*3.2. Electromagnetically-Induced Transparency Analogue Resonances*

Optical resonances in photonic structures can be created due to interference between coupled resonances of varying quality factor. For example, a high-quality factor and low-quality factor resonance can couple together to result in the observation of analogues of electromagnetically-induced transparency (EIT) in photonic structures [18]. The electromagnetically-induced transparency effect can be understood based on quantum interference between two different path-ways to excite an upper energy level in a three-level system, as shown in Figure 11a. This system consists of a lower level, an upper level and a meta-stable state, as denoted in the figure. A probe field can control the transition from the lower to upper state, while a control field can control the transition from the excited state to metastable state. The direct transitions from the lower state to metastable state are forbidden. In this context, the transitions induced by the probe field can interfere with the indirect transitions from lower to upper state through interaction with the metastable state, in the presence of a strong control field. This can result in destructive interference of the probe field and results in the observation of peaks at

the probe field frequency in an otherwise expected dip in transmission spectrum, as shown in Figure 11b. This phenomenon is hence termed as electromagnetically induced transparency. In the context of photonic systems, the low-quality factor and high-quality factor resonant modes are equivalent to the direct and indirect lower level to upper level transition respectively, as discussed above.

**Figure 11.** (**a**) The representative energy diagram for electromagnetically-induced transparency (EIT). (**b**) Schematic of the expected transmission spectrum for the EIT effect.

Photonic structures in the form of coupled Fabry-Perot cavities, coupled ring resonators, coupled photonic crystals etc. have been explored to be studied as EIT analogues [18]. In the context of arrayed metasurfaces and guided-mode resonant structures, there is also interest in coupling bright-mode and dark-mode resonances to observe EIT effects. Few examples of such implementations, supporting simulations and experimental studies are shown in Figure 12. Figure 12a–c shows an array of bar-ring structures in which the incident linearly polarized light preferentially couples to the bar and indirectly couples into the ring through the bar excited mode [67,68]. This results in excitation of dark modes in the ring, as shown in Figure 12b. The measured transmission spectra show peak in the middle of a broad transmission dip with strong field localization in the ring structures with maximum quality factor of the EIT resonance of ~300 times. Asymmetry in the array elements also results in coupling between bright and dark modes. Figure 12d–f shows one such example of rectangular bar dimer array with slight asymmetry added to one of the bars [69,70]. In this case, the coupling between the bright dipolar mode with the dark quadrapolar modes results in peaks in the transmission spectra (shown in Figure 12f) in the mid infrared wavelength region. Such structures have been used for enhanced infrared sensing [71]. Asymmetry can be introduced to the shape of a single monomeric unit as well to achieve bright-to-dark mode coupling. This is shown in Figure 12g–i, for the case of a nanocube with symmetry breaking protrusion [72]. As shown in Figure 12h, this protrusion results in the coupling of the electric dipole excitation, denoted as *px* to the magnetic dipole mode, denoted as *mz*. 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.

**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]).

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

Bound-states in continuum (BIC) represent another interesting class of high-quality factor resonance. BIC are bound states in an otherwise continuum of states, considered as a trapped state with embedded eigenvalue. Figure 13 shows a schematic representation of BIC states in quantum-well systems and comparison with analogous waveguide grating systems (figure adapted from ref. [76]). Figure 13a compares a conventional quantum well with a slab-waveguide. It is found that in a conventional quantum well, the allowed states within the quantum well are bounded by the steep

potential walls of the quantum well. This results in discrete, bounded states within and a continuum of states outside. This can be compared with a straight, unperturbed slab waveguide with optical mode confinement achieved by the core-cladding refractive index profile. This results in discrete, bound modes within the waveguide and a continuum of radiation modes outside. When the sharp edges of the quantum well potential profile are replaced by a modulated potential profile, it is found that there can exist bound states even within the continuum, as shown in Figure 13b. In similar lines, the introduction of a periodic modulation to the waveguide using a grating structure results in the creation of bound-states within the continuum of radiation states. These states end up being forbidden from external excitation/ coupling due to symmetry considerations [77]. This lack of in- and out-coupling or non-radiating characteristic can result in ideal infinitely high-quality factor for these bound states within the continuum resonances. However, in practice one would consider quasi-BIC states which tend towards the ideal situation and can result in large, but finite quality factors with external excitation mechanism [78,79].

**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]).

The BIC states in the context of periodic grating structures can be broadly classified as symmetry protected BIC and accident BIC [76]. Symmetry protected BIC occurs at the high symmetry, zero wave-vector points (Γ-point in the case of photonic band structures) with direct excitation forbidden for normal incidence by the BIC resonant field symmetry. An example of such symmetry protected states in a simulated periodic dielectric constant modulated structure is shown in Figure 14a [80,81]. The schematic shows the periodic modulated structure with dielectric constant difference of Δε = ε*H*−ε*L*. The structure supports multiple resonant modes (TE0 and TE1 modes) as shown in the figure. The bandstructure for the TE0 and TE1 resonances in the vicinity of kz = 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 TE0 and TE1 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 kz, 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−<sup>2</sup> [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.

**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]).
