIC-enhanced coupling. The upper limit [shaded region; 2D analogue of Eq. (4.9), see 4.2.51 attains extremely large values due to the minute material loss(|x/Imx 105); nevertheless, BIC-enhanced coupling enables the radiation intensity to closely approach this limit at several resonant velocities. In the presence of absorptive channel, the maximum enhancement occurs at a small offset from the BIC where the Q-matching condition (see 4.2.5) is satisfied, i.e., equal absorptive and radiative rates of the resonances.

4.4 Experimental verification

4.4.1 Measurement in the visible wavelengths

We perform quantitative experimental measurement of Smith-Purcell radiation to directly probe the upper limit. Fig. 3-5(a) shows our experimental setup. A one- dimensional 50%-filling-factor grating (Au-covered single-crystalline Si)-the quintessential Smith-Purcell setup-is chosen as a saniple, and shown by SEM images in Figs. 4-4(b-c). Free electrons pass above and impinge onto the sample at a grazing angle of 1.50 under 10 to 20 kV acceleration voltages.

Fig. 4-4(d) depicts our measurements of first order m= 1 Smith-Purcell radiation appearing at wavelengths between 500 and 750 nm. In quantitative agreement with Eq. (4.1) evaluated at normal emission angle (solid lines), the measured radiation spectra (dots) blueshift with increasing electron velocity. Notably, we experimentally obtain the absolute intensity of the collected radiation via a calibration measurement.

The upper limits [Eq. (4.9)] for the surface-normal emission wavelengths (A = a/#) are evaluated at the center of the interaction region [height 140 nm (kd a 1.5),

100 Optics (enlosure bood es LenksI

(d) x10-8

2.5 201keV 18 keV I il4 keV 13 keV 12keV 11keV tokev I E S2 P S1.5 JI Al /1 A 05 KXA -I i~ 1 S0 550 600 650 700 750 Wavelength (nm)

Figure 4-4: Experimental probing of the upper limit. (a) Experimental setup. OBJ, objective (NA = 0.3); BS, beam splitter; SP, spectrometer; CAM, camera. (b-c) SEM images of the structure in (b) top view and (c) cross-sectional view. (d) Quantitative measurement of Smith-Purcell radiation (inset: camera image of the radiation). Solid lines mark the theoretical radiation wavelengths at the normal angle [Eq. (4.1)]. The envelope (peak outline) of the measured spectra (dots) follows the theoretical upper limit (shaded to account for fabrication tolerance; calculated at each wavelength with the corresponding electron velocity for surface-normal radiation).

101 varying with beam energy], and is shown with shading in Fig. 4-4(d) to account for the thickness uncertainty (±1.5 nm). The envelope spanned by the measurement peaks follows the upper-limit lineshape across the visible spectrum: both the theoretical limit and the measured intensities peak near 550 nm and decrease in a commensur- ate manner for other wavelengths. This lineshape originates from two competing factors. At shorter wavelengths, the material factor |x2 / Imx decreases significantly for both Au and Si[see Fig. 4-1(c)], which accounts for the reduced radiation intensity. At longer wavelengths, the major constraint becomes the less efficient interaction between the electrons and the structure, as the electron-beam diameters increase for the reduced brightness of the electron gun (tungsten) at lower acceleration voltages. These experimental evidences for the upper limit are at kd a 1.5 (estimated from a geometrical ray-tracing model; see 3.9.3), where fast electrons are still preferred [Fig. 3-4(a)].

4.4.2 Measurement in the infrared wavelengths

We also conduct near-infrared experiment to further confirm out theory with the same experimental setup and a near-infrared spectrometer. A one-dimensional grating (Au-covered patterned-Si, see Fig. 4-5 inset; LightSmyth Technologies) with a longer periodicity (272 nm) is used such that the Smith-Purcell radiation moves to near- infrared. Adopting the same methods of data acquisition, calibration, and analysis [as those of our initial experiment in the visible (as described in 4.4.3)], we are able to obtain the absolute emission probabilities for the near-infrared Smith-Purcell radiation. The new experimental results are shown in Fig. 4-5, where the envelope lineshape of the emission spectra again follows our theoretical prediction. The measured currents and the calculated electron structure separations are shown in Fig. 4-6. In addition to the agreement between our theory and each of the experiment, the comparison between the visible and the infrared experiment gives rise to interesting observations that further confirm our theory. Two key observations can be made from the comparison. First, the absolute emission probabilities increase by about two

102 x10-6 8

20 19 18 '17 61 13 12 11 10 kev keV kev keV vv kev kev kev E 6

900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 Wavelength (nm)

Figure 4-5: Smith-Purcell radiation observed in the near-infrared regime and the comparison with the upper limit theory.

(kV) 10 (k 10 (a) 200 20 Voltage (b) 20 Voltage 230

15150 * 00 0

170 ' 100 140 900 1000 1100 1200 1300 1400 900 1000 1100 1200 1300 1400 Wavelength (nm) Wavelength (nm)

Figure 4-6: (a) Measured current of the near-infrared experiment. (b) Electron structure separations d obtained from the model (dots; see 4.4.3) and their polynomial fitting (curve) for calculating theoretical upper limits.

103 orders of magnitude from the visible to the near-infarred regime-consistent with the same order of increase in the material factor of Au [see Fig. 4-1(b)], which confirm the material factor dependence explicitly. Second, although the two experiments are both in the fast-electron-efficient regime, the measured emission probabilities feature a peak for the visible experiment, while exhibit monotonic decrease for smaller electron energies (except for a small increase between 17 keV to 16 keV) for the near-infrared experiment. Such a difference arises because the material response is much less dispersive in the near-infrared, which implicitly corroborates the functional impact-parameter dependence within our upper limit.

4.4.3 Experimental methods and data analysis

1 BS Lens

Calibrated source Lens 2

Mirror

Figure 4-7: Experimental setup of the calibration measurement.

We are able to obtain the absolute intensity of Smith-Purcell radiation by implementing a calibration measurement using a broadband (visible and near infrared) calibrated source (AvaLight-HAL-CAL). The experimental setup for celibration is shown in Fig. 4-7. All the optics remain the same as Fig. 3-5(a) except that we replace the SEM with the calibration source. The spectral intensity So(w) of the calibrated source is already known from the manufacturer. Passing through all the optics, the radiation from the calibrated source enters the spectrometer and generates a signal count Co(w). With So(w) and Co(w), we are able to gauge Smith-Purcell radiation intensity S(w) by reading the corresponding signal count C(w). The relation is given by

So(w)_ S(w) CO = .S) (4.36) COMw CMw 104 This relation is valid for two reasons. First, the generated photons into the sample substrate is negligibly small compared to the total radiation (see Fig. 4-8). Second, the optics and spectrometer configurations remain unchanged for Smith-Purcell radiation measurement and calibration measurement. This approach allows us to obtain the absolute radiation intensity of the collected Smith-Purcell radiation, without knowing the loss functions of each individual optical elements or the quantum efficiencies and EM gains of the spectrometer at each wavelength, since all these factors will cancel

out if inserted into Eq. (4.36). To calculate the number of photons generated per electron, measurement of the current from the SEM is necessary. The currents are measured using a picoammeter connected to a built-in Faraday cup inside the SEM chamber. The measured currents are shown in Fig. 4-9(a).

10 20 keV

18 keV 10 -2

10 keV

103 6keV 14keV 12 keV 13 keV 11 keV

10-4 500 550 600 650 700 750 Wavelength (nm)

Figure 4-8: Fraction of the generated photons into the substrate for different acceler- ating energies at normal emission angle (A = a/,3) for the first-order Smith-Purcell radiation.

For comparisons with the analytical limits, we also need to evaluate the number of unit cells Nuc of interaction and consider the beam diameters (spatial spread) of the electron beams. We estimate the electron beam diameter D with the equation [86]

2 2 2 2+ A)2]ap- +C2 a6 +(cAE )2 2 D2= D2+D +D± D = [C2 +(0.6A) 2 ]a 2 + + a. (4.37)

Here Do is the aberration-free Gaussian probe diameter, Ddcorresponds to aperture diffraction, D, corresponds to spherical aberration, and De corresponds to chromatic aberration. Our SEM uses a tungsten thermionic cathode, for the energy regime

105 (10-20 keV) we use, Dd and De are negligible [188] C2a D 2 1 Do2 + D.2 = C02-2 + a P, (4.38) where C.= v4I/b7r2, (4.39) b is the electron gun brightness, I is the probe current, a, is the convergence semi- angle of the electron beam, and C, is the spherical aberration coefficient. For the 2 brightness b of the source, we choose 1 x 10 A/cm /sr for the acceleration energy 20 keV (typical value for a tungsten source [86, 188, 125]) and scale it linearly [86, 188, 125] for other voltages. The focal length (working distance) of our SEM is 28 mm, which corresponds to a spherical aberration coefficient C, ~ 300 mm [188, 125]. For each measurement, we adjusted the SEM to achieve the smallest possible beam 3 8 1 4 diameter. In theory, this corresponds to Dmi = (4/3) / (CoC,) / for the optimal 1 8 4 convergence semi-angle aopt = (4/3) / (Co/C,)1/ [derived from Eq. (4.38)].

In our experiment, the electron beams grazingly impinges onto the sample at an

nonzero angle of 0 = 1.5, which leads to a finite number of unit cells where electrons strongly interact with the structure such that the radiation contribution from other areas are negligible. The backscattering coefficient 1 of the SEM can be generally estimated as [86] 77=1/(1 + sin 0)P, (4.40)

(a) 20 Voltage 10 (b)0 Voltage (k) 0

400-----. 0 E 200

200 H 1

U 100 L - 2D32/sin 0

100 - - 0 700 750 500 550 600 650 700 750 Sample 500 550 600 650 Wavelength (nm) Wavelength (nm)

Figure 4-9: (a) Measured current of the experiment. (b) Schematic of the model to evaluate the interaction length of the electron beam with the structure. (c) Electron structure separations d obtained from the model (dots) and their polynomial fitting calculating theoretical (curve; the 20 kV outlier data point dropped from fitting) for upper limits.

106 where p = 9/VZ and Z is the atomic number. In our case, 0 = 1.50 and Z = 79 (Au), and thus r ~ 0.974, meaning that most electrons get elastically scattered and maintain their initial momenta, which correspond to the scenario shown in Fig. 4- 9(b). The highlighted rectangle is treated as the region where electrons strongly interact with the structure. The number of unit cells is consequently determined via the length of the interaction region Nc = L/a = 2D/a sin. After obtaining Nu, the measured radiation spectral density S(w) can be translated into emission probability per electron per frequency per unit propagation length

drexpt(W)_ eS(w) dx hwINuca' which produces the measured emission probability shown in Fig. 3-5(d). On the theory side, the upper limit in Fig. 3-5(d) is calculated for Smith-Purcell radiation at the surface-normal emission angle (i.e., # = a/A). The limit is evaluated at the center of the interaction region with separation d = H/2 = D tan 0/4 sin9 {see

Fig. 4-9(c)] by generalizing Eq. (4.18). The generalization of Eq. (4.18), analogous to the expression of Eq. (4.9), is straightforward for the inhomogeneous Au-Si grating sample: move |x| 2/Imx into the integrand, and account for different materials:

IF(w) < (T KIxmt1 ([pp)+Kr k K (pp)] dV, (4.42) 27rw 2 '-"Im Xa mat mat Xmat whereVmat andXmat are the occupied volume and susceptibilities of the materials

(mat E {Si, Au}).

4.5 Discussion

We have theoretically derived and experimentally probed a universal upper limit to the energy loss and photon emission from free electrons. The limit depends crucially on the impact parameter rpd, but not on any other detail of the geometry.

Hence, our limit applies even to the most complex metamaterials and metasurfaces, given only their constituents. Surprisingly in the near field slow electrons promise

107 stronger radiation than relativistic ones. The limit predicts a divergent radiation rate as the material loss rate goes to zero, and we show that BIC resonances enable such staggering enhancements. This is relevant for the generation of coherent

Smith-Purcell radiation [231, 15, 133]. The long lifetime, spectral selectivity, and large field enhancement near a BIC can strongly bunch electrons, allowing them to radiate coherently at the same desired frequency, potentially enabling low-threshold Smith-Purcell free electron lasers. The combination of this mechanism and the optimal velocity prediction reveals prospects of low-voltage yet high-power free-electron radiation sources. In addition, our findings demonstrate a simple guiding principle to maximize the signal-to-noise ratio for EELS through an optimal choice of electron velocity, enabling improved spectral resolution.

The predicted slow-electron-efficient regime still calls for direct experimental val- idation. We suggest that field-emitter-integrated free-electron devices (e.g. [145]) are ideal to confirm the prediction due to the achievable small electron-structure separation and high electron beam quality at relatively large currents. Additionally, the microwave or Terahertz frequencies could be suitable testing spectral ranges, where the subwavelength separation requirement is more achievable.

The upper limit demonstrated here is in the spontaneous emission regime for constant-velocity electrons, and can be extended to the stimulated regime by suitable reformulation. Stronger electron-photon interactions can change electron velocity by a non-negligible amount that altersthe radiation. If necessary, this correction can be perturbatively incorporated. In the case of external optical pumping [203], the upper limit can be revised by redefining the incident field as the summation of the electron incident field and the external optical field. From a quantum mechanical perspective, this treatment corresponds to stimulated emission from free electrons, which multi- plies the limit by the number of photons in that radiation mode. This treatment could also potentially translate our limit into a fundamental limit for particle acceleration [179, 34], which is the time-reversal of free electron energy loss and which typically incorporates intense laser pumping. In the multi-electron scenario, the radiation upper limit will be obtained in the case of perfect bunching, where all electrons

108 radiate in phase. In this case, our single-electron limit should be multiplied by the number of electrons to correct for the superradiant nature of such coherent radiation.

109 110 Chapter 5

Synthesis and Observation of Non-Abelian Gauge Fields

5.1 Introduction

In the previous chapters, we investigate the scattering of electromagnetic waves with real structures. In this chapter, we will go beyond this scope and discuss the scattering of electromagnetic waves with synthetic gauge fields, and consequently, synthetic magnetic fields.

Gauge fields are the backbone of gauge theories, the earliest example of which is classical electrodynamics. However, until the seminal Aharonov-Bohm effect [5], the scalar and vector potentials of electromagnetic fields have been considered as a convenient mathematical aid, rather than objects carrying physical consequences. It has been realized by Berry [26] that the Aharonov-Bohm phase imprinted on electrons can be interpreted as a real-space example of geometric phases [26, 176], which in fact appear in versatile physical systems. For charge-neutral particles, such as photons [215, 256] and cold atoms [50, 58, 82], synthetic gauge fields can be created in real, momentum, or synthetic (i.e. other parameters besides position or momentum) space. These synthetic gauge fields enable engineered, artificial magnetic fields in systems of either broken or invariant time-reversal symmetry; and thus play a pivotal role in the realizations of topological phases [82, 149, 174, 9], quantum simulations [29,

111 19], and optoelectronic applications [229, 64].

Gauge fields are classified into Abelian (commutative) and non-Abelian (non- commutative), depending on the commutativity of the underlying group. Synthetic Abelian gauge fields have been realized in various platforms including cold atoms [144, 7, 164, 217, 6, 8, 115, 184, 139], photons [66, 65, 93, 230, 138, 163, 204, 187, 95, 237], phonons [246, 255, 1], polaritons [143], and superconducting qubits [207, 190, 191]. The synthesis of non-Abelian gauge fields is more challenging, due to the requirements of degeneracy and non-commutative, matrix-valued gauge potentials. So far, they have been achieved only in the momentum and synthetic spaces. Specifically, non- Abelian gauge fields have been realized in the momentum space using two-dimensional spin-orbit coupling [105, 244] in cold atoms. In the synthetic space, non-Abelian geometric phases [240, 239], initially observed in nuclear magnetic resonances [265, 259, 157, 158], have enabled non-Abelian geometric gates [2] and the simulation of an atomic Yang monopole [218].

There have been complimentary efforts for synthesizing non-Abelian gauge fields in the real space. Two initial proposals, using atoms, were based on tripod couplings in spatially varying laser fields [196] and laser-assisted, state-dependent tunneling in optical lattices [173]. Since then, real-space non-Abelian gauge fields have been predicted to enable numerous intriguing phenomena, such as the quantum anomalous hall effect [83], topological insulators in shaken lattices [98], and real-space non- Abelian monopoles in superfluids [247]. To observe non-Abelian gauge fields, Wu and Yang [242] conceived the non-Abelian Aharonov-Bohm effect, which has been widely discussed in gauge theory. Moreover, there have been several proposals in atomic [110, 173] and photonic [107, 40] systems that aim at implementing this effect with synthetic gauge fields. Despite of these theoretical advances, the synthesis and observation of non-Abelian gauge fields in real space remain experimentally elusive.

112 Realspace ibetspace

Abelian

I.j

c d

non-Abelian

Figure 5-1: Corparison between SU(2) Abelian and non-Abelian gauge fields in real space and in Hilbert space. a-d. Along a closed loop inside an Abelian gauge field A oc a,, (a) or a, (c), the sate evolves by rotating around the z (b) or y (d) axis of the Poincar6 sphere. Within each case (a-b or c-d), the state evolution are always commutative. e-f. In non-Abelian gauge fields, the evolution operators for different loops are no longer commutative, which leads to different final states, so and sf, for the same initial state si. The non-commutativity can be tested by an Aharonov-Bohm interference of the two final states.

113 5.2 Non-Abelian gauge fields and state evolution

Synthetic non-Abelian gauge fields demand a degeneracy of levels, which can, for example, be achieved by utilizing the internal degrees of freedom in quantum gases or exploiting the polarization/mode degeneracy and electromagnetic duality in photons. For a particle moving along a closed path in a non-Abelian gauge field, its evolution operator reads W = P exp if A dl, where P represents path-ordered integral and A is the matrix-valued gauge field. Its trace, W = Tr W, is gauge-invariant, and is also known as the Wilson loop 241]. For particles with N-fold degeneracies, the non-Abelian gauge fields can take forms of U(N). Here we focus on the SU(2) gauge fields, since our photonic system enables the definition of a pseudospin-a two-fold degeneracy in the polarization states. Crucially, we focus on the situations where the involved gauge fields break T-symmetry and state transport becomes nonreciprocal. In what follows we illustrate the consequence of real-space guage fields on the pseudospin evolution in Hilbert space [i.e. the Poincar6 (or Bloch) sphere].

The difference between how a state evolves in Abelian gauge fields versus in non- Abelian gauge fields is shown in Fig. 5-1. In a uniform Abelian gauge field oc o, the evolution operator along a closed loop can be simplified as W = eioz, where oz is the z component of the Pauli matrices and # is the flux of the gauge field through this closed loop (Fig. 5-1a). Consequently, the state rotates by 2# around the z axis of the

Poincar6 sphere (Fig. 5-1b). If the state evolves along two consecutive closed loops, the two evolution operators are commutative, which reflects the Abelian nature of this gauge field. Similarly, a homogeneous gauge field oc o, in real space (Fig. 5-1c) is also Abelian, as the state always evolve around the y axis in the Hilbert space (Fig. 5-1d).

In contrast, non-Abelian gauge fields require inhomogeneous gauge structures. Fig. 5-lef illustrate such an example where two different oz and ay gauge structures are concatenated into one compound closed loop. The same initial state si can now evolve into different final states: sf or 8 f (Fig. 5-1g), depending on the different ordering-# and then 6, or alternatively, 0 and then #-of the two gauge structures.

114 The interference between the two final states sq and so is known as the non-Abelian Aharonov-Bohm effect [242, 100, 12, 38, 107, 40, 50, 31, 110, 173]. This effect, that we will experimentally demonstrate later, is the most direct manifestation of non-Abelian gauge fields in real space.

5.3 Synthesis of non-Abelian gauge fields

To observe the non-Abelian Aharonov-Bohm effect, we synthesize non-Abelian gauge fields in real space in a photonic system. Exploiting a degeneracy in photonic modes, we create non-Abelian gauge fields by cascading multiple non-reciprocal optical elements that break the time-reversal symmetry (T) in orthogonal bases of Hilbert space. We demonstrate the genuine non-Abelian condition of our gauge fields in a fiber-optic Sagnac interferometer. The observed interference patterns show the signature features of the non-commutativity between a pair of time-reversed, cyclic evolution operators. We also demonstrate that our synthetic magnetic fluxes are fully tunable, enabling controlled transitions between the Abelian and the non-Abelian regimes.

In our photonic implementation, we experimentally synthesize the inhomogeneous gauge potentials in a fiber-optic system, which is conceptually illustrated in Fig. 5- 2a. We identify the horizontal and vertical transverse modes (denoted by 1h) and |v) respectively) in optical fibers as the pseudospin. Crucially, we synthesize two types of gauge fields, #o, andOo , using two distinct methods to break T-symmetry. To construct a gauge field of #oz, we first employ dynamic modulations that dress 1h) and |v) with nonreciprocal phase shifts of ±i, respectively. Specifically, four LiNbO3 phase modulators-two (labeled 1 and 2) for 1h) and two (labeled 3 and 4) for jv)-are driven by arbitrary waveform generators that create phase shifts in the form of sawtooth functions in time (Fig. 2b). Modulators 1 and 4 are positive in slope: # 1 ,4 = Wt mod2ir; and modulators 2 and 3 are negative in slope, 2 ,3 = -Wt mod 27r. The delay line between modulators 1 and 2 (3 and 4) corresponds to a delay time T. As a result, besides dynamic phases, h) (1v)) picks up an extra phase

# = wr (-wr) in the forward (i.e. left-to-right) direction, but an opposite phase

115 aZ -- - 0Gy

h> -, T T A B

93 1 94au

Forward Backward b c

d"V

PM m PMS

Dd" AAAG

~Ver

Figure 5-2: Synthesis of non-Abelian gauge fields. a.Non-Abeliangauge fields for photons. Temporal modulation and the Faraday effect, which break T-symnmetry in two orthogonal bases of theHilbert space, are used to synthesizeoeando«,gauge fields, respectively. b. Pseudospin-dependent non-recijrocal phase shifts are created through sawtooth phase modulations, which corresponds asynthetic gauge field along oe.c. Non-reciprocal rotation of the pseudospin is achieved via the Faraday effect in a terbium gallium garnet crystal, which corresponds to asynthetic gauge field along o«,. d. Experimental setup. The interference between different final pseudospin states-originated from reversed ordering of the gauge structures (CW and COW, e)-is read outthrough aSagnac interferometer, which gives rise tothe non-Abelian Aharonov-Bohm effect. PBS/C: polarization beam splitter/combiner; PM: phase modulator;AW :arbitrarywaveformgenerator;COL:collimator; TG: Terbium GalliumGarnet;PD: photodetector.

116 -# (+#) in the backward direction. This pair of opposite nonreciprocal phases for opposite pseudospin components (Ih) and |v)) correspond to a #oz gauge field, which is continuously tunable by varying the modulation frequency w.

A second, orthogonal type of gauge field, 0oy, is created using the Faraday effect. Specifically, light is coupled out of the fiber, sent through a Terbium Gallium Garnet crystal placed in an external magnetic field, and then coupled back into the fiber.

Through the Faraday effect, pseudospin of light is rotated in a nonreciprocal way, which corresponds to a gauge field of OoY. This gauge field is also continuously tunable through the external magnetic field.

We then concatenate the two non-Abelian gauge fields to demonstrate the non- Abelian Aharonov-Bohm effect via Sagnac interferometry (Fig. 5-2e). In such Sagnac configuration, the two sites A and B in Fig. 5-2a are combined into the same physical location to enable well-defined non-Abelian gauge fluxes. Evolved from the clockwise (CW) and counter-clockwise (CCW) paths of the Sagnac loop, the two final states are s 0 = oze"'veiozsi and s 0 = e-eoze-'OUoozsi, where the oz term maintains a consistent handedness of the polarization for counter-propagating states. The interference of the two final states is given by (Sec. 5.5.8)

s= s o+ sX = -o, 0e Sf =Sf +-Sf ~(ei' +eei*zeiO'rY Yize si, (5.1) ~ where 0' = 0 + 7r/2 and ox is a global spin flip. This interference describes a Sagnac-type realization of the non-Abelian Aharonov-Bohm effect [50]-the interference between two final states, which originate from the same initial state, but undergo reversely-ordered, inhomogeneous path integrals (Fig. 3-le-g) in the CW and CCW directions.

Fig. 5-2d details our experimental setup (Sec.5.5.1). We place a polarization synthesizer in front of the Sagnac loop, to prepare any desired pseudospin state as the input in a deterministic manner. After exiting the Sagnac loop, the two final states s 0 and sOO interfere with each other. The associated interference intensity is projected onto the horizontal and vertical bases, which are then measured separately. Within

117 the Sagnac loop, a solenoid-driven by tunable pulsed currents (peak current ~ 2kA, duration ~ 10 ms)-provides a magnetic field between 0 and ~ 2 T (Sec. 5.5.2) for the Faraday rotator. The solenoid also provides a temporal trigger signal for the detection. For the dynamic modulation, we assign four different modulation frequencies (i.e. slopes of the temporal sawtooth functions) + 1 , - 2 , - 3 , and +W4 to each of the modulators with wo defined to be positive. We impose an additional constraint that w = (W1 + W 2)/2 = (W4 + W 3)/2. This modified arrangement from Fig. 5-2b maintains the same nonreciprocal phases and thus the gauge fields #ao (Sec. 5.5.10). The advantage of this modification is an experimental one: it relocates the relevant interference fringes from zero to a nonzero carrier frequency Q - w1 - w2 + 3 - W 4 , which is less sensitive to environmental or back-scattering noises.

5.4 Observation of Non-Abelian Aharonov-Bohm ef-

fect

We next define our experimental observable in the interference measurement and ex- plain its relevance to non-Abelian gauge fields. In the original U(1) Abelian Aharonov- Bohm effect, the observable is the interference intensity as a function of the Abelian magnetic flux. In our case, analogously, for each given set of non-Abelian gauge fluxes (0,), we measure the contrast p between the interference intensities projected onto the horizontal and vertical bases. Specifically, we measure p(, #, a,3) =I/I where (a,#) are the latitude and longitude of the input pseudospin state on the Poincar6 sphere and In is the intensity of jh) and |v) component of the output pseudospin state at the carrier frequency Q, respectively. Therefore, p is defined on a manifold of S2 x T2 , which is spanned by the Hilbert space of the input pseudospin

S2 and the synthetic space of the gauge fluxes (0, #) that is T2.

For a fixed set of magnetic fluxes (0, #), the contrast function p(a, 3) always exhibits two pairs of first-order zeros and poles on the Poincar6 sphere (Fig. 5-3a; also see Sec. 5.5.11). Within each pair, the zero and the pole are always antipodal and thus

118 Latitude a (deg) Longitude 0 (deg) WisonloopW

- 0 90 -2 0 2 a C d

,M tunable fbec 0

West East -1 ------J ------hemisphere hemisphere -1 0 1 -1 0 1 -1 0 1 O/n 0/n 9/n

Abelian f Non- Abelian (P)=(0,0) (-0.21,0) (-0.21n,-0.SOT) (-021n,-0.30n) (024n,-0.30n) 90

45 0

-45

-90 Q U -10 0 10 20 X Y 90 ontrast p(d)

-45

-90 -180 -90 0 90 180 -180 -90 0 90 180 -180 -90 0 90 180 -180 -90 0 90 I80 -180 -90 0 90 180 Longitude A(deg) Longitude P (deg)

Figure 5-3: Non-Abelian Aharonov-Bohm interference. a. Contrast function p on the Poincare sphere, featured by a fixed zero/pole pair on the equator, and a tunable zero/pole pair (which indicates the consequence of gauge fluxes). The two pairs of zeros and poles are always antipodal. b-c. Location (latitude and longitude) of the tunable pole on the Poincar6 sphere as a function of the gauge fluxes (0, #). Abelian gauge fields correspond to on-equator poles (red dashed lines); non-Abelian gauge fields correspond to off-equator poles-both of which are experimentally demonstrated. d. Wilson loops W on the synthetic torus (0, #). |WI = 2 (red dashed lines) is a necessary but insufficient condition for non-Abelian guage fields (cf. b) . e-f. Ex- amples of predicted and observed contrast functions p for Abelian (Q, U, and V) and non-Abelian (X and Y) gauge fields.

119 represent orthogonal pseudospins. One pair, being linear polarizations (1,0) (zero) and (0,1) (pole), is fixed on the two ends of the equator, regardless of the choice of (0, #). The other orthogonal pesudospin pair, however, is tunable on the entire sphere via the synthetic gauge fluxes (0,#). These zeros and poles are conserved quantities on the Poincare sphere and dictate the behavior of the contrast p function. Their generation, evolution, and annihilation are directly related to the transitions between the Abelian and non-Abelian regimes. Fig. 5-3bc show the latitude a and longitude # of the tunable pole on the Poincar6 sphere, as a function of magnetic fluxes (0,#). When 0 = mr/2 or # = nr/2 (m and n are integers), the tunable zero-pole pair appears on the equator (red dashed lines in Fig. 5-3b). This key feature-an on/off- equator zero/pole-can be used to straightforwardly differentiate between Abelian and non-Abelian gauge fields synthesized in our experiment (see Fig. 5-3ef).

The necessary and sufficient condition for gauge fields to be non-Abelian is as follows. There exists two loop operators, W 1 andW2 , both starting and ending at the same site in space, such that they are non-commutative, i.e.W1W2 W2W1 [82].

In an Aharonov-Bohm interference, whether Abelian or non-Abelian, W 1 andW 2 can be identified as a pair of time-reversal partners that share the same physical

1 path. We first examine W1 = l - it andW 2 = ltI i in the Abelian Aharonov- Bohm experiment, whose two distinct top and bottom paths are denoted by lt and

1b, respectively. Under time-reversal, both momentum and vector potential flip sign, rendering W 1 = W2= ea that are clearly commutative and exhibit identical, scalar Berry phases - (Sec. 5.5.9). In our non-Abelian Aharonov-Bohm experiment, the time-reversal pair W 1 andW 2 can be analogously defined by replacing it andlbwith CW and CCW paths (Fig. 5-2e), which yields (Sec. 5.5.9)

W =P exp i j A dl = oze**Ye***`zei've'0oz, (5.2)

W2= Pexp ifW -lccAdl e= ze* ozei0zei*evoaz. (5.3)

The condition for W 1 andW 2 to be non-commutative is satisfied when 0 , mir/2 and # / nr/2 (Sec. 5.5.9)-the same condition also guarantees the existence of a zero and

120 a pole of the contrast function away from the equator (Fig. 5-4b). W 1 and W 2 are also connected via a unitary gauge transformation (Sec. 5.5.9); therefore they always

2 2 share the same Wilson loop (Sec. 5.5.9) W = Tr W 1 = Tr W2 = 2 - 4 cos 6sin 4o.

Fig. 5-3d shows this Wilson loop on the 2 space of gauge fluxes. Generally speaking, in an N-fold degenerate system, |WI= N means the state evolution can be trivially understood by decoupling the system into the product of N Abelian subsystems [821. In our case, such trivial configurations are shown with red dashed lines [9 = (m + 1/2)r, # = n, or = mir and # = (n +1/2)ir] in Fig. 5-3d. Nevertheless, IWI N is only a necessary but insufficient condition for gauge fields to be non-Abelian [821, as evident from the comparison between Fig. 5-3b and Fig. 5-3d: some configurations with IWI N are still Abelian.

-2 0 0.2 -01 0 0.2

Figure 5-4: Tunability of the non-Abelian gauge fields. Predicted (a) and measured (b) contrast function p for a fixed incident pseudospin state (a, f) ~~ (-51- 12. The gauge fields #u2 and O6, are continuously tuned by respectively varying the modulation frequencies in the arbitrary waveform generators and the voltages applied to the solenoid.

In Fig. 5-3ef, we characterize our synthetic gauge fields by measuring the contrast function p. We present the comparison between theoretical predictions (top row) and experimental measurements (bottom row) for five sampling points on the synthetic space T2: Q, U, V are Abelian; and X, Y are non-Abelian. In the Abelian case Q [(,#) ~ (0,0)], the tunable pole and the fixed zero annihilate each other at (a,#) = (00; so do the tunable zero and the fixed pole at (a,/#) = (0180. As a result, the contrast remains a constant p = 1 regardless of the input pseudospin state. This is a direct consequence of the preserved T-symmetry in the absence of gauge fluxes. In case U [(, 4) ~ (-0.21r, 0)], the annihilation of poles with zeros are lifted; nevertheless, both poles and zeros appear on the equator, and the gauge

121 structure remains Abelian, since we only break T-symmetry once. In case V [(0, #) ~ (-0.217r,0.50S)], which is still Abelian, the two poles (zeros) coalesce and produce a second-order pole (zero) on the equator. In cases X [(0, #) ~ (-0.21r, -0.30r)] and Y [(0, #) ~ (0.24r, -0.30)], our synthesized gauge fields become non-Abelian, as indic- ated by the observed off-equator zeros and poles. For all the cases, our observations show agreement with the associated predictions. In our interferometer, the two spin basis 1h) and jv), are not perfectly degenerate due to the difference in their refractive indices (~ 104). This difference leads to a reciprocal, linear birefringent phase (i.e. a dynamic phase contribution), which is calibrated and consistently applied to all measurements (Sec. 5.5.4). Up to this point, we have measured the contrast p for fixed gauge fluxes, while changing the input states. In a complementary manner, we can now fix the input state (a,#) and demonstrate the tunability of the synthesized non-Abelian gauge fields by measuring the contrast p for different synthetic gauge fluxes (0, #). As shown in Fig. 5-4, we reach similar agreeement between the theoretical prediction and the measurement.

5.5 Materials and Methods

5.5.1 Experimental setup

In this section, we describe the details of our experimental setup. Laser (Ando AQ4321D) light at A=1.55 pm, initially polarized with a polarizer (Thorlabs ILP155PM- APC), was sent into a polarization synthesizer (Thorlabs TXP5004 system installed with an in-line deterministic polarization controller DPC5500) to prepare arbitrary initial pseudospins. The prepared initial pseudospin was sent into the Sagnac loop after passing two 50:50 couplers (Thorlabs PN1550R5A2). Inside the Sagnac loop, the sawtooth modulation applied to the four LiNbO 3 modulators (Thorlabs LN65S-FC) enables the preparation of A = #a-, guage fields. Concretely, the pseudospin was projected onto

122 the linear basis, underwent MHz sawtooth modulation, and was recombined using polarization maintaining beam splitter/combiner (Thorlabs PBC1550PM-APC). The sawtooth modulation was generated by two arbitrary waveform generators (Tektronix

AFG3252 and Agilent 33600A) and was amplified to V2 with an RF amplifier (Mini- Circuits ZPUI30P), powered by DC voltage (Mastech HY3002D-2).

On the other side of the Sagnac loop, light was coupled to free space and underwent the Faraday rotation in the Terbium gallium garnet (TGG) crystal (Northrop Grumman). The magnetic field applied to the TGG crystal was provided by a solenoid system (Woodruff Scientific). An RF circuit, driven by a DC voltage (Circuit Special- ists CSI12001X), applied pulsed currents (peak currents ~ 2kA, duration ~ 10ms) to the solenoid. By tuning the DC voltage, tunable magnetic fields between 0 and ~ 2 T) were obtained (Sec. 5.5.2). A magnetic field sensor was embedded inside the solenoid and produced a synchronization signal upon detecting the maximal magnetic field.

After exiting the Sagnac loop, the final pseudospins that evolved from the clockwise (CW) and counter-clockwise (CCW) loops interfered with each other. The interference was projected onto the linear basis and the intensities were detected by two photodectors (Thorlabs FPD610-FC-NIR) and an oscilloscope (Teledyne Lecroy WavePro 760Zi). The oscilloscope were triggered by the synchronization provided by the solenoid system.

The time scales in the whole setup satisfied the relation Tphoton < Tsamping <

Tmod

Tsampiing (inverse of GHz) is the sampling rate of the oscilloscope, Tmod (inverse of MHz) is the modulation period, and Tscope (inverse of 2 kHz) is the time window of the oscilloscope, and Tpuis (inverse of 100 Hz) is the pulse duration. This relation justifies the approximation of a DC magnetic field in the solenoid (therefore a static gauge potential A = Oo; see Sec. 5.5.2) and enables detection of interference intensity in the frequency domain.

123 5.5.2 Magnetic field characterization

a 2

1.5 -OV

circuit measurement *ischarge e time window b 1 trigger 0.5 signal 0 0.5 0 2 4 6 8 10 Time (ms)

Figure 5-5: Characterization of pulsed magnetic fields of the solenoid system. a. Pulsed Magnetic field at the center of the solenoid as a function of charged DC voltage. The duration of the magnetic field 10 ms. b. Synchronization signal for triggering and the associated measurement time window (shaded area) Tope = 0.5 ms.

Fig. 5-5 shows the pulsed magnetic field applied to the Faraday rotator, and the associated trigger signal applied to the oscilloscope. The measurement time window Tscope and the pulse duration Tpuis satisfy Tscope < Tuise, which justifies a constant magnetic field approximation.

5.5.3 Rotation angle characterization

In our experiment, the spatial distribution of the longitudinal magnetic field is inhomogeneous in the solenoid. Therefore, the polarization rotation angle is given by 0 = V f B dl, where V is the Verdet constant. We characterize the rotation angle 0 as a function of the DC driving voltage of the pulsed current source experimentally. The characterization setup is shown in Fig. 5-6a, where we adjust the polarization angle difference between the two polarizers to achieve maximal power detection. The

124 a bC100 .b . DC voltage 0 measured --- linear fit ~60

Laser Polarizer 1 Solenoid PolarIzer 2 Power and meter 20 crystal 0 20 40 60 80 100 Voltage (V)

Figure 5-6: Characterization of the rotation angle of the crystal in solenoid. a. Characterization setup. b. Measured rotation angle as a function of driving DC voltage. characterization curve is shown in Fig. 5-6b. We observe linear relation between the rotation angle and the DC driving voltage of the solenoid.

5.5.4 Reciprocal dynamic phase characterization

1 I I "0

0 1 2 3 4 5 6 Time (h)

Figure 5-7: Temporal fluctuation of the calibrated dynamic birefringent phase factor.

In our interferometer, the difference (10-) of the refractive indices of thejh) and |v) renders quasi-degenerate internal degrees of freedom. Therefore, the states accumulate reciprocal, dynamic phase difference.

As long as photons exit the Sagnac loop, such reciprocal linear birefringence has no consequence on the results of the interference, because the interference is an intensity projection detection in the linear basis. Inside the Sagnac loop, nonreciprocal phases synthesized by the temporal modulation is commutable with this reciprocal birefringence. Therefore, as long as a fiber is spatially adjacent to the temporal modulation components, the associated reciprocal birefringence can be mathematically casted into #rh and #,in Eq. (5.16). As shown in Eq. (5.22), #rh and #,do not

125 affect the results of interference.

The only remaining reciprocal linear birefringence-that affects the interference results-is from the fibers that connects the pseudospin synthesizer and the Faraday rotator (which includes a few patch cables, two couplers, and the associated pig- tails). Although these fibers contains a few optical elements, their reciprocal birefringence can be parametrized as a single scalar phase factor 'b, which is the accumulated birefringent, dynamic phase difference between |h) and Iv) in these fibers. For the Abelian cases Q and case V, the contrast function p does not depend on Ob. For the rest of our measurements (Abelian case U, and non-Abelian cases X and Y), the consequence of @b is a transformation on the incident pseudospin si = (a, beio) - (a, bei(0+0)). We calibrate @b ~ -7r/2 via polarimetric measurement (Thorlabs PAX1000IR2) in the aforementioned relevant subsystem, by measuring the phase difference between the input and output states. The stability of b is shown in Fig. 5-7, which justifies its scalar parametrization. The calibrated O' is then consistently applied to all measurements as a initial state transformation.

5.5.5 Breaking reciprocity and time-reversal symmetry

The Faraday rotation and temporal modulation break the time-reversal symmetry in two orthonormal basis of the Hilbert spape-they produce nonreciprocal circular and linear birefringence for photons, respectively. We briefly summarize their breaking of time-reversal symmetry below.

5.5.6 Faraday rotation

For Faraday rotation with the TGG crystal (negative Verdet constant) shown in

Fig. 3-3c, its Jones matrix P in the forward (+) and backward (-) directions are

P+ = exp(iOo)and P_ =exp(-i6oy). The associated scattering matrix of Faraday

126 rotation is

0 (5.4) SP PT

Since Sp # ST, i.e. Sp is asymmetric [112], we break time-reversal symmetry using nonreciprocal phases in the circular basis.

5.5.7 Dynamic modulation

For the modulation scheme shown in Fig. 3-3b, the Jones matrix Q in the forward (+) and backward (-) directions are Q+ = exp(ioaz). and Q_ = exp(-iouz). The associated scattering matrix of dynamic modulation is

0 Q Sq= Q (5.5) 0)

Since SQ # ST, we break time-reversal symmetry using nonreciprocal phases in the linear basis.

5.5.8 non-Abelian Aharonov-Bohm effect

In this section, we show that the interference between two final states s = ozeiOeveiasi and s? = e-Ocze-iOOYozsi correspond to the non-Abelian Aharonov-Bohm effect.

127 Their interference is given by

sf = s 00+8s0 (5.6a)

= (ozei'Oel'O + e-+42eieOa'oz)Si (5.6b)

=oz (eioaveiOz" + aze-i'"zeOo'owaz)si (5.6c)

= oz(e0vei0`2 + eidei"Y)si (5.6d)

= oz (eOa~e'Oz+ oYeqei0 O.Y) si (5.6e)

OzaY (uyei'vel'' + e'Oaz ei'Ofy) S i (5.6f)

= -o, [ei(O+/2)Ueiw0, + ei'Oazei(+7/2)"y] si. (5.6g)

We note that the o term only flips the pseudospin globally and does not affect the gauge structure in the parenthesis of Eq. (5.6g). Therefore, the interference is in the standard form of the non-Abelian Aharonov-Bohm effect [50]: sf describes the interference between two final pseudospins that originate from the same initial pseudospin but undergo reversely-ordered, path integrals (clockwise and counterclockwise) that contain inhomogeneous non-Abelian gauge fields. The interference state sf is then projected into the linear basis for intensity detection and observable calculation (see Sec. 5.5.11).

5.5.9 Criteria for non-Abelian gauge fields

Non-Abelian gauge fields may refer to different definitions under different contexts. In this section we analyze our experiment with a genuine non-Abelian criterion. We also briefly clarify the relation among various definitions of non-Abelian gauge fields.

Genuine non-Abelian gauge fields

The precise definition of non-Abelian gauge fields is given as follows [821. Consider two Berry holonomies (also known as loop operators) W 1 andW 2 from two contours

128 Abelian non-Abelian

a b c d 1U_ CW CW-1

Id Id CCW_1 CCW

Berry holonomy W 1 Berry holonomy W, Berry holonomy W Berry holonomy W,

Figure 5-8: Time-reversed Berry holonomies in Abelian and non-Abelian

Aharonov-Bohm experiments. The two Berry holonomies W 1 andW 2 share the same physical loop, but are path-ordered reversely.

that start and end at a same point. Non-Abelian gauge fields emerge if and only if

W 1 W 2 = W2W1. (5.7)

This condition is guage invariant although the Berry holonomy in non-Abelian systems is generally not gauge invariant. Next we concretely discuss this criterion in both Abelian and non-Abelian Aharonov-Bohm effects (Fig. 5-8).

Abelian Aharonov-Bohm interference

In the U(1) Abelian Aharonov-Bohm setup with the Mach-Zehnder configuration

(Fig. 5-8ab), we denote A and B to be the source and interference point and It and lb are the top and bottom interference paths, respectively. The contour l1 - lt-which starts and ends at point A-produces a Berry holonomy

W1= exp i A dl) exp (i A dl) (5.8)

We can also consider the time-reversal process in the contour lt-1 -lb with the loop operatorW 2 given by

W2 = exp (i f-A dl) exp (i f-A dl). (5.9)

129 W 1 andW 2 differ by the sign-flips in momentum and vector potential (that are both odd upon time-reversal). Under U(1) gauge,W 1 = W2 = exp(i7), where Y, the well-known Berry phase, is the enclosed magnetic flux of the contours. Evidently,

W1W 2 = W 2W 1,which confirms the associated guage fields are Abelian. non-Abelian Aharonov-Bohm interference

In a similar manner, we analyze the non-Abelian Aharonov-Bohm interference with a Sagnac configuration (Fig. -8cd). In a Sagnac setup, the source point A and interference point B overlaps but we still keep the notation for clearness. Also, the distinct paths l andlbare replaced with clockwise (CW) and counter-clockwise (CCW) loops.

The contour lb' -l (Fig. 5-8a) thus becomes CCW-1 -CW (Fig. 5-8c), which produces

a Berry holonomy W1

W = Pexp ij5 A d1, (5.10a)

= ozela-Ye da.1ozelavel00z (5.1Ob)

e2 i cos 2 0 + sin2 0 (1 - e2ie) cosO sin 0 (5.1Oc) 2 (-1S + e2 ) cos sinO e -2io cos 0 + sin2 0 J

The time-reversed process gives rise to another contour CW.- CCW (Fig. 5-8d) whose Berry holonomy is given by

W 2 = Pexpi j (-A) dl, (5.11a)

= eida ed""orze O*r e0 "oz (5.11b)

2 2 2 ei cos 0 + sin 0 1 - e214) cos 0 sin 0 (5.11c) ( (-1+e-2i4) cos 0 sin 0 e-21 cos2 0 + sin2 0

Applying the criteria Eq. (5.7) to the two Berry holonomies

2 2 2 2 [W 1 , W 2 ] = -8 sin 0 cos 0 sin cos # (sin #cos #I+ sin #cos # + cos o -Y+ i sin 0( (5.12)

130 which provides the necessary and sufficient condition for genuine non-Abelian gauge

fields in our experiment: 0 , mr/2 and # 4 n7r/2. This condition corresponds to our experimental observable-an off-equator tunable pole/zero on the Poincare sphere (Fig. 3-4c).

The two holonomies W 1 andW 2-a time-reversal pair-can be simultaneously

manifested in our measurement. For the W1 holonomy, the 4-component incident

state is x = (si, Si)T. For theW 2 holonomy, the synthetic fluxes flip sign and

the 4-component incident state should be revised as x = (Tsi,Tsi)T, where T is the bosonic time-reversal operator T = o K (K is complex conjugate). These two configurations correspond to different Berry holonomies but give rise to identical physical observable-the same interference contrast in our experiment.

By attesting the commutability between a pair of time-reversed Berry holonomies, our approach provides a minimal one-dimensional scheme to examine two-dimensional

non-Abelian gauge connections or curvatures in real space. This scheme can also be generalized to parameter space and momentum space (e.g. to differentiate one-

dimenional or two-dimensional spin-orbit coupling in quntum gases).

Loose definition of non-Abelian gauge fields

Generally, the non-commutability [A,, A,] 0 is loosely referred to as non-Abelian gauge fields [50, 82]. The commutator of the two types of gauge potentials prepared in our experiment is

[AO, AO] = [0oy, #oz] = 2iBo_-z. (5.13)

If they are arranged as link variables along different directions on a lattice (for example, using symmetric gauge), 0 / 0 and## 0 correspond to non-Abelian guage fields.

131 Wilson loop

Another criterion for non-Abelian gauge fields is the gauge-invariant Wilson loop, i.e. the trace of the Berry holonomy. For N-fold degenerate system, a trivial Wilson loop |W|= N generally corresponds to Abelian gauge fields. For U(2) matrices, |W|= 2 if and on if W = e1I [84}, where I is the identity matrix andy is a U(1) phase factor (i.e. the scalar Berry phase). For SU(2) matrices, -y reduce to 0 or r.

The Berry holonomies W 1 [Eq. (5.10)] andW 2 [Eq. (5.11)] are linked via unitary gauge transformation W2 = UW UI,1 where U eUi(++h/2)ee(r+/2)". Therefore, W1 andW 2 share the same Wilson loop

2 2 W = TrW1 = TrW 2= 2 - 4cos sin #. (5.14)

W = 2-that corresponds to Abelian gauge fields-is given by the (6,#) configurations:

1. 6 = (m + 1/2)7r

2. # = n7

3. 6=m7 and #= (n+1/2)r, where m and n are integers.

It should be noted that the criterion based on non-trivial Wilson loops |WI N (for N-fold degenerate systems) is a necessary but insufficient condition for non- Abelian gauge fields [821. In our case, this is manifested by comparing the conditions for |W # N (Fig. 3-4d) and [W ,1 W 2 ] , 0 (Fig. 3-4b)-some (0,#) configurations with |WI / N still satisfy [W 1 , W 2] = 0 and are thus still Abelian.

Criteria relation

It would be helpful to illustrate the relation among the three aforementioned criteria for non-Abelian gauge fields. Based on our analysis above, the associated Venn diagram is shown in Fig. 5-9.

132 Figure 5-9: Venn diagram illustration of the relation among criteria for U(N) Abelian and non-Abelian gauge fields. The loose definition [A,, A,] # 0 (orange color) and a nontrivial Wilson loop |WI $ N (blue color) are both necessary but insufficient conditions for genuine non-Abelian gauge fields, which is defined by the non-commutability between two Berry holonomies W 1 andW 2 (green color).

Note that [W 1 ,W 2 ] 0 is not the intersection set of [A,, A,] h 0 and |WI N.

5.5.10 Multifrequency modulation scheme

If the four modulators adopt sawtooth modulation of the same frequency, the fringes of non-Abelian gauge fields will be manifested in the zero frequency of the projection detection. Since the zero frequency is more vulnerable to back scattering and environmental noises, we attribute 4 different modulation frequencies wi (i=1,2,3,4) to the sawtooth modulation described below for better experimental characterization

133 #h+ =wlt 2t - T)= (W1 - W2 )t + W 2 = (W1 - W2 )t + 22(W - w1)r12 +(W 2±+w)T12, (5.15a)

h-= -W 2t + W1(t -- ( 1 - w 2 )t - Wr= (W1 - W 2 )t + (W2 - wi)T/2-(W2 + w)12, (5.15b)

,+ = -w 3 t + w 4 (t - r) = (w4 - w 3 )t - w4r = (w4 - w 3 )t - (w4 - w 3)r/2 - (W4 + w3 )r/2, (5.15c)

w 4 t - W3 (t- T) = (W4 - W3)t + ±W3 = (W4 - w 3 )t - (W4 - w 3)r/2 + (W4 + w 3)r/2. (5.15d)

Here, #., (p E {h, v}, v C {+, -}) are the modulation phases for the |h) or Iv) component of the pseudospin in the forward (+) or backward (-) directions. We define the overall non-zero phase modulation frequency for horizontal and vertical polarization as Wh I1 - W2 and wv W4 - W3 . By comparing the forward and backward modulation phases for both polarizations, we find the reciprocal phases #rp are #rh = (2 - w1)r/2 and #rv = (W4 - W3)r/2. The nonreciprocal phases nrl, are enrh = (2 + wi)T/2 and nrv = (W3 + 4 )T/2. We impose an additional constraint-

W1+ W2 = W 3+w4-on the modulation frequency such that the o-, gauge structure can be maintained. By comparing with the definition of o-, gauge field strength # =_ r in the single frquency scheme (Fig. 3-3b), we find that in this multi-frequency scheme, W simply genearlizes as o = (W1 +W 2 )/2 = (W3 + 4 )/2. As a result, for either forward or backward directions, ih) and Iv) still obtain nonreciprocal phases of same magnitude but opposite signs.

134 The four phase terms can be thus rewritten as

Wh+ ht + rh + #, (5.16a)

Wh- ht + rh - #, (5.16b)

v+ Wvt - #rv - #, (5.16c)

v_ =Wvt - #rv + ,. (5.16d)

Under such a multifrequency modulation scheme, the scattering matrix of dynamic modulation [Eq. (5.5)] should be revised as

0 0 eidh+ 0

SQ (0 Q) 0 0 0 ei'v+ (5.17) QT o eidh- 0 0 0

0 el- 0 0

5.5.11 Contrast function

In this section, we derive the observable, the contrast function p, in our experiment

(setup shown in Fig. 3-3d). Here the analysis neglects various imperfections in the system, such as backscattering, polarization cross-talk, 1/f noise, etc.

We denote the initial pseudospin (Jones vector) as si = (a, be*) in the linear basis

1h) and |v), where a and b are positive, real numbers and 4E[-7r, 7) is the relative phase delay between the two components. The Jones vector relates to the latitude a and longitude # on the Poincar6 sphere as [200]:

2ab sin - 2 b2 sin4, (5.18a) 2ab tan #= 2 bVcos . (5.18b)

The incident state x is given by x = (si, si)T, which includes a pair of identical initial pseudospins si for the clockwise (CW) and counter-clockwise (CCW) paths in

135 the Sagnac loop. The scattering matrix S of the Sagnac loop is given by

0 P+Q S =( Q ) (5.19) QTPT 0

The output state y is

QTPTs, azs 0 y = Sx =f 07f (5.20) P+Q+si 0'_s50 where sf and sf are the final states exiting from the CCW and CW loops and are defined in Sec. 5.5.8. After that, the two states interfere and undergo projection detection of intensity in the linear basis. We define the intensity contrast function as

_F (|soq'|h) + so |h)|12 )6(W - Q) p = Ih" /I =f (5.21) F (Isp*v) +s Iv)12) 6(W Q)

Here F denotes the Fourier transform. Under the multifrequency modulation scheme (Sec. 5.5.10), the intensities Ih(v) contains multiple frequency components.

We are only interested in the intensities at the frequency component Q Wo - w ,, which provides us the observable for non-Abelian gauge fields.

Combining Eq.(5.20) and (5.21) yields

bsino [bsino + 2acos~cosocos(#+ ?p)] 2 + 4a2 cos 2 0cos 2 #sin2 # +#) p = a sin Ov[a sin 0 - 2b cos 0 cos # cos(# + 0)]2 + b2 cos2 0 cos2 sin 2 (#+0') (5.22)

As can be seen, The dynamic phases are not manifested in pp is only affected by the initial pseudospin and the synthetic magnetic fluxes (0, #). p is characterized by finite number of zeros and poles. Outside the square root in Eq. 5.22, the fixed zero and pole (Fig. 3-4b)-not affected by the synthetic magnetic fluxes (0, #) E

[-7r,ir) x [-r,7r)-are respectively given by b = 0 and a = 0, which correspond to

136 two orthogonal linearly-polarized pseudospins. The tunable zero and pole are given by the expressions inside the square roots. The condition for the tunable zero is

a_ tanO a- =co (5.23a) b 2 cos#0' S=7r -. (5.23b)

The condition for the tunable pole is

a 2cos# b tanGtan ' (5.24a) (5.24b)

Therefore, the tunable zero and pole are antipodal on the Poincar6 sphere. The corresponding latitude and longitude can be obtained by applying these conditions to Eq. 5.18. We next discuss the properties of the contrast function p along some high-symmetry lines-that correspond to Abelian gauge fields (Sec. 5.5.9)-on the synthetic torus.

1. 0 = mr, where m is an integer.

In this case, the tunable zero (pole) annihilates with the fixed pole (zero), since we have lim p= 1 for all possible pseudospins. This correspond to Abelian 0-+myr gauge fields and case Q in Fig. 3-4.

2.#= nr, where n is an integer. For linearly polarized light (i.e. 1r where 1

is an integer)

I b sinO(b sin0 k 2a cos 0) - a sin 0(a sin9 -F 2b cos 0). (5.25)

It means that the annihilation is lifted and there are two pairs of zeros and poles on the equator. (Fig. 3-4. This correspond to Abelian gauge fields and case U

in Fig. 3-4.

3. 0 = (m + 1/2)7r or= (n + 1/2)7r

137 p = b 2/a 2, meaning that two pairs of poles and zeros coalesce respectively, producing two second-order zero and pole on the two ends of the equator. This correspond to Abelian gauge fields and case V in Fig. 3-4.

5.6 Discussion

In summary, we demonstrate an experimental synthesis of non-Abelian gauge fields in the real space, which is confirmed by our observation of the non-Abelian Aharonov- Bohm effect using classical particles and classical fluxes. The realized gauge fields demonstrate a viable way to engineer the Peierls phase in the simulation of topological systems, such as the non-Abelian Hofstadter models [173, 84]. Our experiment also introduces non-Abelian ingredients for realizing high-order topological phases [23, 180, 210] and topological pumps [262, 147]. Besides, recent advances in on-chip modulation [235] and magneto-optical materials [27] could enable future observations of non-Abelian topology in integrated photonic platforms. Towards the quantum regime, non-Abelian gauge fields may be utilized to help generate non- Abelian anyonic excitation [35, 118, 257] to offer an alternative, synthetic approach for topological quantum computation. Finally, the synergy of non-Abelian gauge fields with engineered interactions (e.g. bosonic blockade and superconducting qubits) may enable the realization of many-body physics such as the non-Abelian fractional quantum hall effect.

138 Chapter 6

Conclusion and Outlook

In this thesis, we studied several electromagnetic scattering phenomena, both in the near- and far-field, as well as for real structures and synthetic gauge fluxes. In Chapter 2, we theoretically proposed strategies for improving radiative efficiency in plasmonics. These strategies include optically thin films and hybrid dielectric plasmonic resonators. In Chapter 3, we presented a general framework for nanoscale electromagnetism and the associated dark-field scattering experiment for the measurement of surface response functions known as Feibelman d parameters. In Chapter 4, we treated spontaneous free-electron radiation as electromagnetic near-field scattering and derived an radiation upper limit, verified by measurement on the Smith-Purcell radiation. In Chapter 5, we observed the non-Abelian Aharonov-Bohm effect as a synthetic scattering phenomena.

A few directions emerge for future study.

For nonclassical plasmonics, it is important to measure the surface response functions of important plasmonic interfaces, using various approaches. These include optical and electronic excitations, such as near-field scanning optical microscopy, electron energy loss spectroscopy, and fluorescence measurements. It would also be useful to develop technologies based on these nonclassical effects. The potential technologies include widely-tunable color generation based on nonclassical spectral shifts achieved by nanomechanics and tunneling-controlled photoemission controlled by external bias.

More broadly speaking, one could apply the mesoscopic framework to other nanoscale

139 electromagnetic problems such as near-field hear transfer and fluctuational electrodynamics.

For free electron radiation, one should aim at approaching radiation upper limit with judiciously designed experiments. One central unanswered question for nan- ophotonic free-electron sources is to overcome the transverse momentum mismatch between electrons and photonic modes to improve the radiation intensity. The recent flourish of topological photonics may provide new insights for this purpose. It is also worthwhile investigating these effects in the stimulated radiation regime, where the influence of optical modes on electrons becomes non-negligible. For synthetic gauge fields, a natural direction is to generalized the interference measurement into lattice models and experimentally study single- and many-body (with effective interaction incorporated) topological phenomena in lattice gauge theories. In the single-particle regime, it would be interesting to construct first-order and high-order photonic topological insulators. It is important to answer the question of whether photons are able to realize fermionic topological phases through the internal degrees of freedom in non-Abelian gauge fields. It is also interesting to enter the quantum optics regime and test how entangled photon pairs interact with non- Abelian gauge fields. In the many-body regime, non-Abelian gauge fields could be used as link variables for a lattice of interacting qubits; this setup may enable many exotic phenomena such as non-Abelian fractional quantum hall effects and synthetic non-Abelian anyons.

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