Modeling and Applications of Nonlinear Metasurfaces
by
Xiaojun Liu
Department of Electrical and Computer Engineering Duke University
Date:______Approved:
______David R. Smith, Supervisor
______Steven A. Cummer
______Qing H. Liu
______Nan M. Jokerst
______Willie J. Padilla
Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Electrical and Computer Engineering of Duke University
2018
ABSTRACT
Modeling and Applications of Nonlinear Metasurfaces
by
Xiaojun Liu
Department of Electrical and Computer Engineering Duke University
Date:______Approved:
______David R. Smith, Supervisor
______Steven A. Cummer
______Qing H. Liu
______Nan M. Jokerst
______Willie J. Padilla
An abstract of a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Electrical and Computer Engineering of Duke University
2018
Copyright by Xiaojun Liu 2018
Abstract
A patterned metasurface can strongly scatter incident light, functioning as an extremely low-profile lens, filter, reflector or other optical devices. Nonlinear metasurfaces‒combine the properties of natural nonlinear medium with novel features such as negative refractive index, magneto-electric coupling‒provide novel nonlinear features not available in nature. Compared to conventional optical components that often extend many wavelengths in size, nonlinear metasurfaces are flexible and extremely compact.
Characterization of a nonlinear metasurface is challenging, not only due to its inherent anisotropy, but also because of the rich wave interactions available. This thesis presents an overview of the work by the author in modeling and application of nonlinear metasurfaces. Analytical methods - transfer matrix method and surface homogenization method - for characterizing nonlinear metasurfaces are presented. A generalized transfer matrix method formalism for four wave mixing is derived, and applied to analyze nonlinear interface, film, and metallo-dielectric stack. Various channels of plasmonic and Fabry-perot enhancement are investigated. A homogenized description of nonlinear metasurfaces is presented. The homogenization procedure is based on the nonlinear generalized sheet transition conditions (GSTCs), where an optically thin nonlinear metasurface is modeled as a layer of dipoles radiating at
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fundamental and nonlinear frequencies. By inverting the nonlinear GSTCs, a retrieval procedure is developed to retrieve the nonlinear parameters of the nonlinear metasurface. As an example, we investigate a nonlinear metasurface which presents nonlinear magnetoelectric coupling in near infrared regime. The method is expected to apply to any patterned metasurface whose thickness is much smaller than the wavelengths of operation, with inclusions of arbitrary geometry and material composition, across the electromagnetic spectrum.
The second part presents the applications of nonlinear metasurfaces. First, we show that the third-harmonic generation (THG) can be drastically enhanced by the nonlinear metasurfaces – film-coupled nanostripes. The large THG enhancement is experimentally and theoretically demonstrated. With numerical simulations, we present multiple ways to clarify the origin of the THG from the metasurface. Second, the enhanced two-photon photochormism is investigated by integrating spiropyrans with film-coupled nanocubes. This metasurface platform couples almost 100% energy at resonance, and induces isomerization of spiropyrans to merocyanines. Due to the large
Purcell enhancement introduced by the film-coupled nanocubes, fluorescence lifetime measurements on the merocyanine form reveal large enhancements on spontaneous emission rate, as well as high quantum efficiency. We show that this metasurface platform is capable of storing information, supports reading and writing with ultra-low power, offering new possibilities in optical data storage.
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To my beloved parents and family, for their love, support and encouragement.
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Contents
Abstract ...... iv
List of Tables ...... x
List of Figures ...... xi
Acknowledgements ...... xv
1. Introduction...... 1
1.1 Metasurfaces ...... 2
1.2 Nonlinearity ...... 3
1.2.1 Nonlinear polarizations ...... 4
1.2.2 Nonlinear processes ...... 5
1.3 Modelling ...... 7
2. Modeling of a nonlinear surface, film, and stack ...... 10
2.1 Analytic expression: an interface ...... 11
2.2 Transfer matrix method: a film ...... 18
2.2.1 Transfer matrix method ...... 19
2.2.2 Nonlinear Polarizations ...... 22
2.2.3 Numerical validation ...... 27
2.2.4 A thin film in the kreschmann configuration ...... 29
2.3 Transfer matrix method: a multilayer stack ...... 32
2.4 Conclusion ...... 35
3. Modeling of nonlinear metasurfaces ...... 40
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3.1 Effective surface description of metasurfaces...... 41
3.1.1 Universal Boundary Conditions ...... 44
3.1.2 Average fields across a surface...... 45
3.1.3 Generalized sheet transition condition ...... 48
3.1.4 Nonlinear generalized sheet transition condition ...... 50
3.2 Nonlinear surface parameter retrieval ...... 55
3.2.1 Retrieval method ...... 55
3.2.2 Connection between surface parameters and bulk parameters ...... 60
3.3 Applications ...... 66
3.3.1 Nonlinear slab ...... 67
3.3.2 Magnetoelectric nonlinear metasurfaces ...... 71
3.4 Conclusion ...... 81
4. Enhanced nonlinear response from metasurface platforms ...... 84
4.1 Third-harmonic generation enhancement ...... 85
4.1.1 Geometry and method ...... 86
4.1.2 Result and Discussion ...... 90
4.1.3 Conclusions ...... 92
4.2 Clarifying the origin of THG from an isolated film-coupled nanostripe ...... 93
4.2.1 Geometry and method ...... 94
4.2.2 Results and discussions ...... 96
4.2.3 Generality ...... 103
4.2.4 Conclusion ...... 106
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5. Enhanced two photon photochromism in metasurface perfect absorbers ...... 108
5.1 Overview ...... 109
5.1.1 Nanoantennas for memory devices ...... 110
5.1.2 Photochromism for memory devices ...... 111
5.2 Enhanced two-photon photochromism in film-coupled nanocubes ...... 113
5.3 Methods ...... 115
5.3.1 Fabrications ...... 115
5.3.2 Simulations ...... 117
5.3.3 Optical measurements ...... 118
5.4 Results and Discussions ...... 120
5.4.1 Purcell Enhancement ...... 120
5.4.2 Enhanced two-photon absorption ...... 123
5.4.3 Power dependence study ...... 124
5.4.4 Capability of storing information ...... 126
5.4.5 Specificity of wavelength ...... 128
5.5 Conclusion ...... 130
6. Conclusions ...... 132
Appendix A...... 134
Appendix B ...... 136
Appendix C ...... 141
References ...... 147
Biography ...... 162
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List of Tables
Table 4-1: Nonlinear sources and their equivalent dipole sources ...... 103
x
List of Figures
Figure 2-1: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) (a) a single air-metal interface; (b) a single metal layer; (c) multi- layer stack with n layers in total and r layers of metal; (d) silver-dielectric-silver structure...... 13
Figure 2-2: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) Norms of the FWM field generated from an air-silver interface as a function of excitation angle...... 17
Figure 2-3: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) (a, b) Norms of the reflected (red, circle) and transmitted (blue, square) FWM fields generated from a silver film in air as the function of film thickness. (c, d) Norms of the reflected (c) and transmitted (d) FWM fields from a 20nm-silver-film in air as the function of the incidents angles (a, b ) of pumping fields...... 27
Figure 2-4: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) (a, b) Norms of the reflected (a) and transmitted (b) FWM fields generated from a silver film in the Kretschmann configuration as the function of incidents angles (a, b ) of pumping fields...... 29
Figure 2-5: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) (a) Norms of the reflected FWM fields generated from a silver- dielectric-silver structure in air as the function of dielectric thickness d2. (b) Norms of the reflected FWM fields from a silver-dielectric-silver structure in air as a function of the incidents angles (a, b ) of pumping fields...... 34
Figure 3-1: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) (a) A nonlinear metasurface. (b) Each metasurface elements are equated to an electric dipole (blue arrow) and a magnetic dipole (red arrow) induced in each of the metamaterial elements. (c) The surface polarization and magnetization are averages of the discrete electric and magnetic dipoles over the metasurface...... 42
Figure 3-2: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Retrieved linear surface susceptibilities of a SiO2 slab...... 67
Figure 3-3: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Retrieved nonlinear surface susceptibilities of a SiO2 slab...... 70
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Figure 3-4: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) A nonlinear metasurface consists of arrays of split ring resonators (SRRs) pairs positioned opposite to each other...... 72
Figure 3-5: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Linear spectrum of the nonlinear metasurface excited by TM polarized wave at (a) normal incidence; (b) oblique incidence (30o)...... 72
Figure 3-6: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Retrieved linear surface susceptibilities of the nonlinear metasurface...... 74
Figure 3-7: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) (a) Reflection and (b) transmission SHG spectrum of the nonlinear metasurface excited by TE polarized wave at 0 degree (orange-dash) and 30 degree (blue-solid). (c) The phase of the SHG transmission (blue-solid) and reflection (orange-dash)...... 75
Figure 3-8: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Retrieved second-order surface susceptibilities as a function of SHG wavelength. 79
Figure 3-9: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier 2 2 B.V.) Second-order surface susceptibilities EEEs, yyy (left), EEMs, yyz (middle), and 2 EMMs, yzz (right) as a function of SHG wavelength...... 80
Figure 3-10: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Comparison between simulation and recovered SHG spectrum at 0 degree (first column), 30 degree (second column), and 45 degree (third column) using retrieved second-order surface susceptibilities. (Top row) SHG reflection; (Bottom row) SHG transmission. (Lines) Simulation; (Circles) Recovered SHG spectrums...... 81
Figure 4-1: (Reproduced with permission from Ref. [24]. Copyright (2015) by Optical Society of America.) (a) Geometry of the film-coupled nanostripe. EV gold: evaporated gold; TS gold: template-stripped gold. (b), (c) Magnetic and electric field distributions at the fundamental resonance of the system (1.5 μm); arrows represent the direction of magnetic and electric fields. (d), (e) Magnetic and electric field distributions at a higher- order resonance...... 86
Figure 4-2: (Reproduced with permission from Ref. [25]. Copyright (2014) by American Chemical Society.) Experimental (blue) and simulated (black) reflectance spectra for each of the stripe samples used in the third-harmonic generation experiment. Each
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panel shows a representative spectrum from the sample with gap size indicated. Insets: SEM images of the stripe samples corresponding to the spectra (scale bar = 500 nm). .... 88
Figure 4-3: (Reproduced with permission from Ref. [25]. Copyright (2014) by American Chemical Society.) Schematic diagram of experimental setup for measurement of THG. Inset: representative THG spectrum measured from film-coupled stripe sample...... 89
Figure 4-4: (Reproduced with permission from Ref. [25]. Copyright (2014) by American Chemical Society.) Third harmonic generation enhancement (as compared to a bare gold film) vs. gap size (g)...... 91
Figure 4-5: (Reproduced with permission from Ref. [24]. Copyright (2015) by Optical Society of America.) (a) – (c) THG enhancement factor for the THG from the Al2O3 spacer, the gold stripe and the gold film, respectively. (d) – (f) Total THG enhancement factor evolving with the ratio of the third-order susceptibilities between the Al2O3 spacer and the gold...... 98
Figure 4-6: (Reproduced with permission from Ref. [24]. Copyright (2015) by Optical Society of America.) (a) – (c) Near field distributions of the THG generated from the Al2O3 spacer. (d) Far field radiation patterns of the THG generated from the Al2O3 spacer radiating in vacuum (red), being reflected by a gold film (black), being coupled to the full structure (blue). (e) – (g) Near field distributions of the THG generated from the gold stripe. (h) Far field radiation patterns of the THG generated from the gold stripe radiating in vacuum (red), being reflected by a gold film, being coupled to the full structure (blue). (i) – (k) Near field distributions of the THG generated from the gold film. (l) Far field radiation patterns of the THG generated from the gold film radiating in vacuum (red), being reflected by a gold film (black), being coupled to the full structure (blue)...... 99
Figure 4-7: (Reproduced with permission from Ref. [24]. Copyright (2015) by Optical Society of America.) (a), (b) The THG enhancement factor for THG from the Al2O3 spacer, the gold stripe and the gold film as a function of gap size, at radiation angles of 90 degree (a) and 45 degree (b). (c), (d) The total THG enhancement factor evolving with the ratio of the third-order susceptibilities between the Al2O3 spacer and the gold, at radiation angles of 90 degrees (c) and 45 degrees (d)...... 104
Figure 5-1: Photochromism of spiropyran...... 112
Figure 5-2: Film-coupled nanocubes (a) Schematic of silver nanocubes deposited on a gold film coated sipropyran/PMMA blend and PE layers. (b) Reflectance spectrum of
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film-coupled 110nm silver nanocubes resonance around 792nm. The reflectance is 3% at resonance. (c) Magnetic field distribution in the gap region when at resonance. (d) Scheme of read and write using spiropyrans...... 113
Figure 5-3: Fabrication procedure of film-coupled nanocubes with spiropyran/PMMA blend within the gap region ...... 116
Figure 5-4: fluorescence lifetime imaging system setup ...... 119
Figure 5-5: IRF by scattering laser light into the PMT...... 120
Figure 5-6: Enhancement of spontaneous emission rate and quantum efficiency relative to a dipole in free space as a function of position under the nanocube...... 120
Figure 5-7: time-resolved emission of merocyanines from (a) film-coupled nanocubes; (b) gold substrate; (c) silicon substrate. Blue lines: before two-photon absorption. Red lines: after two-photon absorption. Yellow lines: after UV exposure...... 122
Figure 5-8: Power dependence study at (a) excitation wavelength (583nm), and (b) pump wavelength (792nm). A = 5.06*103, B = 1.30*105...... 124
Figure 5-9: (a) SEM image of the pattern sample. (b) Fluorescent intensity image in log scale before two-photon absorption. (c) Fluorescent intensity image in log scale after two-photon absorption...... 126
Figure 5-10: Specificity of wavelength of the system. (a) The reflectance spectrums the film-coupled nanocubes with various thickness of the PE layers. (b) Absorptions at 792nm (circles) and 650nm (stars) of samples with different resonance wavelengths in (a). (c) Fluorescent enhancement of various samples with different resonance wavelengths in (a). The enhancement is calculated by the ratio between the fluorescent intensity after and before the two-photon absorption...... 128
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Acknowledgements
I first want to thank my PhD supervisor, David Smith, who provided me with the opportunity and guidance to pursue this research.
I also want to acknowledge Stéphane Larouche, Alec Rose, Britt Lassiter, and
Xiaomeng Jia. These works would not have been possible without their inventive support and contributions.
The works presented here were supported by the Air Force Office of Scientific
Research.
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1. Introduction
With the development of information technology and computer engineering, there is an increasing demand on high-performance computing unit, interconnects, transistors, and data storage devices. Simultaneously, these demanding applications require reduced dimension. To date, microprocessor chips have kept the pace of the
Moore’s law. In 1985, 1μm in CMOS fabrication process was the state of the art. By 2004, it was moved down to 90nm. However, the doubling starts tapering off. Electronic components suffer from heat, limiting their application in densely integrated computing unit and next generation of super-computers. Substituting electric components with optical components provides an alternative route to high-performance, high-throughput computing chips. Light pulses are capable of transmitting large amount of data at high speed with diminished thermal issue. Integrating optical components side-by-side with the electric chips paves the highway toward extreme compact, super-fast devices.
Furthermore, due to their high speed, wide bandwidth, and low energy consumption, on-chip optical devices, such as photonic integrated circuit, even have the possibility to completely replace the electronics.
Optical components and devices often extends many wavelengths in size. The light-matter interactions take place and accumulate along the optical path, leading to the modulation of the wave-front or the generation of new frequencies that facilitates certain photonic functionality. Metamaterials offer an alternative to ultra-compact optical
1
components. With artificially designed meta-atoms, light are squeezed and engineered in a scale much shorter than the working wavelength. By properly arranging the meta- atoms into metamaterials, exotic electromagnetic properties that are not available in nature can be achieved. For example, the negative refractive index, zero-refractive index and hyperbolic anisotropy have raised great interest in the community. Furthermore, by integrating tunable components into the meta-atoms, the dynamic properties of natural materials are combined with the exotic electromagnetic properties from the metamaterials, leading to much greater flexibility in engineered materials.
1.1 Metasurfaces
Though metamateirals have drawn large amount of research interest, their properties mostly rely on bulk response from spatially arranged meta-atoms. For on- chip applications, a single layer or several layers of meta-atoms are desirable.
Metasurfaces are unique platforms for controlling the propagation of electromagnetic waves that can break this limitation and scale the light-matter interactions down to subwavelength regime. A metasurface generally consist of an optically thin layer of metamaterials or nanoscatterers such as nanoparticles, whose sizes are much smaller than the wavelength. Each meta-atom or nanoscatterer functions as an optical antenna that is able to confine light and introduce abrupt amplitude and phase change in wavefront. The meta-atoms and nanoscatterers are flexible in shape, composite materials, and periodicity, resulting at enough freedom in controlling of the local field
2
distribution and wave propagation. Similar to metamaterials, novel features, such as reconfigurable light pathways, beamforming and on-chip optical radiation, can be achieved by integrating dynamically tunable components, gain medium, or nonlinear materials with metasurfaces.
1.2 Nonlinearity
Nonlinear optical effects play an important in the implementation of various photonic functionalities, such as ultra-fast switching, optical signal processing, optical communications and optical data storage. Any natural material possesses some nonlinearities when interacting with strong enough electromagnetic field. However, nonlinearities from nature are inherently weak. To obtain high efficiency, the nonlinear response from a medium that extends over many wavelengths in size is often required.
Integrating inherently nonlinear materials into otherwise linear metasurfaces can produce nonlinear metamaterial composites, which are compelling due to their ability to enhance [1-4] and control nonlinear processes in ways not possible with conventional nonlinear crystals, polymers and other materials. Furthermore, the inherent large nonlinear susceptibilities of the materials constitute the meta-atoms, enables new possibilities for phase matching [5-8] and many other nonlinear phenomena [9-15].
Particularly, metals, which are the major constitute of meta-atoms, offer an interesting trade-off in the context of nonlinear metasurfaces. On the one hand, the already substantial nonlinear susceptibilities of metals can be further boosted by the
3
incredibly large localized enhancements of local fields near metallic surfaces. On the other hand, bulk metals do not support propagating waves by themselves and showcase significant optical losses. In short, the wave propagation characteristics of metals make long interaction lengths impossible at optical frequencies, but their localized and nonlinear properties are promising for achieving high-efficiencies at sub-wavelength scales.
1.2.1 Nonlinear polarizations
To describe the nonlinearities of materials, we start by introducing the nonlinear polarizations and magnetizations. The induced polarization and magnetization of a material interacting with strong electric or magnetic field are given by [16]:
1 2 3 t t t t ..., (1.1)
1 2 3 t t t t ..., (1.2) where the superscripts indicate the dependence of the order of the fields. For natural materials, the magnetic responses are generally diminished:
t1 t 2 2 t 3 3 t ... , 0 (1.3)
t 0 t, (1.4)
n where are n-th order susceptibility.
Nonlinear metasurfaces exhibit richer wave interactions. The linear polarization and magnetization of a metamaterial or a metasurface are expressed as [17]
4
1 1 1 0 e i em H , (1.5)
1 1 1 m i em , (1.6)
1 where the fields in Eq. (1.1) and (1.2) are expanded in frequency domain. ’s are
1 rank-2 linear susceptibility tensors, and ’s are rank-2 linear magneto-electric coupling tensors. With wave interactions between electric and magnetic fields, the nonlinear polarizations and magnetizations presents more nonlinear tensor elements than natural materials. For example, the second-order polarization of a nonlinear metasurface is
2 ;,: 2 ;,: 2 eeesqr q r emmsqr q r , (1.7) 0 2 2 qr ;,: ;,: eemsqr q r emesqr q r and second-order magnetization is
2 ;,: 2 ;,: 2 meesqr q r mmmsqr q r , (1.8) 2 2 qr ;,: ;,: memsqr q r mmesqr q r
2 where ’s are rank-3 second-order susceptibility tensors describing second-oder interactions as well as the second-order magneto-electric couplings. A detailed expression for all possible nonlinear interactions can be found in Chapter 3.
1.2.2 Nonlinear processes
Harmonic generations refer to nonlinear processes that doubles, triples, or multiples the frequency of the fundamental field. Second-order nonlinear response, and
5
second-harmonic generation, in particular, are associated with non-centrosymmetric materials, or materials without inversion symmetry. They have been extensively studied for metallic surfaces and nonlinear metasurfaces consist of meta-atom with broken symmetry [18-20]. The third-order nonlinear response of metals has equally received considerable attention due to the large bulk nonlinear coefficients. As the meta-atoms or metallic nanostructures have the potential for strongly localize field in proximity of their surface, the large bulk nonlinearity can also be utilized in nonlinear metasurfaces.
Recent studies have demonstrated enhanced third-order responses from a wide range of nonlinear metasurfaces, including nanoantenna arrays[21, 22], plasmonic gratings[23], and film-coupled nanostructures[24, 25].
Nonlinear frequency mixing results from the coherent summation of the nonlinear response of individual atoms or molecules. Metasurfaces can mimic this procedure by overlapping the pumping fields within the meta-atoms. The ability to independently design the anisotropy, frequency dispersion and field enhancement of a metasurface at the fundamental and harmonic or mix frequencies enables new possibilities for phase matching [5-8] and many other nonlinear phenomena [9-15]. If the field enhancement is large enough, metasurfaces can provide a significant nonlinear response, potentially serving as a multifunctional optical device. Numerous studies have revealed the advantages of metasurfaces – allowing control over the wavefront phase, amplitude, and polarization over a drastically reduced propagation length [26-30].
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Nonlinear metasurfaces not only preserve the advantages associated with their linear properties, but also introduce large nonlinear enhancement in a reduced interaction scale [18, 19, 23, 24, 31-33].
1.3 Modelling
When the metasurface is patterned uniformly, its linear optical properties can be expressed using effective surface electric and magnetic polarizabilities obtained through a homogenization procedure. The interaction between electromagnetic waves and a nonlinear metasurface has been extensively studied in many different contexts. Pior theoretical studies of waves interacting with nonlinear metasurfaces have relied primarily on numerical simulations, limiting the generalized characterization of nonlinear efficiency and hindering the prediction of nonlinear phenomena. The homogenized description of a nonlinear metasurface, however, presents challenges both because of the inherent anisotropy of the medium as well as the much larger set of potential wave interactions available, making it challenging to assign effective nonlinear parameters to the otherwise inhomogeneous layer of metamaterial elements.
Analytical methods recently applied for characterizing nonlinear metamaterials and metasurfaces include microscopic coupled-mode theory and effective medium theory [10, 34-36]. Microscopic coupled-mode theory, which requires the numerical integration of all local fields within the repeated cell of a periodic array, has been successful for evaluating the effective nonlinear susceptibility of well-known
7
metamaterial structures, such as arrays of spheres, and many other systems [10]; however, the use of coupled-mode theory for arbitrary metamaterial elements hinges on the numerical simulation and subsequent integration of all local fields.
Alternatively, effective medium theory, which provides a homogenized description of metamaterials and metasurfaces, avoids the need to integrate the detailed local fields, instead relying on the simulation of scattering parameters from a metamaterial layer of finite thickness. Retrieval methods for linear materials based on effective medium theory, in which the properties of planar metamaterial slabs are determined by comparing their scattered fields with those from equivalent homogeneous materials, have been well-studied and are now commonplace, often referred to as scattering (S-) parameter retrieval methods [37-39]. S. Larouche et al. and A.
Rose et al. extended the S-parameter retrieval method to nonlinear metamaterials [35, 36], showing that both effective bulk second- and third- order susceptibilities could be determined. However, this method has only been demonstrated at normal incidence, for which those tensor elements of the nonlinear susceptibilities that describe wave scattering at oblique incidence are left undetermined. Furthermore, it introduces an ambiguity for metasurfaces consisting of just one structured layer. Though the scattered fields can be calculated for a metasurface, the assignment of bulk constitutive parameters (permittivity and permeability) based on inverting the S-parameters cannot be uniquely defined without assigning an effective thickness [40, 41]. Though it is
8
possible to assign an arbitrary thickness to the layer—often the same distance as the lattice constant within the plane of metamaterial elements—there is an inevitable ambiguity in the intrinsic properties caused by trying to apply a volumetric description to a metasurface.
To address the complexity of nonlinear metasurfaces, in the thesis I discuss my effort toward modelling the nonlinear metasurfaces. I start by modeling the nonlinear process from a metallic surface, a metallic film, and a metallo-dielectric stack in chapter
2. A step-by-step analysis of four-wave mixing based on transfer matrix formalism is developed. With analytic expressions as well as numerical simulations, a complete analysis of various channels of plasmonic and Fabry-Pérot enhancement is presented. In chapter 3, I develop a homogenization procedure of nonlinear metasurfaces based on generalized sheet transition conditions. The analytic expressions can be inverted to retrieve the effective nonlinear surface susceptibilities of the nonlinear metasurfaces.
Next, I apply the principles of nonlinear metasurfaces engineering by studying various nonlinear procedures in nonlinear metasurfaces. In chapter 4, the enhanced third- harmonic generation in film-coupled nanostripes is investigated. Additional efforts are made to clarity the origin of the nonlinearity of this metasurface platform. In chapter 5, I present the experimental study of enhanced nonlinear absorption of spiropyrans in film- coupled nanocubee, which have potential application in optical data storage. Chapter 6 summarizes the presented material and my outlook for nonlinear metasurfaces.
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2. Modeling of a nonlinear surface, film, and stack
Even a system as minimal as a bare metallic film can possess rich and counter- intuitive dynamics. Before turning to more complex and fabrication-intensive geometries, such as corrugated and structured surfaces, it can be beneficial to understand the nonlinear optics of metal films, and the related problem of laminar stacks, in greater detail. Despite their simplicity, such one-dimensional systems are capable of supporting surface plasmons, Fabry-Pérot resonances, and photonic bandgaps, while remaining analytically approachable, offering a convenient starting point for evaluating the potential for metals in integrated nonlinear devices.
The goal of this chapter to develop an intuitive and convenient description of four-wave mixing (FWM) in one-dimensional stacks of metals and dielectrics. In detail, section 2.2 begin with the derivation of FWM at a single interface. Though surface plasmons cannot be directly excited by plane waves incident on a solitary surface, their nonlinear interaction can generate frequency mixed surface plasmons at greatly enhanced conversion rates. Section 2.3 extends these expressions into a transfer-matrix based formalism to quantitatively study FWM excited at oblique angles in metallo- dielectric multilayers. This method assumes continuous wave excitation in the non- depleted pump limit, and is ideal for parametrical investigation of various laminar systems. As a demonstration, we study a thin metal film both in air and over a substrate,
10
highlighting various plasmonic features of the FWM spectrum. Section 2.4 takes a brief look into more complex multilayer systems, investigating Fabry-Pérot enhancements by inserting a dielectric between two metal films. In this way, we offer a unified approach to understanding the channels of localization-induced enhancement of nonlinear activity in layered metallic structures.
This chapter is reproduced with permission from Xiaojun Liu, Alec Rose,
Ekaterina Poutrina, Cristian Ciracì, Stéphane Larouche, and David R. Smith, JOSA,
11(30), 2999-3010 (2013). Copyright (2013) by Optical Society of America.
2.1 Analytic expression: an interface
Before considering FWM, it is important to clarify our notation and conventions.
Throughout this chapter, the exp j t time convention is assumed. Frequencies are either defined in parentheses, or by “a, b, c, nl” superscripts. For clarity, we drop the explicit frequency labels in equations where all parameters are in terms of the same frequency. All the inhomogeneous source fields and corresponding transfer matrices are notated with subscript “s”, except the wave vectors of source fields are denoted by
QQx x Q z z .
We start by expanding the real electric field and polarization in the frequency domain,
1 t 0 qexp jt q (2.1) 2 q
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1 t 0 qexp jt q (2.2) 2 q
* * * where q q , q q , q q and kq k q , 0 and 0 are the zero-frequency components, and the summations are taken over all negative and positive frequencies.
Next, we consider the third-order interaction between three monochromatic
plane waves. Labeling the generated frequency nl and the pumping fields a ,
b , and c , we can write the third-order polarization as
3 1 3 Pi nl 0 ijkl nl;,,, a b c EEE j a k b l c (2.3) 4 abc jkl
3 where ijkl,,,,, xyz , ijkl is the corresponding tensor element of the third-order
3 3 nonlinear susceptibility tensor, , and nl a b c . In general, has 81 elements. For an isotropic nonlinear medium, only 21 non-zero elements exist, with only
3 being independent. In particular, from group theory, the following is true[42]:
(3) (3) (3) xxxx yyyy zzzz (3) (3) (3) (3) (3) (3) yyzz zzyy zzxx xxzz xxyy yyxx (3) (3) (3) (3) (3) (3) yzyz zyzy zxzx xzxz xyxy yxyx (2.4) (3) (3) (3) (3) (3) (3) yzzy zyyz zxxz xzzx xyyx yxxy (3) (3) (3) (3) xxxx xxyy xyxy xyyx
Thus, in an isotropic nonlinear medium, the third-order polarization at nl reduces to
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3 D 3 a b c a c b b c a Pi nl 0 i i i , (2.5) 4
1 where we have taken 3 (3) (3) (3) (3) , and D is the total number of 3 xxxx xxyy xyxy xyyx permutations of the frequency indices.
Figure 2-1: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) (a) a single air-metal interface; (b) a single metal layer; (c) multi-layer stack with n layers in total and r layers of metal; (d) silver-dielectric-silver structure.
Now we are in a position to consider FWM at a single interface, as in Figure 2-1
(a). As shown in the Appendix B, the magnetic and electric source fields produced by
3 the third order polarization nl are given by
3 jnl r, nl sr,, nl 2 2 (2.6) Q nl nl nl
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3 3 r, 2 r, nl nl nl nl nl sr,, nl 2 2 (2.7) Q nl nl nl where Q is the wave vector of the source fields and the nonlinear polarizations. These fields represent the inhomogeneous solutions to the nonlinear wave equation. From the
a b c boundary condition and momentum conservation, Q has components: Qx k x k x k x ,
a b c Qz k z k z k z .
We can simplify our analysis by considering the TE and TM polarizations separately. For a nonlinear polarization in the plane of incidence (generated by TM
linear field), s is along the y axis and s lies in x-z plane. Considering both source
fields and the subsequent homogenous fields at frequency nl , the boundary conditions at the interface of media 1 and 2 can be expressed as
nl nl H1y H 2 y H sy , (2.8)
nl nl E1x E 2 x E sx . (2.9)
The spatial variation of the fields can be decomposed into forward and backward plane waves, such that
r,q A exp j krA exp j kr exp jt q . (2.10)
In this configuration, there are no forward propagating nonlinear fields in medium 1 and no backward fields in medium 2. Dropping the “nl” superscript, Eqs. (2.8) and Eq.
(2.9) reduce to 14
1kE 1zx 1 2 kEH 2 zy 2 sy , (2.11)
E1x E 2 x E sx . (2.12)
3 3 As H sy , Esx can be expressed in terms of Px and Pz , we can derive the expression for
the x component of the reflected third-order nonlinear field E1x by solving Eq. (2.11) and
Eq. (2.12):
2 kPk3 PQPQQ 3 3 PQPQk 3 3 1zx 2 xxzzxxzzxz 2 E 1x 2 2 (2.13) 2k 1z 1 kQk 2 z 2
Subsequently, we calculate H y and derive the z component of the reflected nonlinear
field E1z from Maxwell-Faraday equation:
2 kPk3 PQPQQ 3 3 PQPQk 3 3 1xx 2 xxzzxxzzxz 2 E . 1z 2 2 (2.14) 2k 1z 1 kkQ 2 z 2
Since we are interested in plasmonic enhancement of FWM, it is instructive to consider explicitly the condition for surface plasmon excitation. For a single interface, the linear reflection coefficient is given by
k k R 1 2z 2 1 z (2.15) 1k 2z 2 k 1 z
Surface plasmon excitation corresponds to the pole in R, or 1k 2z 2 k 1 z 0 . For silver-
air interface, 2 0 , and this is just the usual surface plasmon dispersion relation
15
1 2 kx (2.16) 1 2 c
This implies a transverse wave vector that is too large to be excited directly from
medium 1. However, for a degenerate FWM process with nl2 a b , the generated wave has a transverse wavevector determined from momentum conservation,
nl a b kx2 kk xx (2.17) which can have a magnitude greater than the free-space wave-vector. Thus, plasmonic enhancement of FWM at a single interface occurs when momentum conservation satisfies the surface plasmon dispersion relation. For the nonlinear polarization parallel to the y axis (generated by TE linear fields), we can follow a similar procedure.
However, no surface plasmons can be excited in this case.
As an example, we consider FWM at an air-silver interface (Fig.1 (a)). The permittivity and permeability of silver are taken from Drude model, and its bulk
2 nonlinear susceptibility 3 is assumed isotropic and is 2.8 / 3 1019 m / V [42]. The
system is excited by two TM-polarized pumping fields whose frequencies are a and b , respectively. From energy conservation, the two pumping fields excite FWM fields at
frequencies nl2 a b and nl2 b a . In our calculation, we consider the FWM
at frequency nl2 a b , and assume field amplitudes Ea E b 40 MV/m , and
wavelengths a 628 nm, b 780 nm. a and b are the incident angles of a and b , respectively. The total number of permutations D=3. 16
Figure 2-2: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) Norms of the FWM field generated from an air-silver nl interface as a function of excitation angle. Solid lines represent k1z 0 Dashed lines denote the surface plasmon excitation condition derived from Eq.(2.16)
Figure 2-2 shows the norms of the generated FWM fields as a function of the incident angles of the pumping fields and analytical curves for the conditions of total internal reflection and surface plasmons excitation, respectively. The solid lines
represent k1z 0 , indicating the transition of generated FWM fields from propagating to
evanescent. The dashed lines correspond to the excitation of surface plasmons atnl , where the strongest FWM fields are seen. However, since the generated wave is bound to the surface, extraction becomes difficult. Moreover, the pumping fields are not localized at all in this configuration and so do not contribute to the enhancement of nonlinear process. In order to address these shortcomings, we must add additional complexity.
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2.2 Transfer matrix method: a film
For two or more interfaces, the analytic approach of the previous section becomes prohibitively cumbersome. Alternatively, a number of mathematic approaches, such as the effective nonlinear susceptibility method [44], the Green function method
[45], and the transfer matrix method [46-48] have been developed to calculate the second-order nonlinear response (especially second-harmonic generation) of one- dimensional multi-layer structures. The calculation for FWM at metal surfaces and metal-dielectric interface using the Green’s function method is given in Ref. [49-51]. The transfer matrix method is extensively used to calculate second-harmonic generation
(SHG) in multi-layer structures such as one-dimensional photonic crystals [48, 52, 53].
Transfer matrix expressions for the calculation of third-harmonic generation (THG) were proposed in Ref. [46], and generalized to FWM for normal incidence in Ref. [35]. As excitation angles and multi-layers are both critical factors that influence the generation of nonlinear fields in plasmonic systems, we employ here a transfer-matrix based formalism for oblique-angle FWM.
In section 2.2.1, we present the transfer matrix method of propagating the linear fields for both TM and TE polarizations. In section 2.3.2, the nonlinear polarization expressions are borrowed from the section and generalized to include multiple reflections. In section 2.3.3, the method is applied to a metal film and verify the results with finite element simulations. In section 2.3.4, we study a thin metal film over a
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substrate, or the so-called Kretschmann configuration, revealing a variety of angle- dependent FWM features. In particular, our method naturally reveals the parameters for which the FWM field can be greatly
2.2.1 Transfer matrix method
The system considered is shown in Figure 2-1 (b). A uniform slab of thickness d,
permittivity 2 and permeability 2 is sandwiched by two semi-infinite layers
of permittivity 1 and 3 , and permeability of 1 and 3 , respectively.
For now, all three layers are assumed to be linear and isotropic, but are free to exhibit dispersion and loss in the form of frequency-dependent complex material parameters.
The system is excited by a plane wave at oblique incidence, travelling in the x-z plane with an angle relative to the surface normal. For oblique incidence, the polarization of the wave should be considered. To simplify the calculation, we present the TM case in terms of magnetic fields and the TE case in terms of electric fields.
For TM polarization, the homogeneous wave equation in layer i (i=1,2,3) has the solution
iq xz, , exp jtHxzHxz qi , i , y , (2.18)
where Hxzi , A i exp jkxkz xizi are the complex amplitude of the magnetic
field. kxi sin ii k and kzi cos ii k are the wave vector components in the x and z direction, and the ± corresponds to forward/backward propagating waves. Since the x
19
and ω dependencies are shared across all layers, we can drop the term exp j t kxix , and represent the field in vector notation consisting of the z dependent complex amplitudes of the forward and backward propagating waves, such that
H z z i . (2.19) i Hi z
The fields on either side of the interface between the layer i and layer i+1 can be related by
i1z ii , 1 K ii 1 iii z , 1 , (2.20) where
iiz1k iiz 1 k 1 1 1 iizk1, iiz k 1, K (2.21) i i 1 2 k k 1iiz1 1 iiz 1 iizk1, iiz k 1, is the magnetic transfer matrix of the interface for TM polarization. Similarly, the magnetic fields at the two ends of layer i can be related by
zii, 1 i z ii 1, , (2.22)
where zi1, i , zi, i 1 are the two ends of layer i,
i 0 i 1 (2.23) 0 i
20
is the propagation transfer matrix for the layer, iexp jk izi d and di is the layer thickness.
The transfer matrix of a composite system is the product of its individual transfer matrices. For the three-layer sample in Figure 2-1 (b), the transfer matrix for the system is
K11 K 12 K KK2 3 2 1 2. (2.24) K21 K 22
Since there is no negatively propagating field in the third layer, the fields in layer 3 and layer 1 are related by
H z H1 z 12 3 23 K . (2.25) 0 H1 z 12
The fields in layer 2 at its interface with layer 1 are given by
Hz Hz 2 12 K 1 23 . (2.26) 1 2 Hz2 12 Hz 1 23
Thus, if the incoming wave amplitude H1 is known, the magnetic fields at any position can be calculated using appropriate transfer matrices. Finally, the corresponding electric fields are given by
1 H y Ex , (2.27) ji z
1 H y Ez . (2.28) ji x
21
The same procedure can be applied for TE polarization, where the electric transfer matrix of the interface is given by
iiz1k iiz 1 k 1 1 1 iizk1, iiz k 1, M . (2.29) i i 1 2 k k 1iiz1 1 iiz 1 iizk1, iiz k 1,
The electric field on either side of the interface between the layer i and layer i+1 can be related by
i1 z ii , 1 M ii 1 iii z , 1 . (2.30)
The electric fields at opposite ends of the same layer follow the relation
zii, 1 i z ii 1, , (2.31)
The transfer matrix the single layer system shown in Fig.1(b) is thus
M1 3 M 2 3 2 M 1 2 , (2.32) accordingly we can calculate electric fields at any layer:
E z E1 z 12 3 23 M , (2.33) 1 3 0 E1 z 12
Ez Ez 2 12 M 1 12 . (2.34) 1 2 Ez2 12 Ez 1 12
2.2.2 Nonlinear Polarizations
In the presence of multiple reflections, the fields can be written in terms of both forward and backward propagating waves, which lead to nonlinear polarizations
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propagating with multiple wave vectors Q at a given frequency. The third-order
polarization in layer 2 is given by the summation of all existing Q at nl :
3 3,Q nl i nl i, (2.35) i xyz,, Q
a b c a b c a b c where Qz has the possible values kz k z k z , kz k z k z , kz k z k z and
a b c a b c kz k z k z , while Qx remains kx k x k x . Dropping the x and nl dependence factor
a b c 3,Q exp jtjknl x k x kx x and rewriting Pi nl in vector notation, we obtain
Qz 3,Q D 3 Ai nl exp jQz z Pi nl 0 , (2.36) 4 Qz Ai nl exp jQz z
AAAabc AAA acb AAA bca , Q kkk abc jji jji jji z zzz j xyz,, AAAabc AAA acb AAA bca , Q kkk abc jji jji jji z zzz j xyz,, Qz Ai nl , AAAa b c AAA a c b AAA b c a , Q kk a b k c jji jji jji z z z z j xyz,, AAAabc AAA acb AAA bca , Q kkk abc jji jji jji z zzz j xyz,,
(2.37)
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AAAabc AAA acb AAA bca , Q kkk abc jji jji jji z zzz j xyz,, AAAabc AAA acb AAA bca , Q kkk abc jji jji jji z zzz j xyz,, Qz Ai nl , AAAa b c AAA a c b AAA b c a , Q kk a b k c jji jji jji z z z z j xyz,, AAAabc AAA acb AAA bca , Q kkk abc jji jji jji z zzz j xyz,,
(2.38)
For each possible Error! Objects cannot be created from editing field codes.Q, the magnetic
3,Q and electric field sources produced by nl are given by
j 3,Q r, Q nl sr,, nl 2 2 (2.39) Q nl nl nl
3,Q 3,Q r, 2 r, nl nl nl Q nl sr,, nl 2 2 (2.40) Q nl nl nl such that each component of the nonlinear polarization produces a corresponding set of source fields with the same Q.
Again, let us consider first the TM polarization, in which the nonlinear polarization is
Q Q perpendicular to the y axis, s is along the y axis and s lies in x-z plane. As shown in the Appendix I, the boundary condition at the interface between media 1 and 2 can be expressed by
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nl Q Q 1 xz,,,,,, 12nl K 2 1 2 xz 12 nl K s 1 s xz 12 nl Q (2.41) Q Nsx1 xz,,, 12 nl Q
Similarly, for the interface between 2 and 3:
nl nl Q Q Q 3 xz,,,,,, 23nl K 2 3 2 2 xz 12 nl K s 3 s s xz 12 nl Q (2.42) Q Q Ns3 s x xz,, 12 nl Q
Q where Ks i and Ns i ( i=1,3) are evaluated as
izQ iz Q 1 1 Q 1 2kzi 2 k zi K , (2.43) s i 2 izQ iz Q 1 1 2kzi 2 k zi
i i k k 1 2iz 2 iz Ns i , (2.44) 2 i i 2kiz 2 k iz
Q and s is given by
Q exp jQ z d 0 s . (2.45) 0 expjQz d
Eliminating 2 x,, z 12 nl from Eq. (2.41) and Eq. (2.42), and dropping the explicit x,
nl dependence gives
3 zK 23 1 3 1 zK 12 2 3W 2 z 12 , (2.46) where we have defined the magnetic source term vector 25
QQ QQ W2zK 12 ss 2 2 K s 2 s z 12 Q (2.47) Q 3, Q Nss2 2 N sx 2 z 12 Q
Q Q Here we have used Ksi KK s 2 2 i and Nsi NK s 2 2 i . Since there are no incident
fields at nl , we can rewrite Eq. (2.47) as
1 K H3 0 1 3 12 H3 K1 3 K 2 3W 2. (2.48) H 0 K 0 1 1 3 22 H1
Thus, the total output magnetic fields at nl generated by the third order nonlinearity are given by
1 H3 z 23 1 K1 3 12 KW z . (2.49) H z 0 K 2 3 2 12 1 12 1 3 22
The corresponding electric fields can be easily calculated by Eq. (2.27) and (2.28):
Ez Hz 3 23Fx F z 3 23 (2.50) x z Ez1 12 Hz 1 12 where
k 3z 0 3 Fx , (2.51) k1z 0 1
k x 0 3 F . (2.52) z kx 0 1
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2.2.3 Numerical validation
Figure 2-3: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) (a, b) Norms of the reflected (red, circle) and transmitted (blue, square) FWM fields generated from a silver film in air as the function of film o o thickness. (a) Propagating FWM fields. Excitation angles: a c 20 , b 30 . o o (b) Evanescent FWM fields. Excitation angles: a c 50 ,b 40 . Lines denote the analytical results, Markers represent the finite element simulations results. (c, d) Norms of the reflected (c) and transmitted (d) FWM fields from a 20nm-silver-film in air as the function of the incidents angles (a, b ) of pumping fields. Black and White dots represent the two sets of chosen angles in (a, b), respectively.
As an example, we consider a single layer system, in which a homogenous silver slab is sandwiched by semi-infinite regions of air (Figure 2-1 (b)), using the same
27
material properties and excitation parameters as in the previous section. For fixed incident angles of the excitation fields, we vary the thickness of the silver film from 0nm
o o to 200nm. We select incident angles of a 20 , b 30 as an example of
o o propagating FWM fields (black dot in Figure 2-3 (c)), and a 50 , b 40 to generate evanescent FWM fields (white dot in Figure 2-3 (c)). The resulting field magnitudes calculated by the transfer matrix method (lines) and by COMSOL finite- element simulations (markers) are shown in Figure 2-3 (a) and (b). Excellent agreement is seen between the two approaches.
Note that the skin depth of silver over the considered frequency range is approximately 10 nm. As shown in Figure 2-3 (a) and (b), when d is much larger than the skin depth, the film behaves like bulk metal: the transmitted fields tend to 0 and the reflected fields are largely unaffected by the second interface. When d is comparable to the skin depth, however, the fields reflected from the second interface generate a standing wave pattern within the silver film. Therefore, the nonlinear field conversion is moderately enhanced, reaching a maximum around d = 10 nm. However, in practice, the growth mechanism of silver films prevents the fabrication of such thin smooth films.
Therefore, we take the optimized thickness d = 20 nm. Figure 2-3 (c) and (d) show the
FWM fields as a function of a and b for this optimized thickness, showing an overall enhancement factor of roughly 3 compared to bulk metal. Once again, however, the
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propagating pumping fields are unable to directly excite surface plasmons, limiting the overall conversion efficiency.
2.2.4 A thin film in the kreschmann configuration
Figure 2-4: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) (a, b) Norms of the reflected (a) and transmitted (b) FWM fields generated from a silver film in the Kretschmann configuration as the function of incidents angles ( a, b ) of pumping fields. Dashed lines denote the surface plasmon excitation condition derived from Eq.(2.16). (c, d) FWM field patterns at points I (c) and II (d).
Next, we consider the Kretschmann configuration, in which medium 1 is taken to
have a dielectric constant greater than medium 3. As an example, we select 1 3 0 ,
29
3 0 , 1 3 0 , as in Figure 2-1 (b). We set the thickness of the silver film to be 40 nm, allowing the fields to penetrate the film, while simultaneously avoiding hybridization of the surface plasmons supported at either interface. To make sure the field intensity in layer 1 the same as that in previous sections, we assume the field
amplitudes Ea E b 52.6 MV / m . The other parameters remain unchanged from subsection D. In this configuration, pumping fields incidenting from the higher dielectric side can possess sufficient transverse components of the wavevectors to directly excite surface plasmons at the interface of media 2 and 3. From Eq. (2.16), we
o can predict pumping fields a and b will couple to surface plasmons at 36.32 and
35.93o , respectively (straight black dashed lines in Figure 2-4 (a) and (b)).
Moreover, since the silver film is sandwiched between two different materials
(dielectric and air), the FWM fields emitted on each side of the silver film are coupled to evanescent fields for different values of the transverse wavevector. Thus, by considering
k1z 0 and k3z 0 , the FWM spectrum naturally divides into three regions: propagating FWM fields on both sides of metal film, propagating FWM fields on the substrate side and evanescent fields on the air side, and evanescent FWM fields on both sides.
Figure 2-4 (a), (b) shows the norms of the reflected and transmitted FWM fields
as a function of a and b . For d = 40 nm, the conditions for surface plasmon excitations
30
nl nl at interface 1-2 and 2-3 can be calculated analytically by 2kz 1 1 k z 2 0 and
nl nl 3kz 2 2 k z 3 0 , respectively. These conditions are also plotted on Fig.4 (a) and (b)
(black dashed curves). Significant enhancements are seen when the conditions for surface plasmon excitation are satisfied, as well as at points I, II and their symmetric points. Thus, Fig.4 reveals that simple calculations of the surface plasmon dispersions for all of the interacting waves are capable of predicting the relevant enhancement features in the four-wave mixing angular spectrum.
The four intersections of the straight dashed lines in Figure 2-4 (a) are of particular interest. Due to the symmetry of the fields, we only look into two points: I
( 36.32o ,35.93o ) and II ( 36.32o , 35.93o ). For points I and II, both of the pumping fields are coupled to surface plasmon, thus confining more energy within the silver film.
Compared to the simple silver-air interface, an overall enhancement factor of roughly 40 is seen at these points. Fig. 4 (c) and (d) shows the spatial distribution of the FWM magnetic fields corresponding to points I and II, respectively. Though the nonlinear conversion rate is higher at point II due to the excitation of FWM surface plasmons,
Point I has the advantage that the generated FWM fields propagate into the dielectric, enabling easy extraction of the converted energy. Thus, of the systems considered thus far, the Kretschmann configuration supports the greatest flexibility in FWM extraction and can achieve plasmonic enhancement through all the interacting waves, thus giving it the greatest potential as a simple but compact nonlinear plasmonic device.
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2.3 Transfer matrix method: a multilayer stack
Multilayer metal-dielectric stacks offer a unique opportunity of accessing the high nonlinearity offered by metals while overcoming the limitation of strong attenuation that would otherwise occur on wave propagation through a metal layer of an equivalent thickness [54].
We show next that the procedure introduced in the previous section can be extended for an arbitrary number of layers, both metallic and dielectric, as shown in
Figure 2-1 (c), and can be used for the analytical prediction and optimization of the
FWM process in such structures. In particular, once the nonlinear polarizations in each layer are given, the resulting problem is in fact a linear boundary value problem. Each nonlinear layer can be considered individually, followed by summation over all contributions [36]. In formal terms, suppose we have n layers in total, and r nonlinear layers. The total transfer matrices for TM and TE polarized linear fields can be written as
KK1n n 1 n n 1 K 1 2 , (2.53) and
M1n M n 1 n n 1 M 1 2 , (2.54) respectively. For a nonlinear polarization perpendicular to the y axis (generated by TM linear field), similar to Eq. (2.46), the field at layer n and 1 can be related by
nnnzKz1, 1 n 1 1,2 K rnrrr W z 1, , (2.55) r
where Wr is the magnetic source term vector at layer r : 32
QQrr QQ rr Wrrrz1, K srs rsrs K z rr 1, Q r (2.56) Qr QNL r Nsrs rsrx NPz rr 1, Qr
The total output magnetic fields at nl are given by
1 Hn z n1, n 1 K1n 12 KW z , (2.57) rn r rr 1, H z 0 K r 1 1,2 1n 22 and the corresponding electric fields can be easily calculated by Eq. (2.50).
For a nonlinear polarization parallel to the y axis, n z n1, n and 1z 1,2 are related by
nnnz1, Mz 1 n 1 1,2 Mz rnrrr S 1, , (2.58) r
where the electric source term vector Sr z rr1, is given by
QQrr QQ rr Srrr z1, M srs rsrs M z rr 1, . (2.59) Qr
Thus, the total output electric fields at nl are
1 En z n1, n 1 M1n 12 MS z . (2.60) rnr rr 1, E z 0 M r 1 1,2 1n 22
As an example, we consider the 3 layer system of Figure 2-1 (d). The first layer and last layer are silver and their thicknesses are chosen to be 20 nm and 200 nm,
respectively. The middle layer is a dielectric medium ( r 5) with thickness d2 varying from 0 nm to 1000 nm. The amplitudes and wavelengths of the pumping fields are the
33
o same as those in previous sections, while the incident angles are set to be a 5 ,
o b 20 . The resulting field magnitude obtained by transfer matrix method (lines) and simulations (markers) is shown in Figure 2-5 (a).
Figure 2-5: (Reproduced with permission from Ref. [43]. Copyright (2013) by Optical Society of America.) (a) Norms of the reflected FWM fields generated from a silver-dielectric-silver structure in air as the function of dielectric thickness d2. Lines denote the analytical results; Circles represent the finite element simulations. o o Excitation angles: a 5 , b 20 . Inset-Reflected pumping fields; red-fa; blue-fb. (b) Norms of the reflected FWM fields from a silver-dielectric-silver structure in air as a o function of the incidents angles ( a, b ) of pumping fields. White dot: a 5 , o b 20 .
When d2 = 655 nm, the reflected nonlinear field is enhanced, while the transmitted nonlinear field is negligible due to the presence of the 200 nm silver film.
The enhancement can be explained as the overlap of the Fabry-Pérot effect of each pumping field. As shown in the inserted graph in Figure 2-5 (a), reflectance minimums at λa = 628nm nm and λb = 780 nm are overlapped when d2 = 655 nm. As a result of the
34
Fabry-Pérot induced localization of the pumping fields, the FWM field can be enhanced by 3 orders of magnitude compared to a bare silver surface.
Note that the enhancement peak can be tuned by other parameters, such as the permittivity of the dielectric layer, incident angles and wavelengths of the excitation
fields. Here we fix the film thickness at 655 nm, and vary the incident angles a and b
o o from -85 to 85 . The field amplitude is plotted versus a and b in Figure 2-5 (b). On the
o o one hand, for a certain range of angles in the propagating region (a , b : -15 to 15 ), the reflected field can be greatly enhanced. On the other hand, even larger enhancements are seen again when the FWM surface plasmon conditions are met, yielding an interesting combination of both plasmonic and Fabry-Pérot enhancements.
Furthermore, as more energy is concentrated in the structure, it is reasonable to consider the third-order nonlinearity introduced by the dielectric layer. For general dielectric mediums whose third-order susceptibilities are several orders of magnitude lower than silver, such as SiO2, their nonlinear effect can be neglected. However, for dielectrics with higher third-order susceptibilities, such as TiO2, their third-order nonlinearities are non-negligible.
2.4 Conclusion
Theoretically, the efficiencies of nonlinear processes can be increased either by finding new materials with large nonlinear susceptibilities or by structuring the material in such a way to increase the interacting fields. Here, we have discussed enhanced FWM
35
through field confinement in one-dimensional metallo-dielectric structures, particularly through the excitement of surface plasmons. At a solitary metal-dielectric interface, there exists no configuration in which the pumping fields can couple to surface plasmons.
However, for sufficiently oblique angles, the FWM wave can be tuned throughout both propagating and evanescent regimes, as visualized in Figure 2-2. For TM polarization, the FWM field reaches its maximum when coupled to surface plasmons, and can be completely predicted by the usual surface plasmon dispersion relation. For metal films embedded in a homogeneous dielectric, the behavior does not change much. A metal film much thicker than its skin depth behaves like bulk metal, reproducing the single- interface results discussed above. For thin metal films, the conversion rate can be moderately increased through the generation of a standing wave pattern between the two interfaces. Nevertheless, the pumping fields are still not localized, and any surface plasmons generated through FWM are bound to the surface, making extraction difficult, and absorption likely.
On the other hand, the Kretschmann configuration, in which waves are launched at a metal film through a high-dielectric substrate, can alleviate many of these issues.
The asymmetry of the system lifts the degeneracy in the wave equations on either side of the interface. In particular, for specific angles of incidence, the pumping fields can directly excite surface plasmons on the far interface, enhancing nonlinear conversion by orders of magnitude. Moreover, the generated FWM fields are free to couple to surface
36
plasmons or propagating fields. The former can further enhance the generated fields, while the latter allows for simple extraction of the mixed waves. Indeed, by varying the thickness of the metal film, one can benefit from both effects simultaneously through the hybridization of the plasmonic and propagating modes supported by each interface individually.
For multilayer stacks, the dynamics are further complicated, although we can still draw some intuitive conclusions. In particular, the inclusion of dielectric layers can lead to even more energy concentration through Fabry-Pérot effects. Similar to surface plasmons, the largest nonlinear enhancements are obtained when the Fabry-Pérot resonance for each pumping field is overlapped. However, with the inclusion of more layers, the system will inevitably suffer from greater sensitivity to fabrication errors, phase mismatch, and result in a less-compact device overall.
This brings to light an interesting tradeoff between Fabry-Pérot and plasmonic enhancements. On the one hand, plasmonic enhancements are achievable in optically- thin films and benefit from the large nonlinearities of metals at the expense of ohmic losses. On the other hand, metallo-dielectric stacks must be optically thick to achieve
Fabry-Pérot effects, but introduce little attenuation compared to bulk metals. Intuitively, the combination of Fabry-Pérot and plasmonic enhancements can navigate the competition between nonlinear conversion rate, loss and device dimensions, allowing
37
the advantages and disadvantages of each to be balanced within the constraints of a particular device.
In summary, we have presented a step-by-step analysis of FWM in simple metal films and multilayers. We find that the role of surface plasmons and Fabry-Pérot resonances in enhancing the nonlinear conversion rates can be visualized through a simple set of analytic expressions and transfer matrix operations. The proposed method, however, is only valid for the non-depleted pump approximation. We also assumed all layers of our system to be isotropic. While the transfer matrix method can be extended for arbitrarily anisotropic nonlinear media, the expressions become more complex and less intuitive. Alternatively, the method employed here can be inverted to calculate the effective nonlinear susceptibilities of structured nonlinear materials, such as nonlinear metamaterials.
While our quantitative analysis is restricted to one-dimensional systems, the dynamics are quite rich, with opportunities for massive plasmonic induced enhancements. Though the examples above were given for instructive purposes, the method can be exploited for parametrical optimization of one-dimensional nonlinear system in terms of geometry, material makeup, incidence angles, and frequencies.
Nevertheless, the concepts and intuition developed here can be applied to more complicated structures, such as surface gratings, nanoparticles, and nonlinear
38
metamaterials, from which we expect even greater flexibility in manipulating the interacting fields and obtaining higher nonlinear conversion rates.
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3. Modeling of nonlinear metasurfaces
In this chapter, we consider corrugated and structured nonlinear surfaces. A homogenization procedure is developed to describe nonlinear metasurfaces, where nonlinear response from the enhanced local fields are raised within the structured plasmonic elements. First, the effective surface description of metasurfaces is discussed in section 3.1. We start by introducing universal boundary condition and generalized sheet transition condition (GSTC), which are the basis for homogenizing a metasurface.
Nonlinear GSTCs are derived, where effective nonlinear surface susceptibilities are assigned to a nonlinear metasurface, and linked averaged macroscopic pumping fields across the metasurface. In section 3.2, a generalized retrieval method for nonlinear metasurfaces is developed based on the nonlinear GSTCs. Section 3.3 presents applications of the nonlinear surface retrieval method. The validity of the retrieval method is demonstrated by retrieving the nonlinear susceptibilities of a SiO2 nonlinear slab. A nonlinear metasurface presenting nonlinear magnetoelectric coupling is also investigated. The method is expected to apply to any patterned metasurface whose thickness is much smaller than the wavelengths of operation, with inclusions of arbitrary geometry and material composition, across the electromagnetic spectrum.
This chapter is reproduced with permission from Xiaojun Liu, Stéphane
Larouche, and David R. Smith, Optics Communication, 410(1), 53 - 69 (2018). Copyright
(2017) by Elsevier B.V.
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3.1 Effective surface description of metasurfaces
The description of a metasurface is contained in the language of surface electric and magnetic polarizabilities [55-58]. Within the context of such a description, each meta-scatterer distributed in the surface, and assumed to be completely characterized by its electric and magnetic polarizability [59-62]. Scattering from such a metasurface depends on the electric and magnetic polarizabilities of each scatterer, as well as their density distribution on the metasurface, giving rise to both phase and amplitude shifts in the incident waves. Linear metasurfaces can be characterized by applying generalized sheet-transition conditions (GSTCs) to the layer, where the averaged local electric and magnetic fields across a surface distribution of small scatterers are calculated and equated to macroscopic fields across a surface having effective electric and magnetic polarizabilities [63-69]. The GSTCs thus allow a set of surface susceptibilities to be assigned to the metasurface that serves as the link between the macroscopic fields and microscopic polarizabilities and distributions. When the GSTCs are inverted, the surface susceptibilities are uniquely determined [70]. Surface retrieval methods based on the inverted GSTCs have been extensively studied and have shown their validity in characterizing many linear metasurfaces [71-73].
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Figure 3-1: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) (a) A nonlinear metasurface. (b) Each metasurface elements are equated to an electric dipole (blue arrow) and a magnetic dipole (red arrow) induced in each of the metamaterial elements. (c) The surface polarization and magnetization are averages of the discrete electric and magnetic dipoles over the metasurface. The nonlinear signal (blue) generated in (a), (b) and (c) are identical.
As is the case for their linear counterparts, nonlinear GSTCs are derived that describe the interaction between electromagnetic waves and a nonlinear metasurface
(Figure 3-1 (a)). For a nonlinear metasurface, each meta-scatterer not only scatters the incident field, but also generates electromagnetic fields at nonlinear frequencies. By replacing these discrete nonlinear scatterers with continuous linear and nonlinear surface electric and magnetic polarizabilities, a relationship is established between the nonlinear polarization densities and the averaged electromagnetic fields at all frequencies across the surface. These averaged electromagnetic fields are further linked to macroscopic nonlinear fields by introducing nonlinear terms in the ordinary GSTCs, making it possible to assign effective nonlinear surface susceptibilities to metasurfaces.
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There are two aspects to the metasurface description that must be addressed. The first is to determine the salient properties of a surface in terms of its prescribed charge and current distributions. The singularities embodied by the Dirac delta functions associated with this description can be formally handled using distribution theory, as presented by Idemen [75, 76]. The second aspect is the use of effective medium techniques to homogenize the metasurface array as an effective surface with electric and magnetic polarizabilities, as presented by C. Holloway, E. Kuester, et al [66, 72]. To be specific, we assume the size and spacing of the metasurface elements are much smaller than the wavelengths of any of the interacting fields. That is, we preserve and average the slowly varying components of the microscopic field, while neglecting any rapidly varying components. The discontinuities of these averaged fields – referred to here as the macroscopic fields – across an interface are then related to the effective continuous distribution of surface polarization and magnetization, while the surface polarization and magnetization themselves represent averages of discrete electric and magnetic dipoles, respectively. Additionally, the distribution of metasurface elements is sparse as compared with the size of the elements, enabling us to replace the fields acting on each metasurface element by continuous polarization and magnetization densities and macroscopic fields. In analogy to the Clausius-Mossotti procedure, the fields acting on each metasurface element can either be the sum of the incident fields plus the
43
contributions of surface polarizations with the considered element removed, or the macroscopic field minus the field acting at the particular element.
Throughout this chapter, an e j t time dependence is assumed. Frequencies are either defined in parentheses, or by “ω, 2ω, nl” superscripts. For simplicity, we drop the explicit frequency labels in equations where all parameters are in terms of the same frequency. The nonlinear interactions are denoted by “EM” subscripts, and the direction of the electromagnetic fields are denoted by “xyz” subscripts. For example, the subscript
“EEMl, xzy” indicates the z component of the electric field interacts with the y component of the magnetic field, resulting in an electric response in the x direction for the scatterer centered at l.
3.1.1 Universal Boundary Conditions
Based on the assumptions outlined above, we characterize the response of a metasurface by replacing it with a surface with continuous surface electric and magnetic polarization densities. To include these surface polarization densities, we introduce the universal boundary condition [65, 75, 76]:
z0 Psz z z j , (3.1) z0 0 st 0
z0 zˆ zMj ˆ , (3.2) z0 sz st
z0 zˆ D = , (3.3) z0 st
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z0 zˆ . (3.4) z0 st
, , D and denote the limiting macroscopic fields across the surface. s and s are the surface electric and magnetic polarization densities, respectively. The subscript z denotes the direction perpendicular to the surface, while the subscript t denotes the
directions tangential to the surface. Note that s and s are attributed to the surface charges, surface currents, and electric and magnetic dipoles on the surface. In the sense
of distributions, s and s are related to the induced dipoles and currents on the metasurface, which are ultimately related to the electric and magnetic polarizabilities of the metamaterial elements and are free to exhibit nonlinear terms if nonlinearities are introduced to the metasurface elements.
3.1.2 Average fields across a surface
In this section, we relate the local electric and magnetic fields to macroscopic fields across a surface with effective electric and magnetic polarizabilities. As shown in
Figure 3-1 (b) and (c), the surface polarization and magnetization are averages of the discrete electric and magnetic dipoles induced in each of the metasurface elements constituting the metasurface; we thus need to identify the local fields acting at the position of a metasurface element to connect the metasurface properties to the surface polarization and magnetization. Because of the periodicity assumed for the metasurface, any of metamaterial elements can be chosen as a reference point for the calculation. In
45
analogy to the Clausius-Mossotti/Lorentz-Lorenz effective medium procedure, a small circular disk of radius R centered at the position of the metasurface element is considered [66]. The radius of the disk, R, is chosen just large enough to enclose the dipole under consideration, but small enough to exclude all other metamaterial elements.
Moreover, the surface magnetization and polarization are considered constant over the small disk. The field corresponding to the sheet with fictitious polarization and magnetization, and excluding the disk region, then is identical to the field determined from summing the contributions of all the discrete elements. Thus, the fields acting on each metasurface element are the sum of the incident fields plus the contributions of the surface polarizations, with the element within the disk removed. Alternatively, the fields may be calculated from the field on the disk subtracted from the macroscopic field. For
the linear case, as shown in Ref.[66], the macroscopic fields and are defined as the
combination of the incident fields ( inc and inc ) and the induced fields radiated from
the entire surface ( sheet and sheet ):
sheet inc , (3.5)
sheet inc . (3.6)
The field acting on the hypothetical disk of an arbitrary metasurface element l is the macroscopic field excluding the contribution of the metasurface element l:
act av diskav, , (3.7)
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act av diskav, . (3.8)
av and av are the average macroscopic fields across the surface. For a surface embedded in a uniform medium, the field averages are defined as
1 Fieldav Field Field , (3.9) 2 z0 z 0 where Field can be either the macroscopic electric field or the macroscopic magnetic field, the subscripts z = 0+ and z = 0- indicate the surface is taken to be at the plane z=0,
and infinite in x and y directions. disk, av and disk, av are the average scattered fields associated with the polarized, magnetized disk for the metasurface element l, which can
be in turn related to s , s through [66]:
P jkP Edisksx jkRe 1 jkR sx , (3.10) x, av z0 4R 2
Psy jkP sy Edisk jkRe 1 jkR , (3.11) y, av z0 4R 2
P Edisksz jkRe 1 jkR , (3.12) z, av z0 2R
M jkM Hdisksx jkRe 1 jkR sx , (3.13) x, av z0 4R 2
Msy jkM sy Hdisk jkRe 1 jkR , (3.14) y, av z0 4R 2
M Hdisksz jkRe 1 jkR . (3.15) z, av z0 2R
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Unlike the Clausius-Mossotti effective medium procedure, R must be selected such that the field produced by the sheet equates to that of the inhomogeneous array of dipoles in the static limit. It is shown that the choice of R=0.6956d for a distribution of dipoles arranged in a square lattice establishes the equality to first order in kR [77].
3.1.3 Generalized sheet transition condition
The linear generalized GSTCs have been carefully studied by C. Holloway et al and E. Kuester, et al, who have shown their validity in many metasurface contexts [66,
70-72]. In this section, we provide a brief overview on the derivation of linear GSTCs, which is important to our later derivations of nonlinear GSTCs.
The induced dipole moments for an arbitrary metasurface element l on a
metasurface characterized by s and M s are given by
pl El act, l , (3.16)
m Ml act, l , (3.17)
where act, l and act, l are defined through Eqs.(3.7) and (3.8). s and M s are the average of all the scatterers per unit area:
nl s N El act , (3.18)
nl s N Ml act , (3.19) where N is the number of scatterers per unit area, and denotes an average over the
local distribution of polarizability densities within the region where s and M s are
48
defined. Note that can be dropped for identical, periodic scatters whose polarizabilities are identical.
Following the derivations in ref.[66], we obtain
, (3.20) s ES av z0
, (3.21) s MS av z0
where ES and MS are the effective surface electric polarizability and magnetic polarizabilities, respectively, which are given by
N El, xx 0 0 N El, xx 1 4R N El, yy ES 0 0 , (3.22) N 1 El, yy 4R N 0 0 El, zz N El, zz 1 2R
N Ml, xx 0 0 N Ml, xx 1 4R N Ml, yy MS 0 0 . (3.23) N 1 Ml, yy 4R N 0 0 Ml, zz N Ml, zz 1 2R
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Inserting Eqs. (3.22) and (3.23) into the universal boundary conditions, we obtain the
GSTCs, which are given by
z0 zˆ Ezj ˆ , (3.24) z0 t ES, zz z , av 0 MS t , av z 0
z0 zˆ z ˆ Ej . (3.25) z0 t ESzzzav,,, z0 ES tav z 0
z0 D = , (3.26) zz0 ES t, av z0
z0 B . (3.27) zz0 MS t, av z0
The linear properties of non-chiral metasurfaces can be characterized by applying the
GSTCs. Different from the S-parameter retrieval method, where there is an ambiguity in
choosing the effective layer thickness d, the effective surface parameters ES and MS are uniquely assigned to the metasurface. Furthermore, if we have a metasurface with
well-defined thickness, for example, a metafilm with thickness d, ES and MS can be
linked to the effective r and r through [72]
xx,, yy xx yy MSd r1 ES d r 1
zz1 zz 1 (3.28) MSd1 ES d 1 . r r
3.1.4 Nonlinear generalized sheet transition condition
As is the case for their linear counterparts, we derive the nonlinear GSTCs that characterize the nonlinear property of metasurfaces (Figure 3-1 (a)). For a nonlinear metasurface, each meta-scatterer not only scatters the incident field, but also generates
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electromagnetic fields at harmonic or mix frequencies. By replacing these discrete nonlinear scatterers with continuous nonlinear surface electric and magnetic polarizabilities, we obtain nonlinear GSTCs.
As nonlinear metasurfaces may support responses that cannot be described by purely electric or magnetic interactions, we consider the all the possible nonlinear interactions, that is, electro-electro interaction, magneto-magneto interaction and magneto-electric interaction. However, we also neglect any linear magnetoelectric response, which is valid for most symmetric metasurfaces. The induced dipole moments at the second harmonic generation (SHG) frequency, for example, can then be expressed as
2 2 2 nl nl nl pl0 EEl act EEEl : act act 2 EEMl : act act EMMl : act act , (3.29)
2 2 2 nl nl nl ml MMl act MMMl : act act 2 MEMl : act act MEEl : act act , (3.30)
nl nl where El , Ml are the electric and magnetic polarizability dyadics for the scatterer
2 2 2 2 2 2 centered at rl, while EEEl , EEMl , EMMl , MMMl , MEMl and MEEl are the second-order polarizabilities for the same scatterer. The first terms in Eqs. (3.29) and (3.30) correspond to the linear response to the fields at ωnl= 2ω1, which are due to the collective radiation and scattering from all the scatterers except the one centered at l. The subsequent terms in Eqs. (3.29) and (3.30) are the nonlinear terms generated by the pumping fields and local second-order susceptibilities within the scatterer. Averaging and summing all the scatterers per unit area give the electric and magnetic polarization densities: 51
nl nl nl s0 N EEl act (3.31) N 2 : 2 2 : 2 : , 0 EEEl act act EEMl act act EMMl act act
nl nl nl sN MMl act (3.32) 2 2 2 NMMMl: act act 2 N MEMl : act act N MEEl : act act , where N is the number of scatterers per unit area, and denotes an average over the
nl nl local distribution of polarizability densities within the region where s and s are defined. Note that can be dropped for identical, periodic scatters whose polarizabilities are identical.
nl nl Our goal is to relate s and s to the averaged macroscopic fields at ω and
ωnl. Assuming a pumping field at ω, the surface electric and magnetic polarizations are linked to the macroscopic average field through Eqs. (3.20) and (3.21). Inserting Eqs.
(3.20) and (3.21) into Eqs. (3.10) - (3.15), the fields acting at the center of the disk are expressed in terms of the averaged macroscopic fields on both sides of the surface in the following manner:
actxˆ1 ESxx,,,,,, 4 Ry avx ˆ 1 ESyy 4 Rz avyˆ 1 ESzz 2 R avz , (3.33)
actxˆ1 MSxx,,,,,, 4 Ry avx ˆ 1 MSyy 4 Rz avyˆ 1 MSzz 2 R avz , (3.34)
where ES, xx , ES, yy and ES, zz are the tensor elements of the effective electric surface
polarizability ES , while MS, xx , MS, yy and MS, zz are the tensor elements of the effective
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nl nl magnetic surface polarizability MS . As shown in Eqs. (3.7) and (3.8), act and act can be expressed as
nl nl nl actav,, av diskav , (3.35)
nl nl nl actav,, av diskav , (3.36)
nl nl nl nl where disk, av and disk, av are linked to s and s through Eqs. (3.10) - (3.15).
nl nl Inserting Eqs. (3.33) - (3.36) into Eqs. (3.31) and (3.32), all that is left are s , s , and
nl the averaged macroscopic fields at ω and ωnl. As it is shown in Appendix I, s and
nl s can be expressed in terms of the averaged macroscopic fields at ω and ωnl:
2 2 2 nl nl nl s0 ES av EEEs: av av 2 EEMs : av av EMMs : act act , (3.37)
2 2 2 nl nl nl s MS av MEEs: av av 2 MEMs : av av MMMs : act act . (3.38)
2 2 2 2 2 2 EEES , EEMs , EMMs , MMMs , MEMs and MEEs are the effective second-order surface tensors, each of which is a tensor containing 18 tensor elements. Specifically, the tensor elements of each of the effective surface tensors are
nl 2 El, ii 2 EEEsijk,,,, EEElijk1 ESjjAR j 1 ESkk AR k , (3.39) nl El, ii
nl 2 2 El, ii 1 2 EEMsijk,,,, EEMlijk1 ESjjAR j 1 MSkk AR k , (3.40) nl El, ii
nl 2 2 El, ii 1 2 EEMsijk,,,, EEMlijk1 ESjjAR j 1 MSkk AR k , (3.41) nl El, ii
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nl 2 2 El, ii 1 2 EMMsijk,,,, EMMlijk1 MSjjAR j 1 MSkk AR k , (3.42) nl El, ii
nl 2 2 Ml, ii 1 2 MEEsijk,,,, MEElijk1 ESjjAR j 1 ESkk AR k , (3.43) nl Ml, ii
nl 2 2 Ml, ii 1 2 MEMsijk,,,, MEMlijk1 ESjjAR j 1 MSkk AR k , (3.44) nl Ml, ii
nl 2 2 Ml, ii 1 2 MEMsijk,,,, MEMlijk1 ESjjAR j 1 MSkk AR k , (3.45) nl Ml, ii
nl 2 2 Ml, ii 1 2 MMMs,,,, ijk MMMl ijk1 MS jjAR j 1 MS kk AR k , (3.46) nl Ml, ii where ijk,,,, xyz , and
4,jk , xy , Aj, k (3.47) 2,jk , z
The surface susceptibilities in Eqs. (3.37) - (3.46) are angle and polarization invariant, indicating that they are indeed the characteristic or constitutive parameters of the metasurface. The expressions for the effective second-order surface susceptibilities are intuitively reasonable as each of them is proportional to the averaged local second- order polarizability, as well as the properties of the metasurface at both the pumping
2 2 2 2 2 2 and generated frequencies. Note that EEES , EEMs , EMMs , MMMs , MEMs and MEEs also
nl nl depend on the density of the scatterers N, which is folded into Elii,, Elii or
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nl nl Mlii,, Mlii . Since averages of the first- and second-order local polarizabilities are required, it would be a lengthy process to demonstrate the accuracy of the nonlinear
GSTCs through Eq.(3.39) - (3.46). Alternatively, by reinserting Eqs. (3.37) and (3.38) back into (3.1) - (3.4), we invert the nonlinear GSTCs and establish a linear system of equations that enable us to retrieve the first- and second-order surface parameters. For this reason, we present a generalized surface retrieval method in the following sections and show the validity of our method via retrieval of a simple two-dimensional slab with defined second-order nonlinear susceptibilities.
3.2 Nonlinear surface parameter retrieval
The nonlinear GSTCs allow a set of nonlinear surface susceptibilities to be assigned to the metasurface that serve as the link between the nonlinear macroscopic fields and nonlinear microscopic polarizabilities and distributions. We show that the
GSTCs can be inverted to retrieve all the effective linear and nonlinear susceptibilities at both normal incidence and oblique incidence, providing a generalized retrieval method for nonlinear metasurfaces.
3.2.1 Retrieval method
As discussed above, we insert Eqs. (3.37) and (3.38) into Eqs. (3.1) - (3.4), and retrieve the first- and second-order surface parameters from the macroscopic fields across the surface. We assume polarization invariance; that is, the polarization of the
SHG wave stays the same as that of the pumping beam, and consider the TM and TE
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polarization separately. This assumption, which is common for non-chiral metamaterials, allows a portion of the tensor elements to be set to zero in each of the second-order surface polarizability tensors, while still leaving us with thirty-six non-trivial tensor elements to solve in total.
First we consider TM polarization, which contains 18 non-trivial tensor elements.
A subset of 6 of these elements is determined by inserting Eq.(3.37) into Eq.(3.2):
Hnl H nl yz0 y z 0 2 2 2 1 EEEsxxx,,,,,,,,,EE avx avx2 EEEsxxz EE avx avz EEEsxzz EE avz avz jnl E nl j . nl0 ES av , x2 nl 0 2 2 2 2EEMsxxy,,,,,,,,,EH avx avy 2 EEMsxzy EH avz avy EMMsxyy HH avy avy (3.48)
nl The linear surface susceptibility ES is obtained separately by application of linear surface retrieval method at ωnl, and treated as a known parameter [70]. The left side of
Eq.(3.48) is linearly proportional to the second-order surface susceptibilities, allowing us to establish a linear system of equations as matrix. As there are 6 unknown second-order surface susceptibilities, we need 6 non-degenerate measurements to fully solve for the system of equations. Writing the 6 equations in a matrix form, we have,
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2 EEmav, x av , x 1 HHm av , y av , y 1 EEEs, xxx EEm HHm 2 av, x av , x 2 av , y av , y 2 EEEs, xxz EEm HHm 2 avxavx, , 6 avyavy , , 6 EMMs, xyy
nl nl, nl nl (3.49) Hm Hm / j E m y1z0 y 1 z 0 nl 0 ES , xx av ,x 1 nl nl, nl nl Hm Hm / j Em y2z0 y 2 z 0 nl 0 ESxxavx , , 2 . nl nl, nl nl Hm Hm / j Em y6z0 y 6 z 0 nl 0 ES , xx av , x 6 where mi (i=1,…,6) represents different measurements. The measurements include sending the pumping beam at different angles of incidence from either side of the metasurface, as will be discussed in the following sections. Once the macroscopic fields—including the limiting fields across the surface at the SHG frequency and the average fields at both pumping and SHG frequencies—are obtained (either through experimental or numerical measurements), the linear and second-order surface susceptibilities are solved from Eq. (3.49). The 12 remaining second-order surface susceptibilities are solved by inserting Eqs. (3.37) and (3.38) into Eq. (3.1):
Enl E nl xz0 x z 0 nl jkx nl nl nl nl ESzz,Ej avz , nl 0 MSyy , H avy , 0 nl 2 2 2 jk EEEszxx,,,,,,,,,EE avx avx2 EEEszxz EE avx avz EEEszzz EE avz avz (3.50) x 2 2 2 2 1 2 0 2EEMszxy,,,,,,EH avx avy 2 EEMszzy EH avz avy EMMszyy,,,H avy H avy 2 2 2 1 MEEs,,,,,,,,, yxxEE av x av x2 MEEs yxz EE av x av z MEEs yzz EE av z av z j , 2 nl 0 2 2 2 2MEMs,,,,,,,,, yxyEH av x av y MEMs yzy EH av z av y MMMs yyy HH av y av y
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nl nl Similar to the previous procedure, the linear surface susceptibilities ES, zz and MS, yy at
the ωnl are solved independently using linear surface retrieval method, and the 12 unknown second-order surface susceptibilities are solved by a linear system of equations which includes 12 different measurements. Expressing Eq. (3.50) in matrix form, we have
nl 2 jkx1 0 EEm av , x av , x 1 jEmEm 0 av , x 1 av , x 1 EEEs, xxx nl 2 jk EEm jEmEm x7 0 av , x av , x 7 0 av , x 7 av , x 7 MEEs, yxx (3.51) nl nlnl nl nl nl nl Em Em jk E mj Hm x1z0 x 1 z 0 x 1 0 ESzzavz , , 1nl 0 MS , yy av , y 1 . nl nlnl nl nl nl nl Emx7 Em x 7 jk x 70,,7 ES zz Emj av z nl 0,,7 MS yy Hm av y z0 z 0
For TE polarization, inserting Eq. (3.38) into Eq. (3.1), we obtain
Enl E nl yz0 y z 0 2 2 2 1 MMMsxxx,,,,,,,,,HH avx avx2 MMMsxxz HH avx avz MMMsxzz HH avz avz jnl H nl j . 0MS avx ,2 nl 0 2 2 2 2MMMsxzz,,,,,,,,,HH avz avz 2 MMEsxzy HE avz avy MEEsxyy EE avy avy (3.52)
Inserting Eqs. (3.37) and (3.38) into Eq. (3.2),
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Hnl H nl xz0 x z 0
nl nl nl nl nl jkx MSzz, H avz , j nl 0 ESyy , E avy , 2 2 2 1 MMMszxx,,,,,,,,,HH avx avx2 MMMszxz HH avx avz MMMszzz HH avz avz jk nl (3.53) 2 x 2 2 2 2MMEszxy,,,,,,,HE avx avy 2 MMEszzy HE avz avy MEEszyy Eav,, y E av y 2 2 2 1 EMMsyxx,,,,,,,,,HH avx avx2 EMMsyxz HH avx avz EMMsyzz HH avz avz j , 2 nl 0 2 2 2 2EMEs,,,,,,,,, yxyHE av x av y 2 EMEs yzy HE av z av y EEEs yyy EE av y av y
Again, the linear surface susceptibility components in Eqs. (3.52) and (3.53) are solved individually by the standard surface retrieval method. The 6 unknown second-order surface susceptibilities in Eq. (3.52) and the 12 unknowns in Eq. (3.53) are solved by making 6 and 12 different measurements, respectively. Expressing Eqs. (3.52) and (3.53) in their matrix forms, we have
2 HHmav, x av , x 1 EEm av , y av , y 1 MMMs, xxx HHm EEm 2 av, x av , x 2 av , y av , y 2 MMMs, xxz HHm EEm 2 av, x av , x 6 av , y av , y 6 MEEs, xyy
nl nl nl nl (3.54) Em Em / j H m av, y 1z0 av , y 1 z 0 nl 0 MS , xx av, x 1 nl nl nl nl Em Em / j Hm av, y 2z0 av , y 2 z 0 nl 0 MS , xx av , x 2 . nl nl nl nl Em Em / j Hm av, y 6z0 av , y 6 z 0 nl 0 MS , xx av , x 6 and
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nl 2 jkHHmx1 av , x av , x 1 j nl 0 HHm av , x av , x 1 MMMs, zxx nl 2 jkHHm j HHm x7 av , x av , x 7 nl 0 av , x av , x 7 EMMs, yxx (3.55) nl nlnl nlnl nl nl Hm Hm jk Hmj E m x1z0 x 1 z 0 x 1 MS , zz av , z 1 nl0 ES , yy av , y 1 . nl nlnl nl nl nl nl Hmx10 Hm x 7 jk xMSzzavz 7,,10 Hmj nl 0,,7 ESyyavy Em z0 z 0
Eqs. (3.49), (3.51), (3.54) and (3.55) provide the complete expressions of the 36 non-trivial second-order susceptibilities under the assumption that the polarization of the SHG signal stays the same as that of the pumping beam. Similar expressions can be developed for the general case, which would increase the number of unknowns and the number of measurements required. However, as the size of the retrieval matrix increases, the matrix may be ill-conditioned, leading to numerical errors in the matrix inversion and final retrieval.
3.2.2 Connection between surface parameters and bulk parameters
To explicit connect surface susceptibilities to bulk susceptibilities, we consider the bulk polarization for a very thin slab in the sense of distribution:
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x,,,,,,,,, y z x y z x y z x y z dxdydz d x,,,, y z x y z dz dxdy d d 2 x, y ,0 x , y ,0 dz dxdy (3.56) d 2 d x, y ,0 x , y ,0 dxdy d x, y ,0 x , y , z z dxdydz dxy , ,0 z , xyz , , , where x,, yz is any test function, and the slab thickness d is much smaller than the wavelength. As a result, we conclude
xyz, , d xy , ,0 z (3.57) s z .
s is the surface polarization, which is given by
s d xy, ,0 d (3.58) x, y , d 2 0 x , y , d 2 0 . 2
For the tangential components of s ,
d P E x, y , d 2 0 E x , y , d 2 0 st2 0 rt t (3.59) d E x, y , d 2 0 E x , y , d 2 0 2 0 rt t
Expressing Pst in terms of effective surface parameters and averaged macroscopic fields across the surface, we have
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1 E d 1 E x , y , d 2 0 E x , y , d 2 0 . (3.60) 0ES , tt av , t 0 r2 t t
1 Simplifying Eq. (3.60) by E E x, y , d 2 0 E x , y , d 2 0 , we obtain av, t2 t t
EStt, d r 1 . (3.61)
For the normal components, the presence of boundary introduces discontinuities in
electric fields. For this reason, r is included in the denominator of the normal components of electric field to account for the discontinuity:
d r Psz E z x, y , d 2 0 E z x , y , d 2 0 . (3.62) 2 r
Expressing Pst in terms of effective surface parameters and averaged macroscopic fields across the surface, we have
r 1 d 1 0ESzzavz ,E , 0 E z x, y , d 2 0 E z x , y , d 2 0 . (3.63) r 2
1 Simplifying Eq. (3.60) by E E x, y , d 2 0 E x , y , d 2 0 , we obtain av, z2 z z
r 1 ES, zz d. (3.64) r
A similar derivation can be applied to the bulk magnetization, where we obtain
MStt, d r 1 , (3.65)
r 1 MS, zz d. (3.66) r
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Eqs. (3.61), (3.64), (3.65) and (3.66) give a complete description of the connection between the surface susceptibilities and bulk susceptibilities at linear regime.
As is the case for their linear counterparts, we consider bulk polarization and surface polarization with nonlinear terms. As the total polarization is the sum of all the contributions of the nonlinear interactions originated from different nonlinear tensor elements, we can simplify our derivation by considering each nonlinear interaction
nl independently. Here, we show the derivation for Ps, xxx , which is a pure tangential term,
nl and Ps, zxz , which is a mixed term that contains interaction between the tangential component and normal component of electric field.
nl Starting from Eq. (3.58), for Ps, xxx , we have
nl nl Psxxx, dP xxx xy, ,0 . (3.67)
nl As Pxxx is a tangential component induced by the nonlinear interaction between
the tangential components of the electrical fields at 1 and 2 , the electric field and the nonlinear polarization are continuous across the boundary. Expressing the right hand side of Eq. (3.67) in terms of bulk second-order susceptibility and local electric fields:
nl 2 1 2 Psxxx,, d EEExxxx ExyExy , ,0 x , ,0 . (3.68)
1 2 Write Ex xy, ,0 and Ex xy, ,0 in terms of the fields at the boundaries and drop their explicit x, y dependence, we have
nl 2 1 1 P d E1 d2 E 1 d 2 E 2 d 2 E 2 d 2 . (3.69) s, xxx 0 EEE , xxx2 x x 2 x x 63
nl Expressing Ps, xxx in terms of effective nonlinear surface parameters and averaged macroscopic fields across the surface:
2 1 2 0 EEEs , xxxE av , x E av , x (3.70) 2 1 1 d E1 d2 E 1 d 2 E 2 d 2 E 2 d 2 . 0EEExxx , 2 x x 2 x x
By comparing the right hand side to the left hand side of Eq. (3.70), we obtain
2 2 EEEs,, xxx d EEE xxx (3.71)
nl As Pzxz is a normal component induced by the nonlinear interaction between the
tangential component of the electrical field at 1 and the normal component of the
electric field 2 , the presence of boundary introduces discontinuities in the nonlinear
nl nl polarization at nl . For this reason, we include 1 r to account for the jump of Pzxz :
2 EEE, zxz Pnl d ExyExy1, ,0 2 , ,0 . (3.72) s, zxznl x z r
Notice that the presence of boundary also introduces a jump in electric fields at 2 , We
1 2 then express Ex xy, ,0 and Ex xy, ,0 in terms of the fields at the boundaries and drop their explicit x, y dependence. Notice that the presence of boundary also introduces
2 a jump in electric fields at 2 . 1 r is included to take into account this discontinuity:
nl Ps, zxz 2 (3.73) EEE, zxz 1 1 d E1 d2 E 1 d 2 E 2 d 2 E 2 d 2 . 0 nl x x 2 z z r2 2 r
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nl Expressing Ps, zxz in terms of effective nonlinear surface parameters and averaged macroscopic fields across the surface:
2 1 1 0 EEEs , zxzE av , x E av , z 2 (3.74) EEE, zxz 1 1 d E1 d2 E 1 d 2 E 2 d 2 E 2 d 2 . 0 nl x x 2 z z r2 2 r
By comparing the right hand side to the left hand side of Eq. (3.70), we obtain
2 2 EEE, zxz d . (3.75) EEEs, zxz nl 2 r r
To link the bulk second-order susceptibility to pure surface susceptibilities, we express
2 nl r and r in terms of surface parameters by using Eq. (3.64) and insert them into Eq.
(3.75):
2 2 2 nl 2 EEEszxz,,1 EEEszxz EEEzxz, r r . (3.76) dnl 2 d 1ESzz,,d 1 ESzz d
For all the other nonlinear tensor elements, we can apply similar derivations as it is shown above. The relation between the nonlinear surface parameters and nonlinear bulk parameters:
2 2 2 2 EEE,,,, xxx EEEs xxxd,, EMM xyy EMMs xyy d 2 2 2 2 EEM,,,, xxy EEMs xxyd,, EME xyx EMEs xyx d
2 22 2 2 1 EEExxz,,,,,, EEEsxxzd ESzz ,, EEExzx EEEsxzx d ESzz (3.77)
2 21 2 2 2 EEM,,,,, xzy EEMs xzyd ES zz , EME xyz EMEs xyz d ES, zz ,
2 2 d 11 2 d , EEE,,,, xzz EEEs xzz ES zz ES zz
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2 2 nl EEEzxx,,, EEEszxxd ESzz ,
2 2nl 2 2 nl EEMzxy,,,,,, EEMszxyd ESzz ,, EEMzxy EEMszxy d ESzz
2 2 d nl 1 2 d , EEEzxz,,,, EEEszxz ESzz ESzz 2 2 d nl 1 1 d , EEE,,,, zzx EEEs zzx ES zz ES zz (3.78) 2 2 d nl 1 1 d , EEM, zzy EEMszzy,,, ESzz ESzz 2 2 d nl 12 d , EME,,,, zyz EMEs zyz ES zz ES zz 2 2 d nl 11 d 1 2 d . EEEzzz,,,,, EEEszzz ESzz ESzz ESzz
2 2 2 2 MEE,,,, yxx MEEs yxxd,, MMM yyy MMMs yyy d 2 2 2 2 MEM,,,, yxy MEMs yxyd,, MME yyx MMEs yyx d
2 22 2 2 1 MEE,,,,,, yxz MEEs yxzd ES zz ,, MEE yzx MEEs yzx d ES zz (3.79)
2 21 2 2 2 MEM,,,,, yzy MEMs yzyd ES zz , MME yyz MMEs yyz d ES, zz ,
2 2 d 11 2 d . MEE,,,, yzz MEEs yzz ES zz ES zz
3.3 Applications
In previous sections, we derive the nonlinear GSTCs that describe the interactions between electromagnetic waves and a nonlinear metasurface. The nonlinear
GSTCs can be inverted to retrieve all the effective nonlinear surface susceptibilities at both normal incidence and oblique incidence. To demonstrate its validity, we retrieve the nonlinear susceptilibities of a SiO2(fused silica) thin slab. Finally, we apply the retrieval method to a nonlinear metasurface that presents nonlinear magnetoelectric coupling in the near-infrared regime.
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3.3.1 Nonlinear slab
We investigate a fused silica slab in vacuum whose linear and nonlinear properties are well known. The refractive index of the SiO2 slab is frequency dispersive; for the simulations presented here, we obtain the refractive index values as a function of wavelength from ref. [78]. To demonstrate the retrieval method, we consider second harmonic generation (SHG) from the SiO2 slab, which has a nonlinear susceptibility of
2 EEE, xxx 0.6 pm V [16]. We assume the thickness of the SiO2 slab to be 100nm, and
TM polarized pumping field (E along x axis).
Figure 3-2: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Retrieved linear surface susceptibilities of a SiO2 slab. Left column:
surface parameters: ES, xx , MS, yy and ES, zz . Right column: (blue-solids) bulk
parameters calculated using Eq. (3.28); (red stars) SiO2 refractive index from ref.[78].
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COMSOL Multiphysics is used to simulate scattering from the SiO2 slab, from which we calculate the average electromagnetic fields at the pumping frequencies, and the average and limiting electromagnetic fields at SHG frequency. By inserting these fields into Eqs. (3.49) and (3.51), we are able to solve for second-order susceptibilities. As the linear properties are also unknowns in Eqs. (3.49) and (3.51), we first retrieve the linear surface parameters using the surface retrieval method [70]. The retrieved results are shown in Figure 3-2, where the left column shows the retrieved surface parameters and the right column presents their corresponding bulk parameters. When the wavelength is much larger than the optical thickness, the bulk parameters are in good agreement with the reference data. For smaller wavelength, the slab behaves less like a surface; unsurprisingly, the retrieved results are off from the reference data. All the magnetic surface parameters converge to 0 while their bulk parameters converge to 1,
corresponding to the fact that r 1 for SiO2.
We then retrieve the nonlinear parameters. As SiO2 only has electric nonlinear response, we neglect all the magnetic nonlinear susceptibilities in Eqs. (3.49) and (3.51).
2 As only xxx is non-zero in Eq. (3.49), Eq. (3.49) reduces to a single equation:
1 2 HHnl nl j nl Ej nl EE , (3.80) yz0 y z 0 nl0 ES av , x2 nl 0 EEEs , xxx av , x av , x
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2 from which we can solve for EEEs, yyy with a single measurement at normal incidence. To be more general, we include all the electric second-order susceptibilities in Eqs. (3.49) and (3.51), which gives two 3-by-3 matrices:
2 EE EE EE EEEs, xxx av, x av , x 1 av , x av , x 1 av , x av , x 1 2 EEavxavx, , 2 EE avxavx , , 2 EE avxavx , , 2 EEEsxxz , 2 EEav, x av , x 3 EE av , x av , x 3 EE av , x av , x 3 EEEs, xzz
, (3.81) Hnl H nl / j nl E nl y1z0 y 1 z0 nl0 ES , xx av , x 1 nl nl, nl nl H H /, j E y2z0 y 2 z 0 nl 0 ESxxavx , , 2 nl nl, nl nl H H / j E y3z0 y 3 z 0 nl 0 ES , xx av , x 3
2 EE EE EE EEEs, zxx av, x av , x 1 av , x av , z 1 av , z av , z 1 nl 2 jkx10,,2 EE av x av x EE av ,,2 x av z EE av ,,2 z av z EEEs , zxz 2 EEav, x av , x 3 EE av , x av , z 3 EE av , z av , z 3 EMMs, zzz (3.82) 2 2 2 2 1 2 2 E E jk E j2 H x 1 z0 x1z0 x 1 0 ESzzavz , , 1 0 MSyyavy , , 1 2 2 2 2 1 2 2 E E jk E j2 H , x2z0 x 2 z 0 x 10,,2 ES zz av z 0,,2 MS yy av y 2 2 2 2 1 2 2 E E jk E j2 H x3z0 x 3 z 0 x 10,,3 ES zz av z 0,,3 MS yy av y
o To solve Eq. (3.81) and (3.82), the pumping fields are incident at angles of 1 30 ,
o o 2 45 and 3 60 . The left column of Figure 3-3 shows the retrieved results from Eq.
(3.81) and (3.82). We see the same trend as the linear surface parameters: for larger
2 20 2 2 2 2 wavelengths, EEEs, xxx converges to 0.6 10 m V , while EEEs, xxz , EEEs, xzz , EEEs, zxx ,
2 2 20 2 EEEs, zxz and EEEs, zzz converge to some small values below 0.03 10 m V . As
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2 2 2 2 2 2 EEEs, xxz , EEEs, xzz , EEEs, zxx , EEEs, zxz and EEEs, zzz are much smaller than EEEs, xxx , we consider them as numerical errors due to the surface approximation in the GSTCs. The
2 right column of Fig.3 shows the bulk second-order susceptibility EEE, xxx calculated from
2 the EEEs, xxx using Eq. (3.77), whose value converges to the known bulk nonlinear
2 susceptibility of SiO2 xxx 0.6 pm V .
Figure 3-3: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Retrieved nonlinear surface susceptibilities of a SiO2 slab. Left column: 2 2 2 (blue-solid) EEEs, xxx ; (orange-solid) EEEs, xxz ; (yellow-solid) EEEs, xzz retrieved from Eq. 2 2 2 (3.81). (purple-dash) EEEs, zxx ; (green-dash) EEEs, zxz ; (light blue-dash) EEEs, zzz retrieved from Eq. (3.82). Right column: nonlinear bulk susceptibility calculated from the 2 nonlinear surface susceptibility EEEs, xxx from the left column.
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3.3.2 Magnetoelectric nonlinear metasurfaces
To apply the surface retrieval method to nonlinear metasurfaces, we analyze a nonlinear metasurface whose unit cell consists of a pair of split ring resonators (SRRs) positioned opposite to each other (Figure 3-4). A similar structure has been well-studied using the bulk retrieval method in the microwave regime, and used to show a variety of interesting nonlinear phenomena such as nonlinear interference, phase matching, harmonic generation and magnetoelectric nonlinearity[6, 8-10]. The structure investigated here has been scaled to exhibit resonant behavior in the near-infrared region. The periodicity of the SRR pairs is 400nm. The length of each SRR is 350nm, and the two arms are 150nm. The widths and thicknesses of all the lines are 50nm. The dual gaps formed between the arms of the two SRRs are 50nm long, and are loaded with a
2 12 nonlinear materials having a nonlinear susceptibility of 10 m V . The nonlinear material could be realized by, for example, a nonlinear polymer deposited on the surface of the entire structure.
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Figure 3-4: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) A nonlinear metasurface consists of arrays of split ring resonators (SRRs) pairs positioned opposite to each other. The incident field is TM polarized. Electric field is always along y axis.
Figure 3-5: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Linear spectrum of the nonlinear metasurface excited by TM polarized wave at (a) normal incidence; (b) oblique incidence (30o). Blue curve: S11; Red curve: S21.
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The incident electric field E is oriented along the y axis. Figure 3-5(a) shows that the ring exhibits a resonance at 1μm when the metasurface is excited by a TE polarized electromagnetic field at normal incidence. The resonance is purely electric as the magnetic field H is along x axis and thus does not induce any current in the SRRs. When the electromagnetic field is incident at oblique incidence, that is, when H has a z component, a purely magnetic resonance is excited at 2.2μm. Simultaneously, a higher order magnetic resonance is excited around 1μm nearly degenerate with the electric resonance at 1μm, leading to plasmon induced transparency, as shown in Figure 3-5(b)
[79-82]. Figure 3-6 shows the linear effective surface susceptibilities of the metasurface.
As no magnetic resonance is excited at normal incidence, the in-plane magnetic surface
susceptibility MS, xx is almost constant over the considered wavelength range. A peak is
seen in the in-plane electric surface susceptibility ES, yy , corresponding to the electric
resonance at 1μm. The out-of-plane magnetic surface susceptibility, MS, zz , exhibits the richest interactions. The peak at 2.2μm is due to the magnetic resonance excited by the z component of the H field. The double peaks around 1μm are due to the effects of the coupled resonances. Note that since these spectral features are not due to a linear superposition of the electric magnetic modes, the two interactions cannot be separated in
MS, zz .
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Figure 3-6: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Retrieved linear surface susceptibilities of the nonlinear metasurface:
MS, xx (top), ES, yy (middle) and MS, zz (bottom).
To probe the nonlinear response of the system, we illuminate the metasurface with a plane wave over a scan of pump wavelengths, and monitor the generated SHG that occurs at half the incident wavelength. Figure 3-7(a) and (b) show that the SHG transmission and reflection spectrums. At normal incidence (dashed lines). There is no significant enhancement since the magnetic resonance of the SRR is not excited at normal incidence. When the metsurface is excited at normal incidence (dashed lines), the transmitted and reflected SHG fields are enhanced by 10 times due to the presence of the magnetic resonance around 2.2μm. Figure 3-7(c) shows the phase of the reflected and transmitted SHG field. The transmitted electric field is in phase with the reflected 74
electric field at the SHG frequency, indicating that the equivalent nonlinear sources are purely electric and polarized along the y axis [9].
Figure 3-7: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) (a) Reflection and (b) transmission SHG spectrum of the nonlinear metasurface excited by TE polarized wave at 0 degree (orange-dash) and 30 degree (blue-solid). (c) The phase of the SHG transmission (blue-solid) and reflection (orange-dash).
We next apply the nonlinear surface retrieval method described in Section IV to retrieve the nonlinear susceptibilities of the metasurface. To minimize the size of the retrieval matrices, we only consider the most significant nonlinear interactions that generate y-polarized electric dipoles at the SHG frequencies. As the magnetic resonance at 2.2μm enhances the SHG when the metasurface is excited at oblique incidence, the
2 first interactions to consider are the magneto-magneto interactions that generate y . As the magnetic pumping field has its x component as well as its z component at oblique 75
incidence, the nonlinear interactions have three terms: in the x direction, in the z direction and their cross term, which correspond to the second-order effective surface
2 2 2 susceptibilities EMMs, yzz , EMMs, yzx , and EMMs, yxx . The electric resonance around 1μm also contributes to the enhancement, as the SHG signal may couple to the resonances arising from the induced transparency effect, leading to higher SHG radiation. The
2 corresponding second-order effective surface susceptibility is EEEs, yyy . Since either the magnetic resonance at 2.2μm or around 1μm may interact with the electric field and subsequently contribute to the SHG enhancement, we need to consider the magento- electric interactions. The corresponding second-order effective surface susceptibilities
2 2 are EMEs, yzy and EMEs, yxy . As a result, we end up with 6 nonlinear surface parameters, all of which can be solved by Eq. (3.53). We set all other nonlinear surface parameters to zero, such that Eq. (3.53) reduces to
H2 H 2 xz0 x z 0 2 2 jnl 0 ES , yy E av , y (3.83) 2 2 2 1 EMMs,,,,,,,,, yxxHH av x av x2 EMMs yxz HH av x av z EMMs yzz HH av z av z j . 2 nl 0 2 2 2 2EMEs,,,,,,,,, yxyHE av x av y 2 EMEs yzy HE av z av y EEEs yyy EE av y av y
Written it in matrix form for 6 different incident angles,
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2 EMMs, yxx 2 EMMs, yxz HHav, x av , x 12 HH av , x av , z 1 EE av , y av , y 1 2 HH2 HH EE EMMs, yzz j av, x av , x 2 av , x av , z 2 av , y av , y 1 0 2 EMEs, yxy 2 HHav, x av , x 62 HH av , x av , z 6 EE av , y av , y 1 EMEs, yzy 2 EEEs, yyy (3.84) 2 2 2 2 H H jE2 x1z0 x 1 z 0 0 ESyyavy , , 1 2 2 2 2 H H jE2 x2z0 x 2 z 0 0 ES , yy av , y 2 , 2 2 2 2 H H jE2 x6z0 x 6 z 0 0 ES , yy av , y 6 where i i 1,2, ,6 represents the six different measurements: sending pumping
o o o o o o fields at 1 0 , 2 0 , 3 30 , 4 30 , 5 45 and 5 45 , respectively. The “+“ sign represents the pumping fields propagating in the forward direction (+ z direction), while the “-”sign represents backward propagating pumping fields (- z direction). Figure 3-8 shows the six retrieved second-order surface susceptibilities. As the magnetic resonance is excited by the z component of the magnetic field, all the terms that involve the interaction with the magnetic field in x
2 2 2 direction, EMMs, yzx , EMMs, yxx and EMEs, yxy , vanish. The most significant second-order
2 surface susceptibility is EMMs, yzz , indicating that the magnetic resonance at 2.2μm is the major contribution to the SHG enhancement. The second-order surface susceptibility
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2 EMEs, yzy that arises from the magneto-electric interaction also has a peak at 1.1μm,
2 which is due to the magnetic resonance at 2.2μm as well. The EEEs, yyy curve shows a peak at 1μm, corresponding to the electric-electric interaction enhanced by the electric resonance at 1μm. Note that the second-order susceptibilities that present magnetic interactions have to be much larger to achieve the same effect, asthe magnetic fields are
377 times smaller than the electric fields. The plots in Figure 3-8 have been corrected for
2 2 this effect. The actual value for EMMs, yzz is 377 larger than the value in the plot, and for
2 EEMs, yyz is 377 larger. Figure 3-9 plots the real retrieved values of non-zero second-order
2 2 2 2 susceptibilities EMMs, yzz , EEMs, yyz and EEEs, yyy separately. We see that EMMs, yzz is 6
2 orders of magnitude larger than EEEs, yyy , which is the major contribution to the large
2 2 nonlinear enhancement. For EMMs, yzz and EEMs, yyz , we also see smaller, broader peaks around 1 μm, which are due to the induced transparency. Again, as the couplings between the magnetic mode and electric mode of plasmon induced transparency is nonlinear, we cannot separate the magnetic and electric resonances by using the retrieval method.
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Figure 3-8: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Retrieved second-order surface susceptibilities as a function of SHG 2 2 2 wavelength: (blue) EEEs, yyy ; (orange) EEMs, yyx ; (yellow) EEMs, yyz ; (purple-dash) 2 2 2 EMMs, yxx ; (green) EMMs, yxz ; (light blue) EMMs, yzz .
To prove the accuracy of the retrieved second-order surface susceptibilities, we plug them back into Eq. (3.52) and (3.83), and solve for the transmitted and reflected
SHG field E 2 and E 2 . Note that the electric fields and magnetic fields in Eq. y z0 y z0
(3.52) and (3.83) are linked through Maxwell’s Equations, and all the second-order susceptibilities in Eq. (3.52) goes to zero. Figure 3-10 plots the recovered SHG spectrum at 0 degree, 30 degree and 45 degree. As a comparison, we also plot the SHG spectrum from the simulation. Perfect agreements are seen between the simulation and the recover
SHG spectrum from retrieved second-order susceptibilities, indicating the retrieved
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second-order surface susceptibilities are indeed the nonlinear parameters of the metasurface, and can be used to predict the nonlinear response.
Figure 3-9: (Reproduced with permission from Ref. [74]. Copyright (2017) by 2 2 Elsevier B.V.) Second-order surface susceptibilities EEEs, yyy (left), EEMs, yyz (middle), 2 and EMMs, yzz (right) as a function of SHG wavelength.
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Figure 3-10: (Reproduced with permission from Ref. [74]. Copyright (2017) by Elsevier B.V.) Comparison between simulation and recovered SHG spectrum at 0 degree (first column), 30 degree (second column), and 45 degree (third column) using retrieved second-order surface susceptibilities. (Top row) SHG reflection; (Bottom row) SHG transmission. (Lines) Simulation; (Circles) Recovered SHG spectrums.
3.4 Conclusion
In summary, we have derived the nonlinear GSTCs that describe the interaction between electromagnetic waves and a nonlinear metasurface, which are valid across the electromagnetic spectrum. Through the definition of nonlinear surface polarization densities at a nonlinear metasurface, we established the connection between the nonlinear macroscopic fields and the microscopic fields across a surface. Simultaneously, 81
the effective nonlinear surface susceptibilities have been uniquely defined. These nonlinear surface susceptibilities can either be related to the local properties of the meta- scatterers, or can be linked to macroscopic fields through the nonlinear GSTCs.
One of the key advantages of the nonlinear GSTCs is that they allow a generalized retrieval method of nonlinear metasurfaces and thin metamaterials, either from numerical simulations or experiments. Unlike the nonlinear bulk retrieval method, which has limited application for oblique incidence, the surface retrieval procedure utilizing the nonlinear GSTCs enables us to retrieve all the nonlinear parameters at oblique incidence. Moreover, for both linear and nonlinear bulk retrieval methods, it is necessary to define an effective thickness to single layers of metamaterial elements, introducing additional ambiguity to the retrieval procedure. This problem is especially critical for retrievals at oblique incidence, where the fields travelling at different angles may experience different lengths of interaction. With the surface retrieval procedure derived from the nonlinear GSTCs, as both the linear and nonlinear surface parameters are uniquely defined for a surface, we avoid the ambiguity of defining the effective thickness of a nonlinear metasurfaces or a thin metamaterial. If an estimate of the nonlinear bulk parameters is critical, the nonlinear surface parameters can be easily linked back to bulk parameters.
Another advantage of the nonlinear GSTCs is that they relate the nonlinear surface polarization densities to the transmission and reflection properties of a nonlinear
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metasurface. As long as the nonlinear surface susceptibilities can be retrieved, they can be inserted back to the nonlinear GSTCs, enabling us to predict the properties of transmission and reflection at nonlinear frequencies. When the elements of nonlinear metasurface elements are properly designed, it is possible to achieve desired transmission and reflection, with appropriate angles of excitations and intensities. This could further enable us to add tunability to one or several specific nonlinear surface parameters, allowing us to achieve controllable nonlinear metasurfaces.
We have validated the nonlinear GSTCs and the nonlinear surface retrieval procedure by retrieving second-order surface susceptibilities from a fused silica slab. We also investigated a nonlinear metasurface that involved nonlinear magneto-electric interactions. The nonlinear surface retrieval method successfully separated the rich magneto-electric nonlinear interactions. Even though the host nonlinear material in the nonlinear metasurface is isotropic, purely electric, and with modest nonlinearity, the dominant effective nonlinear susceptibility is magneto-electric, and is 6 orders of magnitude larger than that of the host nonlinear material. Finally, we successfully recover the SHG reflection and transmission spectrum by using the retrieved nonlinear susceptibilities, which provides a more intuitive way to design and predict the performance of the nonlinear metasurfaces.
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4. Enhanced nonlinear response from metasurface platforms
The collective oscillations of conduction electrons in metal nanostructures give rise to plasmonic resonances at optical frequencies. These resonances, which are typically accompanied by large enhancement and localization of electromagnetic fields, have been exploited in many nonlinear processes [31, 32, 83, 84]. Various plasmonic structures, including films, nanoparticles, and metasurfaces, have been investigated for second-harmonic generation (SHG) [20, 85-88], third-harmonic generation (THG) [21, 22,
25, 89-92], four-wave-mixing (FWM) [43, 93-97], and high harmonic generation [98-100].
Coupled plasmonic nanostructures are of particularly interesting, as when such nanostructures are positioned in close proximity to each other, the hybridization of the plasmonic resonances can lead to extremely large localized fields between the nanostructures. The gaps between nanostructures, often referred to as “hot spots,” have the potential to dramatically boost the nonlinear response of embedded materials that interact with the strongly enhanced fields. The optical properties of many coupled geometries, such as gap-antennas [21, 22], bowties [90, 92, 98], and film-coupled nanostripes [25], have been investigated for their scattering and field enhancement characteristics.
Among the wild range of hybrid nanostructures being investigated, film-coupled nanostripes have recently attracted significant interest. Film-coupled nanostripes were
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found to behave as near-perfect absorber when they were excited [101, 102]. Such a near- perfect absorber induces large field enhancement within the coupling region (gap region), allowing large nonlinear response when nonlinear materials are integrated. The gap region of film-coupled nanostripes are easily controllable down to the sub- nanometer scale, which make it a promising plasmonic platform to study quantum effects.
4.1 Third-harmonic generation enhancement
In this section, my research on the third-harmonic generation (THG) enhancement from the film-coupled nanostripe resonator is discussed. This section is reproduced with permission from J. Britt Lassiter, Xiaoshu Chen, Xiaojun Liu, Cristian
Ciracì, Thang B. Hoang, Stéphane Larouche, Sang-Hyun Oh, Maiken H. Mikkelsen, and
David R. Smith, ACS Photonics, 1 (11), 1212 - 1217 (2014). Copyright (2014) by American
Chemical Society.
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4.1.1 Geometry and method
Figure 4-1: (Reproduced with permission from Ref. [24]. Copyright (2015) by Optical Society of America.) (a) Geometry of the film-coupled nanostripe. EV gold: evaporated gold; TS gold: template-stripped gold. (b), (c) Magnetic and electric field distributions at the fundamental resonance of the system (1.5 μm); arrows represent the direction of magnetic and electric fields. (d), (e) Magnetic and electric field distributions at a higher-order resonance.
The geometry used in this study is shown in Figure 4-1 (a). A 30nm-thick gold film was fabricated by template stripping, which resulted in an ultra-smooth gold film on a hardened optical epoxy [103]. A thin layer of Al2O3 was then deposited on the gold film by atomic layer deposition (ALD) [104]. On top of the Al2O3 coated gold film, periodic gold nanostripes were fabricated using electron beam lithography. This geometry results in a coupled plasmonic structure of which each unit element behaves like a two-dimensional optical patch antenna, where a cavity resonator is defined between the stripe and film in the Al2O3 gap. When the system is excited at resonance and normal incidence by a TM-polarized wave, with the electric field oriented along the width of the stripe, most of the incident energy is coupled to the system and a close-to- zero reflectance can be observed [101, 105]. The electromagnetic field in the gap region is 86
localized and enhanced by the cavity effect: confined gap plasmons propagate along the width and are reflected at the edges. As a result, a maximum in the magnetic field is created at the center of the gap region and a maximum in the electric field is created at either edge of the gap region (Figure 4-1 (b) and (c)). The electric fields in the stripe and the film are horizontal and anti-parallel to each other. The resonance frequency of this structure is determined by the stripe width, W, the gap size, g, the optical properties of the Al2O3 spacer layer and those of the gold [106]. Note that this structure also supports higher order resonances, where the confined plasmons propagating along the gap have multiple nodes, as shown in Figure 4-1 (d) and (e).
In order to study the THG enhancement, a set of film-coupled nanostripes samples with different gap sizes g are considered. To ensure the minimal effect of spectral dispersion in both the linear and nonlinear optical properties of the materials, we design each sample to have the same resonance wavelength at 1.5 μm, but vary the stripe widths W to compensate the changing gap size g. The period of the film-coupled nanostripes is chosen to be 250 nm, which is two times smaller than the wavelength of the THG to eliminate any higher order diffractions from the periodic structures. The width of the stripes measured from SEM are 104 nm, 119 nm, 127 nm, 153 nm, 166 nm, and 188 nm, respectively (Figure 4-2, insets). The corresponding gap sizes are 2.83 nm,
3.98 nm, 4.40 nm, 6.80 nm, 8.60 nm, and 11.40 nm, respectively. These gap sizes are determined by fitting the simulated reflectance spectrums with experimental results
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from Fourier transform infrared spectroscopy (FTIR), as it is shown in Figure 4-2. In numerical simulations, the permittivity of the gold stripes (evaporated gold) and the ultra-flat gold film (template-stripped gold) were taken from empirical data [107]. We choose the refractive index of the Al2O3 to be 1.45, which was determined by ellipsometry measurements. The accuracy of the fitted gap sizes is also guaranteed by ellipsometry measurements.
Figure 4-2: (Reproduced with permission from Ref. [25]. Copyright (2014) by American Chemical Society.) Experimental (blue) and simulated (black) reflectance spectra for each of the stripe samples used in the third-harmonic generation 88
experiment. Each panel shows a representative spectrum from the sample with gap size indicated. Insets: SEM images of the stripe samples corresponding to the spectra (scale bar = 500 nm).
Figure 4-3: (Reproduced with permission from Ref. [25]. Copyright (2014) by American Chemical Society.) Schematic diagram of experimental setup for measurement of THG. Inset: representative THG spectrum measured from film- coupled stripe sample.
The THG were both experimentally measured, and numerically computed.
Figure 4-3 represents a schematic diagram of experimental setup for measurement of
THG, where a 1.5 μm ultrafast laser pulses (~200 fs) was focused onto the sample using a broadband-corrected microscope objective, and the reflected third-harmonic signal from the sample were collected using the same objective and directed to the spectrometer. To obtain an absolute enhancement factor, the generated third-harmonic signal from the samples was normalized by that from a 30nm template stripped gold film. In numerical simulations, the third-harmonic fields were computed using COMSOL Multiphysics - a 89
finite-element based solver - under the undepleted pump approximation. The fields at the pumping wavelength (1.5 μm) were used as the source term for computing the response at THG. Both the gold and the Al2O3 spacer layer are considered and treated as isotropic nonlinear materials, whose third-order nonlinear polarizations are expressed as
1 3 r 0 rrr . (4.1) 4
The χ(3) used for calculating the THG from the bare gold film was 2.45×10-19 (m/V), which was an orientational average of the χ(3) tensor of polycrystalline gold film [108]. The real
χ(3) for the gold was uncertain and could be complex [109]. As an approximation, we used χ(3) =2.45×10-19 (m/V)2 in numerical simulations. The χ(3) for crystalline Al2O3 is known to be 3.1×10-22 (m/V)2 [42]. However, an amorphous Al2O3 ALD spacer layer may not necessarily have the same properties as the bulk crystals. The χ(3) of the ALD spacer layer was then found to be 2.3×10-23 (m/V)2 by fitting the simulation results to the experimental results, which was in reasonably good agreement with the χ(3) (~10-24 m2/V2) estimated in Ref. [42].
4.1.2 Result and Discussion
As we have discussed in Ref. [25], both the gold film and nanostripe possess a relatively large 3 that could potentially give rise to the observed THG. We performed a numerical study to investigate the THG that would arise from the gold alone (Figure
4-4 green squares). In this case, the trend of the simulated THG as a function of gap size 90
is distinct from that measured in the experiments. The trend observed in the simulations results from the limited field penetrating into the gold. When decreasing the gap size, the fields confined within the gap are stronger, as are the fields inside the gold; however, due to the large losses in gold, only the field near the gold surface is significantly enhanced. The tradeoff between the field enhancement and loss mechanisms due to the gold nonlinearity results in a much flatter curve.
Figure 4-4: (Reproduced with permission from Ref. [25]. Copyright (2014) by American Chemical Society.) Third harmonic generation enhancement (as compared to a bare gold film) vs. gap size (g). (Blue) experimentally measured THG enhancement, (Red) numerically simulated THG enhancement that only take into account the nonlinearity from the Al2O3 spacer layer. (Green) numerically simulated THG enhancement that only take into account the nonlinearity from gold, (Orange) numerically simulated THG enhancement taking into account the nonlinearities of both the gold and Al2O3 spacer layer.
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The simulation that only considers the THG arises from the spacer layer (Figure
4-4 (red squares)) is in good agreement with that measured from experiment. However,
3 as the of the gold is about fourth power larger than that of the Al2O3 ALD spacer layer, it is unreasonable to neglect the nonlinearity of the gold. For this reason, we show the simulation result including both nonlinearities from the gold and the spacer layer
(orange squares). The THG from the gold and the spacer interfere constructively, resulting in a total THG enhancement exceeding that from either the gold or spacer alone.
4.1.3 Conclusions
In summary, none of the simulated cases (green, orange, and red squares) could fit with the experimental results (blue circles). One possible cause of this discrepancy is that we did not account for other possible nonlinear effects that can occur in this system.
It is possible, for example, that competition between enhancement by the Al2O3 spacer layer and reduction by nonlinear absorption and other processes could account for the observed experimental enhancement values. It is also possible that nonlinear absorption could severely decrease the observed THG signal in simulation and hence bring both the green and the red curve down below the experimental curve. However, as the THG responses were measured at single wavelength, which inherently possessed large errors due to the coarse tune in plasmonic resonance of each sample, it was thus not possible to
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independently identify the contribution from the metallic component versus that from the dielectric component, though a large enhancement of the THG was observed.
4.2 Clarifying the origin of THG from an isolated film-coupled nanostripe
Motivated by our THG experiments on film-coupled nanostripes described in section 4.1, we propose a method to identify the origin of the THG from an isolated film- coupled nanostripe [24]. By considering the THG from each nonlinear source separately in numerical simulations, we investigate the far field radiation pattern of each nonlinear source respectively, and show that their far field radiation patterns are distinguishable due to the fundamental difference in their radiation properties. We also find a simple relation that connects the total THG and the THG from each individual nonlinear source. To illustrate the underlying mechanism, we divide our analysis into two steps.
First, we only consider the THG before it couples to the film-coupled nanostripe, whose field distribution is directly related to that of the third-order nonlinear polarization.
Second, we take into account the film-coupled nanostripe and study the coupling between the THG and the structure. Finally, we demonstrate the generality of our method by investigating film-coupled nanostripes over a wider range of parameters.
This section is reproduced with permission from Xiaojun Liu, Stéphane Larouche,
Patrick Bowen, and David R. Smith, Optics Express, 23 (15), 19565 - 19574 (2015).
Copyright (2015) by Optical Society of America. 93
4.2.1 Geometry and method
The geometry used in this section is the same as that in section 4.1, except we consider an isolated nanostripe positioned on top of a 100 nm-thick, infinitely wide gold film coated with an aluminum oxide (Al2O3) layer. In numerical simulations, we choose the refractive index of the Al2O3 to be 1.45, which was determined by ellipsometry measurements previously reported [25]. The permittivity of the gold stripes (evaporated gold) and the ultra-flat gold film (template-stripped gold) were taken from empirical data [107]. By choosing the stripe width W to be 115 nm and the gap spacing to be 3 nm, we are able to tune the resonance wavelength to 1.55 μm. This structure also supports higher order resonances, where the first-order gap plasmon mode is around 720 nm, and the second-order gap plasmon mode is around 600 nm. As we are going to show in the following text, while the fundamental resonance ensures the enhancement of nonlinear signal by strongly localizing the pumping field in nonlinear materials, these higher order resonances are beneficial as they facilitate the radiation of the nonlinear signal through coupling.
The third-harmonic fields were computed using COMSOL Multiphysics. The
3 of the gold and of the amorphous Al2O3 spacer layer are unknown. In previous
3 19 2 work, a value for the susceptibility of Au 2.45 10 m / V was used as an
3 approximation. The of the amorphous Al2O3 spacer layer was found to be
2 2.3 1023 m / V by fitting simulation results with experimental measurements. Here,
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3 to make our discussion general, instead of assigning specific values to Au and
3 3 3 Al2 O 3 , we focus on the ratio between Au and Al2 O 3 . We maintain the bulk nonlinearity of gold in simulation, but ignore any possible surface nonlinearity. In order to study the THG arising from different nonlinear sources, we performed separate
THG simulations, taking into account the nonlinear contribution from the Al2O3 spacer layer, the gold stripe, and the gold film, respectively. The far-field radiation of each nonlinear source is calculated as a function of radiation angle in COMSOL Multiphysics, where
farlimr sca . (4.2) r
To avoid the influence of the near-field, we choose our simulation domain to be much larger than the pumping wavelength.
To describe the THG enhancement by the film-coupled nanostripe, we define a
THG enhancement factor of the form
far (4.3) 3 3 0
where 0 is the amplitude of the pumping field at normal incidence, far is the amplitude of the THG far field measured at , and 3 is the third order nonlinear susceptibility of the component (e.g., gold, dielectric) of interest. depends neither on the field amplitude at the pumping frequency, nor the specific values of the source
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3 3 nonlinearities, Au or Al2 O 3 . That is, presents the overall THG enhancement of the system, rather than the THG from a specific bulk 3 pumped by certain amount of energy.
4.2.2 Results and discussions
For nonlinear enhancement, the nanostructure plays a dual role as a coupler for the incident field, and an antenna for the harmonic field. In this section, we probe in detail the behavior of the nanostripe structure, providing a potential path for future optimization of nonlinear nanostructures.
First, consider the case that only the Al2O3 spacer, the gold stripe, or the gold film has nonlinear optical properties. Figure 4-5 (a) – (c) shows the THG enhancement factor as a function of radiation angle for the THG generated from the Al2O3 spacer layer, the gold stripe, and the gold film, respectively. The patterns of the three THG enhancement factors are completely different. The enhancement factors for the THG from the gold stripe and the gold film are more omnidirectional, while that from the Al2O3 spacer layer is more directional. This difference in the shape of the enhancement factor indicates different enhancement mechanisms for the three nonlinear sources. The total THG that includes all the nonlinear contributions is found empirically to be approximately
THG3 3 3 3 Efartotal Au stripe film EE 0 Al 2 O 3 spacer 0 (4.4)
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This equation indicates that the contribution from the spacer layer and that from the stripe interfere constructively, while that from the gold film interferes destructively with the previous two. Equation (4.4) also indicates that the far field radiation pattern of the total THG completely depends on the enhancement factor of each of the nonlinear
3 sources and their relative weights. In other words, the ratio between the Au and
3 Al2 O 3 , and the enhancement factor of each nonlinear source are the only factors that influence the shape of the far field radiation pattern of the total THG. To demonstrate this idea and the validity of Eq. (4.4), we numerically and analytically calculated the total THG using different ratios of the third-order susceptibilities between the Al2O3 spacer and the gold. Figure 4-5 (d) - (f) show the total THG normalized by
3 3 3 4 4 Au and E0 for Al2 O 3 Au 0, 0.5 10 and 5 10 , respectively. The red circles represent simulated data, while the black curves are calculated by inserting the simulated THG enhancement factors into Eq. (4.4). All the data in Figure 4-5 (d) – (f) are
3 normalized by Au and E0 , which are constants and do not influence the results.
3 3 When the nonlinear responses are only from the gold, that is, Al2 O 3 Au 0 , the radiation pattern of the total THG is similar to the shape of stripe . This is caused by the destructive interference between the THG signal from the gold stripe and the gold film, where the influence of the film is small when compared to that of the stripe.
3 3 4 When Al2 O 3 Au 0.5 10 , the enhancement from the gold and Al2O3 spacer
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are comparable to each other. As a result, an intermediate far field radiation pattern is seen between the shape of stripe and spacer , as shown in Fig. 3.5 (e). For
3 3 4 Al2 O 3 Au 5 10 , the signal from the spacer layer dominates, resulting in a shape similar to that of the spacer .
Figure 4-5: (Reproduced with permission from Ref. [24]. Copyright (2015) by Optical Society of America.) (a) – (c) THG enhancement factor for the THG from the Al2O3 spacer, the gold stripe and the gold film, respectively. (d) – (f) Total THG enhancement factor evolving with the ratio of the third-order susceptibilities between the Al2O3 spacer and the gold; (curve) calculated from Eq. (4.4); (circles) simulated data. (d) ratio = 0; (e) ratio = 5×10-5; (f) ratio=5×10-4.
These results highlight two important observations. Firstly, even though the
3 of the Al2O3 spacer layer is four orders of magnitude smaller than that of the gold,
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the THG from the spacer may dominate over the THG from the gold. This is not only due to the large field enhancement in the gap, but also the different enhancement mechanism. Secondly, by investigating the far field radiation of the total THG as a function of the radiation angle, we can infer which nonlinear source dominates. This will be helpful for experiments where it is often difficult to determine the source of the nonlinear signal.
Figure 4-6: (Reproduced with permission from Ref. [24]. Copyright (2015) by Optical Society of America.) (a) – (c) Near field distributions of the THG generated from the Al2O3 spacer, (a) radiating in vacuum, (b) being reflected by a gold film, (c) being coupled to the film-coupled nanostripe. (d) Far field radiation patterns of the THG generated from the Al2O3 spacer radiating in vacuum (red), being reflected by a gold 99
film (black), being coupled to the full structure (blue). (e) – (g) Near field distributions of the THG generated from the gold stripe, (e) radiating in vacuum, (f) being reflected by a gold film, (g) being coupled to the film-coupled nanostripe. (h) Far field radiation patterns of the THG generated from the gold stripe radiating in vacuum (red), being reflected by a gold film, being coupled to the full structure (blue). (i) – (k) Near field distributions of the THG generated from the gold film, (i) radiating in vacuum, (j) being reflected by a gold film, (k) being coupled to the film- coupled nanostripe. (l) Far field radiation patterns of the THG generated from the gold film radiating in vacuum (red), being reflected by a gold film (black), being coupled to the full structure (blue).
To understand the enhancement mechanism for each nonlinear source, we divide our analysis into three steps. First, we consider the THG before it couples to the film- coupled nanostripe, whose field distribution is directly related to the distribution of the
nl third-order nonlinear polarization . In the following text, we call the distribution of
nl the THG considered in this step as the far-field radiation of the to distinguish it from the actual THG radiated from the structure. In the numerical simulations, we use the field distribution of the film-coupled nanostripe at the pumping wavelength as the source to compute the THG response, but assume that there is no structure present at the THG wavelength. We do this by setting the optical properties of all materials at 3ω to be those of vacuum.
nl Figure 4-6 (a), (e) and (i) show the near-field distributions of the considering that only the Al2O3 spacer layer, the gold stripe, or the gold film are nonlinear, respectively. For clarity, we only show the norms of magnetic near fields. The film- coupled nanostripe supports a resonance at the pumping wavelength (Figure 4-1 (b),
nl (c)), such that the generated by nonlinearity within the Al2O3 spacer layer resembles 100
an electric quadrupole mode, with electric field perpendicular to the film (Figure 4-6 (a)).
The corresponding far field radiation consists of two lobes in the horizontal direction, as plotted in Figure 4-6 (d) in red. By contrast, for nonlinearity in the gold stripe or the gold film, the pumping fields penetrating into the gold are along the stripe width and the film
nl nl (Figure 4-1 (c)). As a result, the far-field radiation patterns of stripe and film are similar to planar electric dipoles, as shown in Fig. 3.6 (e) and (i). The corresponding far field radiation patterns are omnidirectional, in agreement with the typical radiation pattern of electric dipoles. Note that there is a π difference in phase between the far-field
nl nl radiations of stripe and film , which is caused by the opposite electric fields in the gold stripe and the gold film at the pumping frequency (Figure 4-6 (c)). This phase difference ultimately causes the THG from the gold stripe and the gold film to destructively interfere in the far-field.
To distinguish the influence of the gold film and that of the nanostripe, we consider that the gold film is present at the third harmonic frequency, but that the optical properties of the spacer and the stripe are still those of vacuum. For the case where the spacer layer is nonlinear with an equivalent quadrupole source distribution, the THG is completely suppressed due to the interaction between its radiation and the reflection of the gold film (black curve in Figure 4-6 (d)). For nonlinearities in the gold stripe or the gold film, whose equivalent electric dipoles are parallel to the gold film, the
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reflection by the gold film enhances the far field by a factor of about four (black curves in Figure 4-6 (h) and (l)).
Finally, we consider how the THG in the first step couples to the entire nanostructure by setting the optical properties of all materials at 3ω to their actual values. The THG far-field generated by the nonlinearity in the gap is now normal to the surface, rather than in the plane of the surface, as shown by the blue curve in Figure 4-6
(d). This change is caused by the coupling between the THG generated in the gap and the higher-order gap plasmon modes supported by the structure at 3 . When examining the near field, nodes are found in the magnetic field distribution of the THG, indicating the presence of gap plasmon modes (Figure 4-6 (c)). By comparing the far field radiation patterns in Figure 4-6 (d), we find that, because of the coupling between the THG field and the higher-order gap plasmon modes, the THG originating from the spacer layer has been enhanced. For THG originating from the gold, the equivalent electric dipoles either couple to the gap plasmon modes supported by the film-coupled nanostripe or the radiation modes. On the contrary, the near-field distributions of
nl nl stripe and film only have a large overlap with the fundamental gap plasmon mode whose resonance is at 1.55μm. However, the THG wavelength is far from 1.55μm and the coupling is so weak that it can be neglected. For this reason, the stripe behaves as a simple scatterer and the film as a mirror, which leads to the coupling of the THG directly to radiation modes, without the intermediary of the gap plasmon mode. By
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examing the far-field radiation patterns in Figure 4-6 (h) and (l), we notice that the radiation originating from the gold stripe and the gold film is suppressed by the nanostripe. The THG radiation from the gold film, corresponding to electric dipoles embedded in gold, experiences larger losses in all three conditions. Note that the phase relation between the THG from the stripe and the film is not changed by the coupling, leading to the destructive interference between the THG from the stripe and the film.
The nonlinear sources, their equivalent dipole sources and how they couple to the nanostructure at the THG wavelength are summarized in Table 4-1.
Table 4-1: Nonlinear sources and their equivalent dipole sources
Equivalent dipole sources Nonlinear sources nl Pumping Strongly couple Strongly couple wavelength to gap plasmon to radiation modes? modes?
Al2O3 spacer layer Magnetic dipole Multipole Yes No Gold stripe Electric dipole Electric dipole No Yes Gold film Electric dipole Electric dipole No Yes
4.2.3 Generality
In this section, we demonstrate the generality of our results by using film- coupled nanostripes of different dimensions. To eliminate any changes in the THG due to the material dispersion, we fix the resonance wavelength at 1.55μm by varying the gap size (g) and stripe width (W) simultaneously. For gap size g=1nm, 2nm, 3nm, 4nm,
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5nm, 6nm, 7nm, 8nm, 9nm, and 10nm, the corresponding stripe widths are 65nm, 94nm,
115nm, 131nm, 145nm, 157nm, 167nm, 177nm, 185nm and 193nm.
Figure 4-7: (Reproduced with permission from Ref. [24]. Copyright (2015) by Optical Society of America.) (a), (b) The THG enhancement factor for THG from the Al2O3 spacer, the gold stripe and the gold film as a function of gap size, at radiation angles of 90 degree (a) and 45 degree (b). Red: η(spacer); blue: η(stripe); η(film). (c), (d) The total THG enhancement factor evolving with the ratio of the third-order susceptibilities between the Al2O3 spacer and the gold, at radiation angles of 90 degrees (c) and 45 degrees (d); (stars) calculated data from Eq. (4.4); (lines) simulated data.
In Figure 4-7 (a), we plot the THG enhancement factor for THG from the Al2O3 spacer, the gold stripe and the gold film as a function of gap size, at a radiation angle of
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90 degrees. For all nonlinear sources, two factors having opposite effects come into play.
As the gap size increases, the field enhancement in the gap region decreases, while more nonlinear material (gold or Al2O3) is included within the gap region due to the increase of the stripe width. For the nonlinear spacer, the former factor predominantly influences the THG, while for the gold, the latter factor is more influential. As a result, spacer reaches its maximum at 2nm and decreases afterward with increasing the gap size, while stripe and film increase all the way to g=10nm.
Figure 4-7 (b) shows the three THG enhancement factors at a radiation angle of 45 degrees. By comparing Figure 4-7 (a) and (b), we notice that although the trends at 45 degrees are similar to those at 90 degrees, spacer decreases more than stripe and
film . This provides evidence that the THG radiated from the Al2O3 spacer is more directional than that radiated from the gold. To demonstrate the validity of Eq. (4.4) for film-coupled nanostripes of different dimensions, we numerically and analytically calculated the total THG evolving with the ratio between the third-order susceptibilities of the gold and the Al2O3 spacer at radiation angles of 90 degrees and 45 degrees, as shown in Figure 4-7 (c) and (d). When the THG signal from the Al2O3 spacer dominates,
3 3 4 that is, Al2 O 3 Au 5 10 , the trend is exactly the same as that of the THG enhancement factor spacer . When the THG signal from the Al2O3 spacer is
3 3 4 comparable to ( Al2 O 3 Au 0.5 10 ) or smaller than that from the gold
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3 3 ( Al2 O 3 Au 0 ), the trend follows the THG enhancement factor gold . In resolving the THG from different nonlinear sources in the film-coupled nanostripes, as the THG enhancement factor of different nonlinear sources are inherent to the plasmonic structure and can be determined using numerical simulations, the ratio between the third-order susceptibilities of the gold and the Al2O3 spacer can be determined by fitting
Eq. (4.4) with the measured THG curve.
4.2.4 Conclusion
In conclusion, we propose a method to determine the origin of the THG from a single nanostripe coupled to infinitely large film. By considering the THG from each nonlinear source separately, the near- and far- fields of the THG radiating from different nonlinear sources are distinguishable due to the fundamental difference in their radiation properties. The THG signal from the gold film and the gold stripe destructively interfere with each other, and the THG signal from the gold is suppressed by the structure itself. However, due to different coupling schemes, the structure enhances the THG signal generated from the Al2O3 spacer and facilitates its radiation.
The total nonlinear radiation pattern is the sum of the far-fields of all the nonlinear sources, which is determined by the ratio between the third-order susceptibilities of the dielectric and the metal. We find that, even though the third-order susceptibility of the gold is four orders larger than that of the Al2O3 spacer, it is still possible for the THG generated by the spacer to dominate.
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Since the THG is a superposition of the contributions from all the nonlinear sources, the THG from each nonlinear source in our structure can be identified in two ways. First, as the THG far-field radiation patterns from different nonlinear sources are distinguishable, the dominating nonlinear source can be determined by measuring the far-field radiation pattern of the THG. Second, by fixing the resonant frequency while varying the structure geometry, the THG enhancement factor of each nonlinear source has a specific trend that is inherent to the plasmonic structure. This trend can be determined using numerical simulations to investigate other nonlinear hybrid plasmonic structures. By fitting the measured total THG with Eq. (4.4), the ratio between the third-order susceptibilities of the metal and the dielectric can be determined.
Our method can be applied to analyze the origin of the nonlinear signal associated with other hybrid plasmonic structures, such as film-coupled nanospheres.
However, our method fails to identify the origin of nonlinear response in periodic structures, whose nonlinear radiation patterns are mostly uniform and indistinguishable in the far-field. A different approach has been developed to address the periodic structures, which will be presented in a future work.
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5. Enhanced two photon photochromism in metasurface perfect absorbers
Light switchable materials are essential to optoelectronic applications in photovoltaics, memories, sensors, and communications. Natural switchable materials suffer from weak absorption and slow response time, preventing them from low-power, ultra-fast applications. Integrating light switchable materials with metasurface perfect absorbers offers an innovative route to achieving desirable features for nanophotonic devices, such as directional emission, low-power and broadband operations, high radiative quantum efficiency and large spontaneous emission rates.
In this chapter, we show an enhanced two photon photochromism based on a metasuface perfect absorber: a film-coupled colloidal silver nanocube. The photochromic molecules—spiropyrans—are sandwiched between the silver nanocubes and the gold substrate. With approximately nearly 100% absorption and an accompanying large field enhancement in the molecular junction, the transformation of spiropyrans to merocyanines is observed under the excitation of infrared laser light. Due to the large
Purcell enhancement in the film-coupled nanocubes, fluorescence lifetime measurements on the merocyanine form reveal large enhancements on spontaneous emission rate, as well as high quantum efficiency. An incident power as low as 10 μW is enough to initiate the two-photon isomerization of spiropyran in the film-coupled nanocubes, and a power of 100 nW is able to excite the merocyanines to emit
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fluorescence. The power consumption is orders of magnitude lower than bare spiropyran thin films on silicon and gold, which is highly desirable for the writing and reading processes relevant to optical data storage. The wavelength specificity is demonstrated by sweeping the plasmonic resonance of the film-coupled nanocubes, which opens up new possibility to for minimizing the cross talk between adjacent bits in nanophotonic devices.
We begin with an overview of the studies of nanoantenna and photochromic materials in the application of optical data storage. Next, we introduce the optical properties of spiropyran - the photochromic material under consideration. Section 5.2 provides an overview of our idea. Section 5.3 discuss in details the simulation, fabrications, and experiments. Section 5.4 shows the results and discussions, where the
Purcell enhancement, enhanced two-photon absorption, power dependence study, capability of storing information, and specificity of wavelength are discussed. The results are concluded in section 5.5, and an outlook is provided for the potential applications.
5.1 Overview
With the development of information technology, there is an increasing demand for high-performance switches and memory devices—especially data storage devices that have immense storage capacity, fast reading and writing rate, small volume, minimal cross talk between adjacent bites, and high reading sensitivity [110, 111].
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Optical data storage, which use laser to read and write on light-sensitive materials, offer a disruptive method to meet these demands [112, 113]. Conventional investigations on optical data storage have mainly focused on phase holograms [114-116] and two-photon processes [117-119], where a laser is focused at various locations to induce photochemical changes of the recording materials during the read and write process.
However, since the storage capacity is ultimately limited by the diffraction limit of light, the response speed is limited by the natural light-switchable materials. Furthermore, power consumption is usually significant as the energy required for two-photon absorption is high.
5.1.1 Nanoantennas for memory devices
Plasmonic nanoantennas, due to their ability to localize and enhance light at the subwavelength scale, are excellent candidates for optical data storage applications. By incorporating the light sensitive materials with nanostructures such as nanoparticles, the photochemical transform can be induced within single, a few, or a cluster of nanoparticles, scaling down the data storage unit to subwavelength scale [120-126].
Associated with plasmonic field enhancement, Purcell enhancement introduced by the nanoantennas leads to significant enhancement of the spontaneous emission rate, high fluorescence enhancement factors, and high quantum yield, which are desirable for fast data access [127-131]. Furthermore, owing to the high one- and two- photon absorption cross sections, nanoantennas facilitate ultra-low power consumption - an important
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featyre in terms of minimal impact on the environment [120, 122]. Coupled nanoantennas are particularly interesting, since the hybridization of the plasmonic resonances associated with each individual antenna can lead to extremely large localized fields between the nanostructures. Coupled nanoantennas, such as bowties, gap- antennas, and film-coupled nanoparticles have been widely investigated for their scattering and field enhancement characteristics [132-135], as well as nonlinear processes
[24, 25, 94, 136] and photoluminescent properties [128, 129, 131, 137]. The film-coupled colloidal silver nanocube platform has been shown to have particular advantages in control of radiative process, since large-area “perfect” absorbing metasurfaces can be created by simple dip-coating methods [132, 138], with additional desirable features such as directional emission; low-power and broadband operaion; and high radiative quantum efficiency and large spontaneous emission rates [127-129].
5.1.2 Photochromism for memory devices
Photochromic materials, which undergo reversible isomerization with absorption of electromagnetic radiation, have been shown to be a promising class of light-sensitive materials for optical switches and memory devices. Various photochromic molecules have been synthesized and investigated, where excellent progress have been made in achieving large two-photon cross-section[139, 140]. Typical photochromic molecules include spiropyrans, polycyclic aromatic compounds, fulgides and fulgimides, and photochromic copolymers.
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Spiropyrans have been widely studied for their reversible transformation to merocyanines under the exposure of UV light, and back to spiropyrans under visible light or heat. As it is shown in Figure 5-1, the exposure to UV light results in the heterolytic cleavage of the C-O bond followed by isomerization to merocyanines. This process corresponds to a change in absorption spectra that enables strong absorption of visible light in the 550-600 nm band, leading to switching of the color from transparent
(spiropyrans) to deep blue (merocyanines). This process also results in a distinct change in their emission behavior: the spiropyrans do not exhibit strong fluorescent emission, while the merocyanines show strong fluorescent emission centered at 650 nm [141]. The absorption and emission properties have been used for nano-writing [142], and in applications of fluorescent imaging and bio-sensing [143, 144].
Figure 5-1: Photochromism of spiropyran.
Spiropyrans also exhibit nonlinear absorptions, which enable the switching of refractive index and absorption spectra with near infrared light. K. Matczyszyn et al
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investigated the two-photon absorption and three-photon absorption of spiropyrans in chloroform solution [145]. The two-photon process have the potential to be applied in many nanophotonic devices, especially optical storage devices, where two-photon absorption is used during the writing process and one-photon absorption is used udring reading/erasing process [110, 111]. However, milliwatt laser power is required for the two-photon process. Furthermore, the fluorescent lifetime of spiropyran/PMMA blends was found to be 5 ns, which limits the speed of the reading process.
5.2 Enhanced two-photon photochromism in film-coupled nanocubes
Figure 5-2: Film-coupled nanocubes (a) Schematic of silver nanocubes deposited on a gold film coated sipropyran/PMMA blend and PE layers. (b) Reflectance spectrum of film-coupled 110nm silver nanocubes resonance around
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792nm. The reflectance is 3% at resonance. (c) Magnetic field distribution in the gap region when at resonance. (d) Scheme of read and write using spiropyrans.
We integrate spiropyran/PMMA blends into a perfect absorbing metasuface: film-coupled colloidal silver nanocubes—which potentially enables ultra-low power consumption with ultra-fast response time for two- and one-photon processes. The metasurface consists of 110 nm colloidally synthesized silver nanocubes densely spread over a gold substrate coated with ~4 nm spiropyran/PMMA blends and ~3 nm polyethylene layers, as shown in Figure 5-2 (a). The resulting geometry corresponds to an optical patch antenna, with a region of field enhancement just below the nanocube.
When the system is excited at resonance, most o the incident light is absorbed, leading to a strong dip in the reflectance. As has been shown in previous work, at resonance strong local electric fields are produced in the junction region, most strongly around the cube edges. With approximately 100% absorption on resonance and a field enhancement of over 200 in the molecular junction, the spiropyran molecule undergo isomerization to merocyanines by the absorption of two 792nm photons. A 583nm laser is used to probe the state of spiropyrans/mecrocyanines, leading to fluorescent emission centered at 650 nm. Figure 5-2 (d) illustrates the energy level diagram of the write and read of one- photon processes of one molecule in the junction.
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5.3 Methods
5.3.1 Fabrications
The fabrication procedure is summarized in Figure 5-3. 4% PMMA in anisole
(microChem 495PMMA-A4) was diluted with pure anisole to obtain 0.2% PMMA. A spiropyran colorant, 1’,3-Dihydro-1’,3’,3’-trimethyl-6-nitrospiro[2H-1-benzopyran-2,2′-
(2H)-indole] (Sigma-Aldrich), was dissolved in 0.2% PMMA solution to obtain a 3mM spiropyran solution. The solution was spin-coated on a 100-nm-thick gold substrates on silion wafer (Platypus technologies), at a speed of 3000 rpm for 2 minutes. The thin layer was hardened by baking the sample in a vacuum oven at 70oC for 30 mins. Four polyelectrolyte (PE) multiplayers were grown on top of the spiropyran/PMMA thin layer. The sample is merged in a 3mM poly(strenesulphonate) (PSS) and 1M NaCl solution for 5 mins, and rinsed with a 1M NaCl solution for 1 min, followed by immersion in a 3mM poly(allylamine) hydrochloride (PAH) and 1M NaCl solution for 5 mins, and rinsing in 1M NaCl for 1 min. After the desired number of polymer layers was reached, the sample was thorough rinsed in water.
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Figure 5-3: Fabrication procedure of film-coupled nanocubes with spiropyran/PMMA blend within the gap region
110 nm Silver nanocubes with a 3 nm polyvinylpyrrolidone (PVP) coating
(Nanocomposix) were deposited on top of the substrate. A 5μL aqueous nanocube solution was dropped on the surface of the substrate, and incubated for 50mins at 4oC.
The negatively charged nanocubes facilitate electrostatic adhesion to the positively charged top polymer layer (PAH) of the substrate, forming a uniformly, densely distributed layer of nanocubes. The nanocubes not adhered to the substrate were removed by rinsing the substrate with water. The final density of the nanocubes on the substrate was ~20 μm-2, which is confirmed by scanning electron microscopy. The samples were stored in dark to avoid fatigue of the spiropyran, and measured within two days of fabrication to avoid silver oxidation. 116
The patterned samples were fabricated using electron beam lithography. 75 nm gold was evaporated on Si wafer with a 5 nm titanium adhesion layer, followed by the same fabrication procedures of spiropyran/PMMA and PE layers, and the same procedure of depositing nanocubes.
5.3.2 Simulations
COMSOL MultiPhysics was used to calculate the field distribution and scattering properties of the film-coupled nanocubes. Since the nanocubes are randomly distributed over a substrate with a density of ~20 μm-2, we assume the collective properties of the surface correspond to the properties of a single 110nm silver nanocube separated from a gold substrate by a 7 nm spacer layer. The refractive index of the spacer layer was chosen to be 1.45, which is a typical value for organic layers. The spiropyran molecule was modeled as a monochromatic dipole with an emission wavelength of 650 nm. To evaluate the emission properties of the dipole coupled to the nanocavity defined by the film-coupled nanocube, the Green’s function of the system was calculated by varying the spatial position of the dipole on a 15-by-15 grid under the nanocube. The total decay was then calculated from the Green’s function. The radiative quantum efficiency was estimated by comparing the radiative decay rate and the total decay rate. The radiative decay rate was found by subtracting the nonradiative decay rate from the total decay rate, where the nonradiative decay rate was evaluated by integrating all metal losses in the system [129].
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5.3.3 Optical measurements
The reflectance of the samples was measured using Fourier transform infrared spectroscopy (FTIR) with a 36X, NA = 0.5 objective. To confirm the uniform distribution of the nanocubes, each sample was measured at three different locations, and the resonances were found to be around 792 ± 5 nm.
The time-resolved fluorescence emission was measured using a fluorescence lifetime imaging system (FLIM), as shown in Figure 5-4. The pump pulses for TPA were from a Spectra-Physics Tsunami (792nm, 80MHz, ~100 fs duration). A portion of the beam was sent to a Coherent Miro OPO and was converted to 583 nm to serve as excitation source for fluorescence. The beams were focused on the sample by a 40X, NA
= 0.7 objective. The estimated focal spot size was nearly diffraction limited, with a half- maximum of 2 μm. The objective was mounted on motorized stages, which allowed longitudinal and transversal scanning in the plane of focus at a rate of 31250 s-1. The resulting field of view was 180 μm, with a 128x128 pixels. To induce the transformation of spiropyrans to merocyanines by two-photon absorption, the pump light at 792 nm was focused on the sample for 1min, with ~3.6 ms average dwell time on each pixel. The pump beam was then blocked, and the excitation source for fluorescence at 583nm was unblocked and focused onto the sample. Fluorescence emission was collected through the objective, reflected by a dichroic mirror (Semrock FF625), passed through a 600nm long-pass filter (Edmound 62985) and a 650±10nm band-pass filter (Edmound 33336) to
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remove the excitation light, and imaged onto a single-photon counting photomultiplier tube (PMC-100). The PMT was connect to a time-correlated single photon counting module (Becker-Hickl TCSPC, SPC-150), which enables the recording of the fluorescence intensity as well as the lifetime at each pixel. The temporal resolution was around 30 ps.
Each image was acquired at a resolution of 128x128 with 5s integral time, repeated 5 times. For the uniform film-coupled nanocube samples, all the pixels from the same image were summed to get the total fluorescence intensity or the lifetime curve. The instrumental response function was measured by directly scattering laser light into the
PMT (Figure 5-5).
Figure 5-4: fluorescence lifetime imaging system setup
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Figure 5-5: IRF by scattering laser light into the PMT.
5.4 Results and Discussions
5.4.1 Purcell Enhancement
Figure 5-6: Enhancement of spontaneous emission rate and quantum efficiency relative to a dipole in free space as a function of position under the nanocube.
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Large field enhancement and accompanying increased photonic density-of-states result in modification of the spontaneous emission rate of the merocyanines in the gap region. The emission from a merocyanine molecule can be modeled as a monochromatic point-dipole emitting at 650nm. The spontaneous emission rate of the dipole is given by
[128]
2 0 spr pr ,, int 30
0 where ω is the emission frequency, p is the transition dipole moment, and int is the internal non-radiative decay rate of the dipole. The first term corresponds to the
radiative spontaneous emission rate, r . Since the dipole is in the gap region of the film- coupled nanocubes, the photonic density of states r, is modified from its free space value by the local electromagnetic environment. Through numerical simulations, we calculate the spontaneous emission rate of a dipole in the gap region of the film- coupled nanocube. Figure 5-6 (left) shows the relative spontaneous emission rate
0 0 sp sp , where sp is the emission rate in free space. The rate enhancement reach 2,000 at the corners of the nanocubes when assuming the dipole is oriented along the electric field. The radiative quantum efficiency, which is given by
QE r sp ,
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is spatially uniform and is larger than 0.49 over the gap region (Figure 5-6 (right)). The color map represents the values of QE, where it reaches its maximum ( > 0.5 ) at the edge of the cube, and is around 0.35 in the center of the cube.
Based on the analysis above, the film-coupled nanocubes exhibit multiple features desirable for enhanced two-photon photochromism: (1) Perfect absorption at pump wavelength; (2) high radiative rate at fluorescent wavelength; (3) high quantum efficiency.
Figure 5-7: time-resolved emission of merocyanines from (a) film-coupled nanocubes; (b) gold substrate; (c) silicon substrate. Blue lines: before two-photon absorption. Red lines: after two-photon absorption. Yellow lines: after UV exposure.
Figure 5-7 shows the time-resolved emission of merocyanines from (a) film- coupled nanocubes, (b) a gold film, and (c) a silicon substrate. For the film-coupled nanocubes, the thickness of the spiropyran/PMMA blends is 4.12 ± 0.25nm, and the thickness of the polyethylene layers is 2.83 ± 0.21nm, as confirmed by ellipsometry (see
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Methods). The nanocubes are uniformly and densely deposited (~20 μm-2). The gold film and the silicon samples are coated with the same spiropyran/PMMA blends and polyethylene layers with the same thickness. Compared to the gold film sample and the silicon sample, a significant shortening of the merocyanines lifetime is seen in the film- coupled nanocubes sample. The lifetime can be determined by fits to the data deconvoluted with the instrument response function (IRF) [146]. For the film-coupled nanocubes sample, the spontaneous decay is so fast that is comparable to the IRF (Figure
5-5); the lifetime is estimated to be close to or smaller than the temporal resolution limit of the single photon counting photomultiplier tube (~30ps). The lifetime for the gold film sample is ~209ps, which exhibits an increased decay rate due to the coupling of the emission to surface-plasmons [147]. The lifetime for the silicon sample is ~574 ps. An at least 20-fold reduction in the fluorescence lifetime is seen in the nanocube sample when comparing to the silicon sample.
5.4.2 Enhanced two-photon absorption
Figure 5-7 also shows the time-resolved emission of merocyanines after the two- photon absorption of the 792nm light. A 68-fold enhancement in the fluorescent emission intensity is seen in the nanocube sample, after a 3.6ms average exposure to
40μW, 792nm laser light focused by a 40X, NA = 0.7 objective (see Methods). This increase indicates massive isomerization of spiropyrans to mecrocyanines, due to the large field enhancement in the gap region of the nanocube sample. By contrast, even as
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increase the power of the 792nm laser is increased to 1.2mW, there is no obvious change in the fluorescence emission intensity for the gold film sample and silicon samples, indicating the absence of two-photon absorption. We also show the fluorescent emission enhancement after exposing the gold film and silicon samples to a 670μW continuous laser centered at 405nm for the same amount of time. The silicon sample shows an 18- fold enhancement in the fluorescentce emission intensity, while the gold film sample has a 6-fold enhancement due to non-radiative quenching close to the gold surface.
5.4.3 Power dependence study
Figure 5-8: Power dependence study at (a) excitation wavelength (583nm), and (b) pump wavelength (792nm). A = 5.06*103, B = 1.30*105.
To confirm the nonlinear absorption is indeed a two-photon process, we perform a power dependence study of the fluorescence at 650nm with the excitation wavelength
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fixed 583nm and pump wavelength at 792nm. As the spiropyran and merocyanine forms are in equilibrium in the layer spiropyran/PMMA blends, the merocyanine form always presents and emits fluorescence centered at 650 nm. We first measured the power dependence at 583 nm before two-photon absorption by varying the power from
100 nW to 15 μW using a neutral density filter. The sample was then exposed to 40μW light at 792nm for ~3.6ms. Due to the two-photon photochromism, spiropyran molecules undergo isomerization, resulting in a large amount of merocyanines and an associated strong fluorescence emission. To avoid the saturation of the detector, we vary the power of 583nm light from 2 nW – 1 μW. The results are detailed in Figure 5-8 (a). Blue circles show the power dependence of 583 nm light before exposing to 792 nm light, while the red circles correspond to measurements after two-photon absorption. The two lines are parallel and are fitted to a line with a slope close to 1, indicating the linear dependence of the fluorescence emission against the excitation power. At least a 40-hold enhancement is seen between the two lines over a large range of excitation power, which can essentially serve as the ON(1) and OFF(0) states in optical switches. For the power dependence study of the pump beam, we fixed the 583 nm light at an optimized power,
100 nW, and varied the power of the 792nm light from 1 - 64 μW. The results are detailed in Figure 5-8 (b), where the data is fitted to a line with slope = 2.15, where A =
5.06*103, B = 1.30*105. B represents the fluorescent emission before two-photon absorption, which is close to the measured value. This measurement verifies the
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nonlinear absorption is indeed a two-photon process. When the pump power is larger than 20 μW, the fluorescence emission no longer follows the power rule. This deviation may result from two competing effects: (1) as the pump power increases, more spiropyrans are isomerized to merocyanines; and (2) the large localized fields in the gap region causes local heating, converting the merocyanines back spiropyrans. As a result, the isomerization reaction is close to equilibrium. In our experiment, we choose 40 μW as the optimized pump power, under which the merocyanines in the gap region are close to maximum.
5.4.4 Capability of storing information
Figure 5-9: (a) SEM image of the pattern sample. (b) Fluorescent intensity image in log scale before two-photon absorption. (c) Fluorescent intensity image in log scale after two-photon absorption.
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To test information storage capacity, patterned “smiley face” samples were fabricated using electron-beam lithography. As shown in Figure 5-9, the dimensions of each of the patterns are 60μm, 50 μm, 40 μm, 30 μm, 20 μm, while widths of the metal lines are 6 μm, 5 μm, 4 μm, 3 μm, 2 μm, respectively. Similar to the uniform nanocube samples, a layer of 4.12 ± 0.25 nm spiropyran/PMMA blend was spin-coated on top of the patterned sample, followed by layer-by-layer deposition of 2.83 ± 0.21nm polyethylene layers. 110 nm silver colloidal nanocubes were spread densely and uniformly on top by a simple dip-coating method. Figure 5-9 (a); the insets show the scaled SEM images of the patterned sample and the distribution of the nanocubes. The patterned samples consist of two parts: (1) spiropyrans in the nanocavities defined by the nanocubes and gold film; (2) spiropyrans between silicon wafter and the nanocubes.
The samples were measured using a 0.5 μW, 583 nm laser with and without the exposure to 40 μW, 792 nm light. Figure 5-9 (b) and (c) show the fluorescence intensity images before and after two-photon photochromism, respectively. Before the two- photon photochromism, the fluorescence intensity is low across the entire sample, and the film-coupled nanocubes provide little enhancement of the fluorescence. After the two-photonchormism, the spiropyrans within the nanocavities are isomerized to merocyanine, leading to an over 40-fold enhancement in fluorescence emission in the patterned region. For the regions without nanocavities, the isomerization from spiropyrans to merocyanine is too weak to detect.
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5.4.5 Specificity of wavelength
Figure 5-10: Specificity of wavelength of the system. (a) The reflectance spectrums the film-coupled nanocubes with various thickness of the PE layers. (b) Absorptions at 792nm (circles) and 650nm (stars) of samples with different resonance wavelengths in (a). (c) Fluorescent enhancement of various samples with different resonance wavelengths in (a). The enhancement is calculated by the ratio between the fluorescent intensity after and before the two-photon absorption.
Spiropyrans have shown a broad wavelength range for one-, two-, and three- photon absorption. However, a narrow absorption band is desirable to minimize the cross talk between adjacent units. The film-coupled nanocubes facilitate a narrow, tunable absorption band over the photochromic molecular spectrum. To demonstrate this effect, a set of film-coupled nanocube samples with different gap sizes were fabricated to resonate at different wavelengths. To make the samples comparable, we fixed the thickness of the spriopyran/PMMA blend to be ~4 nm, but varied the thickness of polyethylene using layer-by-layer deposition[148]. The resulting film-coupled nanocubes samples are resonant at 702.9 nm, 722.5 nm, 756.5 nm, 787.4 nm, 811.8 nm, and 842.5 nm, as it is shown in Figure 5-10 (a). Figure 5-10 (b) shows the absorption
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characteristics of these samples at 792 nm – the pump wavelength, and 650nm – the fluorescence wavelength. The absorption at 792 nm reaches a maximum for the sample resonant at 792 nm, while the absorption at 650 nm decreases as the resonance red-shifts.
The samples were measured using a 0.1μW 583 nm laser with and without exposure to
40 μW, 792 nm light. The fluorescence enhancement was calculated by comparing the fluorescence intensity before and after the two-photon absorption. Figure 5-10 (c) summarizes the fluorescence intensity enhancement for each sample. The sample resonant exactly at 792n m exhibits the highest fluorescent enhancement, due to the near
100% coupling of the 792nm light into the nanocavity. As the resonance is blue-shifted or red-shifted, less energy from the 792 nm light is coupled to the nanocavity, and consequently fewer spiropyrans within the nano-gap regions are isomerized to merocyanines. As a result, the fluorescence enhancement decreases when the resonance of the sample is tuned red or blue. This trend is due to two factors: (1) the absorption at the pump wavelength—792nm; (2) the absorption at the fluorescence wavelength—
650nm. The most blue shifted sample shows the lowest fluorescence enhancement, since it couples the least energy at pump wavelength, but absorbs most energy at the fluorescence wavelength. The most red shifted sample couples only 70% of the energy at the pump wavelength; however, due to the low absorption at 650 nm, the fluorescence intensity dropped by only 20% when compared to the sample resonant at 792 nm.
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5.5 Conclusion
In summary, we have shown that film-coupled colloidal silver nanocubes can dramatically enhance the two photon photochromism of spiropyrans embedded in the gap region of a plasmonic nanostructure. The nanocavities defined by the silver nanocubes and the gold film are capable of coupling nearly 100% of the pump light, thus inducing massive isomerization of spiropyrans to merocyanines. The film-coupled nanocubes produce large Purcell enhancements, with directional emission and high quantum efficiency. For merocyanines embedded within the nanocavities, this enhancement translates to a lower excitation power and significantly shorter emission lifetime. These effects facilitate low power reading and writing if applied to optical data storage. We show that a power as low as 10 μW is sufficient to induce the two-photon isomerization of spiropyran in the film-coupled nanocubes, while a power of 10 nW is able to excite the merocyanines to emit fluorescence. By sweeping the plasmonic resonance of the film-coupled nanocubes, we demonstrate the two-photon absorption is maximized at resonance. This spectral filtering creates a selectable wavelength region within the spectrum that the photochromic molecules can absorb, and can consequently minimize the cross talk between adjacent bits when applied to optical data storage devices. Merocyanines are sensitive to heat and are thus not stable enough to act as a storage material; however, the metasurface platform can also enhance the two-photon absorption of other photochromic materials—many of which may have better
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characteristics for storage application. Film-coupled metasurfaces may also be extended into 3D platforms by distributing coupled colloidal clusters within the photochromic materials for functional photonic devices.
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6. Conclusions
The benefits of nonlinear metasurfaces can be divided into three aspects. First, nonlinear metasurfaces bring the nonlinear interactions down to subwavelength regime, which is desirable for compact all-optical devices. Conventional nonlinear medium, such as nonlinear crystals, has to be many wavelengths in size such that the nonlinear response can accumulate while traveling along the optical path. In addition, phase matching conditions should always be met so that the nonlinear signal accumulates constructively. By integrating nonlinear materials into metasurfaces, the nonlinear interactions are localized in a scale of nanometers, and the phase matching conditions can be neglected.
Second, accompanied with the reduction in size, nonlinear metasurfaces are capable of enhancing nonlinear response from natural matters. Nonlinearities of matter are inherently weak. Seeking nonlinear materials with high nonlinear responses, high damage threshold and low loss has long been common interest in the nonlinear community. Due to their ability to drastically enhance the electromagnetic field, nonlinear metasurface, nonlinear metasurfaces have drawn significant research interest.
As shown in Chapter 4, the film-coupled nanostripes are able to enhance the third- harmonic generation from the Al2O3 ALD spacer layer and the gold structure by five orders of magnitudes.
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Third, nonlinear metasurfaces combines linear and nonlinear properties are not available in natural nonlinear mediums. For example, the nonlinear magneto-electronic coupling shown in Chapter 3 provides intuition of how the linear properties of a nonlinear metasurface would influence the nonlinear signal. The enhanced two- photonchormism demonstrated in Chapter 5 presents an alternative path: the enhanced two-photon absorption modifies the linear property of the material. The experimental results for the Purcell enhancement and enhanced two-photon absorption in the film- coupled nanocubes offer demanding features in optical data storage: ultra-compact, ultra-fast read and write and ultra-low power consumption.
Nonlinear metasurfaces will continue to be an exciting field. As the field is moving towards real world applications, it is critical to have the ability to design, analyze, and optimize nonlinear metasurfaces so that their properties can be tailored to work with other optical or electric components. This thesis provided analytical models to characterize nonlinear metasurfaces and understand their underlying physics. It also presented some applications that exhibit potentials in ultra-compact optical components.
Hopefully it will provide intuition for future development of the field.
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Appendix A
Clarification of contributions
All the contents in this thesis, including the reproduced work from published articles, are original and are my own work. Smith provided advices for all presented work.
The transfer matrix method for nonlinear metamaterials was conceived and developed by Smith, Larouche and Rose [35, 36]. I extended the formalisms to off- normal incidence and to nonlinear multi-layers, and performed all the numerical simulations used in verifying the accuracy of the analytical expressions. Rose advised its application to Kretschmann configurations and metallic nonlinear multi-layers, and contributed to the analysis and interpretation.
I conceived and derived the homogenized description and retrieval method of nonlinear metasurface, with advices from Larouche and Smith. Based on the surface description and homogenization of a metasurface [58], I extended the homogenization formalisms to include the nonlinear terms, and assigned nonlinear metasurfaces with effective nonlinear surface susceptibility tensors. I performed all the numerical comparisons that was used to validate the analytical equations, and provided analysis and interpretation to the method.
The enhanced third-harmonic generation (THG) from metasurface platforms was conceived by Smith, Ciraci, Lassiter and myself [25]. Chen fabricated all the film-coupled
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nanostripe samples. Lassiter conducted all the measurements. I performed all the numerical simulations and analyzed the experiments. In the process of analyzing the results, we realized it was impossible to identify the origin of the enhanced THG. To attack this problem, I purposed two numerical methods that are able to clarify the origin of the THG from an isolated film-coupled nanostripe and periodic film-coupled nanostripes [24]. I performed all the simulations and deviations, and analyzed all the data. Larouche and Smith supervised the theory of the work, and contributed to the interpretation.
I proposed the enhanced two-photon photochromism by a perfect-absorbing metasurface. I performed all the pattern designs, experiment designs, and numerical simulations. I fabricated the film-coupled nanocube samples with dip coating, and Z.
Huang fabricated the patterned sample by electron beam lithography. I conducted the absorption measurement and ellipsometry. X. Jia and I conducted all the fluorescent measurements. Warren S. Warren provided the experimental apparatus for fluorescent measurements. Smith supervised the theory and the measurements.
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Appendix B
Wave equation and boundary conditions for nonlinear medium
In frequency domain, the Maxwell’s equation for charge-free and current-free nonlinear medium are
j , (1.1)
jD, (1.2)
D 0, (1.3)
0, (1.4)
where
D NL , (1.5)
. (1.6)
NL is the summation of the nonlinear polarizations. and are assumed to be time and space invariant but free to exhibit dispersion and loss in the form of frequency-dependent complex numbers.
Inserting Eq. (5), (6) into Eq. (1), (2), gives
j , (1.7)
j j NL . (1.8)
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Taking appropriate manipulation on Eq. (7) and (8), we obtain
2 2 2 NL , (1.9)
NL 2 2 2 NL , (1.10) where
2 , (1.11)
0, (1.12)
2 , (1.13)
NL (1.14) are applied.
We can obtain the inhomogeneous solutions by solving Eq. (9) and (10).
According to momentum conservation, the phase factor of a nonlinear polarization of order α can be defined as exp j t Q r , where Q is the wave vector determined by
NL ,Q pumping fields. Thus, for , the source magnetic and electrical fields generated by this nonlinear polarization is given by
j ,Q r, ,Q nl s 2 2 , (1.15) Q nl nl nl
,Q ,Q r, 2 r, ,Q . (1.16) s Q2 2
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We note that s is not always perpendicular to Q due to the existence of ,
,Q thus it is possible to simplify the problem by considering separately in two orthogonal directions.
According to Eq. (7) and (8), the relation between the electric and magnetic source terms are
,Q ,Q ,Q s , (1.17) s j
,Q ,Q s . (1.18) s j
,Q If is in x-z plane, s will be in s plane while y will be along y axis. For interface between a linear layer i and a nonlinear layer j, considering both of the source field and the subsequent homogenous field, the boundary conditions are given by
nl nl ,Q Hiy H jy H sy , (1.19)
nl nl ,Q Eix E jx E sx . (1.20)
Expressing Eq. (19) and (20) as the sum of forward and backward waves, and dropping the “nl” labels, we have
,,Q Q HHHHHiy iy jy jy sy H sy , (1.21)
,,Q Q EEEEEix ix jx jx sx E sx . (1.22)
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According to Eq. (2.21) and Eq. (17), Eix and E jx can be expressed in terms of
,Q ,Q ,Q Hiy and H jy respectively, and Esx can be expressed by H sy and Px . Thus, by
,Q ,Q solving Eq. (21) and (22), we can express Hiy in terms of H jy , H sy and Px .
Writing everything in vector notation, we have
Q Q Hiy H jy Q H sy P x Kji K si N si (1.23) H H H Q Q iy jy sy Px
Q Where K j i is the linear transfer matrix defined by Eq. (2.21), Ks i and Ns i are corrected matrix interface transfer matrices for the magnetic field:
izQ iz Q 1 1 k k Q 1 jzi jzi Ks i , (1.24) 2 Q Q 1iz 1 iz jk zi j k zi
i i 1 jzik jzi k N . (1.25) s i 2 i i jzik jzi k
Here, Qz is the z component of Q. kzj is the z component of homogenous wave vector at
nl . Therefore, Eq. (19) – (25) constitute the corrected transfer matrix method for the
,Q nonlinear polarization in x-z plane.
,Q When is along y axis, s is along y axis, while s lies in x-z plane. It is easy to find transfer matrices in terms of electrical fields. Again, for the interface
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between layer i and j, the second term in Eq. (16) vanishes, hence both s and s are
,Q perpendicular to Q. We can express Eiy in terms of E jy and Esy . Written in vector notation:
Q Eiy E jy Q Esy M M . (1.26) Eji E si Q iy jy Esy
Q where M j i is the linear transfer matrix defined by Eq. (2.80), M s i is the corrected matrix interface transfer matrices for the electric source field:
izQ iz Q 1 1 k k 1 jzi jzi M s i . (1.27) 2 Q Q 1iz 1 iz jzik jzi k
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Appendix C
Nonlinear surface polarization and magnetization
As shown in Eqs.(3.31) and (3.32), the surface polarization and magnetization are the average of all the induced electric and magnetic dipoles per unit area, which are related to the local nonlinear polarizabilities and the acting fields at both pumping frequency and the nonlinear frequency. Inserting Eqs. (3.33) - (3.36) into Eqs. (3.31) and
(3.32), we have BNonlinear surface polarization and magnetization
nl nl nl nl Psi0 N El , ii EE avi disk , avi
2 1 1ARE 1 1 2 ARE 2 EEElijk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 EEMlijk,,,,, ESjj j avj MSkk k avk (1.28) 0 N , 2 jk,,, xyz 1 1AR H 1 1 2AR E 2 EMElijk,,, MSjj j avj ES,, kk k av k 2 11ARH 1 1 2 ARH 2 EMMlijk,,,,, MSjj j avj MSkk k avk
nl nl nl nl MNsi Hl,,,, ii HH av i disk av i
2 1 1ARE 1 1 2 ARE 2 MEElijk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 MEMlijk,,,,, ESjj j avj MSkk k avk (1.29) N , 2 jk,,, xyz 11AR H 1 1 2AR E 2 MMElijk,,, MSjj j avj ES,, kk k av k 2 11ARH 1 1 2 ARH 2 MMMlijk,,,,, MSjj j avj MSkk k avk where
4,ik , xy , Ai, k . (1.30) 2,ik , z
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nl nl Substituting Edisk,, av i and H disk,, av i with the expressions in Eqs.(3.10) - (3.15) and moving
nl nl all the terms that have Psi and M si to the left side of the equation, we obtain
Pnl1 N nl 1 jkRe jkR 2 jkRR 4 N nl E nl st Eltt,,, Elttavt 2 1 1ARE 1 1 2 ARE 2 EEEltjk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 EEMltjk,,,,, ESjj j avj MSkk k avk (1.31) N , 2 jk,,, xyz 1 1ARH 1 1 2 ARE 2 EMEltjk,, MSjj j avj,,, ESkk k avk 2 11ARH 1 1 2 ARH 2 EMMltjk,,,,, MSjj j avj MSkk k avk
Mnl1 N nl 1 jkRe jkR 2 jkRRN 4 nl E nl st Mltt,,, Mlttavt 2 1 1ARE 1 1 2 ARE 2 MEEltjk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 MEMltjk,,,,, ESjj j avj MSkk k avk (1.32) N , 2 jk,,, xyz 1 1ARH 1 1 2 ARE 2 MMEltjk,, MSjj j avj,,, ESkk k avk 2 11ARH 1 1 2 ARH 2 MMMltjk,,,,, MSjj j avj MSkk k avk where t xy, . For z components,
Pnl1 N nl 1 jkRe jkR 2 RN nl E nl sz Elzz,,, Elzzavz 2 1 1ARE 1 1 2 ARE 2 EEElzjk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 EEMlzjk,,,,, ESjj j avj MSkk k avk (1.33) N , 2 jk,,, xyz 1 1 A RH1 1 2 ARE 2 EMElzjk,, MSjj j av,,, j ES kk k av k 2 11ARH 1 1 2 ARH 2 EMMlzjk,,,,, MSjj j avj MSkk k avk
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Mnl1 N nl 1 jkRe jkR 2 RN nl E nl sz Mlzz,,, Mlzzavz 2 1 1ARE 1 1 2 ARE 2 MEElzjk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 MEMlzjk,,,,, ESjj j avj MSkk k avk (1.34) N . 2 jk,,, xyz 1 1 AR H 1 1 2AR E 2 MMEl,, zjk MS jj j avj,,, ESkk k avk 2 11ARH 1 1 2 ARH 2 MMMlzjk,,,,, MSjj j avj MSkk k avk
Note that all the tensor elements of the local second-order susceptibilities are included in
Eqs. (1.31) - (1.34), some of which are folded into the summation. For small kR, both
1 jkR e jkR and 1jkRe jkR 2 jkR on the left hand sides of Eqs. (1.31) - (1.34) are approximately 1. As a result, Eqs. (1.31) - (1.34) become
PNnl1 nl 4 RN nl E nl st Eltt,,, Eltt avt 2 1 1ARE 1 1 2 ARE 2 EEEltjk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 EEMltjk,,,,, ESjj j avj MSkk k avk (1.35) N , 2 jk,,, xyz 1 1AR H 1 12AR E 2 EMEltjk,,, MSjj j avj ES,, kk k av k 2 11ARH 1 1 2 ARH 2 EMMltjk,,,,, MSjj j avj MSkk k avk
Mnl1 N nl 4 RN nl E nl st Mltt,,, Mltt avt 2 1 1ARE 1 1 2 ARE 2 MEEltjk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 MEMltjk,,,,, ESjj j avj MSkk k avk (1.36) N , 2 jk,,, xyz 1 1AR H 1 12AR E 2 MMEltjk,,, MSjj j avj ES,, kk k av k 2 11ARH 1 1 2 ARH 2 MMMltjk,,,,, MSjj j avj MSkk k avk
143
PNnl1 nl 2 RN nl E nl sz Elzz,,, Elzz avz 2 1 1ARE 1 1 2 ARE 2 EEElzjk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 EEMlzjk,,,,, ESjj j avj MSkk k avk (1.37) N , 2 jk,,, xyz 1 1AR H 1 12AR E 2 EMElzjk,,, MSjj j avj ES,, kk k av k 2 11ARH 1 1 2 ARH 2 EMMlzjk,,,,, MSjj j avj MSkk k avk
Mnl1 N nl 2 RN nl E nl sz Mlzz,,, Mlzz avz 2 1 1ARE 1 1 2 ARE 2 MEElzjk,,,,, ESjj j avj ESkk k avk 2 1 1ARE 1 1 2 ARH 2 MEMlzjk,,,,, ESjj j avj MSkk k avk (1.38) N . 2 jk,,, xyz 1 1AR H 1 12AR E 2 MMElzjk,,, MSjj j avj ES,, kk k av k 2 11ARH 1 1 2 ARH 2 MMMlzjk,,,,, MSjj j avj MSkk k avk
According to Ref. [66],
N nl El, ii ,,i xy nl 1 N El, ii 4 R nl ES, ii , (1.39) nl N El, ii , i z 1 N nl 2 R El, ii
N nl Ml, ii ,,i xy nl 1 NMl, ii 4 R nl MS, ii . (1.40) nl N Ml, ii , i z 1 N nl 2 R Ml, ii
nl nl nl We note the denominators 1 N El, ii 4 R , 1 N El, ii 2 R , 1 N Ml, ii 4 R and
nl 1 N Ml, ii 2 R on the right hand sides of Eqs. (1.39) and (1.40) are the factors on the
144
left hand sides of Eqs. (1.31) - (1.34). Moving these factors to the right hand sides of Eqs.
(1.31) - (1.34), respectively, and applying Eqs. (1.39) and (1.40), we obtain
nl nl nl Psi ES,, ii E av i
2 1 1ARE 1 1 2 ARE 2 EEElijk,,,,, ESjj j avj ESkk k avk
2 1 1 2 2 nl 1 ARE 1 ARH EEMlijk,,,,, ESjj j avj MSkk k avk (1.41) ES, ii , nl 2 1 1 2 2 El, ii jk,,, xyz 1 AR H 1AR E EMEl,,, ijk MS jj j av j ESkk,, k avk 2 11ARH 1 1 2 ARH 2 EMMl,,,,, ijk MS jj j av j MS kk k av k
nl nl nl Msi MSii,, E avi
2 1 1ARE 1 1 2 ARE 2 MEElijk,,,,, ESjj j avj ESkk k avk
2 1 1 2 2 nl 1 ARE 1 ARH MEMlijk,,,,, ESjj j avj MSkk k avk (1.42) MS, ii , nl 2 1 1 2 2 Ml, ii jk,,, xyz 1 AR H 1 AR E MMElijk,,, MSjj j avj ES,, kk k av k 2 11ARH 1 1 2 ARH 2 MMMlijk,,,,, MSjj j avj MSkk k avk where i xyz,, . At this point, all the microscopic fields are eliminated in Eqs. (1.41) and (1.42). When there is no nonlinear term, Eqs. (1.41) and (1.42) reduce to
nl nl nl Psi ESii,, E avi , (1.43)
nl nl nl Msz MSzz,, E avz , (1.44) which are the same as the linear effective surface expressions. The nonlinear terms can
2 2 2 2 be simplified into formats of s: av av , s: av av or s: av av , where s is the effective second-order surface susceptibility tensor that includes the local nonlinear
145
nl nl terms. As a result, s and s can be expressed in terms of the averaged macroscopic fields at ω1, ω2 and ωnl:
2 2 2 2 nl nl nl 1 2 1 2 1 2 1 2 s0 ES av EEEs::::, av av EEMs av av EMEs av av EMMs act act (1.45)
2 2 2 2 nl nl nl 1 2 1 2 1 2 1 2 s MS av MEEs::::. av av MEMs av av MMEs av av MMMs act act (1.46)
146
References
[1] A. Rose, S. Larouche, D.R. Smith, Quantitative study of the enhancement of bulk nonlinearities in metamaterials, Phys. Rev. A, 84 (2011).
[2] F.B. Niesler, N. Feth, S. Linden, J. Niegemann, J. Gieseler, K. Busch, M. Wegener, Second-harmonic generation from split-ring resonators on a GaAs substrate, Opt. Lett., 34 (2009) 1997-1999.
[3] B. Wang, J. Zhou, T. Koschny, C.M. Soukoulis, Nonlinear properties of split- ring resonators, Opt Express, 16 (2008) 16058-16063.
[4] S. Tang, D.J. Cho, H. Xu, W. Wu, Y.R. Shen, L. Zhou, Nonlinear responses in optical metamaterials: theory and experiment, Opt Express, 19 (2011) 18283- 18293.
[5] H. Suchowski, K. O'Brien, Z.J. Wong, A. Salandrino, X. Yin, X. Zhang, Phase mismatch-free nonlinear propagation in optical zero-index materials, Science, 342 (2013) 1223-1226.
[6] A. Rose, D. Huang, D.R. Smith, Controlling the second harmonic in a phase- matched negative-index metamaterial, Phys. Rev. Lett., 107 (2011) 063902.
[7] A. Rose, D.R. Smith, Broadly tunable quasi-phase-matching in nonlinear metamaterials, Phys. Rev. A, 84 (2011).
[8] A. Rose, D.R. Smith, Overcoming phase mismatch in nonlinear metamaterials [Invited], Optical Materials Express, 1 (2011) 1232-1243.
[9] A. Rose, D. Huang, D.R. Smith, Nonlinear interference and unidirectional wave mixing in metamaterials, Phys. Rev. Lett., 110 (2013) 063901.
[10] A. Rose, S. Larouche, E. Poutrina, D.R. Smith, Nonlinear magnetoelectric metamaterials: Analysis and homogenization via a microscopic coupled-mode theory, Phys. Rev. A, 86 (2012).
[11] Y.S. Kivshar, Tunable and nonlinear metamaterials: toward functional metadevices, Advances in Natural Sciences: Nanoscience and Nanotechnology, 5 (2013) 013001.
147
[12] M. Liu, Y. Sun, D.A. Powell, I.V. Shadrivov, M. Lapine, R.C. McPhedran, Y.S. Kivshar, Nonlinear response via intrinsic rotation in metamaterials, Physical Review B, 87 (2013) 235126.
[13] A.R. Katko, S. Gu, J.P. Barrett, B.-I. Popa, G. Shvets, S.A. Cummer, Phase Conjugation and Negative Refraction using Nonlinear Active Metamaterials, Phys. Rev. Lett., 105 (2010) 123905.
[14] A.B. Kozyrev, I.V. Shadrivov, Y.S. Kivshar, Soliton generation in active nonlinear metamaterials, Appl. Phys. Lett., 104 (2014) 084105.
[15] A. Rose, D.A. Powell, I.V. Shadrivov, D.R. Smith, Y.S. Kivshar, Circular dichroism of four-wave mixing in nonlinear metamaterials, Physical Review B, 88 (2013) 195148.
[16] R.W. Boyd, Chapter 1 - The Nonlinear Optical Susceptibility, in: R.W. Boyd (Ed.) Nonlinear Optics (Third Edition), Academic Press, Burlington, 2008, pp. 1- 67.
[17] D.R. Smith, J.B. Pendry, Homogenization of metamaterials by field averaging (invited paper), Journal of the Optical Society of America B, 23 (2006) 391-403.
[18] S. Kruk, M. Weismann, A.Y. Bykov, E.A. Mamonov, I.A. Kolmychek, T. Murzina, N.C. Panoiu, D.N. Neshev, Y.S. Kivshar, Enhanced Magnetic Second- Harmonic Generation from Resonant Metasurfaces, ACS Photonics, 2 (2015) 1007-1012.
[19] M.W. Klein, C. Enkrich, M. Wegener, S. Linden, Second-harmonic generation from magnetic metamaterials, Science, 313 (2006) 502-504.
[20] M. Scalora, M.A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, M.J. Bloemer, Second- and third-harmonic generation in metal-based structures, Phys. Rev. A, 82 (2010) 043828.
[21] B. Metzger, M. Hentschel, T. Schumacher, M. Lippitz, X. Ye, C.B. Murray, B. Knabe, K. Buse, H. Giessen, Doubling the efficiency of third harmonic generation by positioning ITO nanocrystals into the hot-spot of plasmonic gap-antennas, Nano Lett, 14 (2014) 2867-2872.
148
[22] H. Aouani, M. Rahmani, M. Navarro-Cia, S.A. Maier, Third-harmonic- upconversion enhancement from a single semiconductor nanoparticle coupled to a plasmonic antenna, Nat. Nanotechnol., 9 (2014) 290-294.
[23] M. Ren, E. Plum, J. Xu, N.I. Zheludev, Giant nonlinear optical activity in a plasmonic metamaterial, Nat Commun, 3 (2012) 833.
[24] X. Liu, S. Larouche, P. Bowen, D.R. Smith, Clarifying the origin of third- harmonic generation from film-coupled nanostripes, Opt. Express, 23 (2015) 19565-19574.
[25] J.B. Lassiter, X.S. Chen, X.J. Liu, C. Ciraci, T.B. Hoang, S. Larouche, S.H. Oh, M.H. Mikkelsen, D.R. Smith, Third-Harmonic Generation Enhancement by Film- Coupled Plasmonic Stripe Resonators, Acs Photonics, 1 (2014) 1212-1217.
[26] N. Yu, F. Capasso, Flat optics with designer metasurfaces, Nat Mater, 13 (2014) 139-150.
[27] N. Yu, P. Genevet, M.A. Kats, F. Aieta, J.P. Tetienne, F. Capasso, Z. Gaburro, Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science, 334 (2011) 333-337.
[28] B. Memarzadeh, H. Mosallaei, Array of planar plasmonic scatterers functioning as light concentrator, Opt. Lett., 36 (2011) 2569-2571.
[29] Y. Nanfang, P. Genevet, F. Aieta, M.A. Kats, R. Blanchard, G. Aoust, J.P. Tetienne, Z. Gaburro, F. Capasso, Flat Optics: Controlling Wavefronts With Optical Antenna Metasurfaces, Selected Topics in Quantum Electronics, IEEE Journal of, 19 (2013) 4700423-4700423.
[30] Y. Yao, R. Shankar, M.A. Kats, Y. Song, J. Kong, M. Loncar, F. Capasso, Electrically Tunable Metasurface Perfect Absorbers for Ultrathin Mid-Infrared Optical Modulators, Nano. Lett., 14 (2014) 6526-6532.
[31] J. Lee, M. Tymchenko, C. Argyropoulos, P.Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.C. Amann, A. Alu, M.A. Belkin, Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions, Nature, 511 (2014) 65-69.
149
[32] M. Kauranen, A.V. Zayats, Nonlinear plasmonics, Nat. Photonics, 6 (2012) 737-748.
[33] M.R. Shcherbakov, A.S. Shorokhov, D.N. Neshev, B. Hopkins, I. Staude, E.V. Melik-Gaykazyan, A.A. Ezhov, A.E. Miroshnichenko, I. Brener, A.A. Fedyanin, Y.S. Kivshar, Nonlinear Interference and Tailorable Third-Harmonic Generation from Dielectric Oligomers, ACS Photonics, 2 (2015) 578-582.
[34] A.A. Sukhorukov, A.S. Solntsev, S.S. Kruk, D.N. Neshev, Y.S. Kivshar, Nonlinear coupled-mode theory for periodic plasmonic waveguides and metamaterials with loss and gain, Opt. Lett., 39 (2014) 462-465.
[35] A. Rose, S. Larouche, D. Huang, E. Poutrina, D.R. Smith, Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials, Phys Rev E Stat Nonlin Soft Matter Phys, 82 (2010) 036608.
[36] S. Larouche, D.R. Smith, A retrieval method for nonlinear metamaterials, Opt. Commun., 283 (2010) 1621-1627.
[37] D.R. Smith, S. Schultz, Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Physical Review B, 65 (2002).
[38] D.R. Smith, D.C. Vier, T. Koschny, C.M. Soukoulis, Electromagnetic parameter retrieval from inhomogeneous metamaterials, Physical Review E, 71 (2005).
[39] D.R. Smith, S. Schultz, P. Markoš, C.M. Soukoulis, Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Physical Review B, 65 (2002) 195104.
[40] D. Smith, D. Schurig, J. Mock, Characterization of a planar artificial magnetic metamaterial surface, Physical Review E, 74 (2006).
[41] S. Arslanagić, T.V. Hansen, N.A. Mortensen, A.H. Gregersen, O. Sigmund, R.W. Ziolkowski, O. Breinbjerg, A Review of the Scattering-Parameter Extraction Method with Clarification of Ambiguity Issues in Relation to Metamaterial Homogenization, IEEE Antennas and Propagation Magazine, 55 (2013) 91-106.
150
[42] R.W. Boyd, Nonlinear Optics, 3rd ed, Academic Press: San Diego2008.
[43] X.J. Liu, A. Rose, E. Poutrina, C. Ciraci, S. Larouche, D.R. Smith, Surfaces, films, and multilayers for compact nonlinear plasmonics, Journal of the Optical Society of America B-Optical Physics, 30 (2013) 2999-3010.
[44] J.D. McMullen, Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates, Journal of Applied Physics, 46 (1975) 3076-3081.
[45] X.-H. Wang, B.-Y. Gu, Nonlinear frequency conversion in 2D χ (2) photonic crystals and novel nonlinear double-circle construction, The European Physical Journal B - Condensed Matter and Complex Systems, 24 (2001) 323-326.
[46] D.S. Bethune, Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques, Journal of the Optical Society of America B, 6 (1989) 910-916.
[47] D.S. Bethune, Optical harmonic generation and mixing in multilayer media: extension of optical transfer matrix approach to include anisotropic materials, Journal of the Optical Society of America B, 8 (1991) 367-373.
[48] J. Yuan, Computing for second harmonic generation in one-dimensional nonlinear photonic crystals, Optics Communications, 282 (2009) 2628-2633.
[49] J. Renger, R. Quidant, N. van Hulst, S. Palomba, L. Novotny, Free-Space Excitation of Propagating Surface Plasmon Polaritons by Nonlinear Four-Wave Mixing, Physical Review Letters, 103 (2009) 266802.
[50] J. Renger, R. Quidant, L. Novotny, Enhanced nonlinear response from metal surfaces, Opt. Express, 19 (2011) 1777-1785.
[51] J. Renger, R. Quidant, N. van Hulst, L. Novotny, Surface-Enhanced Nonlinear Four-Wave Mixing, Physical Review Letters, 104 (2010) 046803.
[52] J.-J. Li, Z.-Y. Li, D.-Z. Zhang, Second harmonic generation in one- dimensional nonlinear photonic crystals solved by the transfer matrix method, Physical Review E, 75 (2007) 056606.
151
[53] P. Szczepański, T. Osuch, Z. Jaroszewicz, Modeling of amplification and light generation in one-dimensional photonic crystal using a multiwavelength transfer matrix approach, Appl. Opt., 48 (2009) 5401-5406.
[54] R.S. Bennink, Y.-K. Yoon, R.W. Boyd, J.E. Sipe, Accessing the optical nonlinearity of metals with metal–dielectric photonic bandgap structures, Opt. Lett., 24 (1999) 1416-1418.
[55] J. Wait, Guiding of electromagnetic waves by uniformly rough surfaces : Part I, IRE Transactions on Antennas and Propagation, 7 (1959) 154-162.
[56] J. Wait, Guiding of electromagnetic waves by uniformly rough surfaces: Part II, IRE Transactions on Antennas and Propagation, 7 (1959) 163-168.
[57] C. Hou-Tong, J.T. Antoinette, Y. Nanfang, A review of metasurfaces: physics and applications, Reports on Progress in Physics, 79 (2016) 076401.
[58] C.L. Holloway, E.F. Kuester, J.A. Gordon, J.O. Hara, J. Booth, D.R. Smith, An Overview of the Theory and Applications of Metasurfaces: The Two- Dimensional Equivalents of Metamaterials, IEEE Antennas and Propagation Magazine, 54 (2012) 10-35.
[59] B.N.J. Persson, A. Liebsch, Optical properties of two-dimensional systems of randomly distributed particles, Physical Review B, 28 (1983) 4247-4254.
[60] R.G. Barrera, M. del Castillo-Mussot, G. Monsivais, P. Villaseor, W.L. Mochán, Optical properties of two-dimensional disordered systems on a substrate, Physical Review B, 43 (1991) 13819-13826.
[61] M.M. Wind, J. Vlieger, Optical properties of thin films on rough surfaces, Physica A: Statistical Mechanics and its Applications, 125 (1984) 75-104.
[62] D. Bedeaux, J. Vlieger, A phenomenological theory of the dielectric properties of thin films, Physica, 67 (1973) 55-73.
[63] T.B.A. Senior, J.L. Volakis, Sheet simulation of a thin dielectric layer, Radio Science, 22 (1987) 1261-1272.
[64] M.A. Ricoy, J.L. Volakis, Derivation of generalized transition/boundary conditions for planar multiple-layer structures, Radio Science, 25 (1990) 391-405. 152
[65] T.B.A. Senior, J.L. Volakis, Approximate Boundary Conditions in Electromagnetics, Institution of Engineering and Technology1995.
[66] E.F. Kuester, M.A. Mohamed, M. Piket-May, C.L. Holloway, Averaged transition conditions for electromagnetic fields at a metafilm, Antennas and Propagation, IEEE Transactions on, 51 (2003) 2641-2651.
[67] C.L. Holloway, E.F. Kuester, Equivalent boundary conditions for a perfectly conducting periodic surface with a cover layer, Radio Science, 35 (2000) 661-681.
[68] C.L. Holloway, E.F. Kuester, Impedance-type boundary conditions for a periodic interface between a dielectric and a highly conducting medium, IEEE Transactions on Antennas and Propagation, 48 (2000) 1660-1672.
[69] C.L. Holloway, E.F. Kuester, Power loss associated with conducting and superconducting rough interfaces, IEEE Transactions on Microwave Theory and Techniques, 48 (2000) 1601-1610.
[70] C.L. Holloway, E.F. Kuester, A. Dienstfrey, Characterizing Metasurfaces/Metafilms: The Connection Between Surface Susceptibilities and Effective Material Properties, Antennas and Wireless Propagation Letters, IEEE, 10 (2011) 1507-1511.
[71] C.L. Holloway, A. Dienstfrey, E.F. Kuester, J.F. O’Hara, A.K. Azad, A.J. Taylor, A discussion on the interpretation and characterization of metafilms/metasurfaces: The two-dimensional equivalent of metamaterials, Metamaterials, 3 (2009) 100-112.
[72] C.L. Holloway, E.F. Kuester, J.A. Gordon, J. O'Hara, J. Booth, D.R. Smith, An Overview of the Theory and Applications of Metasurfaces: The Two- Dimensional Equivalents of Metamaterials, Antennas and Propagation Magazine, IEEE, 54 (2012) 10-35.
[73] A.I. Dimitriadis, N.V. Kantartzis, T.D. Tsiboukis, Consistent Modeling of Periodic Metasurfaces With Bianisotropic Scatterers for Oblique TE-Polarized Plane Wave Excitation, Magnetics, IEEE Transactions on, 49 (2013) 1769-1772.
[74] X. Liu, S. Larouche, D.R. Smith, Homogenized description and retrieval method of nonlinear metasurfaces, Optics Communications, 410 (2018) 53-69.
153
[75] M. Idemen, The Maxwell's equations in the sense of distributions, Antennas and Propagation, IEEE Transactions on, 21 (1973) 736-738.
[76] M.M. Idemen, Discontinuities in the electromagnetic field [Advertisement], Antennas and Propagation Magazine, IEEE, 56 (2014) 300-300.
[77] S.I. Maslovski, S.A. Tretyakov, Full-wave Interaction Field in Two- dimensional Arrays of Dipole Scatterers, International Journal of Electronics and Communications - AEü, 53 (1999) 135-139.
[78] I.H. Malitson, Interspecimen Comparison of the Refractive Index of Fused Silica*,†, J. Opt. Soc. Am., 55 (1965) 1205-1209.
[79] S. Zhang, D.A. Genov, Y. Wang, M. Liu, X. Zhang, Plasmon-Induced Transparency in Metamaterials, Phys. Rev. Lett., 101 (2008) 047401.
[80] X. Liu, J. Gu, R. Singh, Y. Ma, J. Zhu, Z. Tian, M. He, J. Han, W. Zhang, Electromagnetically induced transparency in terahertz plasmonic metamaterials via dual excitation pathways of the dark mode, Appl. Phys. Lett., 100 (2012) 131101.
[81] P. Tassin, L. Zhang, T. Koschny, E.N. Economou, C.M. Soukoulis, Low-Loss Metamaterials Based on Classical Electromagnetically Induced Transparency, Phys. Rev. Lett., 102 (2009) 053901.
[82] R. Singh, C. Rockstuhl, F. Lederer, W. Zhang, Coupling between a dark and a bright eigenmode in a terahertz metamaterial, Physical Review B, 79 (2009) 085111.
[83] J. Butet, O.J. Martin, Nonlinear plasmonic nanorulers, ACS Nano, 8 (2014) 4931-4939.
[84] H. Harutyunyan, S. Palomba, J. Renger, R. Quidant, L. Novotny, Nonlinear dark-field microscopy, Nano Lett, 10 (2010) 5076-5079.
[85] W.L. Mochán, J.A. Maytorena, B.S. Mendoza, V.L. Brudny, Second-harmonic generation in arrays of spherical particles, Physical Review B, 68 (2003) 085318.
154
[86] S. Kujala, B.K. Canfield, M. Kauranen, Y. Svirko, J. Turunen, Multipole interference in the second-harmonic optical radiation from gold nanoparticles, Phys. Rev. Lett., 98 (2007) 167403.
[87] B.K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, M. Kauranen, Local field asymmetry drives second-harmonic generation in non- centrosymmetric nanodimers, Nano Lett, 7 (2007) 1251-1255.
[88] Y. Zhang, N.K. Grady, C. Ayala-Orozco, N.J. Halas, Three-dimensional nanostructures as highly efficient generators of second harmonic light, Nano Lett, 11 (2011) 5519-5523.
[89] B. Lamprecht, J.R. Krenn, A. Leitner, F.R. Aussenegg, Resonant and off- resonant light-driven plasmons in metal nanoparticles studied by femtosecond- resolution third-harmonic generation, Phys. Rev. Lett., 83 (1999) 4421-4424.
[90] M. Hentschel, T. Utikal, H. Giessen, M. Lippitz, Quantitative modeling of the third harmonic emission spectrum of plasmonic nanoantennas, Nano Lett, 12 (2012) 3778-3782.
[91] T. Hanke, J. Cesar, V. Knittel, A. Trugler, U. Hohenester, A. Leitenstorfer, R. Bratschitsch, Tailoring spatiotemporal light confinement in single plasmonic nanoantennas, Nano Lett, 12 (2012) 992-996.
[92] K.D. Ko, A. Kumar, K.H. Fung, R. Ambekar, G.L. Liu, N.X. Fang, K.C. Toussaint, Nonlinear optical response from arrays of Au bowtie nanoantennas, Nano Lett, 11 (2011) 61-65.
[93] J. Renger, R. Quidant, N. van Hulst, L. Novotny, Surface-enhanced nonlinear four-wave mixing, Phys. Rev. Lett., 104 (2010) 046803.
[94] H. Harutyunyan, G. Volpe, R. Quidant, L. Novotny, Enhancing the Nonlinear Optical Response Using Multifrequency Gold-Nanowire Antennas, Physical Review Letters, 108 (2012) 217403.
[95] M. Danckwerts, L. Novotny, Optical frequency mixing at coupled gold nanoparticles, Phys. Rev. Lett., 98 (2007) 026104.
155
[96] Y. Zhang, F. Wen, Y.R. Zhen, P. Nordlander, N.J. Halas, Coherent Fano resonances in a plasmonic nanocluster enhance optical four-wave mixing, Proc Natl Acad Sci U S A, 110 (2013) 9215-9219.
[97] P. Genevet, J.P. Tetienne, E. Gatzogiannis, R. Blanchard, M.A. Kats, M.O. Scully, F. Capasso, Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings, Nano Lett, 10 (2010) 4880-4883.
[98] S. Kim, J. Jin, Y.J. Kim, I.Y. Park, Y. Kim, S.W. Kim, High-harmonic generation by resonant plasmon field enhancement, Nature, 453 (2008) 757-760.
[99] I.Y. Park, S. Kim, J. Choi, D.H. Lee, Y.J. Kim, M.F. Kling, M.I. Stockman, S.W. Kim, Plasmonic generation of ultrashort extreme-ultraviolet light pulses, Nat. Photonics, 5 (2011) 678-682.
[100] B. Fetić, K. Kalajdžić, D.B. Milošević, High-order harmonic generation by a spatially inhomogeneous field, Annalen der Physik, 525 (2013) 107-117.
[101] A. Moreau, C. Ciraci, J.J. Mock, R.T. Hill, Q. Wang, B.J. Wiley, A. Chilkoti, D.R. Smith, Controlled-reflectance surfaces with film-coupled colloidal nanoantennas, Nature, 492 (2012) 86-89.
[102] A. Moreau, C. Ciracì, D.R. Smith, Impact of nonlocal response on metallodielectric multilayers and optical patch antennas, Physical Review B, 87 (2013).
[103] P. Nagpal, N.C. Lindquist, S.-H. Oh, D.J. Norris, Ultrasmooth Patterned Metals for Plasmonics and Metamaterials, Science, 325 (2009) 594-597.
[104] X. Chen, H.-R. Park, M. Pelton, X. Piao, N.C. Lindquist, H. Im, Y.J. Kim, J.S. Ahn, K.J. Ahn, N. Park, D.-S. Kim, S.-H. Oh, Atomic layer lithography of wafer- scale nanogap arrays for extreme confinement of electromagnetic waves, Nat. Commun., 4 (2013).
[105] J.B. Lassiter, F. McGuire, J.J. Mock, C. Ciraci, R.T. Hill, B.J. Wiley, A. Chilkoti, D.R. Smith, Plasmonic waveguide modes of film-coupled metallic nanocubes, Nano Lett., 13 (2013) 5866-5872.
156
[106] C. Ciracì, J. Britt Lassiter, A. Moreau, D.R. Smith, Quasi-analytic study of scattering from optical plasmonic patch antennas, J. Appl. Phys., 114 (2013) 163108.
[107] R.L. Olmon, B. Slovick, T.W. Johnson, D. Shelton, S.H. Oh, G.D. Boreman, M.B. Raschke, Optical dielectric function of gold, Phys. Rev. B, 86 (2012).
[108] W.K. Burns, N. Bloember, Third-Harmonic Generation in Absorbing Media of Cubic or Isotropic Symmetry, Physical Review B, 4 (1971) 3437-&.
[109] R.W. Boyd, Z.M. Shi, I. De Leon, The third-order nonlinear optical susceptibility of gold, Opt. Commun., 326 (2014) 74-79.
[110] A.S. Dvornikov, E.P. Walker, P.M. Rentzepis, Two-Photon Three- Dimensional Optical Storage Memory, The Journal of Physical Chemistry A, 113 (2009) 13633-13644.
[111] D.A. PARTHENOPOULOS, P.M. RENTZEPIS, Three-Dimensional Optical Storage Memory, Science, 245 (1989) 843-845.
[112] M. Gu, X. Li, Y. Cao, Optical storage arrays: a perspective for future big data storage, Light Sci Appl, 3 (2014) e177.
[113] M. Gu, Q. Zhang, S. Lamon, Nanomaterials for optical data storage, 1 (2016) 16070.
[114] M. Wuttig, H. Bhaskaran, T. Taubner, Phase-change materials for non- volatile photonic applications, Nat Photon, 11 (2017) 465-476.
[115] Z. Shirakashi, T. Goto, H. Takagi, Y. Nakamura, P.B. Lim, H. Uchida, M. Inoue, Reconstruction of non-error magnetic hologram data by magnetic assist recording, Scientific Reports, 7 (2017) 12835.
[116] T. Nobukawa, T. Nomura, Linear phase encoding for holographic data storage with a single phase-only spatial light modulator, Appl. Opt., 55 (2016) 2565-2573.
[117] C.C. Corredor, Z.L. Huang, K.D. Belfield, Two-Photon 3D Optical Data Storage via Fluorescence Modulation of an Efficient Fluorene Dye by a Photochromic Diarylethene, Advanced Materials, 18 (2006) 2910-2914. 157
[118] B.H. Cumpston, S.P. Ananthavel, S. Barlow, D.L. Dyer, J.E. Ehrlich, L.L. Erskine, A.A. Heikal, S.M. Kuebler, I.Y.S. Lee, D. McCord-Maughon, J. Qin, H. Rockel, M. Rumi, X.-L. Wu, S.R. Marder, J.W. Perry, Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication, Nature, 398 (1999) 51-54.
[119] K. Iliopoulos, O. Krupka, D. Gindre, M. Sallé, Reversible Two-Photon Optical Data Storage in Coumarin-Based Copolymers, Journal of the American Chemical Society, 132 (2010) 14343-14345.
[120] P. Kunwar, J. Hassinen, G. Bautista, R.H.A. Ras, J. Toivonen, Sub-micron scale patterning of fluorescent silver nanoclusters using low-power laser, 6 (2016) 23998.
[121] S. Fu, X. Zhang, Q. Han, S. Liu, X. Han, Y. Liu, Blu-ray-sensitive localized surface plasmon resonance for high-density optical memory, 6 (2016) 36701.
[122] K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, M. Notomi, Ultralow-power all-optical RAM based on nanocavities, Nat Photon, 6 (2012) 248-252.
[123] H. Ditlbacher, J.R. Krenn, B. Lamprecht, A. Leitner, F.R. Aussenegg, Spectrally coded optical data storage by metal nanoparticles, Opt. Lett., 25 (2000) 563-565.
[124] A. Bouhelier, R. Bachelot, G. Lerondel, S. Kostcheev, P. Royer, G.P. Wiederrecht, Surface Plasmon Characteristics of Tunable Photoluminescence in Single Gold Nanorods, Physical Review Letters, 95 (2005) 267405.
[125] P. Zijlstra, J.W.M. Chon, M. Gu, Five-dimensional optical recording mediated by surface plasmons in gold nanorods, Nature, 459 (2009) 410-413.
[126] H. Ren, X. Li, Q. Zhang, M. Gu, On-chip noninterference angular momentum multiplexing of broadband light, Science, 352 (2016) 805-809.
[127] C. Ciracì, A. Rose, C. Argyropoulos, D.R. Smith, Numerical studies of the modification of photodynamic processes by film-coupled plasmonic nanoparticles, Journal of the Optical Society of America B, 31 (2014) 2601-2607.
158
[128] G.M. Akselrod, C. Argyropoulos, T.B. Hoang, C. Ciracì, C. Fang, J. Huang, D.R. Smith, M.H. Mikkelsen, Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas, Nat Photon, 8 (2014) 835-840.
[129] A. Rose, T.B. Hoang, F. McGuire, J.J. Mock, C. Ciracì, D.R. Smith, M.H. Mikkelsen, Control of Radiative Processes Using Tunable Plasmonic Nanopatch Antennas, Nano Letters, 14 (2014) 4797-4802.
[130] R.M. Bakker, H.-K. Yuan, Z. Liu, V.P. Drachev, A.V. Kildishev, V.M. Shalaev, R.H. Pedersen, S. Gresillon, A. Boltasseva, Enhanced localized fluorescence in plasmonic nanoantennae, Applied Physics Letters, 92 (2008) 043101.
[131] T.B. Hoang, G.M. Akselrod, C. Argyropoulos, J. Huang, D.R. Smith, M.H. Mikkelsen, Ultrafast spontaneous emission source using plasmonic nanoantennas, 6 (2015) 7788.
[132] A. Moreau, C. Ciraci, J.J. Mock, R.T. Hill, Q. Wang, B.J. Wiley, A. Chilkoti, D.R. Smith, Controlled-reflectance surfaces with film-coupled colloidal nanoantennas, Nature, 492 (2012) 86-89.
[133] L. Lin, Y. Zheng, Optimizing plasmonic nanoantennas via coordinated multiple coupling, 5 (2015) 14788.
[134] B. Paolo, H. Jer-Shing, H. Bert, Nanoantennas for visible and infrared radiation, Reports on Progress in Physics, 75 (2012) 024402.
[135] T. Hanke, J. Cesar, V. Knittel, A. Trügler, U. Hohenester, A. Leitenstorfer, R. Bratschitsch, Tailoring Spatiotemporal Light Confinement in Single Plasmonic Nanoantennas, Nano Letters, 12 (2012) 992-996.
[136] A. Nevet, N. Berkovitch, A. Hayat, P. Ginzburg, S. Ginzach, O. Sorias, M. Orenstein, Plasmonic Nanoantennas for Broad-Band Enhancement of Two- Photon Emission from Semiconductors, Nano Letters, 10 (2010) 1848-1852.
[137] B. Lee, J. Park, G.H. Han, H.-S. Ee, C.H. Naylor, W. Liu, A.T.C. Johnson, R. Agarwal, Fano Resonance and Spectrally Modified Photoluminescence Enhancement in Monolayer MoS2 Integrated with Plasmonic Nanoantenna Array, Nano Letters, 15 (2015) 3646-3653.
159
[138] G.M. Akselrod, J. Huang, T.B. Hoang, P.T. Bowen, L. Su, D.R. Smith, M.H. Mikkelsen, Large-Area Metasurface Perfect Absorbers from Visible to Near- Infrared, Advanced Materials, 27 (2015) 8028-8034.
[139] M. Irie, Diarylethenes for Memories and Switches, Chemical Reviews, 100 (2000) 1685-1716.
[140] S. Kawata, Y. Kawata, Three-Dimensional Optical Data Storage Using Photochromic Materials, Chemical Reviews, 100 (2000) 1777-1788.
[141] R. Klajn, Spiropyran-based dynamic materials, Chemical Society Reviews, 43 (2014) 148-184.
[142] C. Triolo, S. Patanè, M. Mazzeo, S. Gambino, G. Gigli, M. Allegrini, Pure optical nano-writing on light- switchable spiropyrans/merocyanine thin film, Opt. Express, 22 (2014) 283-288.
[143] M.-Q. Zhu, G.-F. Zhang, Z. Hu, M.P. Aldred, C. Li, W.-L. Gong, T. Chen, Z.- L. Huang, S. Liu, Reversible Fluorescence Switching of Spiropyran-Conjugated Biodegradable Nanoparticles for Super-Resolution Fluorescence Imaging, Macromolecules, 47 (2014) 1543-1552.
[144] B. Liao, P. Long, B. He, S. Yi, B. Ou, S. Shen, J. Chen, Reversible fluorescence modulation of spiropyran-functionalized carbon nanoparticles, Journal of Materials Chemistry C, 1 (2013) 3716-3721.
[145] K. Matczyszyn, J. Olesiak-Banska, K. Nakatani, P. Yu, N.A. Murugan, R. Zaleśny, A. Roztoczyńska, J. Bednarska, W. Bartkowiak, J. Kongsted, H. Ågren, M. Samoć, One- and Two-Photon Absorption of a Spiropyran–Merocyanine System: Experimental and Theoretical Studies, The Journal of Physical Chemistry B, 119 (2015) 1515-1522.
[146] J. Enderlein, R. Erdmann, Fast fitting of multi-exponential decay curves, Optics Communications, 134 (1997) 371-378.
[147] R.R. Chance, A. Prock, R. Silbey, Comments on the classical theory of energy transfer, The Journal of Chemical Physics, 62 (1975) 2245-2253.
160
[148] Y. Guo, W. Geng, J. Sun, Layer-by-Layer Deposition of Polyelectrolyte−Polyelectrolyte Complexes for Multilayer Film Fabrication, Langmuir, 25 (2009) 1004-1010.
161
Biography
Xiaojun Liu
Born on May 6, 1989, in Tianjin, China
BS – Optoelectronic Technical Science, Tianjin University
MS – Electrical and Computer Engineering, Duke University
MS – Biomedical Engineering, Duke University
PhD – Electrical and Computer Engineering, Duke University
Publications:
1. X. Liu, S. Larouche, D. Smith, “Homogenized Description and Retrieval Method
of Nonlinear Metasurfaces” Opt. Comm., 400(1), 53-69(2018)
2. X. Jia, P. Bowen, Z. Huang, X. Liu, C. Bingham, and D. R. Smith, “Clarification of
Surface Modes of a Periodic Nanopatch Metasurface” Opt. Exp., 26(3), 3004-
3012(2018)
3. T. Frometeze, X. Liu, M. Boyarsky, et al, “Phaseless computational imaging with
a radiating metasurface”, Opt. Exp., 24 (15), 16760 (2016)
4. X. Liu, S. Larouche, P. Bowen, et al, “Clarifying the Origin of Third-harmonic
Generation from Film-coupled Nanostripes” Opt. Exp., 23 (15), 19565(2015).
5. J. B. Lassiter, X. Chen, X. Liu, et al, “Third-Harmonic Generation Enhancement
by Film-Coupled Plasmonic Stripe Resonantor” ACS Photon., 1(11), 1212 (2014)
162
6. C. Ciraci, X. Chen, J. J Mock, F. McGuire, X. Liu, et al, “Film-coupled
nanoparticles by atomic layer deposition: Comparison with organic spacing
layers” Appl. Phys. Lett., 104, 023109 (2014)
7. X. Liu, A. Rose, E. Poutrina, et al, “Surfaces, films, and multilayers for compact
nonlinear plasmonics” J. Opt. Soc. Am., 30(11), 2999 (2013)
8. J. Gu, X. Liu, Z. Tian, et al, “Modulation of metamaterial induced transparency
via competition between dual excitation pathways of dark mode” Laser and
Tera-Hertz Science and Technology, Coference (2012)
9. X. Liu, J. Gu, R. Singh, et al, “Electromagnetically induced transparency in
terahertz plasmonic metamaterials via dual excitation pathways of the dark
mode” Appl. Phys. Lett., 100, 131101 (2012)
10. J. Gu, R. Singh, X. Liu, et al, “Active control of electromagnetically induced
transparency analogue in terahertz metamaterials,” Nat. Comm., 3, 1151 (2012)
11. Y. Liu, X. Liu, B. Qi, et al, “δ-P1 approximation model of biological tissue,” Acta
Phys. Sin., 60, 7 (2011) (in Chinese)
163