# Nonlinear Optics in Dielectric Guided-Mode Resonant Structures and Resonant Metasurfaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design Considerations for Resonant Dielectric Grating Structures

#### 2.1. Isolated Particle Versus Array

_{i}, b

_{i}are the electric and magnetic mode coefficients respectively, which are expanded in terms of Bessel, Hankel, Ricatti-Bessel and Ricatti-Hankel functions, x = ka refers to the modified dimension parameter, and $m=\frac{\sqrt{{\epsilon}_{1}{\mu}_{1}}}{\sqrt{{\epsilon}_{host}{\mu}_{host}}}$ refers to the contrast parameter [7]. Simplified forms of the scattering expansion for specific structures can be found in ref. [32]. Typical field profile obtained close to the electric/magnetic dipolar and quadrapolar resonances for isolated spherical particle are also shown in Figure 2a.

#### 2.2. Effect of Refractive Index Contrast on Resonant Interaction

## 3. Physical Mechanisms behind Resonances in Arrayed Structures

#### 3.1. Guided-Mode Resonances

#### 3.2. Electromagnetically-Induced Transparency Analogue Resonances

_{x}to the magnetic dipole mode, denoted as m

_{z}. This coupling results in characteristic dips in the measured reflection spectra, as shown in Figure 12i. Such structures have been utilized for nonlinear optical process enhancement, as discussed in Section 4. In the context of guided mode resonances, there is interest in coupling a low- and high- quality factor resonant waveguide structure. Some examples of such structures are shown in Figure 12j–l. Some of the early simulation studies of these structures consisted of a top grating-waveguide layer coupled to another bottom waveguide layer, as shown in the inset of Figure 12j. In this case, the direct coupling of the incident light to the bottom waveguide and indirect coupling through interaction with the top-layer guided-mode resonant structure can occur. This results in a sharp transmission peak, which is different from the transmission spectra of an equivalent refractive index homogenous medium. Some of the early work [73] was not even called an EIT analogue at that time. The EIT analogue ideas have been extended recently to one-dimensional and two-dimensional structures [74,75]. The spectral width of the EIT resonance and hence its overall quality factor can be engineered by choosing the separation between the top-layer grating-waveguide structure and the bottom waveguide layer. An example of such an engineered resonance shape is shown in Figure 12k,l. Such resonance shapes can potentially be used as narrow band-pass filters when compared to the more complex multi-layer dichroic filters.

#### 3.3. Bound-States in Continuum Resonances

_{H}−ε

_{L}. The structure supports multiple resonant modes (TE

_{0}and TE

_{1}modes) as shown in the figure. The bandstructure for the TE0 and TE1 resonances in the vicinity of k

_{z}= 0 point shows two different types of resonances, one which is the leaky GMR and the other is the non-leaky BIC resonance [82], as shown in Figure 14b. The mode profile of the GMR and BIC resonances show odd and even order symmetry respectively and this inherently determines the ability to excite or couple into these modes through normal incidence plane wave excitation. The odd-symmetry profile can be excited with a normal incident wave, while the even-symmetry profile is forbidden from excitation. Furthermore, the GMR and BIC resonances are found to flip with change in Δε [80]. The same band dynamics are observed for both TE

_{0}and TE

_{1}resonant modes. The symmetry protected BIC resonances strictly remain protected only at normal incidence. With off-axis illumination, the symmetry can be broken resulting in quasi-BIC resonances with finite quality factor. BIC resonances can also be observed for non-zero k

_{z}, which are called accidental BIC resonances [76]. Quasi-BIC resonances can also be excited at normal incidence by the introduction of asymmetry in the periodic structures [83]. Figure 14c shows schematic of such asymmetric structures. The resonant metasurfaces can be modelled by the amount of asymmetry introduced into the structure, denoted by α parameter [83]. Figure 14d shows that the asymmetry parameter can represent angular tilt, addition/ removal of material in split-ring, rectangular and bar-dimer structures in normalized units [83]. The asymmetric resonant metasurfaces discussed in Figure 12g–i which exhibit EIT-like coupling between the electric and magnetic dipolar modes can also be considered as an asymmetric structure in which quasi-BIC modes are observed. The quality factor of the BIC resonance is found to be directly related to the asymmetric parameter, with the quality factor scaling as α

^{−2}[83]. In addition to resonant metasurfaces, BIC resonances are also predicted for isolated sub-wavelength particles in the form of narrow spectral features in the scattering spectra. These are termed as super-cavity modes [46]. Such high-quality factor BIC resonances in periodic grating structures, asymmetry metasurfaces and even isolated objects are finding innovative applications in BIC metasurface lasers [84,85], sensing [86], and nonlinear optics [87]. Few of the nonlinear optics applications are discussed below in Section 4.

## 4. Nonlinear Optical Studies of Resonant Dielectric Grating Structures

#### 4.1. Second- and Third-Harmonic Generation

^{n}where, Q is the quality factor of the resonance under consideration and V is the cavity volume and n is the order of the nonlinearity [91]. With reduced cavity volumes in sub-wavelength metasurfaces and the ability to achieve moderately high-quality factors (few 100 s to 1000 s), the resonant nonlinear optical process can be enhanced by 10

^{3}to 10

^{5}times. This field enhancement can counteract the effect of reduced interaction length in sub-wavelength thick metasurface, which is potentially promising for realizing high efficiency nonlinear photonic devices. Second order nonlinear optical processes are observed in materials which lack inversion symmetry and in material interfaces, while third-order nonlinear optical processes are observed in all optical media [1]. This leads to the careful selection of the nonlinear media to build resonant metasurface platforms for study various nonlinear optical processes. In general, the second- and third- harmonic generation processes satisfy the frequency relationships, ${\omega}_{out}=\omega +\omega \text{}$ and ${\omega}_{out}=\omega +\omega +\omega $ respectively. The need for momentum or wave-vector matching is relaxed in sub-wavelength metasurface platforms in most implementations due to the reduced length resulting in negligible phase mismatch. Here, we broadly divide the second and third-harmonic generation studies in periodic dielectric structures into guided-mode resonance type and resonant metasurface type platforms. Few examples under each of these categories are listed in Figure 15 and Figure 16, respectively.

**Figure 15.**Various guided-mode resonance structures studied for nonlinear optical enhancement studies. (

**a**) Schematic of glass-grating with PMMA layer used for phase-matched second-harmonic generation studies. (

**b**) Schematic and scanning electron microscopy image of the Azo-polymer coated titanium oxide gratings used for second-harmonic generation enhancement. (

**c**) Silicon nitride grating structures and simulated field profiles used for UV-third-harmonic generation. (

**d**) AlGaAs high-contrast grating structures and second harmonic generation microscopy studies for different incident/ detection polarizations. (Figure a is reproduced with permission from ref. [92], b is reproduced with permission from ref. [93], c is reproduced with permission from ref. [95] and d is reproduced with permission from ref. [96]).

^{5}has been reported in this work with an overall conversion efficiency of 10

^{−4}. Silicon nanodisks in ordered two-dimensional arrangement have been used to leverage magnetic dipolar resonances from the unit cell elements to enhance third-harmonic generation [34]. Figure 16b shows one such arrangement of nanodisks with the corresponding linear and nonlinear spectral measurement results. Maximum enhancement of close to two orders of magnitude with conversion efficiencies of ~10

^{−7}has been reported in this work. Similar third-harmonic enhancement studies have been extended to dimer and more complex oligomeric unit cells to study the collect interaction of the individual elements in the unit cells [98,99]. There has also been interest in understanding the effect of disorder in the particle arrangement [100,101]. Figure 16c shows the arrangement of the nanodisks with controlled disorder introduced during fabrication. In this work, it has been found that the third-harmonic signal and its spatial localization are robust against disorder added to the nanodisk arrangement, making it topologically protected. Gallium Arsenide metasurfaces have been used for second-harmonic generation enhancement [102]. Asymmetric metasurfaces with high quality factor utilized for one such work with the corresponding linear and nonlinear optical spectra are shown in Figure 16d. It is found that the common [1 0 0] oriented Gallium Arsenide results in negligible second-harmonic emission along the optical axis due to the dominant longitudinally polarized nonlinear polarization, thus resulting in poor collection efficiency. One way to alter the far-field emission profile is to change the Gallium Arsenide orientation [103]. Figure 16e shows one such work on nanodisk arrays of [1 1 1] Gallium Arsenide metasurfaces. It is found from the far-field angular distribution that [1 1 1] metasurface does result in strong second-harmonic emission parallel to the optical axis when compared to [1 0 0] metasurface.

**Figure 16.**Various implementations of resonant metasurface for second- and third-harmonic generation studies. (

**a**) Fano-resonant silicon bar-nanodisk structures for third-harmonic generation enhancement. (

**b**) Silicon nanodisk array for third harmonic generation enhancement relying on magnetic dipolar modes. (

**c**) Disorder robust third-harmonic generation from silicon nanodisks which are shown to be topologically protected from disorder in arrangement of the structures. (

**d**) Gallium Arsenide asymmetry resonant metasurface for second-harmonic generation enhancement. (

**e**) Dependence of the resonant second-harmonic far-field signal on [1 1 1] oriented Gallium Arsenide metasurface. (

**f**) Spatial mapping of intensity dependent saturation of third-harmonic signal from silicon nanodisk array. (Figure a is reproduced with permission from ref. [97], b is reproduced with permission from ref. [34], c is reproduced with permission from ref. [100], d is reproduced with permission from ref. [102], e is reproduced with permission from ref. [103] and f is reproduced with permission from ref. [104]).

#### 4.2. Wave Mixing Processes

_{4}= ω

_{1}+ ω

_{2}− ω

_{3}[1]. Two pump frequencies (ω

_{1}and ω

_{2}) being unique or identical are termed as non-degenerate and degenerate FWM processes respectively. The second-order SFG process satisfies the frequency relation: ω

_{3}= ω

_{1}+ ω

_{2}, while degenerate third-order SFG process satisfies frequency relation of the form: ω

_{3}= 2ω

_{1}+ ω

_{2}or ω

_{3}= ω

_{1}+ 2ω

_{2}[1]. Figure 17a–c shows FWM enhancement observed for closely spaced pump-signal wavelengths in the telecom range for silicon-on-insulator based fully etched high-contrast gratings [105]. The sub-wavelength dimension high-contrast grating structures are found to support resonances with intensity enhancement of more than 8000 times and experimentally measured quality factor of ~7300. The signal and pump photons in close vicinity to this resonance results in FWM with the generation of idler with conversion efficiency of −19.5 dB as shown in Figure 17c. The use of high aspect ratio germanium (Ge) nanodisks to observe anapolar resonances [106] and the enhancement of third-order sum-frequency generation processes is shown in Figure 17d–f [107]. The higher order anapolar mode profiles are chosen with good spatial overlap to ensure enhancement of the SFG process by about two-orders of magnitude, as shown in the SFG spectrum in Figure 17f. Silicon (Si) nanodisks that support magnetic and electric-dipole resonances have also been utilized for doubly-resonant enhancement of FWM process as shown in Figure 17g–i [108]. The individual resonance spectra and the corresponding resonance for the FWM are also shown, with approximately two-orders of magnitude enhancement. Doubly-resonant structures are promising to increase the FWM efficiency using both pump and signal resonances. However, the best enhancement can be obtained only when good overlap is ensured between the interacting resonant mode profiles. Detailed spatially-resolved imaging of four-wave mixing process in singly resonant partially etched zero-contrast grating structures is shown in Figure 17j–l [109]. The structures are designed to support resonance at the signal wavelength in the 1550 to 1600 nm wavelength range. Four-wave mixing images acquired across an area of 300 × 300 microns show clear dependence of the FWM image contrast on the incident signal wavelength. A maximum FWM enhancement of 450 times has been experimentally obtained [109].

#### 4.3. Optical Switching

#### 4.4. Photon Acceleration

#### 4.5. Higher Order Wave-Mixing Processes

#### 4.6. Nonlinear Optics with Hybrid Metasurface

^{5}pm/V, one of the highest nonlinear optical susceptibility reported for any solid-state material systems. The plasmonic structures have recently been replaced by dielectric metasurfaces, as shown in Figure 21d–f [117]. The structure consists of a one-dimensional germanium guided-mode resonant structure integrated on top of the multi-quantum layer stack. The designed optical resonances of the germanium guided-mode resonance structures and the associated resonant nonlinear optical susceptibility are shown in Figure 21e. 100s of nano-watt level second harmonic generation signal has been experimentally measured from these all-dielectric hybrid metasurfaces. Such structures with the fundamental wavelength in the mid infrared wavelength range are best suited for frequency up-conversion with high conversion efficiencies from wavelength regions where there is scarcity of high efficiency detectors to wavelength regions in the near-infrared and visible region where mature, high efficiency detectors are readily available.

**Figure 21.**Various implementations of hybrid resonant metasurfaces for nonlinear optical applications. (

**a**–

**c**) Hybrid plasmonic-dielectric metasurfaces in which resonant nonlinearities in multi-quantum well structures are coupled with plasmonic resonances to study second harmonic generation. (

**d**–

**f**) All dielectric implementation of the hybrid resonant metasurface consisting of Germanium guided-mode resonance structures on top of the multi-quantum well structures. (

**g**–

**i**) Hybrid structures consisting of multi-layer Gallium Selenide on top of the asymmetric metasurface used for resonant enhancement of second-harmonic and sum-frequency generation. (Figures a–c are reproduced with permission from ref. [116], d–f are reproduced with permission from ref. [117], g–i are reproduced with permission from ref. [119]).

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The simulated scattering cross section (in arbitrary units–a.u.) for silicon nanowires of varying diameter with incident light polarization oriented: (

**a**) parallel and (

**b**) perpendicular to the nanowire. (

**c**) Experimentally obtained dark field images of nanowires showing light scattering for various width. (

**d**) Experimentally obtained scattering spectra of nanowires of varying width. (Figures c and d are reproduced with permission from ref. [31]).

**Figure 2.**(

**a**) Simulated scattering cross section (in arbitrary units–a.u.) for silicon nanospheres of fixed diameter of 150 nm (shown in black) and the result of decomposing the scattering spectra into magnetic dipole—MD (blue), electric dipole—ED (red), magnetic quadrapole—MQ (green) and electric quadrapole—EQ (brown). The sum of the MD, ED, MQ and EQ spectra is also shown (grey dashed). The field profiles for the MD, ED, MQ and EQ modes are also shown. (

**b**) Experimentally obtained dark field scattering images and spectra for varying diameters of silicon sub-wavelength nanoparticles. (Figure b reproduced with permission from ref. [33]).

**Figure 3.**Scattering spectra from an isolated sub-wavelength cylinder separated into: magnetic dipole —MD (red), electric dipole—ED (orange), magnetic quadrapole—MQ (purple) and electric quadrapole—EQ (green). The sum of the MD, ED, MQ and EQ scattering spectra is shown in black.

**Figure 4.**(

**a**) Transmission spectra contour map for varying array pitch (Λ). The top view and side view profile of the hexagonal arrangement of sub-wavelength cylinders shown in the inset. The height and diameter of the structure are: H = 710 nm, h = 0 nm, and D = 625 nm. (

**b**) Selected transmission spectra for pitch, Λ = 0.8 μm (blue curve), 1.2 μm (red curve) and 1.6 μm (green curve) are shown.

**Figure 5.**Scattering efficiency spectrum from isolated sub-wavelength dielectric sub-wavelength disk with dimensions same as in Figure 3 as a function of varying refractive index. Refractive index of 1.9 (blue curve), 2.2 (red curve), 3.5 (black curve) and n = 4.2 (green curve) are shown. The refractive index is assumed to be constant across the spectral range shown for each curve.

**Figure 6.**(

**a**) Schematic of fully-etched dielectric one-dimensional grating structure. The simulated transmission spectra for (

**b**) silicon high contrast grating and (

**c**) silicon nitride medium contrast grating as a function of varying height for fixed incident TE polarization. The dimensions and wavelength are normalized by the pitch of the grating structure. (Figure is reproduced with permission from ref. [49]).

**Figure 7.**Cross-section view of various guided-mode resonance based grating structures with waveguide and grating layer refractive indices n

_{H}and n

_{G}respectively. (

**a**) The grating and waveguide are made of different materials with the waveguide of higher index below the fully-etched grating structure. (

**b**) Fully etched grating structures which can act as the effective waveguide. (

**c**) Partially etched grating structures with the waveguide layer made of same material as the etched gratings. (

**d**) Substrate grating structures is coated with a high-index waveguiding layer on top.

**Figure 8.**(

**a**) A cross-section schematic of the guided-mode resonance structure showing the resonant coupling of incident light into the waveguide region through interaction with the grating structure. (

**b**) Simulated filter response for the guided-mode resonance structure for parameters: pitch = 330 nm, height = 330 nm, dielectric constant difference (normalized), ∆ε/ε

_{avg}= 0.05 and center wavelength of 547 nm and dependence of the filter response on the angle of incidence. (Figure b is reproduced with permission from ref. [38]).

**Figure 9.**(

**a**) Reflection spectra contour for free-standing silicon high contrast gratings with dimensions: n

_{grating}= 3.48, duty cycle = 70% and thickness varied. The various regions of operation of the grating are also shown. (

**b**) The reflection spectra overlapped with the solutions to the eigen-mode equations of the resonant modes (white curves). The overlap regions of the white curves result in anti-crossing and crossing type resonance. (

**c**) Field intensity profile at anti-crossing resonance. (

**d**) Field intensity profile at crossing resonance. (Figures are reproduced with permission from ref. [50]).

**Figure 10.**The simulated zeroth order reflection (R

_{0}) and transmission (T

_{0}) spectra for silicon zero-contrast gratings. The inset shows the cross-section of the zero-contrast gratings with the dimensions optimized using particle swarm combined with inverse-design algorithm. The optimized dimensions of the structure obtained are: etched grating height of 490 nm, unetched slab thickness of 255 nm, pitch of 827 nm, and fill factor of 0.643. (Figure reproduced with permission from ref. [60]).

**Figure 11.**(

**a**) The representative energy diagram for electromagnetically-induced transparency (EIT). (

**b**) Schematic of the expected transmission spectrum for the EIT effect.

**Figure 12.**Different implementations of EIT-like resonance. (

**a**) SEM image of bars-ring array. (

**b**) Schematic of the coupling between the bar excited by the incident light and coupling to the ring structure creating magnetic dipolar type mode. (

**c**) Experimentally measured transmission spectrum for the bar-split ring array showing EIT-like resonance. (

**d**) Schematic of the achiral bar-dimer structures. (

**e**) SEM image of asymmetric dimer achiral structures. (

**f**) Coupling between the dipole and quadrapolar modes in the asymmetric dimer structures. (

**g**) Perspective view of the asymmetric nanocube unit-cell. (

**h**) Schematic showing the coupling of electric and magnetic dipole modes in asymmetric nanocubes. (

**i**) Measured reflection spectrum for the asymmetric nanocube array. (

**j**) Cross-sectional view of the structure showing upper layer grating-waveguide structure coupled to lower waveguide structure. (

**k**,

**l**) Simulated EIT-like resonance spectra and associated field profiles from GMR structures. (Figures a–c are reproduced with permission from ref. [68], d–f are reproduced with permission from ref. [70], g–i are reproduced with permission from ref. [72], k,l are reproduced with permission from ref. [74]).

**Figure 13.**(

**a**) Schematic of a quantum well with sharp potential edges and the corresponding photonic analogue showing an optical waveguide with confined modes, (

**b**) Schematic of quantum well with modulated edges and the corresponding photonic analogue showing grating-modulated waveguide. (Figure adapted from ref. [76]).

**Figure 14.**(

**a**) Schematic of the periodic dielectric constant modulated grating structure with typical TE0 and TE1 modes supported by such structure. (

**b**) Photonic band-structure calculation corresponding to the TE0 and TE1 modes showing the GMR and BIC states at either band-edge and their corresponding mode profiles. The GMR and BIC states are found to flip by changing the dielectric constant difference between the grating materials. (

**c**) Examples of asymmetric resonant metasurfaces which support quasi-BIC resonances. (

**d**) Modeling the asymmetricity using an asymmetry parameter, α. (

**e**) Variation of quality factor of the quasi-BIC resonance with change in asymmetry parameter. (Figures a–b are reproduced with permission from ref. [81], c–e are reproduced with permission from ref. [83]).

**Figure 17.**Various implementations of resonant FWM processes. (

**a**) Schematic of high-contrast grating resonance, (

**b**) Electron microscopy images and measured reflectivity spectrum of the resonance. (

**c**) Measured FWM spectrum for the high-contrast grating structure. (

**d**) Schematic of the high-aspect ratio Ge nanodisks for SFG studies. (

**e**) Mode profiles of the nonlinear polarization for the two different SFG processes. (

**f**) Measured SFG and THG spectra for the high-aspect ratio Ge nanodisks. (

**g**) Schematic of Si nanodisk structures used for doubly-resonant FWM studies. (

**h**) Comparison between measured and simulated scattering spectra for the two resonant modes under consideration. (

**i**) Comparison of the measured FWM signal and simulated pump intensity enhancement as a function of wavelength. (

**j**) Electron microscopy image of the partially etched zero-contrast grating structures used for FWM studies. (

**k**) Simulated and measured transmission spectra for the zero-contrast grating structures. (

**l**) FWM microscopy images for varying signal wavelength. The enhancement spectrum is also shown. (Figure a–c are reproduced with permission from ref. [105], d–f are reproduced with permission from ref. [107], g–i are reproduced with permission from ref. [108] and j–l are reproduced with permission from ref. [109]).

**Figure 18.**(

**a**) Schematic of the nanodisk array used for ultrafast-optical switching studies. (

**b**) Electron microscopy image and (

**c**) transmission spectra for the nanodisk array with the corresponding dipolar modes marked. (

**d**) Experimental results of pump-probe studied showing fast recovery or switching of probe in the presence of a resonant pump. Various pump laser wavelengths relative to the resonance are shown in the right plot. (Figures are reproduced with permission from ref. [110]).

**Figure 19.**(

**a**) Schematic of the silicon rectangular metasurface used for photon acceleration studies. (

**b**) Measured transmission spectrum and field profile at resonance (inset). (

**c**) The fundamental laser spectra transmitted through the metasurface for varying laser fluence. (

**d**) Comparison of the third-harmonic signal generated for varying fundamental laser fluence for the un-patterned silicon film and silicon metasurface. (Figures are reproduced with permission from ref. [112]).

**Figure 20.**(

**a**) Schematic of the Gallium Arsenide nanodisk array used for nonlinear wave mixing studies. (

**b**) Experimentally measured spectra of various nonlinear wave mixing processes. The name of the various processes and their frequency relationship are labelled. (

**c**) The dependence of the nonlinear wave-mixing spectra on the time-delay between the interacting excitation pulses. (Figures are reproduced with permission from ref. [113]).

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**MDPI and ACS Style**

Raghunathan, V.; Deka, J.; Menon, S.; Biswas, R.; A.S, L.K.
Nonlinear Optics in Dielectric Guided-Mode Resonant Structures and Resonant Metasurfaces. *Micromachines* **2020**, *11*, 449.
https://doi.org/10.3390/mi11040449

**AMA Style**

Raghunathan V, Deka J, Menon S, Biswas R, A.S LK.
Nonlinear Optics in Dielectric Guided-Mode Resonant Structures and Resonant Metasurfaces. *Micromachines*. 2020; 11(4):449.
https://doi.org/10.3390/mi11040449

**Chicago/Turabian Style**

Raghunathan, Varun, Jayanta Deka, Sruti Menon, Rabindra Biswas, and Lal Krishna A.S.
2020. "Nonlinear Optics in Dielectric Guided-Mode Resonant Structures and Resonant Metasurfaces" *Micromachines* 11, no. 4: 449.
https://doi.org/10.3390/mi11040449