Multifunctional Gratings for Multiband Spatial Filtering, Retrore ection, Splitting and Demultiplexing Based on C2 Symmetric Photonic Crystals

The concept of multifunctional reflection-mode gratings based on rod-type photonic crystals with C2 symmetry is introduced and examined. The specific modal properties lead to the vanishing dependence of the first-negative-order maximum on the angle of incidence within a wide range, and the nearly sinusoidal redistribution of the incident-wave energy between zero order (specular reflection) and first negative diffraction order (deflection) at frequency variation that are the key features enabling various functionalities in one structure and functionality merging. The elementary functionalities offered by the studied structures, of which multifunctional scenarios can be designed, include but are not restricted to multiband spatial filtering, multiband splitting, and demultiplexing. The proposed structures are shown to be capable in multifunctional operation in case of an obliquely incident polychromatic wave. The generalized demultiplexing is demonstrated for the case when several polychromatic waves are incident at different angles. The same deflection properties yield multiband splitting, and merging demultiplexing and splitting functionalties in one functionality, which may contribute to various multifunctional scenarios. The proposed gratings are also studied in transmissive configuration.


Introduction
Finite-thickness slabs of a photonic crystal (PhC) known as PhC gratings started attracting the interest more than one decade ago [1][2][3][4] . PhC gratings and metamaterial based gratings with one-side corrugations have been well known, first of all, due to the structurally nonsymmetric designs enabling asymmetric transmission, which is connected with one-way deflection yielded by incident-wave energy redistribution in favor of a higher diffraction order(s) [5][6][7] . Therefore, deflection can serve as the main enabler of diverse functionalities in PhC gratings. The common effect of dispersion of a Floquet-Bloch mode and diffractions at the corrugated interfaces of PhC gratings and metamaterial based gratings may lead to advanced functionalities. It is noteworthy that in the early-stage studies of the deflecting (blazed) gratings, the emphasis has been put on the operation at the Bragg condition [8][9][10] . Later, this restriction has been mitigated in blazed gratings and PhC gratings [11][12][13][14] . Together with the novel blazed gratings 12,13,15 and recently proposed metasurfaces 16,17 and meta-gratings 18 that are capable in deflection at rather arbitrary choice of geometric parameters and angle of incidence, PhC gratings suggest significant extension to the variety of approaches to wavefront manipulation.
Structures capable in multifunctional operation become trendy due to prospective solutions offered for device miniaturization and system integration [18][19][20] . Multifunctionality in PhC gratings assumes that different wave processes or groups of processes can be separated in space 21 and/or in the frequency domain 5 . The recently proposed metasurfaces offer multifunctional scenarios that include but are not restricted to the ones with different functionalities or different manifestations of the same functionality in the neighboring frequency ranges [22][23][24] , at different incidence angles 17,25 , or at different polarization states [26][27][28] . Two functionalities can be merged into one at a fixed frequency [29][30][31][32] . The multifunctional scenarios that have been recently studied by numerous research groups include the elementary functionalities like focusing, deflection, polarization manipulation, splitting, and vortex and Bessel beam generation. Very recently, periodic meta-arrays with wideband and simultaneously wide-angle deflection that enables spatial filtering and wide-angle splitting and other functionalities in one structure have been demonstrated in reflection mode 18 . The earlier examples of PhC gratings with functionality integration include those yielding one-way splitting, spatial separation of two wave processes, and diodelike asymmetric transmission with opposite directions of strong transmission at two close frequencies 5,21,30 . Examples of similar integration have also been known for other configurations based on PhCs 23,33,34 . Although metasurfaces can suggest electrically thinner performances than PhCs, it is still unclear whether they may replicate all functionalities achievable in PhCs or not. Owing to the diverse modal properties, PhCs may serve as a good platform for multifunctional devices.
In the present paper, the concept of PhC gratings backed with a metal reflector, which are capable of new multifunctional scenarios, is proposed and numerically validated. The goal of this study is demonstration of multifunctional scenarios achievable by using a specific Floquet-Bloch mode of the PhCs. The desired modal propertiescan be obtained in the PhCs with a rectangular lattice, i.e. with the C 2 symmetry, for which diffractions are achieved due to a larger lattice constant along the interface direction (i.e. a x > a y ). The focus in this study will be put on the multifunctional scenarios that are expected to be unachievable in quasi-planar structures and the earlier proposed PhC gratings. It will be demonstrated that (i) the incidence-angle insensitive -1st-order deflection and (ii) the frequency selective -1st-order deflection together with the 0th-order specular reflection enable multiband bandpass and bandstop spatial filtering, partial or full demultiplexing, and multiband splitting in one structure, within the same frequency range. Noticeably, spatial filtering 35−44 , demultiplexing 45−49 and splitting 50−56 belong to the basic functionalities which are commonly demanded in photonics, optical communications, etc. while PhCs suggest a suitable platform for their realization. We will demonstrate how the above listed functionalities can be efficiently integrated in one structure. Moreover, multiband retroreflection and multiplexing can be obtained in the same structure, as well as the generalized demultiplexing and merging of demultiplexing and splitting functionalities into one functionality, by using the same physics and the same modal regime. The transmission properties of the same PhC grating but without a reflector will also be discussed. It should be noted that even if some of the elementary functionalities studied here can be obtained in structures of other types, low-symmetric PhCs are used here in order to enable all or most of the considered elementary functionalities at a given frequency range in one device, according to the purposes of this study. For instance, some of demultiplexing scenarios might be obtained by using volume Bragg gratings 57,58 . However, it can then be a challenge to obtain multiband spatial filtering, like that proposed in this paper, in the same structure and at the same frequency range. To calculate reflectance and transmittance, the integral equation technique with well justified and controllable accuracy and convergence has been used for the calculations 59 . Since the general dispersion properties of PhCs with C 2 symmetry are known, a study of dispersion is beyond the scope of this paper. It is organized as follows. The Results section that follows Introduction is divided by four subsections which present, consequently, proposed geometry, multiband spatial filtering and retroreflection, partial and generalized demultiplexing and multiband splitting, and PhC gratings without reflector. Then, the short Discussion and Methods sections are presented.

Proposed Geometry
The general geometry of the proposed grating is shown in Fig. 1. It represents a finite-thickness slab of PhC that has C 2 symmetry, which is placed above the metal reflector. Structures with such a symmetry are known for self-collimation and other fascinating propagation effects 50,60−64 . In this context, the canalization of electromagnetic waves should also be mentioned 65 . The studied structure is assumed to be periodic and infinitely long in the x direction. It is illuminated from the upper half-space by a plane wave incident obliquely at angle θ . Consideration is restricted to TE polarization, i.e. electric field vector is along the z axis. The rods are assumed to be made of a dielectric material with relative permittivity ε r . Besides, it is assumed that a wide variety of the natural dielectric materials can be used to fabricate the PhC.
The x and y components of the PhC lattice vector components are denoted by a x and a y , respectively; here a x = 2a y . Thus, the grating period L is equal to a x . The ratio d/a y = 0.4 where d is rod diameter is used throughout the paper. For instance, at L = a x = 1.5 µm, we have a y = 750 nm and d = 300 nm. The other values of d/a y have also been examined. The host medium is assumed to be air. The only requirement to the back-side reflector is that it should be weakly absorptive and weakly transmissive that is achieved for a very wide variety of materials and geometrical parameters. For the sake of definiteness, it is assumed here that the reflector is made of a Drude metal with permittivity ε m = 1 − ω p /[ω(ω − iγ)], where ω p is plasma frequency and γ is collision frequency; ω p a y /c = 20π and γ/ω p = 0.01.

Multiband Spatial Filtering and Retroreflection
In this section, we demonstrate the potential of the proposed structures that are based on C 2 symmetric PhCs in multiband spatial filtering and retroreflection. These functionalities belong to the elementary functionalities, which serve as the building blocks for multifunctional scenarios. Since far-field operation is considered, we use zero-order and higher-order reflectances to quantify the wave reflection, see Methods. For plane-wave illumination, spatial filtering can be considered as an analogue of frequency-domain filtering, but in the θ -domain, assuming that f = const, see Ref. 35. Various mechanisms and structures have been proposed for reflection and transmission modes that enable low-pass, bandpass, and bandstop spatial filtering. At beam illumination, the contribution of the individual angular components is modified, so that the beam reshaping takes place, e.g. see Ref. 66. In reflection mode, rather simple but carefully designed gratings have been suggested to obtain wideband and simultaneously wide-angle plane-wave deflection by using the reflected order m = −1 to yield wideband spatial filtering 15,18 . Here, the goal is different. Instead of obtaining a single and large area of r −1 ≈ 1 in (kL,θ )-plane, it is obtaining multiple regions of r −1 ≈ 1, which are intermittent with the regions of r 0 ≈ 1 (i.e. small r −1 ).
The results for the first-negative-order reflectance, r −1 , vs. kL (k = ω/c is the free-space wavenumber, ω = 2π f is the angular frequency, f is frequency, c is the speed of light), are presented in Fig. 2. The "mountains" of r −1 ≈ 1 and "valleys" of small r −1 are the main features observed here. The "valleys" of small r −1 correspond to the "mountains" of r 0 (not shown), since transmission is negligible. From the spatial filtering perspective, it is important that we have non-bent "mountains", i.e.  Fig. 2(b). Interestingly, similar features have been observed for transmission-mode spatial filtering in diffraction-free PhCs, where the intermittence occurs for t 0 and r 0 35 . In our case, the reflected order r −1 plays the same role in spatial filtering as t 0 does in Ref. 35.

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Note that φ −1 may vary in a very wide range for each "mountain". For instance, at kL = 5.486, we obtain Such a wide range of the output-wave angle makes possible multiband retroreflection regime that is achieved when φ −1 = −θ and sinφ −1 = −π/kL [67][68][69] . This regime is known as Littrow mounting. As observed in Fig. 2, retroreflection occurs within each "mountain", so that the number of the retroreflection bands depends on N.
For more evidence, Fig. 3 presents r 0 and r −1 at the selected values of kL, which correspond to the tops of the "mountains" in Fig. 2(b). As observed, we obtain bandstop spatial filtering for r 0 and bandpass spatial filtering for r −1 . It is seen that r −1 > 0.9 in a wide θ -range. The boundaries of the θ -domain bands are moderately sharp, and the the upper-θ boundary is more blurred than the lower one. The width and location of the θ -domain band depends on the choice of the "mountain". The location of the lower-θ boundary in Fig. 3(b) can vary from θ = 10 • to 60 • . One can see that the band widening is possible at the price of a decrease of maxr −1 and a less smooth shape of r −1 vs. θ , e.g., compare to the cases of kL = 5.486 and kL = 5.546 in Fig. 3(b). This remains true for all four "mountains". The features observed in Figs. 2 and 3 can be obtained for a wide variety of the PhC rod materials, e.g. for 5.6 < ε r <12.5. As shown in the next section, partial and generalized demultiplexing and multiband splitting can be obtained in the same structure and at the same frequency range as multiband spatial filtering and retroreflection, i.e. without additional parameter adjustment, that makes the proposed PhC gratings highly capable in multifunctional operation.

Partial and Generalized Demultiplexing and Multiband Splitting
In this section, we demonstrate the potential of the same and similar structures in demultiplexing and multiband splitting, which represent the second part of the selected elementary functionalities. Figure 4 presents zero-order and first-negative-order reflectance, r 0 and r −1 , vs. normalized frequency kL for a PhC grating created by a finite-thickness slab of PhC with C 2 symmetry, which is backed by a metal reflector. Structures with different numbers of the rod layers, N, were compared. As expected, the nearly sinusoidal energy exchange between the orders m = 0 and m = −1 occurs at varying frequency. Notably, this feature is rather uncommon, because it needs the specific modal properties. For the energy exchange of such a kind, the maxima of r −1 ≈ 1, at which the almost total conversion of the incident-wave energy into a deflected beam outgoing at the angle φ m (m = −1, φ −1 = θ ) takes place, are intermittent with the specular reflection regimes of r 0 ≈ 1 (φ 0 = −θ ), when 4/12   The observed intermittence reminds us about Fabry-Perot interferences in a finite-thickness slab of a homogeneous dielectric material, but here we have r 0 and r −1 instead of r 0 and t 0 . Figure 5 presents electric field distribution within one period of the structure along the x axis. The values of kL are chosen that correspond to the maxima of r 0 ≈ 1 and r −1 ≈ 1 in Fig. 4(c). Figures 5(a), 5(c), and 5(e) correspond to r −1 ≈ 1, Figs. 5(b) and 5(d) correspond to r 0 ≈ 1. The observed features give evidence of the interferential nature of the mechanism of energy exchange between the orders m = 0 and m = −1. Here, the equivalent grating thickness D ≈ lλ PhC /2 is needed to obtain r 0 ≈ 1, while D ≈ λ PhC /4 + lλ PhC /2 leads to r −1 ≈ 1 (λ PhC is the wavelength in PhC along the y direction, l is integer). Thus, these two conditions correspond to the constructive and destructive interferences in a non-backed homogeneous dielectric slab where they lead to t 0 = 1 and a maximum of r 0 . Note that the various effects of interferences have been studied in PhCs of different types 5,70,71 . On the other hand, the observed features may remind a Gires-Tornouis interferometer. However, in contrast with its standard version, we utilize here the orders m = 0 and m = −1 together with the specific modal properties enabling the desired multifunctional scenarios.
The observed behavior of r 0 and r −1 in Fig. 4 and that of r −1 in Fig. 2 can be used for partial demultiplexing, as illustrated by the schematic in Fig. 6. Let the wave incident at a given angle θ represents a sum of several spectral components, which correspond to (nearly) perfect deflection, r −1 ≈ 1 (frequencies f d 1 , f d 2 , ... f d P ), and several spectral components corresponding 5/12 Figure 6. Schematic illustrating the merging of partial demultiplexing and splitting functionalities at given θ . Weak dependence of the maxima of r −1 on θ makes possible the use of several polychromatic waves incident at different angles, for each of which the merging of partial demultiplexing and splitting functionalities can be obtained. to the specular reflection regime, r 0 ≈ 1 (frequencies f sr 1 , f sr 2 , ... f sr Q ), i.e. the incident wave is presented as where A p and A q are the amplitudes, k is the wave vector, and r is the generalized coordinate. Then, the components with f = f d p will be deflected at the kL-dependent angles φ m given by Eq. (6) at m = −1. In turn, all of the components with f = f sr q will be specularly reflected, so that a new, reduced sum of spectral components will be formed. In such a way, the first part of the incident wave is demultiplexed, while the second part is not. Clearly, the demultiplexed (deflected) and nondemultiplexed (specularly reflected) parts are forwarded to the different quadrants of the upper half-space. Thus, the discussed functionality can also be understood in the context of sorting, i.e. separation of the spectral components of the incident wave by redirecting them to different sectors. For instance, for the regimes of r −1 ≈ 1 in Fig. 4(c), we obtain φ −1 ≈ −59 • , −38 • , and −25 • , respectively, at kL = 4.01, 4.75, and 5.54, while φ −1 ≈ 45 • for the regimes of r 0 ≈ 1. The incident wave can only be a sum of the spectral components with f = f d i , or a sum the specular-reflection components with f = f sr i , see Eq. (6). Then, in the former case, we obtain a full demultiplexing, i.e. all spectral components correspond to the different values of φ −1 . In the latter case, we have conventional specular reflection. Accordingly, the retroreflection regime is excluded from the set of the outgoing waves.
The same structures suggest multiband splitting when r 0 = r −1 ≈ 0.5. If the incident wave represents a sum of the spectral components, for each of which splitting may occur (frequencies f sp where A s is amplitude, then the splitting and demultiplexing functionalities can be merged in one step. In other words, half of the incident-wave energy is converted into specular reflection, and the other half is spatially demultiplexed. Each of the halfs 6/12 . Also here no one of the regimes of r −1 ≈ 1 corresponds to the Bragg condition. Moreover, the input signal may represent a sum of the components with f sr q , f d p , and f sp s , leading to even more complex multifunctionalty. It is noteworthy that the full demultiplexing can also be obtained in reflective quasi-planar meta-arrays like the ones in Ref. 15,18, but rather for a single wide band of large r −1 , so that some of the selected elementary functionalities, i.e. partial demultiplexing and multiband splitting are not achievable therein. Partial demultiplexing and multiband splitting can be tuned in the simplest way (i.e. without tunable materials) by varying θ . Since both φ −1 and the kL threshold for |sinφ −1 | ≤ 1 depend on θ [see Eq. (6)], the width of the φ −1 range that is achievable in the used energy exchange regime can be varied with θ . The weak dependence of kL-values corresponding to the maxima on θ , like in Fig. 2(b), makes possible generalization of demultiplexing, for which several polychromatic waves (instead of one polychromatic wave like that shown in Fig. 6) are incident at different angles, each comprising multiple spectral components [like in Eq. (1) for one wave]. Such generalization may be difficult when using structures of other types, e.g., volume Bragg gratings, while the desired modal properties can be offered, for instance, by C 2 symmetric PhCs. The possibility of demultiplexing in the framework this scenario is demonstrated in Fig. 7 for the PhC grating with ε r = 9.6. Figure 7(a) presents r 0 and r −1 vs. kL at θ = 45 • . The results are similar to those in Fig. 4, in accordance with the fact that the observed energy exchange between the orders m = 0 and m = −1 takes place in a wide range of ε r variation, e.g., from 5.6 to 12.5. Next, Fig. 7(b) shows r −1 vs. kL for four selected values of θ . The feature common for all values of θ is the intermittence of the maxima of r −1 ≈ 1 and r 0 ≈ 1, see Fig. 2(b) for comparison. The maxima of r −1 of two types can be distinguished in Fig. 7(b). For the first one, spectral locations are the same for the maxima that correspond to different θ ; for the second one, spectral locations are slightly different. Nevertheless, they both may contribute to the demultiplexing scenarios, in which several polychromatic waves are incident at different angles. For the case shown in Fig. 7(b), we obtain φ −1 = −58.9 • and −44. that exceeds a desired threshold value can be obtained. On the other hand, an adjustment may lead to the case when two or more waves incident with different pairs of (kL, θ ) are deflected to the same direction, so that multiplexing/beam combining can be obtained. Clearly, not only conventional but also generalized demultiplexing can be a part of the complex multifunctional scenarios which involve spatial filtering, retroreflection, and/or splitting.

PhC grating without reflector
In this section, we briefly discuss the transmission properties of the designed PhC gratings. The structure studied here is easily obtained from the basic geometry in Fig. 1 by removing the back-side reflector. Figure 8 presents the results for r 0 , r −1 , t 0 , and t −1 vs. kL for the PhC grating with the same parameters as in Figs. 4(c), 2(b), 3. The most interesting feature is that all four diffraction efficiencies vary weakly in the kL-range, in which we observed the energy exchange between -1st and 0th orders in case with the reflector. Moreover, their values are close to each other for N = 6, say, at 4.2 < kL < 5.2 (−52 • < φ −1 < −30 • ), and that enables a four-beam wideband splitter.
Finally, Fig. 9 presents the maps of t −1 and r −1 in (kL,θ )-plane. As observed, the features initially found in Fig. 8 remain the same within the large region, although t −1 varies weaker than r −1 . Thus, the nearly equal efficiencies are not a result of an accidental choice of θ and kL. Rather, it reflects the nature of the used modal regime. It is noteworthy that a similar behavior has been observed in a wide range of ε r variation (results are not shown). Interstingly, the behavior of r 0 , r −1 , t 0 , and t −1 outside the range 3.8 < kL < 5.7 is completely different. It results from the difference in the properties of Floquet-Bloch modes, whose analysis is beyond the scope and can be a subject of a future study.

Discussion
We studied the PhC gratings backed with a reflector that are capable of multifunctional operation, so that several functionalities can be achieved in one structure, in the same frequency range. The specific modal properties that result from the C 2 symmetry of the used PhCs yield co-existense of angle-independent deflection and specific energy exchange between zero and first negative orders while varying frequency that serve as the main enablers of the multifunctional scenarios. Thus, there may be no full analogue of the proposed combination of functionalities in the earlier works. An important advantage is that the two feature mentioned above are kept within a large region in (kL, θ )-plane. In particular, this allows to select proper widths of the passbands in the θ -domain and achieve quite a large angular difference between the individual demultiplexed (outgoing) waves or input waves (in case of multiplexing). The most interesting manifestation is probably the nearly perfect, multiband, wide-angle deflection in the first negative diffraction order regime that enables multiband bandpass filtering in the θ -domain. Up to three spectral regimes of the nearly ideal deflection that are robust to the θ variations and intermittent with the spectral regimes of specular reflection have been numerically demonstrated, while the energy exchange between the propagating zero and first negative orders in reflection is similar to that between reflectance and transmittance in case of the interferences in a homogeneous transmissive dielectric slab. A larger number of the bands within the given frequency and/or incidence-angle ranges can be obtained by an increase of the rod layers. A polychromatic incident wave can experience either full or partial demultiplexing or specular reflection, depending on which spectral components are comprised. The generalization of demultiplexing for the case of several polychromatic waves incident at different angles is proposed that can also contribute to multifunctional scenarios. A multiband splitter is one more functionality enabled by first-negative-order deflection in the same frequency range. Moreover, demultiplexing and splitting functionalities can be merged in one functionality. The same structure can function as a multiplexer or a multiband retroreflector. The proposed structures capable in multifunctional operation can be used just for a single functionality, if required. The same structure but without the back-side reflector can show nearly equal efficiencies for all four propagating orders (zero and first-negative orders in transmission and reflection). The use of the both orthogonal polarizations can further reinforce the multifunctionality potential. The studied PhC gratings can be appropriate for fabrication, since fabrication techniques for two-dimensional PhCs are well developed 73,74 . The concept transfer to the PhC slabs will be considered at the next steps of this research program. The obtained results are expected to be important, first of all, from the multifunctionality perspective. They may be useful for the development of new infrared devices with a high degree of miniaturization and integration, and indicate a route to new applications for low-symmetric PhCs and PhC gratings.

Methods
The studied structure is assumed to be illuminated by a plane incident wave that is given by

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where E 0 is the amplitude, α 0 = ksinθ , and β 0 = kcosθ . The electric field above the PhC grating is presented as follows: where β m = k 2 − α 2 m , Imβ m ≤ 0, α m = α 0 + 2πm/L, ρ m is the amplitude of mth-order reflected wave. The transmitted electric field below the reflector is given by where τ m is the amplitude of the mth-order transmitted wave. For a reflective configuration, E − may be assumed to be zero at the appropriate choice of the material and the thickness of the reflector, but it is nonzero in the general case, i.e. when the reflector is semi-transparent or removed. An integral equation technique is used for the purposes of numerical study. The problem is formulated and solved in the spectral domain with respect to the unknown amplitudes of spectral harmonics of the electric field within one structural period over x. Once the amplitudes at the upper and lower boundaries of the calculation region are found, the values of ρ m and τ m , including those at m = 0 and m = −1, are directly calculated. Details of the problem formulation and iterative numerical procedure can be found in Ref. 59. Throughout the present paper, we use mth-order reflectances, r m = ρ m ρ * m Reβ m /W , and mth-order transmittances, t m = τ m τ * m Reβ m /W , in order to quantify far-field behavior. Here, W is the energy of the incident wave and asterisk means complex conjugate. In the ideal (i.e., lossless) case, R + T = W where R = ∑ ∞ m=−∞ r m is the overall reflectance and T = ∑ ∞ m=−∞ t m is the overall transmittance. In the reflective configuration, T ≈ 0. The angle of the mth-order outgoing wave can be found from the grating equation 72 : sinφ m = sinθ + 2πm/kL (6) where m = 0, ±1, ±2, ... . Therefore, the structure can be designed so that a desired number of the diffraction orders formally may propagate, while the remaining ones are evanescent. In our PhC gratings, the orders m = 0 and m = −1 may propagate, whereas the ones with m > 0 and m < −1 remain evanescent. In this case, R = ∑ 0 m=−1 r m and T = ∑ 0 m=−1 t m .