1. Introduction

Chalcogenide glasses are phase change materials (PCM) that can be switched without volatility between the amorphous (glassy) and a crystalline phase. GST (GeSbTe) and similar alloys have transformed data storage giving rise to the familiar re-writable disks (CD-RW, DVD-RW) [1–4] we use to this day. These chalcogenide glasses remain very much current in the electronics industry with the “holy grail” being bridging the disconnect between storing information and computing/manipulating information [3,4]. Related architectures, such as 3D Xpoint [5], have already emerged in the market [3].

Indeed, chalcogenide PCMs have revolutionized the electronics industry and in doing so they inspired further efforts in photonics. In the last decade, research in integrated photonics interfaced with chalcogenide glasses has flourished, investigating reconfigurability/tunability with different components such as waveguides [6–8], multi-pronged ridge-waveguide systems for side-coupling and wave-redirection [9–12], optical attenuators [13], as well as optical switches [11,14,15]. Free-space optical reflectors [16] with directional reconfigurability have also been reported. Furthermore, there is a strong focus around metasurfaces with chalcogenide PCMs [17,18] shown to function as dynamic meta-displays with a high-saturation color switching [19,20], dynamic reflective displays [21,22] or demonstrating strong reconfigurable beam steering in reflection [23]. A combination of non-volatile switching together with controlled volatile switching by exploiting the dynamics during crystallization in GST has also been reported [24]. All these aforementioned photonics-focused chalcogenide PCM platforms are motivated by broader overarching goals for programmable photonics [1,25], including optical FPGAs (field programmable gate arrays) [26] and site-specific addressability [27], photonic memories [28,29], photonic synapses or neurons [24], photonic neuromorphic/brain-inspired computing [29–31], combined photonic memory-computing architectures [32,33] and photonic-based quantum computing [34].

The aforementioned studies focus on the visible or near-IR part of the spectrum and most incorporate GST or another chalcogenide alloy re-purposed from the electronics industry. There are different material aspects in chalcogenide PCMs that come into play with some being more important than others depending on the specific reconfigurable photonic device and application regime [1,35]. The range of relevant PCM characteristics for photonics involve their switching properties between the crystalline and amorphous phase (energy needed, switching speed, material endurance, phase stability etc.) as well as their optical parameters (loss tangent in either or both phases, refractive index values in crystalline and amorphous phases, and the respective value of the refractive index shift between the two phases) [1]. This in its own right underlines the limitation for wide usage of GST or other electronic PCM in photonics. The limitation becomes even more apparent as we move to longer device operating wavelengths. Because of phase instability in the amorphous phase, requiring a fast cooling rate after heating near melting point, GST can be effectively switched only at small thicknesses, typically around 100 nm [35], with some reports of reconfigurability with somewhat thicker GST of 300 nm [17]. This thickness range is too sub-wavelength past the near-IR spectrum making it hard to achieve a strong beam control. For example, a perforated enhanced optical transmission (EOT) hole array type of structure [36], interfaced with GST, yielded only a small transmission modulation when GST is switched from the amorphous to the crystalline phase [37].

Nonetheless, there is a strong need for extreme and reconfigurable beam manipulation in the spectral regime of mid-wavelength to long-wavelength infrared (MWIR-LWIR). Relevant applications are integrated infrared photonics [38], long-range lidar [39] as well as spectrometers [14]. This stresses the need for new PCM materials designed specifically for light control that targets longer MWIR/LWIR wavelengths. A. Hessler et al. [40] have designed a new chalcogenide alloy In$_3$SbTe$_2$ (IST) that switches from a dielectric behavior in the amorphous phase to a metallic one in the crystalline phase, with negative permittivity, $\varepsilon$, thus allowing for a strong transmission modulation even with a small IST material thickness of 200 nm. However, having fully dielectric infrared PCM responses in a material that can be realized at significant thicknesses would enable a wider parameter space and possibilities to explore for infrared reconfigurable platforms for different applications.

Recently, Y. Zhang et al. [35] developed a new chalcogenide PCM family targeting optical applications. This optical PCM (O-PCM) family are quaternary alloys formed as an evolution from GST by substituting the Te atoms with Se atoms with different stoichiometry. The addition of the Se atom introduces shorter stronger bonds in the amorphous phase making it more stable [41], not demanding a fast switching time, thus permitting realization at larger thicknesses of about 1 $\mu$m [35,42]. The substitution of the heavier chalcogenide with a lighter chalcogenide (i.e. substituting Te with Se) also widens the electronic band gap, thus widening the spectral transparency range of the material [1,35,41]. Metavalent bonding has been identified in chalcogenide alloys as a special hybrid bonding with metallic and covalent characteristics that appears to be responsible for the high refractive index attribute in their crystalline phase [41,43]. Tellurium rich alloys have stronger metavalent contributions in comparison to Sellenium ones and thus a higher refractive index. However, this comes simultaneously at the cost of a stronger free-carrier absorption contribution [35].

In search of an O-PCM for all-dielectric MWIR/LWIR reconfigurable platforms, we found that one particular alloy of the Ge-Sb-Se-Te family reported in Ref. [35] has in fact all the required characteristics. This is Ge$_2$Sb$_2$Se$_4$Te$_1$, abbreviated as GSS4T1, which with a wide bandgap and no residual free-carrier absorption has a fully dielectric, essentially lossless, response for free space wavelengths above 5 $\mu$m [35]. This is true for both the amorphous and crystalline phase of the material. While GSS4T1’s refractive index in the crystalline phase is lower than its Tellurium-rich counterparts, it is still very high. In all-dielectric extreme light control avenues, loss can be detrimental. In other words, a cost-benefit analysis favors the lossless attribute rather than the higher value of refractive index in the crystalline phase. Actually, although lower than the other alloys of the family, GSS4T1 has still an impressive refractive index shift of $\Delta n\sim 1.5$ when being switched from the amorphous to the crystalline phase. The powerful potential of this new O-PCM has been already demonstrated in the near-IR, SWIR and MWIR spectral ranges, where related optical filters for optical communications [44,45] and smart night vision devices [46] as well as metasurface-based lenses with dynamic zoom [42] have been reported.

Here, we propose a new paradigm system, functional in the LWIR-MWIR spectrum, capable of exhibiting both reconfigurable beam-steering and beam-splitting. This paradigms takes advantage of the dielectric and near-lossless responses of GSS4T1 in both phases in combination with the large refractive index shift between the two phases. The proposed system relies on the versatile power-redirection capabilities of diffractive meta-gratings [47,48]. Specifically, simple periodic all-dielectric diffractive metagratings were shown to be able to near-completely suppress power output to the zeroth-order diffracted beam, thereby channeling either all transmitted [47] or all reflected power [49] to the back-bent first-order diffracted beam. We discuss the underpinning physical principles of our proposed paradigm that are responsible for the reconfigurable beam steering and splitting phenomena. We further analyze the key attributes that are important for the practical realization of these phenomena. Finally, we confirm the theoretical predictions determined via the semi-analytical integral equation method [50–52] with a first-principles numerical experiment [53]. We note, as new O-PCM are being researched and developed the design principles of our GSS4T1-based paradigm would be transferable to similar systems and possibly other spectral regimes, comprising other new O-PCM’s sharing similar attributes with GSS4T1 as we discussed above.

Specifically, our paper is organized as follows: In Section 2, we present the new metagrating paradigm with its structural characteristics and functional spectrum, while discussing our hypothesis behind its conception for a reconfigurable beam-steering and beam-splitting behavior. In Section 3, we study the metagrating paradigm with the O-PCM in the amorphous phase and analyze the physics behind its transmission response. Conversely, we study the metagrating paradigm with the O-PCM in the crystalline phase and analyze the respective physics behind its transmission response in Section 4. We additionally discuss the key attributes for the reconfigurable beam-steering and beam-splitting phenomenon respectively and how these are affected by factors coming to play in practical implementations. In Section 5, we focus on a selected design based on the analysis of Section 4, and verify the theoretically predicted reconfigurable beam-steering and beam-splitting phenomenon with a first-principles numerical experiment. In Section 6, we discuss how to scale the sizes of the structural parameters of the system to obtain the reconfigurable beam-steering and beam-splitting behavior at desired free-space wavelengths. Finally, we present our conclusions and outlook in Section 7.

2. MWIR-LWIR metagrating paradigm for reconfigurable beam manipulation

2.1 Metagrating structure

We show the schematics (side view) of the metagrating paradigm in Fig. 1(a). This is a periodic grating of period, $a$, and thickness, $b$, with its unit cell comprising an O-PCM of width $d_p$, and a widely-used typical semiconductor of width $d_s$. We will consider responses to a TE-polarized plane wave, incident at an angle $\theta _{\textrm {inc}}$ with the surface normal, i.e., with electric field perpendicular to the plane of incidence [see Fig. 1(a)]. As we discussed in the introduction, the O-PCM will be the newly-developed GSS4T1 [35] which does not require a fast switching time to amorphous phase allowing its implementation at larger thicknesses of $\sim$1 $\mu$m. The targeted wavelength of operation would be above 5 $\mu$m, where GSS4T1 responds as an essentially lossless dielectric for both the amorphous and crystalline phases [35]. We will restrict the spectrum until 10 $\mu$m, which is a spectral region of available practical set-ups with transparent components, such as CaF$_2$ [42], BaF$_2$ [54] etc.

 

Fig. 1. (a) The metagrating paradigm (side view) with its structural parameters designated. TE-polarization incidence will be considered as depicted. (b) Targeted reconfigurable beam-steering. (c) Targeted reconfigurable beam-splitting.

Download Full Size | PDF

The permittivity of GSS4T1 remains almost constant throughout the 5 $\mu$m to 10 $\mu$m free-space wavelength range for both the crystalline and amorphous phases with only a slight variation [35]. For the purpose of this work, aiming to uncover the principles and design route to obtain the reconfigurable beam manipulation behavior with the metagrating system, we will consider the GSS4T1 permittivity responses constant at $\varepsilon _{\textrm {crys}}=21.3$ for the crystalline phase and at $\varepsilon _{\textrm {amor}}=10.2$ for the amorphous phase (see also Appendix A). The chosen material for the widely-used typical semiconductor constituent is GaAs, see discussion in 2.2, which is taken with a constant permittivity response at $\varepsilon _{\textrm {s}}=10.8$ as well (see also Appendix A).

2.2 Operation principle hypothesis

The choice for GaAs as the second material constituent of the grating, is related to our hypothesis for the underpinning operating principle of the reconfigurable beam manipulation behavior. Namely, we chose GaAs because its permittivity at the spectral range of interest is very close to the permittivity of GSS4T1 when at the amorphous phase. Basically, we hypothesize that when the O-PCM is at the amorphous phase an impinging wave will effectively “see” a homogeneous material slab. Hence, after passing through the metagrating, it will continue to propagate at its incident direction as shown in the left panels of Fig. 1(b) and Fig. 1(c). We hypothesize that it is the switching to the crystalline phase that will “switch on” the metagrating behavior. Then, it would be possible to control the distribution of the output power into the emerging diffracted beams, as was reported for the metagrating structure of Tsitsas and Valagiannopoulos of Ref. [47]. Our first target is to determine a case where output power emerges predominantly in the back-bent diffracted beam, in this manner giving rise to a reconfigurable beam steering, as depicted in Fig. 1(b). Conversely, the second target is to determine a case where the output power is evenly split between the two oppositely bent diffracted beams, that would give rise to a reconfigurable beam splitting, as depicted in Fig. 1(c). We discuss more details around the hypothesis of the operating principle of our proposed paradigm in the following section where we analyze the spectral regime where these behaviors can be expected.

2.3 Spectrum of operation and unit cell size

We explained in Section 2.1 why the appropriate spectrum of operation should lie between free-space wavelengths of 5 $\mu$m and 10 $\mu$m. Additionally, we show here that metagrating functionalities that are depicted in the right panels of Figs. 1(b) and 1(c) can be realized only for a certain free-space wavelength range which depends on the unit-cell size, $a$, and also on the incident angle $\theta _{\textrm {inc}}$. In the following we determine such a dependence that will guide making a suitable choice for the unit cell size, $a$.

When an electromagnetic (EM) wave hits a homogeneous structure, with its interface lying along the $x-$axis, the parallel to the interface component of the wavevector, $k_x$ is conserved. On the other hand, when an EM wave impinges on a periodic interface, then the wavevector $k_x$ picks up an additional factor that is an integer multiple of the unit reciprocal lattice vector along the interface [55–57], this being equal to $\frac {2 \pi }{a} \hat x$ for the metagrating of Fig. 1, namely,

The functionality of our proposed metagrating paradigm relies on having only two Bragg beams, of orders $m=0$ and $m=-1$. Specifically, the beam-steering target [Fig. 1(b)] requires to have nearly-all transmitted power into the $m=-1$ Bragg channel. Conversely, the beam-splitting target [Fig. 1(c)] requires having the transmitted power evenly split between the $m=0$ and $m=-1$ Bragg channels. These functionalities concern the metagrating with the O-PCM in the crystalline phase. For these functionalities to be reconfigurable, there should still be only one outgoing beam, with its path unaltered, continuing along the incident direction when the O-PCM is in the amorphous phase.

For the grating of Fig. 1, which is embedded in vacuum and has both interfaces lying along the periodic direction, whether or not a certain Bragg channel of order, $m$, exists, i.e is not “dark”, depends solely on the following parameters: the grating’s unit-cell size, $a$, the angle of incidence, $\theta _{\textrm {inc}}$, and the frequency/free-space wavelength of the impinging wave. That would mean that if a Bragg channel exists when the O-PCM is in the crystalline phase, then it also exists when the O-PCM is in the amorphous phase. Now, this might appear to contradict our hypothesis discussed above where we need to have only one transmitted beam for the amorphous O-PCM grating. Nonetheless, there is no contradiction here. Just because an outgoing Bragg channel is in principle available to the EM wave, this does not mean the wave amplitude in that channel needs to be non-zero. Our hypothesis is, and we will further show this in Section 3, that when the O-PCM is in the amorphous phase, the $m=-1$ Bragg channel can be totally suppressed with zero transmitted power, because the grating is effectively behaving as a homogeneous medium in this case.

Accordingly, the envisioned reconfigurability behaviors of Fig. 1(b) and Fig. 1(c) can be accomplished if the following two conditions are simultaneously satisfied:

We set $a=5 \mu$m and choose a moderate range for incident angles $\theta _{\textrm {inc}}$, to span between 20 to 50 deg. (to avoid near-normal incidence as well as larger angles where transmission tends to be overall lower in periodic structures [58], and are also more difficult to implement in practical set-ups). We show in Fig. 2 the spectral operating region, as determined from Eq. (4), designated with the yellow shading. We notice most of the yellow-shaded region is within the 5 $\mu$m to 10 $\mu$m spectral range; part of it however falls below the 5 $\mu$m, where losses in the GSS4T1 material onset. We note, since the bounds of the operation spectrum are incident-angle dependent, it is difficult to align the entire operational spectral domain for the full angle range from 20 to 50 deg. to all lie within the 5 $\mu$m to 10 $\mu$m spectral range.

 

Fig. 2. Operating spectrum for the reconfigurable beam steering/splitting phenomena for the metagrating of Fig. 1(a) with $a=$ 5 $\mu$m (yellow-shaded area), where only the two Bragg channels with $m=0$ and $m=-1$ exist. Above the red line only the primary ($m=0$) Bragg channel is available; hence in that case there is no possibility to obtain a completely back-bent Bragg beam which has $m=-1$, or split power between the $m=0$ and $m=-1$ Bragg beams. Then, below the blue line an additional Bragg channel can outcouple, which will also take power and introduce an additional beam.

Download Full Size | PDF

2.4 Unit-cell size choice and other structural parameters

In the following sections (Sections 3–4), we aim to understand better the Physics behind the grating’s responses when the O-PCM is in the amorphous and crystalline phase respectively, as well as the key characteristics for utilization of this Physics for the reconfigurable beam-steering/splitting capabilities and their demonstration with a numerical experiment. We take in these sections the unit cell size to be $a=5$ $\mu$m. In Section 3, in particular, we aim to explore the applicability of an effective medium response for the case when O-PCM is in the amorphous phase. For this purpose only, we span the entire yellow-shaded spectrum of Fig. 2 that is common for the entire 20 deg. to 50 deg. incident-angle range, even though part of it falls below 5 $\mu$m, where the permittivities stated in Section 2.1, may not be as representative. In the subsequent sections, after Section 3, we will restrict the operating-wavelength range between 5 $\mu$m and 6.5 $\mu$m in search of the targeted metagrating behaviors; this regime is fully within the yellow-shaded region of Fig. 2 for all incident angles in the 20 deg. to 50 deg. range, with also the considered O-PCM and GaAs permittivities being representative of the actual material. We note to the reader that we will be using $a=5$ $\mu$m throughout this paper, except for Section 6, where we discuss how we can observe the metagrating functionalities reported in Section 5, in other wavelength regimes as well, which involves scaling the unit-cell size $a$.

In Sections 3–5, we take $d_p=$0.8 $\mu$m and $d_s=$4.2 $\mu$m. We will investigate trends and behaviors of the grating system with O-PCM in amorphous and crystalline phase as we vary slightly the gratings thickness, $b$ around the value of 1 $\mu$m. The latter is significant with respect to the free-space wavelength, yet feasible with reliable switching between the two phases [35,42] for the GSS4T1 O-PCM. In Section 6, we will discuss different parameters with the aim to explore the reported behaviors in other wavelength regimes.

3. Grating response when the O-PCM is in the amorphous phase

We calculate the transmission response of the grating of Fig. 1(a) with the integral equation method [50–52]. We show the results in Fig. 3 for three different incident angles, $\theta _{\textrm {inc}}$= 20 deg., $\theta _{\textrm {inc}}$= 35 deg., and $\theta _{\textrm {inc}}$= 50 deg. The total transmission, $T$, is designated with a black-solid line, while the transmission into the $m=-1$ Bragg channel, $P_{\textrm {t},-1}$, is indicated with the red-solid line. The results indicate that the transmission to the $m=-1$ Bragg channel is zero for most part of the spectrum, except for within a few ultra-narrow spectral regions which correlate to ultra-sharp asymmetric Fano-like features [59] in the total transmission spectrum. Outside these ultra-sharp features, the transmission mostly follows a broad Fabry-Perot-like spectral oscillation, typical of a homogeneous slab, indicating interference of two counter-propagating plane waves coming from the back-and-forth bounces at the interfaces. We discuss below in more detail the Physics underpinning this transmission response.

 

Fig. 3. Transmission response for the amorphous O-PCM grating of Fig. 1(a), for thickness $b=~$1 $\mu$m. The angle of incidence, $\theta _{\textrm {inc}}$, in each case, is designated within each panel. The black-solid line represents the total transmission, $T$, while the red-solid line represents the transmission into the $m=-1$ Bragg channel, $P_{t,-1}$, both calculated with the integral-equation method [50–52]. The blue-dashed line represents the total transmission, $T_{\textrm{eff}}$, assuming the grating behaves as a homogenized medium with the effective permittivity of Eq. (5). The blue-dashed line falls on top of the black line in the areas between the ultra-sharp peaks because of the impressive matching between the effective-medium-theory result and the integral-equation-method calculation.

Download Full Size | PDF

3.1 Transmission features and underlying Physics

We hypothesized in Section 2.2 that the amorphous O-PCM grating will respond as an effectively homogeneous medium, thus transmitting only into the $m=0$ Bragg channel. The results of Fig. 3 suggest that our previous hypothesis is not entirely correct; it appears that the wave still “sees” the grating structure. We assert here, and will establish this later in this section, that we were not wrong in assuming that this low-permittivity contrast grating, having the O-PCM in the amorphous phase, acts as an effectively homogeneous slab. A homogeneous slab always supports both propagating modes and slab-waveguide modes [60]. Now, in actual homogeneous media the waveguide modes are inaccessible by an incident plane-wave source. However, the grating structure, albeit with low-permittivity contrast between the constituents, can still convert an incident plane wave to an evanescent wave, because of the generalized $k_x$ conservation relation at the grating interfaces as given in Eq. (1) [61]. This facilitates coupling to the slab-waveguide-modes which are leaky modes in this case [62]. Then, the interference between the broad Fabry-Perot propagating modes and the discrete slab-waveguide modes give rise to the sharp Fano-resonances in the spectrum [63]. Since these modes are leaky, they can get re-emitted back out into the $m=-1$ Bragg transmitted channel; hence the emergence of the peaks in the $m=-1$ spectrum.

Accordingly, except for the regions of the ultra-narrow Fano features, the transmission is into the $m=0$ Bragg channel in accordance with our original hypothesis. Now we check further that whether indeed the broad Fabry-Perot oscillations we observed in the transmission spectrum come as a result of an effective-medium response. An effective-medium response of a periodic dielectric system is generally expected when the free-space wavelength, $\lambda _0$, is much larger than the unit-cell size, $a$. This is not the case here. For shorter wavelengths, the wave typically propagates inside the structure in the form of a Floquet-Bloch wave [57,64]; this complex wave form is a sum of many plane waves and is generally inconsistent with an effective-medium picture. However, it was shown in Ref. [57] that in special cases where the permittivity contrast between the material constituents of the periodic medium is very low, the Floquet-Bloch sum ends up having only one predominant plane-wave contribution. In other words, the wave propagation inside a periodic medium having constituents with an ultra-low permittivity contrast is similar to the wave propagation in a homogeneous medium.

When the O-PCM is in the amorphous phase the permittivity contrast between the material constituents in the grating is ultra-low; 10.8:10.2 i.e. $\sim 1.06$. So based on the previous findings we discussed above, we can expect a single plane-wave type of propagation even at shorter wavelengths [57] and hence an effective medium response. We proceed in putting the effective-medium hypothesis to test. We apply the normal mixing formula for an effective permittivity, $\varepsilon _{\text {eff}}$, corresponding to cases where the electric field is tangential to the boundary between the two constituents [65]:

3.2 Considerations for practical implementations

The results of Fig. 3 support that we can generally expect for the amorphous O-PCM grating to behave as an effective medium with only the primary beam ($m=0$ Bragg channel) being transmitted. We will go with this expectation, but also check specifically whether indeed for a chosen design the operating wavelength is away from a Fano feature. In other words, for the targeted behavior of the amorphous PCM metagrating we will be operating in wavelengths that are away from the ultra-sharp Fano peaks. We stress that the results of Fig. 3 concern responses to an incident plane wave. Practical implementations involve sources that have a finite extent and generally cannot be expected to have perfect-collimation attributes. Hence, typical practical sources have an angle span around the targeted incident angle. Because of such angle span and because the Fano features are so sharp, we presume the sharp features will not end-up having a strong impact to the transmission response, even if the operating wavelengths ends-up in the vicinity of a Fano feature.

Now the broad Fabry-Perot features also depend on both the thickness, $b$, and angle of incidence $\theta _{\textrm {inc}}$. For the reconfigurable beam manipulation, not only do we need to have just the primary beam ($m=0$) transmitting through the amorphous-phase grating, but also this transmission should be rather high. We further look into how transmission changes with thickness, $b$, across the spectrum of Fig. 3, for the particular case with incident angle of $\theta _{\textrm {inc}}$ of 35 deg., –falling in the middle of the 20 deg. to 50 deg. angle range. The focus here is just to check the background transmission variation, without regard to any ultra-sharp Fano features. So, we perform this calculation using the effective medium response with Yeh’s TMM [66] and show the results in Fig. 4. The results in Fig. 4 confirm that there is a broad spectral range where transmission is high which additionally can be adjusted by slight variations of the thickness $b$ around the 1 $\mu$m value.

 

Fig. 4. Effective-medium transmission, $T_{\textrm {eff}}$, for the amorphous O-PCM grating versus thickness, $b$, and free-space wavelength, $\lambda _0$. The incident angle is $\theta _{\textrm {inc}}$= 35 deg. [Data are plotted in a flat-shading format].

Download Full Size | PDF

4. Grating response when the O-PCM is in the crystalline phase

In this section we investigate our assertion that the metagrating response is triggered when the O-PCM is in the crystalline phase. Here, by metagrating response we mean a response by which the output power in each of the Bragg channels can be dramatically controlled, to the extent that it is even possible to have nearly all-power transmitted via the $m=-1$ Bragg channel with near-zero transmission in the primary ($m=0$) channel. The possibility for this type of a metagrating behavior was previously demonstrated in periodic all-dielectric one-dimensional (1D) gratings by Tsitsas and Valagiannopoulos in Ref. [47]. We discuss below the Physics related to the grating’s response when O-PCM is in the crystalline phase as well as factors and attributes that can affect its metagrating behavior and are relevant to practical implementations for reconfigurable beam manipulation.

4.1 Transmission features and underlying physics

To understand how the grating of Fig. 1(a) responds when the O-PCM is in the crystalline phase we first calculate the transmission spectrum for various thicknesses $b$ around 1 $\mu m$ and incident angles $\theta _{\textrm {inc}}$ around 35 deg., while keeping $d_p$ and $d_s$ constant (as given in Section 2.4). The behavior is very different from what we observed when the O-PCM is in the amorphous phase. In the latter case, discussed in Section 3, the transmission response was qualitatively consistent for different structural parameter for the amorphous O-PCM grating. What we mean is that for the amorphous O-PCM grating, the total transmission had always a broadly varying Fabry-Perot response with some ultra-sharp asymmetric Fano spectral features. In stark contrast, we have not observed such consistent behavior for the transmission response for the case of the crystalline O-PCM grating.

We select to highlight in Fig. 5 two cases that are representative of the varied spectral features we found for the crystalline O-PCM grating. Specifically, we calculate the transmission response with the integral equation method [50–52], for incident angle $\theta _{\textrm {inc}}=$ 35 deg., and two thicknesses: $b=0.90$ $\mu$m [results shown in Fig. 5(a)] and $b=1.15$ $\mu$m [results shown in Fig. 5(b)]. The total transmission, $T$, is plotted with the black-solid line, while the transmissions to the $m=0$ and $m=-1$ Bragg channels, $P_{t,0}$ and $P_{t,-1}$, are shown respectively with a red-dashed and blue-solid line. We observe Fano features [59] in the transmission response, some strongly asymmetric such as PA2, PB1 some with moderate asymmetry, such as PB3, and some almost symmetric like PA1 and PB2. Of the latter near-symmetric ones PA1 appears broader in comparison to PB2. Also for the PA1 feature the associated $P_{t,0}$ dip and $P_{t,-1}$ peak are also near-symmetric. On the other hand, for the PB2 feature the associated $P_{t,-1}$ peak is near-symmetric but the $P_{t,0}$ feature has a distinctly asymmetric Fano profile. For some features, like PA2 or PB1, all three spectra, $T$, $P_{t,0}$ and $P_{t,-1}$ display the distinctly asymmetric Fano profile.

 

Fig. 5. Transmission response of the crystalline O-PCM grating. The total transmission, $T$, the transmission to the $m=0$ Bragg channel, $P_{t,0}$, and the transmission to the $m=-1$ Bragg channel, $P_{t,-1}$, are shown for an angle of incidence , $\theta _{\textrm {inc}}$, of 35 deg. and two different thicknesses, $b$, given in each panel. Some interesting spectral features have been identified and labeled.

Download Full Size | PDF

We attribute this rich spectral response, which changes not only quantitatively but also qualitatively with changing parameters, to the different type of leaky modes that can exist in the high-permittivity-contrast grating, having the O-PCM in the crystalline phase. The crystalline-O-PCM grating can still support leaky guided modes but these would be Floquet-Bloch guided waves [67,68], more similar to surface waves at the interface of a photonic crystal [55] rather than the guided modes of a dielectric slab [60], which were relevant to the amorphous O-PCM grating case. Additionally, we cannot really expect the simple Fabry-Perot effect we observed for the amorphous O-PCM grating case either. Here, the Fabry-Perot effect involves two counter-propagating Floquet-Bloch waves [57,69] rather than two counter-propagating plane waves. It is not surprising that the Floquet-Bloch nature of the waves inside the grating would give rise to these rich features in the transmission spectra. Additionally, it is possible to simultaneously have different Floquet-Bloch waves inside the grating at spectral regions with more than one photonic band solutions [57,70]. Given the complex profile of these modes, de-convolving the dominant underpinning mechanisms is a challenge that may not be met (e.g. coupling to leaky Floquet-Bloch guided modes versus Floquet-Bloch Fabry-Perot interference); in all likelihood most features could be a result of a combination of these effects and several associated modes.

To gain more physical insight we also calculate with the integral equation method [50–52] how the $P_{t,0}$ and $P_{t,-1}$ transmissions change with changing thickness $b$, varying the thickness between $b=0.85$ $\mu$m and $b=1.15$ $\mu$m with a step of $b=0.001$ $\mu$m. We show the results in Fig. 6. We identify a mode branch, we designate in Fig. 6 as B1, which is particularly interesting. Along branch B1 we observe the near-suppression of transmission to the $m=0$ Bragg channel and maximization of the transmission into the $m=-1$ channel. The emergence of the branch B1 in Fig. 6 appears to be an “accidental” effect, changing with the thickness $b$, and bears some similarities with the accidental emergence of bound states in continuum (BIC) that have been reported in several different all-dielectric platforms [71–76].

 

Fig. 6. Transmission response for the crystalline O-PCM grating versus thickness, $b$, and free-space wavelength, $\lambda _0$. The transmission into the $m=0$ Bragg channel, $P_{t,0}$, is shown in the left-panel, while the transmission into the $m=-1$ Bragg channel, $P_{t,-1}$, is shown in the right panel. The incident angle is $\theta _{\textrm {inc}}$= 35 deg. We identify a branch exhibiting a near-zero transmission into the $m=0$ Bragg channel and maximized transmission into the $m=-1$ Bragg channel and designate this as B1. [Data are plotted in a flat-shading format].

Download Full Size | PDF

BICs are bound states existing within the radiation continuum but whose radiation is completely suppressed to any channel due to destructive interference between two leaky modes [71,75], known as the Friedrich-Wintgen scenario to BIC [77]. It is an accidental effect in the sense that it is not controlled by a special symmetry of a leaky-mode, as in symmetry-protected BIC [71], but emerges “accidentally” from interference for certain parameters. BIC has been also identified in systems supporting many leaky modes as a result of interference between a dominant pair of them within a certain spectral region [76]. True BIC modes are not accessible by plane waves, since they are purely dark modes with an infinite quality factor, Q [78]. Practically, only modes that are close to BIC modes are observed and these have a high quality factor and are typically referred to as quasi-BIC [72]. Very recently, it was found that quasi-BIC evolve with changing parameters from Fano resonances with the respective asymmetry parameter collapsing resulting in a quasi-symmetric spectral response [79].

To get a more complete picture we also calculate with the integral equation method [50–52] how the transmissions $P_{t,0}$ and $P_{t,-1}$ change with changing incident angle, $\theta _{\textrm {inc}}$, for thickness $b$=0.90 $\mu$m. We show the results in Fig. 7 (top panels). We observe the same branch here B1, where transmission is nearly-suppressed to the $m=0$ Bragg channel and peaking for the $m=-1$ Bragg channel. But we also observe here a second branch we designate as B2, that appears “interrupted” around $\theta _{\textrm {inc}}\sim$ 35 deg. with the $P_{t,-1}$ appearing near-zero (see area around the dashed circle). To understand this “interrupted” branch better, we also show the corresponding reflections for the $m=0$ and $m=-1$ Bragg channels, $P_{r,0}$ and $P_{r,-1}$, in the bottom panels of Fig. 7. The reflection in the “interrupted” part of branch B2 for the $m=-1$ Bragg channel is actually high (see also more detailed plot in Fig. 13 Appendix B zoomed around this area). This means that the area around the dashed-line circle in branch B2 becomes a dark mode, that does not radiate neither to the $m=0$ nor the $m=-1$ Bragg channel, i.e. becomes a BIC.

 

Fig. 7. Crystalline O-PCM grating response versus incident angle $\theta _{\textrm {inc}}$ and free-space wavelength, $\lambda _0$, for thickness $b$=0.90 $\mu$m. Transmission into the $m=0$ Bragg channel, $P_{t,0}$, and into the $m=-1$ Bragg channel, $P_{t,-1}$, are shown respectively in the top left and right panels. The branch observed in Fig. 6 is identified here as well and designated with B1. The corresponding reflection results are also shown in the bottom left and right panels. An additional “interrupted” branch is observed (designated as B2) that passes through a BIC (designated with a dashed-white circle). [Data are plotted in a flat-shading format].

Download Full Size | PDF

A similar evolution from Fano resonances to BIC with angle of incidence has been previously reported for a plasmonic-dielectric hybrid systems [80]. As in Ref. [80] we also observe here the characteristic anti-crossing of branches B1 and B2, with also branch B2 having a distinctly higher associated quality factor, Q. This behavior is a signature of the Friedrich-Wintgen interference between two leaky modes [76]. This means that the transmission suppression into the $m=0$ Bragg channel is an accidental effect coming from destructive interference between the leaky modes supported in the crystalline O-PCM grating. It appears this accidental effect coincides with the accidental constructive interference for radiation into the $m=-1$ Bragg channel, yielding a high transmission to this channel. In other words, we argue that the transmission re-direction into only the $m=-1$ Bragg channel is an accidental effect emanating from a Friedrich-Wintgen interference between two dominant leaky modes that are supported by the crystalline O-PCM grating in this spectral region.

Now, that we understood how these features emerge in the crystalline O-PCM grating’s transmission response we will further analyze in the following what characteristics are important to the practical implementation of the reconfigurable beam manipulation.

4.2 Metagrating behaviors and practical considerations

4.2.1 Metagrating beam back-bending behavior

For the reconfigurable beam steering functionality, as depicted in Fig. 1(b), the crystalline O-PCM grating needs to exhibit a beam back-bending behavior, which can occur when nearly all transmitted power is directed into the $m=-1$ Bragg channel. We first check the degree of the spectral coincidence between the transmission suppression into the $m=0$ Bragg channel and the transmission peaking into the $m=-1$ Bragg channel. The more aligned the respective transmission minima and maxima are, the higher the quality of the back-bending phenomenon exhibited by the crystalline O-PCM grating will be. We show in Fig. 8 the spectral location of the $m=0$ Bragg channel transmission minima and the $m=-1$ Bragg channel transmission maxima versus angle, for the B1 and B2 branches designated in Fig. 7. We also show these versus thickness, $b$, for the branch B1 designated in Fig. 6. We note, the “interrupted” region of branch B2, is omitted in Fig. 8, because of the change of behavior within this branch that signifies the emergence of a BIC. Figure 8 confirms that the aforementioned minima/maxima are almost coinciding, with a tiny, yet identifiable, detuning between the two, which varies slightly with varying incident angle and thickness.

 

Fig. 8. Location of the free-space wavelength, $\lambda _0$, as a function of the incident angle, $\theta _{\textrm {inc}}$ (left panel) and thickness $b$ (right panel) for the minimum (maximum) of the transmission to the $m=0$ ($m=-1$) Bragg channels, $P_{t,0}$ ($P_{t,-1}$). This is shown with a black-solid (red-dashed) line for branch B1 and with a green-solid (blue-dotted) line for branch B2. The region around the BIC state for branch B2, where behavior within the branch changes, is omitted.

Download Full Size | PDF

The aforementioned detuning comes into play in two ways. Firstly, it influences the spectral profile of the total transmission. We found that along the branches B1, B2 the Fano profiles for $P_{t,0}$ and $P_{t,-1}$ tend to be quasi-symmetric or with only a low asymmetry. The total transmission however appears in some cases quasi-symmetric but in others it does show a distinctly asymmetric Fano response. This asymmetry is a compounded result of the degree of asymmetries in the underlying $P_{t,0}$ dip and $P_{t,-1}$ peak but also of their mutual detuning. The detuning appears to have a stronger impact along branch B2 where the $P_{t,0}$ dip and $P_{t,-1}$ peak are sharper. Practically, the more spectrally symmetric the total transmission is, with the underlying $P_{t,0}$ dip and $P_{t,-1}$ peaks being as little detuned as possible, the better. This is because in these type of cases the response will be less sensitive to the degree of monochromaticity and angle span of the input source.

Therefore, we focus from now on only on branch B1 which is broader, i.e. has a lower Q-factor and yields a more symmetric total transmission peak. The detuning is important here as well because it influences how strong the intensity of the back-bent beam can be. Specifically, for the beam back-bending phenomenon the transmission into the $m=0$ Bragg channel needs to be near-zero. Accordingly, what becomes practically relevant is not the maximum value of $P_{t,-1}$ [black solid line in Fig. 9], but rather the value of $P_{t,-1}$ at the wavelength where $P_{t,0}$ takes the minimum value. The latter is shown with the red-dashed line in Fig. 9 and shows fluctuations, which come from the fact that the relevant $(\theta _{\textrm {inc}}, \lambda _0)$ and $(b, \lambda _0)$ parameter spaces were swept in a step-wise fashion for the $P_{t,0}$ and $P_{t,-1}$ transmission calculations rather than continuously. In other words, $P_{t,0}$ and $P_{t,-1}$ values are available at discrete value sets $(\theta _{\textrm {inc}}, \lambda _0)$ and $(b, \lambda _0)$. We have added in Fig. 9 a corresponding fit (blue line) as a guide to the eye through the fluctuating red-line curves. We observe that the maximum power output in the back-bent direction, when there is the least power leaking out at the primary Bragg channel ($m=0$) is $\sim 80\%$, a bit reduced from its maximum value of $\sim 90\%$.

 

Fig. 9. (a) The black-solid line shows the transmission into the $m=-1$ Bragg channel, $P_{t,-1}$, versus the incident angle $\theta _{\textrm {inc}}$ at its maximum (in branch B1). The red-dashed line shows $P_{t,-1}$ at the wavelength where $P_{t,0}$ takes the minimum value. The blue line represents a corresponding fit to guide the eye of the reader. (b) Same as in (a) but versus the thickness $b$. Note: In (a) a piece-wise fit is performed for angles smaller and greater than 35 deg. respectively.

Download Full Size | PDF

4.2.2 Metagrating beam-splitting behavior

For the reconfigurable beam splitting functionality, as depicted in Fig. 1(c), the crystalline O-PCM grating needs to exhibit a beam splitting behavior, which can occur when the transmitted power is evenly split between the $m=0$ the $m=-1$ Bragg channels. For this behavior as well, sharp features would give rise to a highly sensitive response to the monochromaticity and angle span of the input source and should be avoided. For the beam splitting behavior the relevant free-space wavelength of operation is where the $P_{t,0}$ and $P_{t,-1}$ transmission spectra cross-over. At the cross-over point the $P_{t,0}$, $P_{t,-1}$ values should be high. Broader and more aligned peaks are more conducive to a robust response of the platform with higher $P_{t,0}$ and $P_{t,-1}$ values.

4.3 Metagrating design choice

Based on Fig. 9 we will choose the design with thickness $b=$0.90 $\mu$m and for incident angle $\theta _{\textrm {inc}}=35$ deg. The remaining structural parameters are as stated in Section 2, i.e, $a=$ 5 $\mu$m, $d_p=$ 0.8 $\mu$m and $d_s=$ 4.2 $\mu$m. These are the parameters that yield the maximum power in the back-bent Bragg channel with $m=-1$ while the power in the primary channel is near-zero. Figure 7 and 8 also suggests that an incident angle of 35 deg. would be more robust with respect to the angle-span of the source, because the peak wavelength is at a stationary point. Furthermore, cases where $P_{t,0}$ becomes abruptly large after its minimum should also be avoided as these would also give too highly sensitive a response to the angle span of the source. Figure 7 also attests that $P_{t,0}$ is relatively low outside its minimum value. Because of relatively lower-Q and broader response this case would also be suitable for the beam splitting behavior.

In Fig. 10, we show the corresponding crystalline O-PCM grating response for the chosen design parameters; the $P_{t,0}$ and $P_{t,-1}$ transmission responses are shown, calculated with the integral equation method [50–52]. We have zoomed the response around the spectral regime of the operating free-space wavelengths, which are designated with solid circles for the reconfigurable beam steering phenomenon and with a solid square for the reconfigurable beam splitting phenomenon. We also show the integral-equation-method calculations for the corresponding amorphous O-PCM grating; these confirm that near-perfect transmission to the primary beam only is expected when the O-PCM is in the amorphous phase.

 

Fig. 10. Transmission responses of the chosen metagrating paradigm design ($b=$ 0.90 $\mu$m, $a=$ 5 $\mu$m, $d_p=$ 0.8 $\mu$m and $d_s=$4.2 $\mu$m; incident angle $\theta _{\textrm {inc}}=35$ deg) for the demonstration of the reconfigurable beam manipulation. The graph legends designate the quantities plotted. The solid circles denote the operating free-space wavelength for the reconfigurable beam steering. Conversely, the solid square denotes the operating free-space wavelength for the reconfigurable beam splitting.

Download Full Size | PDF

In the following we will implement the chosen metagrating paradigm in a numerical experiment to demonstrate the reconfigurable beam steering and splitting functionalities.

5. Verification of the reconfigurable beam manipulation with a first-principles numerical experiment

5.1 Numerical experiment

We employ the Finite Difference Time Domain (FDTD) method [53], that solves Maxwell’s equations in space and time, with perfect-matched-layer (PML) boundary conditions [81,82]. A slanted line source is implemented in FDTD with a Gaussian spatial profile and a beam waist of $\sim 25 a$, with $a$ being the grating’s unit cell size. This is a two-directional source emitting normally to the slanted line in both directions. The source is implemented as a soft source [53], that is transparent to fields arriving at its location. The simulation is performed in the 2D plane, $xz$, with translational symmetry assumed in the $y$-direction. The 2D space is discretized with a rectangular grid with sides $\Delta x=a/100$ and $\Delta z=a/72$ along the $x$-direction and $z$-direction respectively, so that a source yielding light propagation at 35 deg. with the normal to the metagrating surface can be implemented. Accordingly, the metagrating’s thickness, $b$, was approximated to the value of 0.90278 $\mu$m. The time step, $\Delta t$, is taken equal to $\sim$ 0.0073 $a/c$, satisfying the Courant condition for numerical stability [53]. The source emits a quasi-monochromatic signal varying in time, $t$, as $f(t) \cos \frac {2 \pi c}{\lambda _0} t$, with $\lambda _0$ being the operating wavelength. $f(t)$ is a slow-varying function that slowly ramps up the source’s amplitude to its maximum value [57,83] to avoid inducing noise in the simulation by a sudden turn on of the source [53].

In order to show the reconfigurable beam manipulation phenomena, the quantity that will be plotted is the normalized time-averaged electric-field intensity, $\overline I(x,z)$, given by:

Because FDTD is solved on a grid, and because of the small difference between intended and numerical thickness $b$, a small offset is expected between the theoretical operational wavelengths as designated in Fig. 10 and the numerical FDTD ones. Although broader, the features in the domain of interest are still quite sharp. For this reason, we run the FDTD numerical experiment for many different operating wavelengths, $\lambda _0$, in order to locate the one where the predicted reconfigurability phenomena occur. We do this for the crystalline O-PCM grating. Once the operating free-space wavelength is determined in the FDTD numerical experiment with the crystalline O-PCM grating, we run the simulation for the amorphous O-PCM grating as well at that free-space wavelength. As seen in Fig. 10 the amorphous O-PCM grating response does not change much in the interval between 5.7 $\mu$m to 5.85 $\mu$m.

Actually, we have spanned through the entire spectrum, 5.1 $\mu$m to 6.4 $\mu$m, performing quasi-monochromatic FDTD simulations at many free-space wavelengths at coarser or finer intervals in regions of broader or finer features respectively. In Table 1 we give the free-space wavelength intervals at which separate FDTD simulations were run within the different spectral regions. The respective FDTD results are included in the form of an animation, designated as Visualization 1. The free-space wavelength at each frame, $\lambda _0$, is designated with a solid circle on the theoretical transmission spectra for the $m=0$ and $m=-1$ Bragg beams, i.e., $P_{t,0}$ and $P_{t,-1}$ respectively, calculated from the integral equation semi-analytical method [50–52]. The colormap of the FDTD results at each frame is the same as the colormap of Figs. 11 and 12. Note, the observed response of the crystalline O-PCM metagrating in the FDTD numerical experiment is red-shifted with respect to the theoretical expectation for about $0.10-0.15\%$.

 

Fig. 11. The reconfigurable beam steering phenomenon, observed in the FDTD numerical experiment at $\lambda _0=5.795$ $\mu$m. The normalized timed-averaged electric-field intensity is plotted. $t_0$ and $t_{-1}$ designate the $m=0$ and $m=-1$ transmitted Bragg channels. The dashed line represents the line of the source in FDTD, while the thin white rectangle is the grating structure.

Download Full Size | PDF

Table 1. The intervals, of free-space wavelength, $\Delta \lambda _0$, at which a separate FDTD numerical experiment is performed for the crystalline O-PCM grating within the different spectral regimes that are designated in the first column. The associated results for the normalized time-averaged electric field intensity can be seen in the corresponding animation, Visualization 1.

View Table

In the sub-sections that follow we show the FDTD results only at the operational wavelength where we observed the predicted reconfigurability phenomena.

5.2 Verification of the reconfigurable beam steering in the FDTD experiment

We observed a near-complete back-bending of the transmitted beam in the FDTD for the crystalline O-PCM grating for an operating wavelength of $\lambda _0=$ 5.795 $\mu$m. We show these FDTD results for the normalized timed-averaged electric-field intensity, $\overline I(x,z)$, in Fig. 11 (bottom panel). The corresponding theoretical prediction for the operating wavelength from the integral equation method results, as also seen in Fig. 10, was $\lambda _0=$ 5.787 $\mu$m. The tiny offset of $\sim 0.15\%$ is due to numerical grid effects as we discussed in Section 5.1. In the top panel of Fig. 11 we show the corresponding normalized timed-averaged electric-field intensity, $\overline I(x,z)$, for the same operating wavelength but for the grating having the O-PCM in the amorphous phase. Figure 11 attests that by switching the O-PCM material from the amorphous to the crystalline phase the beam steers into the opposite direction with respect to the grating’s surface normal.

Aside from griding effects we stress that the FDTD result is also influenced by the unavoidable angle span of the Gaussian source. This is a condition that not only affects the numerical experiment but will be present in real set-ups as well; only perfect collimation conditions can emulate a plane-wave source. Moreover, as discussed in Section 5.1, the source in FDTD is two-directional and emitting a propagating wave normal to its line. In other words the source is emitting a propagating wave normal to its line and along the +z direction, in the region below the dashed line, and along -z direction, in the region above the dashed line. Thus, there will be interference between the source field and the reflected beam of the $m=-1$ Bragg channel for the crystalline O-PCM grating, as this reflected channel exists for this case; and indeed this is what we observe in Fig. 11.

5.3 Verification of the reconfigurable beam splitting in the FDTD experiment

We observed a near-even beam split in transmission in the FDTD for an operating wavelength of $\lambda _0=5.82$ $\mu$m and show the result for the normalized timed-averaged electric-field intensity, $\overline I(x,z)$, in Fig. 12 (bottom panel). The theoretical prediction from the integral equation results for the phenomenon, as seen in Fig. 10, was at $\lambda _0=5.8135$ $\mu$m. The tiny offset of $\sim 0.10\%$ is due to numerical grid effects as we discussed in Section 5.1. Additionally, we show the normalized timed-averaged electric-field intensity from the FDTD simulation at the same operating wavelength but for the amorphous O-PCM grating in the top panel of Fig. 12. Figure 12 attests that by switching the O-PCM material from the amorphous to the crystalline phase the beam splits into two parts, each propagating at opposite sides of the grating interface normal. Like in the case of Fig. 11, the observed interference above and below the source line for the crystalline O-PCM grating case is due to the $m=-1$ Bragg reflected beam. As also in the case of Fig. 12, aside from numerical griding effects, the result here is also influenced by the angle span of the Gaussian source.

 

Fig. 12. Same as in Fig. 11 but at operating wavelength $\lambda _0=$ 5.82 $\mu$m where we observe the reconfigurable beam splitting phenomenon.

Download Full Size | PDF

6. Tunability of the operating wavelength for the reconfigurable beam manipulation phenomena

The metagrating paradigm is a fully-dielectric system with a frequency-independent permittivity for all constituents in their idealized form. Maxwell’s equations are scalable for such fully dielectric systems. Specifically, for these systems it can be shown [55] that if a certain behavior is observed at an operating wavelength $\lambda _{0,1}$ but is desired for another operating wavelength $\lambda _{0,2}= s \lambda _{0,1}$, then such behavior can be achieved at wavelength $\lambda _{0,2}$ by just considering a new structure with all its structural features multiplied by a factor $s$.

Hence, the spectral response for the metagrating with the structural parameters of Fig. 5(a) can be replicated exactly with another metagrating for another spectral region when its features are being suitably scaled. For example, if we multiply all grating structural parameters by a factor of $s=1.3$ we will get the identical response but each feature will be observed at a longer wavelength, $\times$ 1.3 the original wavelength. We demonstrate this in Fig. 14 in Appendix C. In practice however, because there is still some frequency variation in the permittivity the transformation to another spectrum would not be as exact. But this recipe demonstrates the powerful capability to move the operating wavelength relatively easily within the 5 $\mu$m to 10 $\mu$m spectral region, as desired for different applications. This is an important advantage of our proposed all-dielectric O-PCM reconfigurable paradigm. On the contrary, a metal-based O-PCM platform would require re-designing from the beginning the system for functionality at another operating wavelength.

7. Conlusions and outlook

We have proposed here a new all-dielectric metagrating paradigm for reconfigurable photonics in the 5 $\mu$m to 10 $\mu$m spectral range. The functionality of this platform relies on the lossless dielectric properties with a high permittivity shift between phases of a newly discovered O-PCM, GSS4T1 [35]. We have predicted theoretically via the integral equation method [50–52] and verified with the FDTD [53,81,82] numerical experiment that this platform can exhibit both reconfigurable beam steering and beam splitting. Our proposed platform’s non-volatile reconfigurable capabilities arise from the distinctly different physical responses of the grating structure when the constituent O-PCM is switched between the two phases, amorphous and crystalline.

Specifically, we have observed an effective-medium response for the amorphous O-PCM grating; hence an incident beam transmits through without any deflection. On the other hand, the crystalline O-PCM grating supports leaky Floquet-Bloch modes [57,64,70] which interfere via a Friedrich-Wintgen scenario [77] giving rise to a BIC mode [71–76,79,80]. As it is typical in Friedrich-Wintgen type of interference effects [76,80], we found the BIC mode is accompanied by a lower-Q modal branch. In our system, modes on this lower-Q branch exhibit simultaneously a constructive interference into the $m=-1$ output Bragg channel and a destructive interference into the $m=0$ output Bragg channel. The lower-Q characteristic makes these interference effects more robust for their numerical or experimental realization. We demonstrated here with an FDTD numerical experiment a negative beam steering effect and a beam-splitting effect for the crystalline O-PCM grating based on a low-Q mode, resulting from the Friedrich-Wintgen type of interference.

Owing to the all-dielectric properties of our proposed metagrating paradigm, the operating wavelength of the demonstrated reconfigurable beam manipulation phenomena can be easily tuned across the 5 $\mu$m to 10 $\mu$m spectrum by properly scaling the metagrating’s feature sizes. We believe our reconfigurable platform can benefit a range of MWIR/LWIR devices and applications, such as programmable integrated photonics [1,6,31,38], lidar systems [39], infrared spectrometers [14] and others. Beyond the relevance to infrared applications we believe our reported findings regarding the Friedrich-Wintgen interference effects that are delineated for the individual Bragg channels can offer transferable insights for quasi-BIC phenomena and associated metasurface design [75,80] relevant to other application domains such as biosensing [84] and higher-harmonic generation phenomena [85–88].

Appendix A: Permittivity responses of metagrating constituents

Permittivity responses of GSS4T1 in crystalline and amorphous phases

The permittivity values of the crystalline and amorphous phase of GSS4T1, $\varepsilon _{\textrm {crys}}$ and $\varepsilon _{\textrm {amor}}$ respectively, are almost constant throughout the 5$\mu$m-10$\mu$m free-space wavelength range with only a slight variation. The values taken in this paper are a wavelength-weighted average in the 5 $\mu$m to 10 $\mu$m range (since available data are not at equal free-space wavelength intervals). These, have been calculated from the optical data of Ref. [35] as follows:

(7)$$\varepsilon_{\textrm{crys}}=\frac{\displaystyle \sum_{j=1}^N \lambda_j n_{{\textrm{crys}},j}^2}{\displaystyle \sum_{j=1}^N \lambda_j},$$

and

(8)$$\varepsilon_{\textrm{amor}}=\frac{\displaystyle \sum_{j=1}^N \lambda_j n_{{\textrm{amor}},j}^2}{\displaystyle \sum_{j=1}^N \lambda_j},$$

where $n_{{\textrm {crys}},j}$ ($n_{{\textrm {amor}},j}$) represents the refractive index (real part) of crystalline (amorphous) GSS4T1 at wavelength $\lambda _j$. The wavelengths $\lambda _1$, $\lambda _N$ represent the data-point wavelengths from the Ref. [35] data being respectively just above 5 $\mu$m and just below 10 $\mu$m. We note the ultra-small imaginary part of the refractive index in the experimental optical data is not taken into account in the calculation above since it is at the level of experimental noise [89].

Permittivity responses of GaAs

The permittivity response of GaAs, $\varepsilon _{\textrm {s}}$ remains almost constant throughout the 5 $\mu$m to 10 $\mu$m free-space wavelength range with only a slight variation. The value taken in this paper is the wavelength-weighted average calculated based on the optical data from Skauli et al. for undoped semi-insulating GaAs [90] (dataset available from Ref. [91]) as follows:

(9)$$\varepsilon_{\textrm{s}}=\frac{\displaystyle \sum_{j=1}^N \lambda_j \varepsilon_{{\textrm{s}},j}}{\displaystyle \sum_{j=1}^N \lambda_j},$$

where $\varepsilon _{{\textrm {s}},j}$ represents permittivity of GaAs at wavelength $\lambda _j$ from the Skauli et al. data [90,91]. The wavelengths $\lambda _1$, $\lambda _N$ represent the data-point wavelengths from the Skauli et al. data that are respectively just above 5 $\mu$m and just below 10 $\mu$m.

Appendix B: Angular and spectral transmission and reflection response of the crystalline O-PCM metagrating around the BIC state

 

Fig. 13. Same as in Fig. 7, but showing calculations with finer $\theta _{\textrm {inc}}$ and free-space wavelength, $\lambda _0$ step with their range focusing only around the region of the “interrupted” branch B2 that is correlated to a BIC state (region designated with dashed-line ellipse). [Data are plotted in a flat-shading format].

Download Full Size | PDF

Appendix C: Demonstrating tunability of the metagrating paradigm’s response to different spectral regions as desired

 

Fig. 14. The top panel is the crystalline O-PCM gratings response with the parameters of Fig. 5(a), calculated with the integral equation method [50–52]. The bottom panel is the corresponding calculation with the same method but for a corresponding grating with all the structural parameters multiplied by a factor of 1.3.

Download Full Size | PDF

Acknowledgments

The authors thank Prof. Juejun Hu (MIT) for kindly providing the optical data of the GSS4T1 material corresponding to Ref. [35] and useful discussions regarding its properties, which enabled this work. SF thanks the UNM Center for Advanced Research Computing, supported in part by the National Science Foundation, for providing the high performance computing resources used in some of the computations of this work.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding author upon reasonable request.

References

1. R. E. Simpson, J. K. W. Yang, and J. Hu, “Are phase change materials ideal for programmable photonics?: opinion,” Opt. Mater. Express 12(6), 2368–2373 (2022). [CrossRef]  

2. N. Yamada, E. Ohno, K. Nishiuchi, N. Akahira, and M. Takao, “Rapid-phase transitions of GeTe-Sb₂Te₃ pseudobinary amorphous thin films for an optical disk memory,” J. Appl. Phys. 69(5), 2849–2856 (1991). [CrossRef]  

3. M. L. Gallo and A. Sebastian, “An overview of phase-change memory device physics,” J. Phys. D: Appl. Phys. 53(21), 213002 (2020). [CrossRef]  

4. W. Zhang, R. Mazzarello, and E. Ma, “Phase-change materials in electronics and photonics,” MRS Bull. 44(09), 686–690 (2019). [CrossRef]  

5. https://en.wikipedia.org/wiki/3D_XPoint.

6. J. Faneca, I. Zeimpekis, S. T. Ilie, T. D. Bucio, K. Grabska, D. W. Hewak, and F. Y. Gardes, “Towards low loss non-volatile phase change materials in mid index waveguides,” Neuromorph. Comput. Eng. 1(1), 014004 (2021). [CrossRef]  

7. J. Faneca, S. G.-C. Carrillo, E. Gemo, C. R. de Galarreta, T. D. Bucio, F. Y. Gardes, H. Bhaskaran, W. H. P. Pernice, C. D. Wright, and A. Baldycheva, “Performance characteristics of phase-change integrated silicon nitride photonic devices in the O and C telecommunications bands,” Opt. Mater. Express 10(8), 1778–1791 (2020). [CrossRef]  

8. M. Rudé, R. E. Simpson, R. Quidant, V. Pruneri, and J. Renger, “Active control of surface plasmon waveguides with a phase change material,” ACS Photonics 2(6), 669–674 (2015). [CrossRef]  

9. Y. Zhang, Q. Zhang, C. Ríos, M. Y. Shalaginov, J. B. Chou, C. Roberts, P. Miller, P. Robinson, V. Liberman, M. Kang, K. A. Richardson, T. Gu, S. A. Vitale, and J. Hu, “Transient tap couplers for wafer-level photonic testing based on optical phase change materials,” ACS Photonics 8(7), 1903–1908 (2021). [CrossRef]  

10. Y. Zhang, C. Ríos, M. Y. Shalaginov, S. An, C. Fowler, J. B. Chou, C. M. Roberts, V. Liberman, S. Vitale, M. Kang, K. A. Richardson, C. Rivero-Baleine, T. Gu, H. Zhang, and J. Hu, “Optical phase-change materials (O-PCMS) for reconfigurable photonics,” in Asia Communications and Photonics Conference/International Conference on Information Photonics and Optical Communications 2020 (ACP/IPOC), (Optica Publishing Group, 2020), p. T2D.1.

11. Z. Fang, R. Chen, J. Zheng, and A. Majumdar, “Non-volatile reconfigurable silicon photonics based on phase-change materials,” IEEE J. Sel. Top. Quantum Electron. 28(3), 1–17 (2022). [CrossRef]  

12. T. Y. Teo, M. Krbal, J. Mistrik, J. Prikryl, L. Lu, and R. E. Simpson, “Comparison and analysis of phase change materials-based reconfigurable silicon photonic directional couplers,” Opt. Mater. Express 12(2), 606–621 (2022). [CrossRef]  

13. T. Martins, B. Gholipour, D. Piccinotti, K. F. MacDonald, A. C. Peacock, O. Frazao, and N. I. Zheludev, “Fiber-integrated phase-change reconfigurable optical attenuator,” APL Photonics 4(11), 111301 (2019). [CrossRef]  

14. V. Liberman, Y. Zhang, M. Shalaginov, C. Rios, P. Robinson, C. Roberts, K. Tibbetts, M. Kang, K. Richardson, J. Hu, and J. B. Chou, “Reconfigurable infrared flat optics with novel phase change materials,” in OSA Advanced Photonics Congress (AP) 2019 (IPR, Networks, NOMA, SPPCom, PVLED), (Optica Publishing Group, 2019), p. NoW3B.2.

15. J. Zheng, A. Khanolkar, P. Xu, S. Colburn, S. Deshmukh, J. Myers, J. Frantz, E. Pop, J. Hendrickson, J. Doylend, N. Boechler, and A. Majumdar, “GST-on-silicon hybrid nanophotonic integrated circuits: a non-volatile quasi-continuously reprogrammable platform,” Opt. Mater. Express 8(6), 1551–1561 (2018). [CrossRef]  

16. Y. Huang, Y. Shen, C. Min, and G. Veronis, “Switching of the direction of reflectionless light propagation at exceptional points in non-pt-symmetric structures using phase-change materials,” Opt. Express 25(22), 27283–27297 (2017). [CrossRef]  

17. A. Karvounis, B. Gholipour, K. F. MacDonald, and N. I. Zheludev, “All-dielectric phase-change reconfigurable metasurface,” Appl. Phys. Lett. 109(5), 051103 (2016). [CrossRef]  

18. C. R. de Galarreta, I. Sinev, A. M. Alexeev, P. Trofimov, K. Ladutenko, S. G.-C. Carrillo, E. Gemo, A. Baldycheva, J. Bertolotti, and C. D. Wright, “Reconfigurable multilevel control of hybrid all-dielectric phase-change metasurfaces,” Optica 7(5), 476–484 (2020). [CrossRef]  

19. O. Hemmatyar, S. Abdollahramezani, T. Brown, and A. Adibi, “Phase-change material micro-displays,” in Conference on Lasers and Electro-Optics, (Optica Publishing Group, 2021), p. JTu2P.6.

20. O. Hemmatyar, S. Abdollahramezani, I. Zeimpekis, S. Lepeshov, A. Krasnok, A. I. Khan, K. M. Neilson, C. Teichrib, T. Brown, E. Pop, D. W. Hewak, M. Wuttig, A. Alu, O. L. Muskens, and A. Adibi, “Enhanced meta-displays using advanced phase-change materials,” https://arxiv.org/abs/2107.12159.

21. P. Hosseini, C. D. Wright, and H. Bhaskaran, “An optoelectronic framework enabled by low-dimensional phase-change films,” Nature 511(7508), 206–211 (2014). [CrossRef]  

22. S. G. Carrillo, L. Trimby, Y. Au, V. K. Nagareddy, G. Rodriguez-Hernandez, P. Hosseini, C. Ríos, H. Bhaskaran, and C. D. Wright, “A nonvolatile phase-change metamaterial color display,” Adv. Opt. Mater. 7(18), 1801782 (2019). [CrossRef]  

23. Y. Zhang, C. Fowler, J. Liang, B. Azhar, M. Y. Shalaginov, S. Deckoff-Jones, S. An, J. B. Chou, C. M. Roberts, V. Liberman, M. Kang, C. Ríos, K. A. Richardson, C. Rivero-Baleine, T. Gu, H. Zhang, and J. Hu, “Electrically reconfigurable non-volatile metasurface using low-loss optical phase-change material,” Nat. Nanotechnol. 16(6), 661–666 (2021). [CrossRef]  

24. N. Youngblood, C. Ríos, E. Gemo, J. Feldmann, Z. Cheng, A. Baldycheva, W. H. P. Pernice, C. David Wright, and H. Bhaskaran, “Tunable Volatility of Ge₂Sb₂Te₅ in Integrated Photonics,” Adv. Funct. Mater. 29(11), 1807571 (2019). [CrossRef]  

25. A. U. Michel, A. Heßler, S. Meyer, J. Pries, Y. Yu, T. Kalix, M. Lewin, J. Hanss, A. D. Rose, T. W. W. Maß, M. Wuttig, D. N. Chigrin, and T. Taubner, “Advanced optical programming of individual meta-atoms beyond the effective medium approach,” Adv. Mater. 31(29), 1901033 (2019). [CrossRef]  

26. J. Schröder, M. A. F. Roelens, L. B. Du, A. J. Lowery, S. Frisken, and B. J. Eggleton, “An optical FPGA: reconfigurable simultaneous multi-output spectral pulse-shaping for linear optical processing,” Opt. Express 21(1), 690–697 (2013). [CrossRef]  

27. A.-K. U. Michel, S. Meyer, N. Essing, N. Lassaline, C. R. Lightner, S. Bisig, D. J. Norris, and D. N. Chigrin, “The potential of combining thermal scanning probes and phase-change materials for tunable metasurfaces,” Adv. Opt. Mater. 9(2), 2001243 (2021). [CrossRef]  

28. C. Ríos, M. Stegmaier, P. Hosseini, D. Wang, T. Scherer, C. D. Wright, H. Bhaskaran, and W. H. P. Pernice, “Integrated all-photonic non-volatile multi-level memory,” Nat. Photonics 9(11), 725–732 (2015). [CrossRef]  

29. N. Peserico, T. F. de Lima, P. Prucnal, and V. J. Sorger, “Emerging devices and packaging strategies for electronic-photonic AI accelerators: opinion,” Opt. Mater. Express 12(4), 1347–1351 (2022). [CrossRef]  

30. B. J. Shastri, A. N. Tait, T. F. de Lima, W. H. P. Pernice, H. Bhaskaran, C. D. Wright, and P. R. Prucnal, “Photonics for artificial intelligence and neuromorphic computing,” Nat. Photonics 15(2), 102–114 (2021). [CrossRef]  

31. T. Y. Teo, X. Ma, E. Pastor, H. Wang, J. K. George, J. K. W. Yang, S. Wall, M. Miscuglio, R. E. Simpson, and V. J. Sorger, “Programmable chalcogenide-based all-optical deep neural networks,” Nanophotonics 0 (2022).

32. J. Feldmann, M. Stegmaier, N. Gruhler, C. Ríos, H. Bhaskaran, C. D. Wright, and W. H. P. Pernice, “Calculating with light using a chip-scale all-optical abacus,” Nat. Commun. 8(1), 1256 (2017). [CrossRef]  

33. C. Lian, C. Vagionas, T. Alexoudi, N. Pleros, N. Youngblood, and C. Rios, “Photonic (computational) memories: tunable nanophotonics for data storage and computing,” Nanophotonics Advanced Online Publication https://doi.org/10.1515/nanoph-2022-0099 (2022).

34. S. Gyger, J. Zichi, L. Schweickert, A. W. Elshaari, S. Steinhauer, S. F. C. da Silva, A. Rastelli, V. Zwiller, K. D. Jöns, and C. Errando-Herranz, “Reconfigurable photonics with on-chip single-photon detectors,” Nat. Commun. 12(1), 1408 (2021). [CrossRef]  

35. Y. Zhang, J. B. Chou, J. Li, H. Li, Q. Du, A. Yadav, S. Zhou, M. Y. Shalaginov, Z. Fang, H. Zhong, C. Roberts, P. Robinson, B. Bohlin, C. Ríos, H. Lin, M. Kang, T. Gu, J. Warner, V. Liberman, K. Richardson, and J. Hu, “Broadband transparent optical phase change materials for high-performance nonvolatile photonics,” Nat. Commun. 10(1), 4279 (2019). [CrossRef]  

36. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]  

37. C. R. de Galarreta, S. G.-C. Carrillo, Y.-Y. Au, E. Gemo, L. Trimby, J. Shields, E. Humphreys, J. Faneca, L. Cai, A. Baldycheva, J. Bertolotti, and C. D. Wright, “Tunable optical metasurfaces enabled by chalcogenide phase-change materials: from the visible to the THz,” J. Opt. 22(11), 114001 (2020). [CrossRef]  

38. H. Lin, Z. Luo, T. Gu, L. C. Kimerling, K. Wada, A. Agarwal, and J. Hu, “Mid-infrared integrated photonics on silicon: a perspective,” Nanophotonics 7(2), 393–420 (2017). [CrossRef]  

39. J. Midkiff, K. M. Yoo, J.-D. Shin, H. Dalir, M. Teimourpour, and R. T. Chen, “Optical phased array beam steering in the mid-infrared on an InP-based platform,” Optica 7(11), 1544–1547 (2020). [CrossRef]  

40. A. Heßler, S. Wahl, P. T. Kristensen, M. Wuttig, K. Busch, and T. Taubner, “Nanostructured In₃SbTe₂ antennas enable switching from sharp dielectric to broad plasmonic resonances,” Nanophotonics Advanced Online Publication, https://doi.org/10.1515/nanoph-2022-0041 (2022).

41. H. Zhang, X. Wang, and W. Zhang, “First-principles investigation of amorphous Ge–Sb–Se–Te optical phase-change materials,” Opt. Mater. Express 12(7), 2497–2506 (2022). [CrossRef]  

42. M. Y. Shalaginov, S. An, Y. Zhang, F. Yang, P. Su, V. Liberman, J. B. Chou, C. M. Roberts, M. Kang, C. Rios, Q. Du, C. Fowler, A. Agarwal, K. A. Richardson, C. Rivero-Baleine, H. Zhang, J. Hu, and T. Gu, “Reconfigurable all-dielectric metalens with diffraction-limited performance,” Nat. Commun. 12(1), 1225 (2021). [CrossRef]  

43. M. Zhu, O. Cojocaru-Miredin, A. M. Mio, J. Keutgen, M. Küpers, Y. Yu, J.-Y. Cho, R. Dronskowski, and M. Wuttig, “Unique bond breaking in crystalline phase change materials and the quest for metavalent bonding,” Adv. Mater. 30(18), 1706735 (2018). [CrossRef]  

44. J. Faneca, L. Trimby, I. Zeimpekis, M. Delaney, D. W. Hewak, F. Y. Gardes, C. D. Wright, and A. Baldycheva, “On-chip sub-wavelength Bragg grating design based on novel low loss phase-change materials,” Opt. Express 28(11), 16394–16406 (2020). [CrossRef]  

45. Y. Wang, J. Zhang, H. Jin, and P. Xu, “Reconfigurable and dual-polarization Bragg grating filter with phase change materials,” Appl. Opt. 60(31), 9989–9993 (2021). [CrossRef]  

46. R. Heenkenda, K. Hirakawa, and A. Sarangan, “Tunable optical filter using phase change materials for smart IR night vision applications,” Opt. Express 29(21), 33795–33803 (2021). [CrossRef]  

47. N. L. Tsitsas and C. Valagiannopoulos, “Anomalous refraction into free space with all-dielectric binary metagratings,” Phys. Rev. Res. 2(3), 033526 (2020). [CrossRef]  

48. Y. Ra’di and A. Alù, “Metagratings for efficient wavefront manipulation,” IEEE Photonics J. 14(1), 1–13 (2022). [CrossRef]  

49. N. L. Tsitsas and C. A. Valagiannopoulos, “Anomalous reflection of visible light by all-dielectric gradient metasurfaces [Invited],” J. Opt. Soc. Am. B 34(7), D1–D8 (2017). [CrossRef]  

50. N. L. Tsitsas, D. I. Kaklamani, and N. K. Uzunoglu, “Rigorous integral equation analysis of nonsymmetric coupled grating slab waveguides,” J. Opt. Soc. Am. A 23(11), 2888–2905 (2006). [CrossRef]  

51. N. L. Tsitsas, N. K. Uzunoglu, and D. I. Kaklamani, “Diffraction of plane waves incident on a grated dielectric slab: an entire domain integral equation analysis,” Radio Sci. 42(6), 1–26 (2007). [CrossRef]  

52. N. L. Tsitsas, “Second-kind Fredholm integral-equation analysis of scattering by layered dielectric gratings,” IET Microwaves Antenna & Prop. 15(10), 1194–1205 (2021). [CrossRef]  

53. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

54. G. C. R. Devarapu and S. Foteinopoulou, “Compact photonic-crystal superabsorbers from strongly absorbing media,” J. Appl. Phys. 114(3), 033504 (2013). [CrossRef]  

55. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, second edition (Princeton University Press, 2008).

56. S. Foteinopoulou, “Breaking transmission symmetry without breaking reciprocity in linear all-dielectric polarization-preserving metagratings,” Phys. Rev. Appl. 17(2), 024064 (2022). [CrossRef]  

57. S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects,” Phys. Rev. B 72(16), 165112 (2005). [CrossRef]  

58. S. Foteinopoulou, “Photonic crystals as metamaterials,” Phys. B 407(20), 4056–4061 (2012). [CrossRef]  

59. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010). [CrossRef]  

60. D. Marcuse, Theory of Dielectric Optical Waveguides, Second Edition (Academic Press, Inc., 2012).

61. G. J. Chaplain and R. V. Craster, “Ultrathin entirely flat umklapp lenses,” Phys. Rev. B 101(15), 155430 (2020). [CrossRef]  

62. F. Monticone and A. Alù, “Bound states within the radiation continuum in diffraction gratings and the role of leaky modes,” New J. Phys. 19(9), 093011 (2017). [CrossRef]  

63. M. V. Rybin, D. S. Filonov, P. A. Belov, Y. S. Kivshar, and M. F. Limonov, “Switching from visibility to invisibility via Fano resonances: Theory and experiment,” Sci. Rep. 5(1), 8774 (2015). [CrossRef]  

64. P. S. J. Russell, “Optics of Floquet-Bloch waves in dielectric gratings,” Appl. Phys. B: Photophys. Laser Chem. 39(4), 231–246 (1986). [CrossRef]  

65. D. E. Aspnes, “Local-efield effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50(8), 704–709 (1982). [CrossRef]  

66. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 2005).

67. D. N. Chigrin, A. V. Lavrinenko, and C. M. S. Torres, “Nanopillars photonic crystal waveguides,” Opt. Express 12(4), 617–622 (2004). [CrossRef]  

68. X. Yu, W. T. Lau, and S. Fan, “Anomalous modal structure in a waveguide with a photonic crystal core,” Opt. Lett. 31(6), 742–744 (2006). [CrossRef]  

69. Z. Ruan, M. Qiu, S. Xiao, S. He, and L. Thylén, “Coupling between plane waves and bloch waves in photonic crystals with negative refraction,” Phys. Rev. B 71(4), 045111 (2005). [CrossRef]  

70. P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Applications, E. Burstein and C. Weisbuch, eds. (Springer US, 1995), pp. 585–633.

71. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljacic, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]  

72. K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum,” Phys. Rev. Lett. 121(19), 193903 (2018). [CrossRef]  

73. M. Sidorenko, O. Sergaeva, Z. Sadrieva, C. Roques-Carmes, P. Muraev, D. Maksimov, and A. Bogdanov, “Observation of an accidental bound state in the continuum in a chain of dielectric disks,” Phys. Rev. Appl. 15(3), 034041 (2021). [CrossRef]  

74. R. M. Saadabad, L. Huang, A. B. Evlyukhin, and A. E. Miroshnichenko, “Multifaceted anapole: from physics to applications [Invited],” Opt. Mater. Express 12(5), 1817–1837 (2022). [CrossRef]  

75. K. Koshelev, A. Bogdanov, and Y. Kivshar, “Engineering with bound states in the continuum,” OPN magazine, January 2020 issue.

76. L. Huang, L. Xu, M. Rahmani, D. N. Neshev, and A. E. Miroshnichenko, “Pushing the limit of high-Q mode of a single dielectric nanocavity,” Adv. Photonics 3(01), 1–9 (2021). [CrossRef]  

77. H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32(6), 3231–3242 (1985). [CrossRef]  

78. Q. Zhou, Y. Fu, L. Huang, Q. Wu, A. Miroshnichenko, L. Gao, and Y. Xu, “Geometry symmetry-free and higher-order optical bound states in the continuum,” Nat. Commun. 12(1), 4390 (2021). [CrossRef]  

79. E. Melik-Gaykazyan, K. Koshelev, J.-H. Choi, S. S. Kruk, A. Bogdanov, H.-G. Park, and Y. Kivshar, “From Fano to quasi-BIC resonances in individual dielectric nanoantennas,” Nano Lett. 21(4), 1765–1771 (2021). [CrossRef]  

80. J. Xiang, Y. Xu, J.-D. Chen, and S. Lan, “Tailoring the spatial localization of bound state in the continuum in plasmonic-dielectric hybrid system,” Nanophotonics 9(1), 133–142 (2020). [CrossRef]  

81. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]  

82. J.-P. Berenger, “Perfectly matched layer for the FDTD solution of wave-structure interaction problems,” IEEE Trans. Antennas Propag. 44(1), 110–117 (1996). [CrossRef]  

83. R. Moussa, S. Foteinopoulou, and C. M. Soukoulis, “Delay-time investigation of electromagnetic waves through homogeneous medium and photonic crystal left-handed materials,” Appl. Phys. Lett. 85(7), 1125–1127 (2004). [CrossRef]  

84. A. John-Herpin, A. Tittl, L. Kahner, F. Richter, S. H. Huang, G. Shvets, S.-H. Oh, and H. Altug, “Metasurface-enhanced infrared spectroscopy: an abundance of materials and functionalities,” Advanced Materials In press, 2110163 (2022).

85. G. Zograf, K. Koshelev, A. Zalogina, V. Korolev, R. Hollinger, D.-Y. Choi, M. Zuerch, C. Spielmann, B. Luther-Davies, D. Kartashov, S. V. Makarov, S. S. Kruk, and Y. Kivshar, “High-harmonic generation from resonant dielectric metasurfaces empowered by bound states in the continuum,” ACS Photonics 9(2), 567–574 (2022). [CrossRef]  

86. S. D. Gennaro, R. Sarma, and I. Brener, “Nonlinear and ultrafast all-dielectric metasurfaces at the center for integrated nanotechnologies,” Nanotechnology (2022).

87. M. R. Shcherbakov, H. Zhang, M. Tripepi, G. Sartorello, N. Talisa, A. AlShafey, Z. Fan, J. Twardowski, L. A. Krivitsky, A. I. Kuznetsov, E. Chowdhury, and G. Shvets, “Generation of even and odd high harmonics in resonant metasurfaces using single and multiple ultra-intense laser pulses,” Nat. Commun. 12(1), 4185 (2021). [CrossRef]  

88. I. Volkovskaya, L. Xu, L. Huang, A. I. Smirnov, A. E. Miroshnichenko, and D. Smirnova, “Multipolar second-harmonic generation from high-Q quasi-BIC states in subwavelength resonators,” Nanophotonics 9(12), 3953–3963 (2020). [CrossRef]  

89. Q. Zhang, Y. Zhang, J. Li, R. Soref, T. Gu, and J. Hu, “Broadband nonvolatile photonic switching based on optical phase change materials: beyond the classical figure-of-merit,” Opt. Lett. 43(1), 94–97 (2018). [CrossRef]  

90. T. Skauli, P. S. Kuo, K. L. Vodopyanov, T. J. Pinguet, O. Levi, L. A. Eyres, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Improved dispersion relations for GaAs and applications to nonlinear optics,” J. Appl. Phys. 94(10), 6447–6455 (2003). [CrossRef]  

91. https://refractiveindex.info/- accessed June 10 2021.