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

In this section, the various nonlinear optical processes studied in the context of guided-mode resonance structures and resonant metasurfaces are discussed. These are broadly classified based on the complexity involved in terms of the nonlinear optical processes studied or the structure being considered for this study. First, the basic nonlinear harmonic generation processes such as second and third harmonic generation are considered, following which wave-mixing processes such as four-wave mixing and sum-frequency generation are considered. This is followed by ultra-fast optical switching, photon acceleration effect, and higher harmonic generation processes. Lastly, nonlinear optical studies

in hybrid metasurfaces are discussed. The discussion is aimed at outlining the salient features of the respective nonlinear processes and specific structures studied. There are few previous review articles written in the areas of nonlinear metasurfaces in plasmonic [9,11], dielectric metasurface [14] and guided-mode resonance [17] platforms. There is also interest in utilizing the metasurface to shape the wavefront of the generated nonlinear signal for beam steering or focusing applications [88–90]. These efforts are not discussed here to keep the focus solely on the resonant enhancement of nonlinear optical processes.

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

Second and third-order nonlinear optical processes are considered as the basic nonlinear optical processes studied in optical media under the influence of a strong incident electric field. The induced polarization or the response of the medium to the incident electric field can be expanded in a perturbative approach into various nonlinear optical processes as follows [1]:

$$\overrightarrow{P}(\boldsymbol{\omega}\_{\text{out}}) = \varepsilon\_{o} \Big( \boldsymbol{\chi}^{(1)} \cdot \overrightarrow{E}(\boldsymbol{\omega}) + \boldsymbol{\chi}^{(2)} : \overrightarrow{E}(\boldsymbol{\omega}) \overrightarrow{E}(\boldsymbol{\omega}) + \boldsymbol{\chi}^{(3)} : \overrightarrow{E}(\boldsymbol{\omega}) \overrightarrow{E}(\boldsymbol{\omega}) \overrightarrow{E}(\boldsymbol{\omega}) + \cdots \Big) \tag{5}$$

The nonlinear interactions which depend quadratically and cubically with the incident electric field gives rise to second- and third-order nonlinear processes respectively. The strength of the nonlinear optical processes can be enhanced significantly by the enhancement of the incident electric field inside the resonant metasurface. The nonlinear process typically scales as the (Q/V)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<sup>3</sup> to 10<sup>5</sup> 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, ω*out* = ω + ω and ω*out* = ω + ω + ω 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 Figures 15 and 16, respectively.

Some of the early sub-wavelength periodic structures studied for nonlinear optical applications are the guided-mode resonance structures leveraging the resonances offered by the dielectric grating structures to enhance nonlinear effects from nonlinear polymer overlayers. Figure 15a,b show two such implementations using PMMA [92] and Azo-polymers [93] as the nonlinear media on top of glass and titanium oxide gratings respectively. In Figure 15a, careful attention is paid to the phase matching of the second-harmonic generation process between the counterpropagating fundamental and second-harmonic slab modes in the presence of the grating structure [92]. Experimentally measured second-harmonic signal shows enhancement corresponding to the phase matched condition when compared to the non-phase matched case. The use of higher refractive index gratings, such as periodically patterned titanium oxide layer is found to enhance the local electric field in comparison to the glass gratings and this is found to enhance second-harmonic by ~3500 times from a Azo-polymer overlayer when compared to a reference sample without the guided-mode resonance structures [93]. There has also been interest in studying nonlinear optical processes from the guided-mode resonance grating structures itself. In this context, silicon nitride gratings have been used for second- and third-harmonic generation studies [94,95]. Even though the nonlinearities in silicon nitride is weak and

the index contrast with the substrate is small, the broad optical transparency window from the visible to mid infrared, makes it attractive for realizing high quality resonant structures. An example for the use of silicon nitride sub-wavelength grating structures for third-harmonic generation in the ultraviolet spectral region is shown in Figure 15c [95]. Aluminum Gallium Arsenide (AlGaAs) high-contrast grating structures with characteristic optical resonances have also been explored for second-harmonic generation studies. Schematic images of such free-standing AlGaAs high contrast grating structure and the corresponding second-harmonic microscopy images obtained for different orientations of the fundamental and second-harmonic polarization are shown in Figure 15d [96].

**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]).

In the context of resonant metasurfaces for second- and third- harmonic generation studies, sub-wavelength spaced arrays of high-index semiconductors such as silicon, germanium and gallium arsenide have been studied. Fano-resonances from silicon bar-nanodisk structures, similar to the bar-ring structures shown in Figure 12a have been utilized to enhance third-harmonic generation [97]. The scanning electron microscopy image and the measured linear and third-harmonic spectra are shown in Figure 16a. Maximum third-harmonic signal enhancement of ~105 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−<sup>7</sup> 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]).

Spatially resolved nonlinear optical studies or nonlinear optical microscopy is also a useful tool to understand the spatial dependence of the nonlinear signal across different regions in the metasurface sample to understand the signal uniformity and can combined with spectral and intensity studies to understand spatial, spectral resonance and intensity saturation behavior of the nonlinear medium. In this context, Figure 16f shows the intensity dependence of the third-harmonic generation microscopy images across a silicon nanodisk array sample at its fundamental resonance wavelength [104]. It is found that the contrast in the third-harmonic microscopy images reverses with increasing intensity. This is attributed to the spatial position dependent onset of saturation of the third-harmonic signal as shown in the intensity dependent third-harmonic plot in Figure 16f.

## *4.2. Wave Mixing Processes*

The wave-mixing processes can be considered as a general case of the above described harmonic generation processes. The processes of interest here are the four-wave mixing (FWM) and sum-frequency generation (SFG). In the case of FWM process, the nonlinear signal frequency is related to the incident light as follows: ω<sup>4</sup> = ω<sup>1</sup> + ω<sup>2</sup> − ω<sup>3</sup> [1]. Two pump frequencies (ω<sup>1</sup> and ω2) being unique or identical are termed as non-degenerate and degenerate FWM processes respectively. The second-order SFG process satisfies the frequency relation: ω<sup>3</sup> = ω<sup>1</sup> + ω2, while degenerate third-order SFG process satisfies frequency relation of the form: ω<sup>3</sup> = 2ω<sup>1</sup> + ω<sup>2</sup> or ω<sup>3</sup> = ω<sup>1</sup> + 2ω<sup>2</sup> [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*

The ultrafast Kerr nonlinearity and multi-photon absorption processes due to the nonlinear interaction of valence electronics in the dielectric medium with incident light can be used to perform fast optical switching at hundreds of femtosecond time scales. Such ultrafast switches have been demonstrated previously in guided-wave systems such as optical fibers and integrated waveguides utilizing self-phase and cross-phase modulation effects [2]. The optical resonances in dielectric resonant metasurfaces can be used to enhance the optical switching process through the enhancement of the nonlinear optical effect [110]. In the context of semiconductor metasurfaces, the presence of parasitic processes such as thermo-optic effects results in additional phase shift, albeit at much longer time scales. Figure 18a,b shows a schematic and scanning electron microscopy image of a silicon nanodisk array used for optical switching studies. The corresponding transmission spectra from the nanodisk array and the associated dipolar resonant modes are shown in Figure 18c. The optical switching process can be studied in a pump-probe configuration with a strong pump on-resonance leading to enhanced electric-field inside the nanodisks which results in a fast transmission dip at the probe wavelength due to enhanced multi-photon absorption. The fastest switching characteristics is obtained with a temporal response time of 65 fs with a slower extended recovery due to thermal and free-carrier recombination effects. Such resonant metasurface with ultrafast all-optical switching capability can find possible applications as fiber-connectorized photonic structures for high speed data communication and pulse-shaping [2].

**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]).
