## Abstract

In this paper, a very simple periodic ridge on a symmetric slab waveguide is used for implementing an on-chip CMOS-compatible second-order spatial differentiator. The reflection and transmission coefficients of this structure show that the second derivative is performed in the transmission when the optical beam normally incidents on the periodic ridge. Simulations confirm that the reason behind the second-order spatial differentiation of the incoming beam is the excitation of the guided mode of the periodic ridge. A Maxwell’s equation solver that utilizes the finite element method (FEM) is used to simulate this structure, and an eigenmode solver is utilized for the validation. The results of both methods have a very good agreement with each other.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Digital computers that are used for many applications have serious restrictions of high power consumption and non-instantaneous response time for solving complex computational problems due to their generality and flexibility in the problems that they can solve [1]. On the other hand, analog computing provides a real-time response with a low power consumption operation in special-purpose computers [2]. Although the electrical and mechanical computers can perform mathematical operations in an analog fashion with high efficiency, they have relatively large sizes and slow responses [3,4]. Recently, analog optical computing has solved these restrictions by using the fast optical phenomena and has triggered a great interest among scientists in the field of photonics and optics for all-optical performing of different real-time computations and mathematical operations in the temporal or spatial domain [5–14].

Spatial analog optical computing such as image processing can be achieved with Fourier optics using lens-based optical systems [15] but the bulky geometry and complexity of these structures prevent their integration with nanophotonic circuits. In [16], suitably designed metamaterial blocks are used for performing mathematical operations such as spatial differentiation, integration, and convolution on optical signals as they propagate through the structure. This concept offers all-optical computation in the scale of a single wavelength or even sub-wavelength using highly compact and integrable structures. The concept of computational metamaterials is categorized into two fundamental approaches: (I) Metasurface (MS) approach and (II) Green’s function (GF) approach. In the MS approach, the Green’s function of the desired operator is implemented in the spatial domain and two extra sub-blocks are required to perform Fourier and inverse Fourier transform. Based on the MS approach, differentiation and integration operators have been implemented using graphene-based metalines [14]. Furthermore, a dielectric metasurface has been designed to solve differential and integrodifferential equations in [17] and a metamaterial-based realization of Grover’s quantum search algorithm has been presented and verified experimentally in [18]. But the GF approach is easier to fabricate due to the fact that it directly implements the Green’s function of the desired operator in the spatial Fourier domain. It can be grouped into two distinct classes based on the resonant and non-resonant nature of the used structure [16]. The plasmonic spatial first-order differentiator with the Kretschmann configuration [19], second-order differentiator based on guided mode resonance (GMR) [20], dielectric slab integrator using the prism coupling technique [21], and photonic crystal slab differentiator [22] are the examples of the resonant-based GF approach. Moreover, Brewster differentiator [23], half-wavelength slab differentiator [24], and differentiator based on photonic spin Hall effect (PSHE) [25] are the examples of the non-resonant-based GF approach. Proposed differentiators can be used for many applications such as edge detection in image processing [19,26].

In many of the planar (integrated) optoelectronic systems, the processed optical signal propagates in some guiding structure, not in the free space. Therefore, the design of planar (on-chip) differentiators and integrators are of great interest. Recently, some operators have been designed in a planar format. For example, a parallel signal processing through on-chip 1D high-contrast transmit array (HCTA) has been explored in [6]. In another work, a simple planar optical differentiator consisting of two grooves on the surface of a slab waveguide has been designed [27]. Additionally, a very simple structure consisting of a single sub-wavelength dielectric ridge on the surface of a slab waveguide has been proposed to perform spatial integration and differentiation of optical beams propagating in the waveguide in the reflection and transmission, respectively when the beam impinging on the ridge at an oblique incident angle in [5].

Extremely fast spatial second-order differentiation is one of the primary operators for optical processing. In this paper, a very simple periodic ridge on a symmetric slab waveguide is proposed for implementing an on-chip CMOS-compatible second-order spatial differentiator. The reflection and transmission coefficients of this structure show that the second derivative is performed in the transmission when the optical beam normally incidents on the periodic ridge. The normal incident has two important advantages, first, there is no need for the alignment of the excitation, and its excitation is much simpler than oblique incident. Second, the second-order derivative at normal incident is very useful for edge detection applications, because the first-order derivative is an odd function and using it at oblique incident is a necessary evil which must be tolerated for edge detection application. The proposed second-order differentiator works on the base of the guided mode resonance (GMR) [20] and its importance is providing the possibility of edge detection at normal incident. Our numerical simulations confirm the presented idea.

## 2. Proposed structure and scientific notation of normal incidence of an optical beam in a symmetric slab waveguide

In Fig. 1, the schematic of the proposed second-order differentiator is shown. By using high contrast material, a simple nanostructure is presented here that enables on-chip second-order derivative of normally incident optical beams in a planar structure. The symmetric slab waveguide in the incident and transmission regions consists of three different layers in the $y$-direction, and there is no variation of the structure in $x$-direction. Cladding and substrate are $SiO_2$ ($n_{cladding}=n_{sub}=1.44\,@\,\lambda _0=1550nm$), and the film is amorphous Silicon $a-Si$ ($n_{film}=3.48\,@\,\lambda _0=1550nm$). In order to have a single mode for both TE and TM modes in a slab waveguide, the height of the film region is set as $h_{f}=80nm$. By considering these values, the effective refractive indices of the fundamental TE and TM modes are $n_{eff,TE}=2.01$ and $n_{eff,TM}=1.47$, respectively. The periodic ridge on the surface of the symmetric slab waveguide helps us to achieve spatially second-order differentiation of incident beam. Parameters of this periodic ridge, namely the height $h$, the length $L$, and the width $W$ are shown in Fig. 1. In this figure, the period of the ridge in the $x$-direction is assumed to be $d$. The filling factor ($FF$) is defined as the ratio between the width of the periodic ridge and the period. The values of the periodic ridge’s parameters at $\lambda _0=1550nm$ are given in Table 1. In the following, it will be shown that the excitation of the periodic ridge’s guided mode (GM) at the normal angle prevents the light beam from passing through the structure.

To explain the physics of our structure that performs the second-order differentiation, we first discuss the input processed signal. Before we go into details of the input processed signal, it should be noted that to realize the spatial computation in such on-chip devices, it needs to consider how to generate the incident field with arbitrary profile. For example, a practical method for generating the incident field with arbitrary profile has been proposed in [6]. The input processed signal is a superposition of the guided TE modes with different directions in the slab waveguide. The wave vector of the first transverse electric (TE) mode of the slab waveguide is $\beta _z$ in the $z$-direction and $k_x$ in the $x$-direction. In this case, the relation between $\beta _z$ and $k_x$ is:

where $n_{eff}$ is the effective refractive index of the slab waveguide for the first TE mode. Like an expansion of a Gaussian beam in free space which expresses a Gaussian beam as a superposition of plane wave components traveling at various angles, by adding some guided TE modes in different angles, our input processed signal can resemble an in-plane confined beam. Therefore, our input processed signal can be modulated on the amplitude of a superposition of the guided TE modes with different directions (in $xz$ plane) in the slab waveguide. The $x$ component of the electric field for the input processed signal in the different regions are given as follows:Because both sides of the periodic ridge (incident region and transmission region) are the same, we expect that the transmitted beam also has the same form as the incident beam. Therefore, we have:

By designing the parameters of the periodic ridge and also tuning for the proper excitation of the GM of the periodic ridge, the structure is simulated and the reflection and the transmission coefficient of the structure are obtained as a function of $k_x$. In these results, we observe the transfer function of the transmitted beam has a parabolic relation with respect to the wave vector in the $x$-direction($k_x$). This observation aligns with the transfer function of a second-order derivative of the input beam in the transmission region [20]:

## 3. Design and simulation of the periodic ridge

To simulate the structure, we utilize two solvers of a commercial multi-physics simulator (COMSOL) to take advantage of both solvers; a frequency-domain solver based on finite element method(FEM) which calculates the reflection and transmission coefficients of the structure, and an eigenmode solver which solves the two Maxwell’s curl equations without any sources in order to calculate the eigenfrequencies and eigenmodes of the periodic ridge. To explain more about eigenmode solver, it should be noted that the eigenmodes and their complex frequencies are the solutions of the eigenvalue equation:

With the aim of careful examination of the extracted results, we plot the transfer function of the optical beam for six cases (determined with stars in Fig. 2) in which the transmission coefficients for the normal incident are near zero in Fig. 3. As shown in Fig. 3, although the simulation results of Case 1 to 3 are not exactly zero at $k_x=0$, they have a parabolic relation near zero in respect to $k_x$, like the transfer function shown in Eq. (6). To be more specific, the value of transfer function of these three cases are 0.05, 0.036, and 0.024 for Case 1 , 2 and 3, respectively. The minimum of the transmittance is almost zero around the resonances but the maximum of reflection does not exactly reach to unity, and there are some losses. One small part of these losses is the coupling of the TE guided mode to TM guided mode due to the grating, however, the amount of this coupling power from TE guided mode to TM guided mode in the transmission region of the slab dielectric is very low and it is negligible. The main part of losses is the scattering loss into the substrate layer below the periodic ridge. To suppress scattering, different techniques have been considered, among which is scattering-free multilayer structures [29]. Let just assume this observation certifies that the second-order derivative is performed in the transmitted beam, the effects of this non-zero value at $k_x=0$ will be discussed in the next. For the cases 4 to 6, these non-zero value are very much and we can not merge second-order derivative to these cases. Let back to Case 1 to 3, when these curves are compared with the transfer function of an ideal second-order differentiator, we define the coefficient multiplying in the transfer function (A) as the gain, and the $\Delta {k_x}$, in which the difference between the ideal and non-ideal cases is less than one percent, as the bandwidth (BW) of the second-order differentiator. For example, the value of BW for Case 3 is shown in Fig. 3. By this definition, as it is clear in this plot, the gain and BW of each case are different. Among these three cases, Case 1 ($FF=0.5, L=0.29\lambda _0$) has the maximum bandwidth for second-order differentiator and Case 3 ($FF=0.72, L=0.19\lambda _0$) has the maximum gain.

In the following, we try to explain the physics behind the resonance. In this regard, through simulation, we determine the origin of resonance occurring in the reflection and transmission coefficients. In the first simulation, we consider the spectrum of $1300-1800 nm$, and choose the value of filling factor from the interval $[0.1 , 0.95]$. Using the frequency domain solver, reflection and transmission coefficients of this structure are shown in Fig. 4. In this figure, we can see that resonances start at wavelength below $1650nm$ for $FF=0.975$ and below $1430nm$ for $FF=0.1$.

Next, we calculate the eigenfrequencies of the periodic ridge waveguide using the eigenmode solver. In Fig. 4, the points marked with stars correspond to these calculated eigenfrequencies. As can be seen, positions of these frequencies match peaks and valleys of reflection and transmission coefficients, respectively. These results imply that excitation of periodic ridge guided mode (GM) makes the transmission coefficient close to zero.

To further understand the resonance, we investigate the first resonance in the frequency spectrum. Specifically, for Case 1 ($FF=0.5, L=0.29\lambda _0$), we illustrate the scattering parameter of the structure and the dispersion diagram of the first two modes in Fig. 5. In Fig. 5(b), it is shown that at normal incident or $k_x=0$, the first two mode of the periodic ridge are excited at $\lambda _0=1550 nm$ and $\lambda _0=1354 nm$. These wavelengths are coincide with two reflection dips in Fig. 5(a).

As an illustration of GMR condition at $\lambda _0=1550 nm$ similar to the condition presented in [20], the $x$ component of the electric fields of the first TE mode of slab waveguide for the normal incident is shown in Fig. 6. As demonstrated, when the propagating wave encounters the grating in the slab waveguide, the mode excitation of the periodic ridge reflects most of the wave into the incident region, a little amount of the wave scattered from the substrate and rest of the wave propagates as the guided mode into the periodic ridges. Hence, it is good to notice that not only the GMR condition is one of the requirements for the performing second-order spatial differentiation through guide modes, but also lowering the scattering loss into the substrate, and coupling loss between different modes of the slab dielectric is very important. As an example of calculating scattering loss, in [30], the scattering loss of a dielectric periodic ridge on the plasmonic metal layer has been investigated. They have shown that, with parasitic scattering suppression, the surface plasmon polariton diffraction is remarkably close to the diffraction of TE-polarized plane waves.

To testify the designed structure for Case 3 which has the maximum gain, the amplitude of the incident beam is modulated to a Gaussian beam as shown in Fig. 1, and the $x$ component of the electric field is calculated. For this purpose, in order to have a reasonable result for the electric field, the bandwidth of the Gaussian beam should be within range of the bandwidth of the second-order differentiator. Hence, for choosing the beam waist of the incident Gaussian beam, we consider the BW of the second-order differentiator. As can be seen in Fig. 3, the bandwidth of the second-order differentiator for Case 3 is $|k_x|<0.03k_0$. The divergence of the Gaussian beam is given by [31]:

Profiles of the $|E_x|$ component of the transmitted beam for our proposed structure (black solid curve), and incident Gaussian beam (red solid curve) are shown in Fig. 7. The results have been calculated with MATLAB using the transfer function obtained via full-wave simulations. In this figure, the blue dotted curve shows the normalized amplitude of the $x$ component electric field of an ideal second-order differentiator. The maximum amplitude of the second-order derivative is about $2.4\%$ of the incident field. By comparing our transmitted field with the output beam of an ideal second-order differentiator, we notice a shift between the dips of these two curves. This discrepancy is caused by the non-zero value of the transfer function of the proposed structure and also non-uniform phase of the transfer function. However, this discrepancy does not hinder our proposed second-order differentiator for its usage in edge detection applications, because, as shown in the Fig. 7, the amplitude of the electric field touches zero.

## 4. Conclusion

In this paper, a periodic ridge on a symmetric slab waveguide was used for designing an on-chip second-order spatial differentiator. The reflection and transmission coefficients of this structure show that the second-order derivative is performed in the transmission when the optical beam normally incidents on the periodic ridge. We demonstrated through full wave simulations that the excitation of the guided modes of the periodic ridge is the reason behind the second-order derivative.

## Funding

Research Office of Sharif University of Technology.

## Disclosures

The authors declare no conflicts of interest.

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