## Abstract

As digital circuits are approaching the limits of Moore’s law, a great deal of effort has been directed to alternative computing approaches. Among them, the old concept of optical signal processing (OSP) has attracted attention, revisited in the light of metamaterials and nano-photonics. This approach has been successful in realizing basic mathematical operations, such as derivatives and integrals, but it is difficult to be applied to more complex ones. Here, inspired by digital filters, we propose a radically new OSP approach, able to realize arbitrary mathematical operations over a nano-photonic platform. Our concept consists in first sampling an optical signal in space through an array of optical antennas and then realizing the desired mathematical operation in discrete space through a network with a discrete number of input and output ports. The design of such network boils down to the design of a structure with a given scattering matrix, which for arbitrarily complex operations can be accomplished through inverse design algorithms. We demonstrate this concept for the case of spatial differentiation through a heuristic design based on a waveguide with periodic arrays of input/output channels at its opposite walls. Our approach combines the robustness and generality of traditional Fourier-based OSP with the compactness of nano-photonics and has the potential of transforming the design of OSP systems with applications in image processing and analog computing.

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

## 1. INTRODUCTION

Image processing is an essential part of modern technology, with applications in autonomous cars, medical devices, computer vision systems, and augmented reality interfaces. An image processing system typically consists of two parts, an optical unit collecting impinging light and converting it to electrical signals and a digital one for processing the collected signals. The optical unit is usually a combination of lenses and photodetectors, and it operates at very high speed, theoretically the speed of light, and minimal power consumption. On the other hand, the digital unit is built up with billions of transistors, which consume power and have operation speed restrictions. Given that integrated circuits are quickly approaching the limits of Moore’s law, it is clear that an all-digital approach can hardly meet the constantly increasing demands for smaller, faster, and power-efficient devices. For this reason, the old field of optical signal processing (OSP) has recently attracted renewed attention [1–3]. Replacing all or some of the digital processing unit with an optical component utilizing only light to carry out the desired processing operation may allow us to largely overcome some of the speed restrictions and power consumption problems of existing digital approaches. This paradigm has quickly been gaining traction and has successfully been explored for applications in edge detection [4–17], optimization [18,19], machine learning [20,21], pattern recognition [22,23], and analog computation [24,25].

Conventional OSP is based on Fourier optics and consists of converting an optical signal to Fourier space through lenses, performing the desired operation in the Fourier space through partially transparent screens or metasurfaces, and finally converting it back to the spatial domain through another set of lenses [26]. Despite its robustness and generality, this approach leads to bulky devices and, therefore, is not very attractive for modern systems. As a solution to this problem, several recent studies have explored metasurfaces, with the goal of tailoring their nonlocal response to the spatial Fourier transform of a desired operation. Such a task can, for example, be achieved through leaky-wave resonances, taking advantage of their strong spatial dispersion [1,6,9,13,15,16]. Another option is using the Brewster effect or the geometrical phase, which leads to broadband responses, but with less flexibility over the realized nonlocal response [5,7,8,11,12,14,17]. Metasurface approaches have so far been limited to simple operations, like differentiation or integration, most likely because they rely on physical mechanisms with specific nonlocal responses that are difficult to be matched to complex mathematical operations. Furthermore, since they often depend on resonant effects, they are subject to limitations in terms of bandwidth versus nonlocality, imposing another restriction on the responses that can be achieved.

Here, we introduce a radically new approach to OSP, with the ability to implement general mathematical operations without the bandwidth limitations of other approaches. To this end we borrow inspiration from digital filters, where a signal is first discretized in time, then the desired operation is performed in the digital domain, and finally the output signal is converted back to the analog domain [27]. Following this paradigm, we propose a system where an optical signal is sampled in space through an array of antennas and subsequently supplied to a nano-photonic network with a discrete number of input and output ports that performs the desired mathematical operation [Fig. 1(a)]. The output of this network is a discrete signal in space and can be directly supplied to an array of photodetectors, thus saving us from having to convert the signal back to the analog domain, as in conventional digital filters. Similar to digital filters, our system offers great versatility for implementing general operations, since its design boils down to the design of a discrete port network with a given scattering matrix, which, as has been recently shown, can be efficiently executed through inverse design algorithms [24,28]. Here, we demonstrate the proposed paradigm for the case of edge detection (differentiation) via a structure consisting of an array of lens antennas for sampling the signal in space and a waveguide with periodic arrays of input and output apertures for implementing the desired mathematical operation. Our approach is characteristic for its generality and opens a new path in OSP with applications in analog image processing and, more broadly, analog computing.

## 2. RESULTS

A simple way to realize sampling in space is through a periodic array of apertures on a metallic layer, as in Fig. 1(b). According to the Nyquist–Shannon theorem, the array periodicity needs to be selected as $W \le \pi /{k_{{\rm{t,max}}}}$, where ${k_{{\rm{t,max}}}}$ is the maximum transverse wavenumber of the impinging optical signal, which is the analogue of angular frequency in the spatial domain. Any value below this limit is permitted, but operating exactly on the limit, leads to maximum efficiency for the antenna array (Supplement 1) and to the simplest form for the spatial filter. Apertures on a metallic layer are effective magnetic dipoles with an effective aperture approximately equal to $\lambda /2$, $\lambda$ being the wavelength in free space [29]. Since ${k_{{\rm{t,max}}}} \lt k$, where $k$ is the wavenumber in free space, we find from $W = \pi /{k_{{\rm{t,max}}}}$ that $W \gt \lambda /2$, meaning that the apertures can only capture a fraction of the impinging power, limiting the system’s efficiency. An array of lenses on top of the apertures solves this problem and improves the efficiency of the whole structure. Using Fermat’s principle, it can be shown that an ellipsoidal lens focuses a bundle of normally incident rays to one of the ellipse’s focal points [30]. Keeping this fact in mind, we position the openings at the lenses’ focal points. A quarter-wavelength coating layer of relative permittivity ${\epsilon _{{\rm{coat}}}} = \sqrt {{\epsilon _{{\rm{lens}}}}}$, where ${\epsilon _{{\rm{lens}}}}$ is the relative permittivity of the lens, is also added on the surface of the lens to eliminate air-lens impedance mismatch and multiple internal reflections. In practice, the ellipsoidal lens is approximated as an extended semi-spherical lens. When the lenses are introduced in the periodic aperture array, they have to be truncated to a width matching the array periodicity. This leads to spillover loss and reduction of the power captured by the apertures. This fact can be better understood by examining the apertures in the transmitting mode and taking into account that due to reciprocity the transmitting and receiving radiation patterns are identical. When the apertures are operated as transmitters, the lenses capture only the portion of the radiated power inside the cones defined by the apertures and the lenses, thus limiting their radiation and receiving efficiency. To overcome this problem, we add a low-index (here air) dielectric slab with a height of approximately half a wavelength between the aperture and the lens, designed to support a broadband leaky-wave resonance with the ability to focus the radiated power from the aperture within the aperture-lens cone [31,32]. From reciprocity, it directly follows that such a strategy also increases the receiving efficiency of the apertures.

Based on the above remarks, we have designed a two-dimensional antenna as in Fig. 2(a), assuming an out-of-plane [transverse-electric (TE)] electric field polarization. The lens is made of a material with a relative permittivity ${\epsilon _{{\rm{lens}}}} = 9$ and is approximated as a spherical sector. The apertures are connected to short channels in the metallic layer with the same width as the apertures, to guide the received signals to the processing part of the system. The channels are filled with a dielectric material ${\varepsilon _{\rm{r}}} = 9$. In order to avoid impedance mismatch between the leaky-wave air cavity and the channels, we slightly extend the dielectric material of the channels to the top of the aperture array. The system is designed for optical fields with angular spread from ${-}{3^ \circ}$ to 3°, which corresponds to ${k_{{\rm{t,max}}}} = k\sin {\theta _{{\max}}}$, where ${\theta _{{\max}}}={ 3^ \circ}$. This value was selected so that at 600 nm (approximately the middle of the optical spectrum), $W = 5.7\;{{\unicode{x00B5}{\rm m}}}$, which is a typical pixel size in commercial imaging systems. The receiving efficiency of the sampling array for an incident plane wave from an angle $\theta$ with respect to the normal direction is given by

where $D(\theta)$ is the radiation pattern of the array elements (Supplement 1). The receiving efficiency becomes maximum (unitary), if the array elements are designed to have a radiation pattern,The second part of the system is a linear optical network performing the desired mathematical operation. Here, this network is a waveguide formed between two metallic walls with an arbitrary material composition depending on the operation that has to be realized, as shown in Fig. 1(b). The waveguide is connected to the antenna array through short channels on its top wall, from where the antenna signals ${x_n}$ are injected. On the bottom wall of the waveguide, there is another array of channels, from where the output signals ${y_n}$ are collected. Both arrays have the same periodicity $W$. Similar to digital filters, the output signals are given by an expression of the form

where $M,N$ are integers and ${h_k}$ is the impulse response of the structure. Equation (3) states that the output signal at the $n$th output aperture is a linear superposition of $M + N$ signals over the top apertures, essentially providing the scattering matrix of the network. Since we are dealing with signals in space, we are not limited by causality; hence, the bottom limit in Eq. (3) can be a negative number. Quite importantly, Eq. (3) describes a translationally symmetric structure, as required to make sure that the response is insensitive to the lateral location of the incident signal with respect to the structure.Taking into account the above remarks, we have designed a system performing a discrete derivative, i.e., difference, operation ${y_n} = {x_{n + 1}} - {x_n}$. To this end, we use a waveguide filled with a uniform dielectric material ${\epsilon _{\rm{r}}} = 9$, which is the same material as for the input/output channels, and output channels displaced by a distance $d$ with respect to the input ones, as shown in Fig. 3(a). Assuming a nearest neighbor approximation, the signal at the $n$th output channel is approximately equal to

where $\beta$ is the wavenumber inside the waveguide and the $\frac{1}{2}$ factor is due to the fact that the input signals are split to two parts propagating in opposite directions when they enter the waveguide. If $d$ is selected such that $\beta (W - 2d) = (2m + 1)\pi$, where $m$ is an integer, the output signal becomes yielding the desired difference operation. From all possible values for $d$, we select the one that is closer to $W/2$, because, as will be shown later, it leads to maximum bandwidth. In order to eliminate reflections at the T-junctions between the input/output channels and the waveguide, we have added carefully designed metallic obstacles in the vicinity of the junctions as shown in Fig. 3(a), following the technique in [33]. A more rigorous analysis beyond the nearest neighbor assumption reveals that the spatial filter is described by the equation (Supplement 1)To gain further insight into the operation of the network, we study its response under plane wave illumination with incidence angle $\theta$ and transverse wavenumber ${k_{\rm{t}}} = k\sin \theta$. Then, the signals at the input channels of the subtracting network have the form ${x_n} = {x_0}{e^{- jn{k_{\rm{t}}}W}}$, where ${x_0}$ is a complex constant that is proportional to the incident field amplitude. Substituting this equation to Eq. (5), we find after some simple algebraic manipulations $|{y_n}{|^2} \approx \sin^2 \left({\frac{{{k_{\rm{t}}}W}}{2}}\right)|{x_0}{|^2}$. We see that for ${k_{\rm{t}}} = 0$, the output signals are zero and increase as ${k_{\rm{t}}}$ increases, as expected from a derivative operation. The output signal takes its maximum value for ${k_{\rm{t}}} = \pi /W$, which, as discussed earlier, is the upper value of the transverse wavenumber dictated by the Nyquist–Shannon sampling theorem. This fact shows that sampling according to the Nyquist–Shannon bound leads to the most efficient use of the structure. If sampling was faster, i.e., $W \lt \pi /{k_{x,{\max}}}$, the output from the subtracting network at the maximum wavenumber of the input image would be less than its maximum possible value, reducing the output signal intensity. Since we are dealing with passive structures, boosting the output signal in this case would require using a larger number of input signals beyond the nearest neighbor ones, increasing the order and, therefore, the complexity of the filter. Following the analysis used for the derivation of Eq. (6), we find that the full expression for the transmission coefficient valid at any frequency is given by (see Supplement 1)

Figure 3(b) presents transmission versus frequency and incident angle obtained through Eq. (7) and full-wave simulations through Comsol Multiphysics for a structure designed to work for a maximum incident angle ${\theta _{{\max}}}={ 3^ \circ}$. The theoretical and numerical results are in excellent agreement with each other. The response is symmetric with respect to $\theta ={ 0^ \circ}$, with transmission being close to zero at $\theta ={ 0^ \circ}$ and increasing as $\theta$ increases, as it should be for a difference operation. The resonant features are due to the standing-wave resonances between the input channels [the sinusoidal terms in the right-hand side in Eq. (6)]. At the center frequency ${f_0}$, i.e., the frequency for which $\beta (W - 2d) = (2m + 1)\pi$ is satisfied, the angle of maximum transmission is $\,\theta = \pm {3^ \circ}$, per design, while as we move further from ${f_0}$, this angle slightly decreases. This response is maintained over a bandwidth of about 5% about ${f_0}$ [roughly speaking, the bandwidth of the center dark region in Fig. 3(b)], which is among the highest among all structures reported so far for edge detection. To gain a better understanding about the factors determining the bandwidth, we calculate the Taylor expansion of Eq. (6) with respect to $\beta$ about its value ${\beta _0}$ at ${f_0}$. Keeping only the lowest-order terms, the result reads

Next, we discuss the response of the complete structure, consisting of both the lens array and the subtracting waveguide. Figure 4(a) shows the magnitude and phase of the transfer function ${T_{{\rm{field}}}} = {E_{{\rm{out}}}}/{E_{{\rm{inc}}}}$ between the incident field ${E_{{\rm{inc}}}}$ and the output field ${E_{{\rm{out}}}}$ at the bottom channels of the waveguide versus the incidence angle at the center frequency ${f_0}$. The response was obtained through full-wave simulation with Comsol Multiphysics. In the same figure, we also plot the response expected for an ideal difference operation. The agreement between the ideal and numerical results is excellent. The intensity of transfer function is symmetric with respect to $\theta ={ 0^ \circ}$, as expected from Eq. (7) and in agreement with the ideal response. On the other hand, the phase is odd symmetric with respect to $\theta ={ 0^ \circ}$ as required for a first-derivative operation. Note that peak transmission is greater than 1, because the system funnels the incident power over one period to the much smaller area of the output channels. Figure 4(b) shows the power transmission coefficient ${T_{{\rm{power}}}} = {P_{{\rm{out}}}}/{P_{{\rm{inc}}}}$ versus the frequency and the incident angle, where ${P_{{\rm{out}}}}$ is the output power at the bottom channel of the waveguide and ${P_{{\rm{inc}}}}$ is the incident power within one period of the antenna array. Note that here the transmission coefficient is smaller than one, because due to passivity, the output power is always smaller than the incident one. The response is almost the same as for the subtracting waveguide in Fig. 3, apart from a scaling to lower values, which is due to the non-unitary efficiency of the antenna array.

To better see the effect of the metasurface on an incident image, Fig. 5 presents the output from the structure when the input is a one dimensional image that includes constant, ramp, parabolic, and sinusoidal functions. The results have been obtained for a broad range of frequencies and for three different image lengths. The image is selected so that for the smaller length, the image’s maximum spatial frequency is equal to the system’s ${k_{{\rm{t,max}}}}$. Obviously, increasing the image length leads to a narrower spatial spectrum. The output images were obtained by multiplying the spatial Fourier transform of the input signal with the system’s transfer function and converting the result to spatial domain through an inverse Fourier transform. The proposed system is capable of performing high-quality edge detection for all lengths and frequencies, with the results being the best at the center frequency ${f_0}$, as expected. Note the proper edge detection performance with this range of frequency bandwidth and over a momentum spectrum with this level of proximity to null-momentum value, as is the case of projected images on typical sensor arrays, is pretty rare in the literature. As will be explained in the following, this fact makes the proposed system particularly suitable for integration with sensor arrays.

## 3. CONCLUSION

The proposed approach can be seamlessly integrated with sensor arrays, if the periodicity of the system is selected to be equal to the pixel size of the array and the output apertures are aligned with the array sensors. In that sense, the proposed system shares similarities with the broadly used Bayer filters, where each sensor is supplied with an extra absorptive layer or more sophisticated nanostructures, e.g., Fabry–Perot resonators, to achieve color separation. Furthermore, it has similarities with the Hartmann–Shack wavefront sensor, which is realized through an array of lenses on top of an array of photodetectors [34]. The proposed system can be considered as a generalization of a Hartmann–Shack wavefront sensor, by adding an OSP block between the lenses and the photodetectors to alter the functionality of the system from simple image registration, as in the case of Hartmann–Shack sensors, to a system that is able of performing mathematical operations.

It is important to mention that our approach is more suitable for integration with arrays of photodetectors than other OSP approaches, like those based on nonlocal metasurfaces. Nonlocal metasurfaces are typically designed with a relatively large numerical aperture (NA), in the order of 0.3, corresponding to a maximum transverse wavenumber ${k_{{\rm{t,max}}}} = 0.3k$. On the other hand, a typical camera with a pixel size $\Delta = 5\;{{\unicode{x00B5}{\rm m}}}$ operating at the middle of the visible spectrum, $0.5\;{{\unicode{x00B5}{\rm m}}}$, can resolve images with a maximum wavenumber ${k_{{\rm{t,max}}}} = 1/(2\Delta) = 0.008k$, more than 2 orders of magnitude smaller than the maximum wavenumber of the metasurface, showing that only a very small fraction of the spatial spectrum of the metasurface around ${k_{\rm{t}}} = 0$ is actually used, which leads to small efficiency, since transmission around ${k_{\rm{t}}} = 0$ is very small. In principle, this problem can be overcome by designing the metasurface to have a small NA or, alternatively speaking, a spatial spectrum matching the one of the sensor array. This would require very strong nonlocality, since transmission would have to change from 0 to 1 over a very small ${k_{\rm{t}}}$ range. For structures based on leaky-wave resonances, such strong nonlocality would necessarily translate to a smaller bandwidth, since the leaky-wave would have to propagate over a longer distance along the metasurface before being converted to radiation, which is only possible by reducing the radiation decay rate and consequently the bandwidth. Our structure is free from this trade-off, because waves propagate as guided modes along the core waveguide of the structure.

Another advantage of the proposed approach is that in principle it can be adapted to any linear or even nonlinear operation, by appropriately designing the core waveguide to yield a scattering matrix corresponding to the desired operation. Such a task could be effectively accomplished through inverse design algorithms [24,28]. It is important to mention that there are different ways one could follow to realize the proposed approach, beyond the structure presented here. For example, one possibility might be to use an array of nano-antennas, such as dipole antennas, on top of a dielectric slab supporting a guided wave mode, with the slab playing the role of the waveguide used in our implementation. Such an implementation might lead to lower loss compared to a plasmonic one. Another possibility might consist in replacing the lenses in our approach with a grating on the top surface of the metallic or dielectric layer to realize a leaky-wave antenna with a directive broadside radiation pattern [35]. In general, our approach combines the generality of Fourier-based OSP with the compactness of nano-photonics and opens new possibilities for analog processing of optical waves in nanometric scale with applications in image processing and, more broadly, analog computing.

## Acknowledgment

The authors would like to thank Helia Moeini for her assistance with the generation of some of the illustrations in the paper.

## Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

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