1. Introduction

Over the past few years, analog optical computing has attracted increasing attention, since it offers high-performance solution of several important computational tasks. Among the basic operations of analog processing of optical signals are spatial differentiation and integration of the profile of an optical beam. Traditionally, for the implementation of these operations, bulky optical systems consisting of lenses and filters are utilized [1]. Recently, spatial differentiators and integrators based on nanophotonic structures having a thickness comparable to the wavelength of the processed optical signal were proposed. In particular, in [2–5], compact analogues of the optical Fourier correlator are considered, in which the spatial filter comprises a metasurface encoding the complex transmission coefficient of a differentiating or an integrating filter. In [2, 6–16], various resonant structures containing systems of homogeneous layers [2,6–12] and diffraction gratings [13–16] are used for spatial differentiation and integration of optical beams. The possibility to utilize such structures for optical differentiation or integration is based on the fact that the Fano profile describing the reflection or the transmission coefficient of the structure in the vicinity of the resonance can approximate the transfer function of a differentiating or an integrating filter.

Development of planar (on-chip) differentiators and integrators, in which the processed optical signal propagates in some guiding structure, is of great interest. In particular, in a wide class of planar (integrated) optoelectronic systems, spectral or spatial filtering of optical signals is performed in a slab waveguide (“insulator-on-insulator” platform) [17, 18]. In this case, the processed signal corresponds to a superposition of slab waveguide modes with different propagation directions (in the case of spatial filtering) or with different frequencies (in the case of spectral filtering). In a recent work, the present authors proposed a simple planar differentiator based on the so-called W-type structure and consisting of two grooves on the surface of a slab waveguide and operating in reflection [19]. The differentiator operation is associated with the excitation of an eigenmode localized between the grooves. A similar graphene-based W-type spatial integrator operating in transmission was considered in [10].

In this work, we for the first time show that an extremely simple structure consisting of a single subwavelength ridge on the surface of a slab waveguide can be used as a spatial integrator and differentiator. The implementation of these operations is associated with the resonant excitation of a cross-polarized eigenmode of the ridge. We demonstrate that the Q-factor of the resonances strongly varies along the dispersion curves, enabling the structure to be used both as an efficient optical integrator and differentiator. In particular, the Q-factor can be arbitrarily high due to the existence of the bound states in the continuum in the investigated structure [20,21]. That makes this structure especially suitable for analog optical integration. In contrast, high-quality differentiation is achieved at relatively low-Q resonances, which are also supported by the proposed structure. The presented results of rigorous numerical simulations confirm high accuracy of spatial integration and differentiation.

2. Diffraction of slab waveguide modes on a ridge

In order to explain the resonant effects that make it possible to implement spatial integration and differentiation, let us first discuss the diffraction of slab waveguide modes on the ridge located on the surface of a slab waveguide. The geometry of the structure is shown in Fig. 1. For the analysis, the following parameters were chosen: refractive index of the waveguide core layer and of the ridge n_c = 3.3212 (GaP at the wavelength λ = 630 nm), refractive indices of the substrate and superstrate n_sub = 1.45 (fused silica) and n_sup = 1, respectively, waveguide thickness h_c = 80 nm, waveguide thickness at the ridge h_r = 110 nm. In this case, at λ = 630 nm the waveguide is single-mode for both TE- and TM-polarizations, and the effective refractive indices of the modes amount to n_wg,TE = 2.5913 and n_wg,TM = 1.6327, respectively. Effective refractive indices of the modes at h_r = 110 nm equal n_r,TE = 2.8192 and n_r,TM = 2.1867.

 

Fig. 1 Geometry of the problem of diffraction of a waveguide mode on a dielectric step.

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Figure 2 shows the reflectance |R_TE (l, θ)|² and the transmittance |T_TE (l, θ)|² of the ridge vs. the angle of incidence θ and the ridge length l in the case of an incident TE-polarized mode, where R_TE (l, θ) and T_TE (l, θ) are the complex reflection and transmission coefficients, respectively. The plots were calculated using an in-house implementation of the aperiodic rigorous coupled-wave analysis (aRCWA) technique [22,23]. The RCWA, also called the Fourier modal method, is an established numerical technique for solving Maxwell’s equations.

It is evident that several resonances (sharp reflectance maxima and transmittance minima) are present in the spectra of Fig. 2. The resonant reflectance peaks (transmittance dips) are located in the angle of incidence range 39.05° < θ < 57.55°, which is marked with dashed lines in Fig. 2. It is important to note that the transmittance strictly vanishes at the resonances, and therefore the reflectance reaches unity.

 

Fig. 2 Reflectance (a) and transmittance (b) of the ridge vs. the ridge length l (horizontal axis) and the angle of incidence θ (vertical axis). Dashed lines show the boundaries of the resonance region and correspond to the cutoff angles of the TM-modes outside the ridge (at h_c = 80 nm, upper line) and inside the ridge (h_r = 110 nm, lower line). Dotted curves show the dispersion of the cross-polarized modes of the ridge. The points marked with white and black asterisks correspond to the spatial integration and differentiation examples considered below, respectively.

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Let us explain these effects. In a general case, at oblique incidence of a TE-polarized guided mode on a ridge located on the waveguide surface, reflected and transmitted TE- and TM-polarized modes are generated, as well as a continuum of plane waves arising from “parasitic” scattering out of the guiding layer to the superstrate and substrate. However, if the angle of incidence exceeds a certain value θ_cr,1, the reflected and transmitted fields contain only the TE-polarized mode, and no out-of-plane scattering and polarization conversion occur [19,24]. Indeed, let us denote by k_x,inc = k₀n_wg,TE sin θ the x-component of the wavevector of the incident TE-mode, which is parallel to the ridge boundaries. Here, k₀ = 2π/λ is the wavenumber. According to the boundary conditions for the Maxwell’s equations, the k_x,inc component has to be conserved, i.e. it is the same for all the waves constituting the reflected and the transmitted fields, including the radiation scattered out of the waveguide. Therefore, at angles of incidence θ > θ_cr,1 = arcsin (n_wg,TM/n_wg,TE) = 39.05°, reflected and transmitted TM-polarized modes become evanescent. At θ > θ_cr,1, k_x,inc also exceeds the wave vector magnitudes of the propagating plane waves over and under the waveguide (in the regions with the refractive indices n_sup and n_sub, respectively) and therefore the incident TE-mode is not scattered out of the waveguide layer. Thus, at θ > θ_cr,1 the reflection and transmission coefficients corresponding to the TE-polarized mode satisfy the equality |R_TE|² + |T_TE|² = 1 according to the energy conservation law, since the materials of the structure are assumed to be lossless. The angle θ_cr,1 is shown with a dashed line in Fig. 2, which corresponds to the upper boundary of the “resonance region”. The lower boundary of this region corresponds to the cutoff angle of the TM-polarized mode in the ridge region, which amounts to θ_cr,2 = arcsin (n_r,TM/n_r,TE) = 57.55°.

The presented analysis suggests that the resonances in Fig. 2 are associated with the excitation of quasi-TM eigenmodes of the ridge, which in this case acts as a leaky rib waveguide. Within the framework of the effective index method (EIM) [25], the quasi-TM modes of a rib waveguide can be approximately described by the dispersion relation of TE-polarized modes of a dielectric slab waveguide with the thickness l, in which the values n_r,TM and n_wg,TM are used as the refractive indices of the core layer and claddings, respectively. The dispersion relation of a symmetric slab waveguide can be written in the following form [26]:

Since in the resonant regime (at θ_cr,1 < θ < θ_cr,2) there is neither out-of-plane scattering nor polarization conversion in the reflected and transmitted radiation, the transmission coefficient strictly vanishes at the resonances [27,28]. Moreover, it is evident from Fig. 2 that at different ridge lengths the resonant peaks (dips) have different angular widths (different quality factor), which vary from units to thousandths of a degree and less. This change in the quality factor is caused by the interaction between the resonances of two types: TE Fabry–Pérot resonance and TM guided-mode resonance. When two types of resonances (two modes) interact, the so-called matrix Fabry–Pérot resonances occur, which can have a very high Q-factor [29]. In fact, the Q-factor in the considered structure can even reach infinity, i.e. the structure supports the so-called bound states in the continuum (BICs) [20,21]. A detailed theoretical investigation of the Q-factor of the resonances and of the BIC formation mechanism in the studied structure will be published elsewhere [21].

3. Transformation of the profile of a beam propagating in the waveguide upon diffraction on a ridge

Let us now consider the transformation of the spatial profile of a TE-polarized beam propagating in the waveguide, which occurs upon reflection from the ridge and propagation through the ridge. In the coordinate system associated with the incident beam (u_inc, v_inc) (Fig. 1), the obliquely incident beam with the angle of incidence θ₀ can be represented as a superposition of slab waveguide modes with different spatial frequencies k_u,inc = k₀n_wg,TE sin θ_inc, where θ_inc is the angle between the propagation direction of the mode and the v_inc axis. We assume that the spatial spectrum of the incident beam G (k_u,inc), |k_u,inc| ≤ g is narrow enough so that g ≪ k₀n_wg,TE. In this case, the field of the beam can be represented as [19]

4. Spatial integration of a beam propagating in the waveguide

The presence of high-Q resonances in the spectra in Fig. 2 makes it possible to use the considered structure as an optical spatial integrator operating in reflection [7,10]. Indeed, in the vicinity of the resonance, the reflection coefficient can be approximated by the Fano lineshape [7]

As an example, solid curves in Figs. 3(a), 3(b) and 3(d), 3(e) show the amplitudes and phases of the TFs H_R (k_u,inc) calculated at two resonant points, at which the reflectance reaches unity: (l_unity,1, θ_unity,1) = (0.355 μm, 53.28°) (point 1) and (l_unity,2, θ_unity,2) = (0.36 μm, 53.37°) (point 2). These points are marked with white asterisks in the inset of Fig. 2(a). Let us note that since the structure works at relatively high angles of incidence, the incident and reflected fields become spatially separated at a reasonable distance from the ridge. Therefore, in the experimental studies, the reflected and transmitted signals can be registered after outcoupling by appropriately designed diffraction gratings located on the waveguide surface. In practical applications, more sophisticated fully integrated optical setups similar to the ones designed for digital planar holograms [17] can be used.

The resonant approximations calculated using Eq. (7) are shown with dashed curves in Figs. 3(a), 3(b) and 3(d), 3(e) and are in a good agreement with the rigorously calculated TFs. The parameters of the approximations γ = γ₁ = 0.0011 μm⁻¹ (point 1) and γ = γ₂ = 0.0018 μm⁻¹ (point 2) were found by fitting Eq. (7) to the computed angular spectra.

Let us now consider the integration of a beam propagating in the waveguide and having the profile $P_{\text{inc}}(u_{\text{inc}})=\left(-2u_{\text{inc}}/{\sigma }^2\right)\text{exp}\left(-u_{\text{inc}}^2/{\sigma }^2\right)$, which corresponds to the derivative of a Gaussian function $g\left(u_{\text{inc}}\right)=\text{exp}\left(-u_{\text{inc}}^2/{\sigma }^2\right)$. The normalized spatial spectrum of the beam $-\text{i}G\left(k_{u,\text{inc}}\right)=\sigma \left[k_{u,\text{inc}}/\left(2\sqrt{\pi }\right)\right]\text{exp}\left(-k_{u,\text{inc}}^2{\sigma }^2/4\right)$ at σ = 15 μm is shown with a dashed curve in Figs. 3(a) and 3(d). The spatial bandwidth of the input beam (the width of its spectrum amplitude) at the 0.1 level with respect to the maximum value amounts to W = 0.052k₀. In order to assess the integration quality, we calculated the profile of the reflected beam P_refl (u_refl) using Eq. (5). Figures 3(c) and 3(f) show the incident beam (dotted curves), analytically calculated integral of the incident beam profile (dashed curves), and absolute values of the profiles of the reflected beams (solid curves) corresponding to the TFs of Figs. 3(a) and 3(d), respectively. The reflected beams are normalized by the maximum amplitude of the incident beam. The analytically calculated integral [the function g (u_refl)] is shown scaled so that its value coincides with the amplitude of the reflected signal at u_refl = 0.

The resonance at the point (l_unity,1, θ_unity,1) [Fig. 3(a)] has a higher Q-factor than the resonance at the point (l_unity,2, θ_unity,2) [Fig. 3(d)]. Therefore, one can expect the quality of integration at the point (l_unity,1, θ_unity,1) [Fig. 3(c)] to be higher than that at the point (l_unity,2, θ_unity,2) [Fig. 3(f)]. As a measure of the integration quality, we use the normalized root-mean-square deviation (NRMSD) of the absolute value of the reflected beam from the analytically calculated integral corresponding to the Gaussian function. For the considered examples, the NRMSD values amount to 1.2% [Fig. 3(c)] and 4.2% [Fig. 3(f)]. Let us note that for the first integrator corresponding to the point (l_unity,1, θ_unity,1) the NRMSD does not exceed 5% for the incident signals with spatial bandwidth in the range 0.013k₀ ≤ W ≤ 0.1k₀. The maximum bandwidth is comparable to the one reported in [10].

 

Fig. 3 Amplitudes (absolute values) and phases (arguments) of the reflection transfer functions H_R (k_u,inc) calculated at the points (l_unity,1, θ_unity,1) = (0.355 μm, 53.28°) (a), (b) and (l_unity,2, θ_unity,2) = (0.36 μm, 53.37°) (d), (e). Dotted curves in (a) and (d) show the spectrum of the incident beam, dashed curves in (a), (b) and (d), (e) show the amplitudes and phases of the TF approximations calculated using Eq. (7). (c), (f) Absolute values of the calculated profiles of the reflected beams corresponding to the TFs shown in (a), (b) and (d), (e), respectively. Dotted curves in (c) and (f) show the incident beam, dashed curves show the absolute values of the analytically calculated integral.

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It is evident from Fig. 3 that the higher the integration quality (the higher the Q-factor), the lower the amplitude of the reflected signal. Maximum amplitudes of the reflected signal normalized by the maximum amplitude of the incident beam in the considered cases equal 0.02 [Fig. 3(c)] and 0.038 [Fig. 3(f)]. Thus, it is possible to achieve the required tradeoff between the quality of integration and the energy (amplitude) of the reflected beam by choosing the Q-factor of the resonance. It is interesting to note that in contrast to the previously proposed optical integrators based on phase-shifted Bragg gratings [7] and W-type structures [10], an increase in the quality factor (which is necessary for an increase in the integration quality) is not necessarily accompanied by an increase in the footprint of the structure in the propagation direction.

The authors believe that higher-order integration can be implemented using several ridges arranged in a configuration similar to the one used for achieving higher-order differentiation in reflection with acoustic metamaterials (see Fig. 4 in [30]).

 

Fig. 4 Amplitudes (absolute values) and phases (arguments) of the transmission transfer functions H_T (k_u,inc) calculated at the points (l_zero,1, θ_zero,1) = (0.2 μm, 48.62°) (a) and (l_zero,2, θ_zero,2) = (0.24 μm, 50.14°) (c). Dotted curves in (a) and (c) show the normalized spectrum of the incident Gaussian beam. (b), (d) Absolute values of the calculated profiles of the transmitted beams corresponding to the TFs shown in (a) and (c), respectively. Dotted curves show the incident Gaussian beam, dashed curves show the absolute value of the analytically calculated derivative of the Gaussian function.

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5. Spatial differentiation of a beam propagating in the waveguide

Let us now discuss the application of the considered structure for the computation of the spatial derivative of the incident beam profile. As shown in Section 2, the transmission coefficient of the ridge strictly vanishes at the resonances, which enables using this structure as an optical differentiator operating in transmission [6, 19]. Indeed, let us assume that the transmission coefficient T_TE (k_x) vanishes at certain angle of incidence θ₀ and ridge length l. Expanding the TF H_T (k_u,inc) of Eq. (3) into Taylor series at k_u,inc = 0 up to the linear term, we obtain:

As an example, Fig. 4(a) shows the amplitude (absolute value) and phase (argument) of the TF H_T (k_u,inc) calculated at the point (l_zero,1, θ_zero,1) = 0.2 μm, 48.62°) marked with a black asterisk in Fig. 2(b), where l_zero,1 and θ_zero,1 are the ridge length and the angle of incidence, at which the transmission coefficient vanishes. Figure 4(a) demonstrates that in the vicinity of the point k_u,inc = 0 (i.e. at θ₀ = θ_zero,1) the TF of the ridge is in good agreement with the TF of an exact differentiator. Let us note that the linear phase of the TF leads only to the shift of the transmitted beam similarly to the Goos–Hänchen effect and does not affect the differentiation quality.

Consider the differentiation of a beam propagating in the waveguide and having a Gaussian profile $P_{\text{inc}}\left(u_{\text{inc}}\right)=\text{exp}\left(-u_{\text{inc}}^2/{\sigma }^2\right)$. The normalized spectrum of the beam $\left(2\sqrt{\pi }/\sigma \right)G\left(k_{u,\text{inc}}\right)=\text{exp}\left(-k_{u,\text{inc}}^2{\sigma }^2/4\right)$ at σ = 25 μm is shown with a dashed curve in Fig. 4(a). The widths of the beam P_inc (u_inc) and of its spectrum G (k_u,inc) (the spatial bandwidth W) at the level 0.1 with respect to the maximum value amount to 75.9 μm and 0.024k₀, respectively.

In order to assess the differentiation quality, we calculated the profile of the transmitted beam P_tr (u_tr) using Eq. (5). Figure 4(b) shows the incident beam (dotted curve), the absolute value of the calculated profile of the transmitted beam (solid curve), and the absolute value of the analytically calculated derivative of the Gaussian function (dashed curve). The analytically calculated derivative is normalized so that its maximum value coincides with that of the transmitted signal. Similarly to the previous section, we use the NRMSD of the transmitted beam from the analytically calculated derivative as the measure of the quality of differentiation. It is evident from Fig. 4(b) that the differentiation quality is high, the NRMSD value in this case is only 1.2%. Note that in the NRMSD calculation, the shift of the transmitted beam was not taken into account (i.e. the minima of the transmitted beam and of the exact derivative were shifted to the same point). The maximum of the amplitude of the transmitted beam for this example equals 0.123. Let us note that for this structure, the differentiation NRMSD does not exceed 5% at spatial bandwidth values of the incident beam W ≤ 0.06k₀.

Similarly to the integration case considered above, by choosing the quality factor of the resonance, one can achieve the required tradeoff between the differentiation quality and the energy (amplitude) of the transmitted beam. For example, let us consider another resonant point (l_zero,2, θ_zero,2) = (0.24 μm, 50.14°) also marked with a black asterisk in Fig. 2(b), at which the transmission coefficient also vanishes. At this point, the resonant dip is narrower (i.e. the resonance has a higher quality factor). Therefore, the corresponding TF shown in Fig. 4(c) has a smaller linearity interval, but a larger amplitude. Thus, in this case, we should expect a decrease in the differentiation quality accompanied by an increase in the amplitude of the transmitted beam. The absolute value of the calculated profile of the transmitted beam corresponding to the point (l_zero,2, θ_zero,2) is shown in Fig. 4(d). In this case, the NRMSD value is indeed higher than in the previous example and amounts to 3.5%. At the same time, maximum amplitude of the transmitted signal is also higher and equals 0.21.

Let us note that in the considered ridge-based differentiators (Fig. 4), the ridge length is of about 3 times smaller than the free-space wavelength of the incident beam λ = 630 nm, while the ridge height h_r − h_c is deep-subwavelength. It follows from Fig. 2 that we can control the ridge length of the designed differentiator or integrator by choosing different (l, θ) points located on the dispersion curves of the resonances. In particular, in the vicinity of the cutoff angle θ_cr,1, the ridge length can be 10–100 nm depending on the required Q-factor of the resonance.

It is worth mentioning that higher-order differentiation can be performed simply by using several consecutive ridges on the waveguide surface (the number of the ridges equals to the order of the computed derivative), similar to how it was done in paper [31] for the case of grating-based optical differentiators also operating in transmission.

6. Conclusion

In the present work, we demonstrated that in the case of oblique incidence of a TE-polarized mode of a slab waveguide on a subwavelength dielectric ridge located on the waveguide surface, resonant changes in the reflectance and the transmittance occur. Using the effective index method, we explained these resonant effects by the excitation of a cross-polarized mode of the ridge.

The discovered resonances enable optical implementation of the operations of spatial integration and differentiation of the profile of an optical beam propagating in the waveguide. The computation of the integral is performed in reflection, whereas the computation of the derivative is performed in transmission. Rigorous numerical simulation results confirm the possibility of high-quality spatial integration and differentiation. The proposed ridge structure has a simpler geometry comparing to the previously proposed integrators and differentiators based on phase-shifted Bragg gratings [7,32] and the so-called W-type structures [10,19].

The obtained results may find application in the design of on-chip systems for all-optical analog computing. The authors also believe that the presented planar structure can be used as an integrated optical spatial (angular) or spectral filter. Moreover, the existence of the bound states in the continuum in the structure also makes it promising for lasing and sensing, among other applications.