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 [13]. 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 [417], 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.

 

Fig. 1. Discrete space optical signal processing. (a) Illustration of the concept. An array of optical antennas is used to sample an impinging optical field in space. Then, the desired operation is performed on the sampled field through a network with a discrete number of input/output ports. (b) Realistic implementation of the proposed concept. The top apertures are used to sample an incoming wave profile. The sampled signals are supplied to a waveguide structure that realizes the desired mathematical operation. The composition of the waveguide may be complex, depending on the operation. Photodetectors collect the output waves at bottom openings. The lenses are used to increase the effective area of the apertures and enhance the efficiency of the system.

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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.

 

Fig. 2. Design of the sampling antenna. (a) Optical fields are sampled in space through apertures on a metallic wall. The apertures are supplemented with a leaky-wave air cavity and a lens to enhance their effective area and consequently maximize the amount of collected power. (b) Radiation pattern of the lens-aperture system. (c) Electric field for excitation with a plane wave with incident angle $\theta ={ 0^ \circ}$ (left) and $\theta ={ 3^ \circ}$ (right). The dimensions of the structures have the following values: $L = 22\lambda$, ${h_1} = 0.038\lambda$, ${h_2} = 0.45\lambda$, $W = \lambda /(2\sin {3^ \circ})$, where $\lambda$ is the wavelength in free space.

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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

$${T_{{\rm{array}}}} = \frac{1}{{kW}}\frac{{D(\theta)}}{{\cos \theta}},$$
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,
$$D(\theta) = \left\{{\begin{array}{*{20}{l}}{kW\cos \theta ,}&{- {\theta _{{\max}}} \lt \theta \lt {\theta _{{\max}}}}\\{0,}&{{\rm{otherwise}}}\end{array}} \right..$$
In reality, the radiation pattern is smoother, as in Fig. 2(b), and is designed so that its value at $\theta ={ 0^ \circ}$ is as large as possible. The antennas designed here exhibit a broadside directivity of 14.2 dB, which is about 3 dB smaller than the broadside directivity of 17.7 dB obtained through Eq. (2). Figure 2(b) shows the simulation results of the electric field intensity for $\theta ={ 0^ \circ}$ and $\theta ={ 3^ \circ}$. For both illumination angles, the extended hemispherical lens and the index-matching layer focus the incoming waves on the leaky-wave cavity. Although the focal point for $\theta ={ 3^ \circ}$ excitation is slightly shifted off to the right from the aperture, the leaky-wave cavity helps the wave to couple to the waveguide, thus mitigating this misalignment effect. Further explanations about the design of the lens antennas are provided in Supplement 1.

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

$${y_n} = \sum\limits_{k = - M}^N {h_k}{x_{n - k}},$$
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.
 

Fig. 3. Discrete port network for implementing a difference operation. (a) The network is based on a parallel-plate waveguide with periodic arrays of input/output channels at opposite walls. The collected signals from the antenna array are injected to the network through the top channels, while the processed signal is retrieved through the bottom ones. The difference operation is achieved by displacing the output channels compared to the input ones by an appropriately selected distance $d$. The obstacles in the vicinity of the T-junctions between the channels and the waveguide are added for matching purposes. The dimensions of the structure are as follows: $d = 0.477W$, ${w_{\rm{g}}} = 0.18\lambda$, $r = \frac{{{w_{\rm{g}}}}}{2} - 0.02{\lambda _{\rm{g}}}$, and ${w_{\rm{o}}} = 0.018{\lambda _{\rm{g}}}$, where ${\lambda _{\rm{g}}}$ is the guided wavelength inside the core waveguide. $W$ is the same as for the sampling array in Fig. 2. (b) Power transmission coefficient through the waveguide versus the incidence angle $\theta$ and the frequency. The center frequency is ${f_0} = 1.1{f_{\rm{c}}}$, where ${f_{\rm{c}}}$ is the cutoff frequency of the core waveguide.

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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

$${y_n} \approx \frac{1}{2}{x_{n + 1}}{e^{- j\beta (W - d)}} + \frac{1}{2}{x_n}{e^{- j\beta W}},$$
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
$${y_n} \approx {e^{- j\beta (W - d)}}({x_{n + 1}} - {x_n}),$$
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)
$$\begin{split} &\left\{{1 - \frac{1}{2}\cos \left[{\beta (W - 2d)} \right]{e^{- j\beta W}}} \right\}{y_n} - \frac{1}{4}{e^{- j\beta W}}\left({{y_{n + 1}} + {y_{n - 1}}} \right) \\& = j{e^{- j\beta W}}\left\{{{x_n}\sin \left[{\beta (W - d)} \right] + {x_{n + 1}}\sin (\beta d)} \right\}.\end{split}$$
The sinusoidal functions in the right-hand side of Eq. (6) are due to standing waves formed between the input channels. Such standing waves do not affect the operation of the structure, as under the condition $\beta (W - 2d) = (2m + 1)\pi$ the input term in Eq. (6) is still proportional to ${x_{n + 1}} - {x_n}$.
 

Fig. 4. Response of the full structure (antenna array and ensuing waveguide). (a) Amplitude and phase for the field transfer function, defined as the ratio of the output and incident electric fields. The results are obtained at the design frequency ${f_0}$. (b) Power transmission coefficient defined as the ratio between the output power at the channels at the bottom of the waveguide and the incident power over a period of the antenna array. The results are presented versus the frequency and the illumination angle.

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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)

$$\begin{split}&{T_{{\rm{wg}}}}({k_{\rm{t}}},\beta) \\[-3pt]&= \frac{{|{y_0}{|^2}}}{{|{x_0}{|^2}}} \\[-3pt]& = \frac{{{{\big| {{e^{j\frac{{\beta W}}{2}}}\cos \big[{\frac{{{k_{\rm{t}}}W\, -\, \beta (W \,-\, 2d)}}{2}} \big] - {e^{- j\frac{{\beta W}}{2}}}\cos \big[{\frac{{{k_{\rm{t}}}W \,+\, \beta (W \,-\, 2d)}}{2}} \big]} \big|}^2}}}{{|\Delta ({k_{\rm{t}}},\beta {{)|}^2}}},\end{split}$$
where
$$\Delta ({k_{\rm{t}}},\beta) = 1 - \frac{1}{2}\left\{{\cos ({k_{\rm{t}}}W) + \cos \left[{\beta (W - 2d)} \right]} \right\}{e^{- j\beta W}}.$$

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

$$\begin{split}{T_{{\rm{wg}}}}({k_{\rm{t}}},\beta) & \approx {T_{{\rm{wg}}}}({k_{\rm{t}}},{\beta _0}) \\& {+}\frac{{\cos^2 \left({\frac{{{k_{\rm{t}}}W}}{2}} \right) \sin^2 \left({\frac{{{\beta _0}W}}{2}} \right)}}{{|\Delta ({k_{\rm{t}}},{\beta _0}{{)|}^2}}}{[\Delta \beta (W - 2d)]^2},\end{split}$$
where $\Delta \beta = \beta - {\beta _0}$. For small ${k_{\rm{t}}}$, as here, $\cos^2 ({k_{\rm{t}}}W/2) \approx 1$, and the second term in Eq. (9) represents a constant shift of the transmission level depending on $\beta$ (frequency). Considering that $\beta = \sqrt {{k^2}{\varepsilon _{\rm{r}}} - {{(\pi /{w_{\rm{g}}})}^2}}$, where ${w_{\rm{g}}}$ is the width of the waveguide, we find $\Delta \beta \approx ({k_0}{\varepsilon _r}/{\beta _0})(2\pi \Delta f/c)$, where $\Delta f = f - {f_0}$ and $c$ is the speed of light. Assuming operation far from the waveguide cutoff frequency, it follows that the perturbation from the ideal response is proportional to $\Delta {f^2}\sqrt {{\varepsilon _r}} {(W - 2d)^2}$. Therefore, the bandwidth increases, if we reduce ${\varepsilon _r}$ or $W - 2d$, with the latter condition achieved by selecting $d$ as close as possible to $W/2$.

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.

 

Fig. 5. Response of the structure under illumination with a generic optical field. Three cases for the length of the input image are analyzed, $500\lambda$, $700\lambda$, and $900\lambda$, with the smallest length yielding an angular spectrum equal to the one of the structure. The results are derived for five different frequencies spanning the entire bandwidth of the structure. In all cases, the response is very close to the ideal difference response.

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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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13. T. J. Davis, F. Eftekhari, D. E. Gómez, and A. Roberts, “Metasurfaces with asymmetric optical transfer functions for optical signal processing,” Phys. Rev. Lett. 123, 013901 (2019). [CrossRef]  

14. J. Zhou, H. Qian, C.-F. Chen, J. Zhao, G. Li, Q. Wu, H. Luo, S. Wen, and Z. Liu, “Optical edge detection based on high-efficiency dielectric metasurface,” Proc. Natl. Acad. Sci. USA 116, 11137–11140 (2019). [CrossRef]  

15. A. Cordaro, H. Kwon, D. Sounas, A. F. Koenderink, A. Alù, and A. Polman, “High-index dielectric metasurfaces performing mathematical operations,” Nano Lett. 19, 8418–8423 (2019). [CrossRef]  

16. Y. Zhou, H. Zheng, I. I. Kravchenko, and J. Valentine, “Flat optics for image differentiation,” Nat. Photonics 14, 316–323 (2020). [CrossRef]  

17. Y. Zhou, W. Wu, R. Chen, W. Chen, R. Chen, and Y. Ma, “Analog optical spatial differentiators based on dielectric metasurfaces,” Adv. Opt. Mater. 8, 1901523 (2020). [CrossRef]  

18. A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8, 937–942 (2014). [CrossRef]  

19. H. Takesue, O. Tadanaga, Y. Haribara, K. Enbutsu, T. Sonobe, P. L. McMahon, K. Inoue, S. Tamate, A. Marandi, T. Honjo, K. Igarashi, H. Takenouchi, S. Utsunomiya, K.-I. Kawarabayashi, T. Umeki, K. Aihara, and T. Inagaki, “A coherent Ising machine for 2000-node optimization problems,” Science 354, 603–606 (2016). [CrossRef]  

20. Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017). [CrossRef]  

21. R. Hamerly, L. Bernstein, A. Sludds, M. Soljačić, and D. Englund, “Large-scale optical neural networks based on photoelectric multiplication,” Phys. Rev. X 9, 021032 (2019). [CrossRef]  

22. X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018). [CrossRef]  

23. T. W. Hughes, I. A. Williamson, M. Minkov, and S. Fan, “Wave physics as an analog recurrent neural network,” Sci. Adv. 5, eaay6946 (2019). [CrossRef]  

24. N. M. Estakhri, B. Edwards, and N. Engheta, “Inverse-designed metastructures that solve equations,” Science 363, 1333–1338 (2019). [CrossRef]  

25. F. Zangeneh-Nejad and R. Fleury, “Topological analog signal processing,” Nat. Commun. 10, 2058 (2019). [CrossRef]  

26. H. Stark, Application of Optical Fourier Transforms (Academic, 1982).

27. A. Antoniou, Digital Filters: Analysis, Design, and Signal Processing Applications (McGraw-Hill Education, 2018).

28. S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018). [CrossRef]  

29. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005).

30. D. F. Filipovic, S. S. Gearhart, and G. M. Rebeiz, “Double-slot antennas on extended hemispherical and elliptical silicon dielectric lenses,” IEEE Trans. Microwave Theory Tech. 41, 1738–1749 (1993). [CrossRef]  

31. A. Neto, “UWB, non dispersive radiation from the planarly fed leaky lens antenna—part 1: theory and design,” IEEE Trans. Antennas Propag. 58, 2238–2247 (2010). [CrossRef]  

32. A. Neto, S. Monni, and F. Nennie, “UWB, non dispersive radiation from the planarly fed leaky lens antenna–part II: demonstrators and measurements,” IEEE Trans. Antennas Propag. 58, 2248–2258 (2010). [CrossRef]  

33. S. Yang and A. E. Fathy, “Synthesis of a compound T-junction for a two-way splitter with arbitrary power ratio,” in IEEE MTT-S International Microwave Symposium Digest, 2005, Long Beach, California, USA, 2005, pp. 4 pp.

34. J. L. Santos and F. Farahi, Handbook of Optical Sensors (CRC Press, 2014).

35. D. R. Jackson, P. Burghignoli, G. Lovat, F. Capolino, J. Chen, D. R. Wilton, and A. A. Oliner, “The fundamental physics of directive beaming at microwave and optical frequencies and the role of leaky waves,” Proc. IEEE 99, 1780–1805 (2011). [CrossRef]  

References

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  1. A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–164 (2014).
    [Crossref]
  2. R. Athale and D. Psaltis, “Optical computing: past and future,” Opt. Photon. News 27(6), 32–39 (2016).
    [Crossref]
  3. D. R. Solli and B. Jalali, “Analog optical computing,” Nat. Photonics 9, 704–706 (2015).
    [Crossref]
  4. A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
    [Crossref]
  5. A. Youssefi, F. Zangeneh-Nejad, S. Abdollahramezani, and A. Khavasi, “Analog computing by Brewster effect,” Opt. Lett. 41, 3467–3470 (2016).
    [Crossref]
  6. W. Wu, W. Jiang, J. Yang, S. Gong, and Y. Ma, “Multilayered analog optical differentiating device: performance analysis on structural parameters,” Opt. Lett. 42, 5270–5273 (2017).
    [Crossref]
  7. T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
    [Crossref]
  8. C. Guo, M. Xiao, M. Minkov, Y. Shi, and S. Fan, “Photonic crystal slab Laplace operator for image differentiation,” Optica 5, 251–256 (2018).
    [Crossref]
  9. H. Kwon, D. Sounas, A. Cordaro, A. Polman, and A. Alù, “Nonlocal metasurfaces for optical signal processing,” Phys. Rev. Lett. 121, 173004 (2018).
    [Crossref]
  10. Y. Hwang, T. J. Davis, J. Lin, and X.-C. Yuan, “Plasmonic circuit for second-order spatial differentiation at the subwavelength scale,” Opt. Express 26, 7368–7375 (2018).
    [Crossref]
  11. D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezus, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26, 10997–11006 (2018).
    [Crossref]
  12. T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11, 034043 (2019).
    [Crossref]
  13. T. J. Davis, F. Eftekhari, D. E. Gómez, and A. Roberts, “Metasurfaces with asymmetric optical transfer functions for optical signal processing,” Phys. Rev. Lett. 123, 013901 (2019).
    [Crossref]
  14. J. Zhou, H. Qian, C.-F. Chen, J. Zhao, G. Li, Q. Wu, H. Luo, S. Wen, and Z. Liu, “Optical edge detection based on high-efficiency dielectric metasurface,” Proc. Natl. Acad. Sci. USA 116, 11137–11140 (2019).
    [Crossref]
  15. A. Cordaro, H. Kwon, D. Sounas, A. F. Koenderink, A. Alù, and A. Polman, “High-index dielectric metasurfaces performing mathematical operations,” Nano Lett. 19, 8418–8423 (2019).
    [Crossref]
  16. Y. Zhou, H. Zheng, I. I. Kravchenko, and J. Valentine, “Flat optics for image differentiation,” Nat. Photonics 14, 316–323 (2020).
    [Crossref]
  17. Y. Zhou, W. Wu, R. Chen, W. Chen, R. Chen, and Y. Ma, “Analog optical spatial differentiators based on dielectric metasurfaces,” Adv. Opt. Mater. 8, 1901523 (2020).
    [Crossref]
  18. A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8, 937–942 (2014).
    [Crossref]
  19. H. Takesue, O. Tadanaga, Y. Haribara, K. Enbutsu, T. Sonobe, P. L. McMahon, K. Inoue, S. Tamate, A. Marandi, T. Honjo, K. Igarashi, H. Takenouchi, S. Utsunomiya, K.-I. Kawarabayashi, T. Umeki, K. Aihara, and T. Inagaki, “A coherent Ising machine for 2000-node optimization problems,” Science 354, 603–606 (2016).
    [Crossref]
  20. Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
    [Crossref]
  21. R. Hamerly, L. Bernstein, A. Sludds, M. Soljačić, and D. Englund, “Large-scale optical neural networks based on photoelectric multiplication,” Phys. Rev. X 9, 021032 (2019).
    [Crossref]
  22. X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018).
    [Crossref]
  23. T. W. Hughes, I. A. Williamson, M. Minkov, and S. Fan, “Wave physics as an analog recurrent neural network,” Sci. Adv. 5, eaay6946 (2019).
    [Crossref]
  24. N. M. Estakhri, B. Edwards, and N. Engheta, “Inverse-designed metastructures that solve equations,” Science 363, 1333–1338 (2019).
    [Crossref]
  25. F. Zangeneh-Nejad and R. Fleury, “Topological analog signal processing,” Nat. Commun. 10, 2058 (2019).
    [Crossref]
  26. H. Stark, Application of Optical Fourier Transforms (Academic, 1982).
  27. A. Antoniou, Digital Filters: Analysis, Design, and Signal Processing Applications (McGraw-Hill Education, 2018).
  28. S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
    [Crossref]
  29. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005).
  30. D. F. Filipovic, S. S. Gearhart, and G. M. Rebeiz, “Double-slot antennas on extended hemispherical and elliptical silicon dielectric lenses,” IEEE Trans. Microwave Theory Tech. 41, 1738–1749 (1993).
    [Crossref]
  31. A. Neto, “UWB, non dispersive radiation from the planarly fed leaky lens antenna—part 1: theory and design,” IEEE Trans. Antennas Propag. 58, 2238–2247 (2010).
    [Crossref]
  32. A. Neto, S. Monni, and F. Nennie, “UWB, non dispersive radiation from the planarly fed leaky lens antenna–part II: demonstrators and measurements,” IEEE Trans. Antennas Propag. 58, 2248–2258 (2010).
    [Crossref]
  33. S. Yang and A. E. Fathy, “Synthesis of a compound T-junction for a two-way splitter with arbitrary power ratio,” in IEEE MTT-S International Microwave Symposium Digest, 2005, Long Beach, California, USA, 2005, pp. 4 pp.
  34. J. L. Santos and F. Farahi, Handbook of Optical Sensors (CRC Press, 2014).
  35. D. R. Jackson, P. Burghignoli, G. Lovat, F. Capolino, J. Chen, D. R. Wilton, and A. A. Oliner, “The fundamental physics of directive beaming at microwave and optical frequencies and the role of leaky waves,” Proc. IEEE 99, 1780–1805 (2011).
    [Crossref]

2020 (2)

Y. Zhou, H. Zheng, I. I. Kravchenko, and J. Valentine, “Flat optics for image differentiation,” Nat. Photonics 14, 316–323 (2020).
[Crossref]

Y. Zhou, W. Wu, R. Chen, W. Chen, R. Chen, and Y. Ma, “Analog optical spatial differentiators based on dielectric metasurfaces,” Adv. Opt. Mater. 8, 1901523 (2020).
[Crossref]

2019 (8)

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11, 034043 (2019).
[Crossref]

T. J. Davis, F. Eftekhari, D. E. Gómez, and A. Roberts, “Metasurfaces with asymmetric optical transfer functions for optical signal processing,” Phys. Rev. Lett. 123, 013901 (2019).
[Crossref]

J. Zhou, H. Qian, C.-F. Chen, J. Zhao, G. Li, Q. Wu, H. Luo, S. Wen, and Z. Liu, “Optical edge detection based on high-efficiency dielectric metasurface,” Proc. Natl. Acad. Sci. USA 116, 11137–11140 (2019).
[Crossref]

A. Cordaro, H. Kwon, D. Sounas, A. F. Koenderink, A. Alù, and A. Polman, “High-index dielectric metasurfaces performing mathematical operations,” Nano Lett. 19, 8418–8423 (2019).
[Crossref]

R. Hamerly, L. Bernstein, A. Sludds, M. Soljačić, and D. Englund, “Large-scale optical neural networks based on photoelectric multiplication,” Phys. Rev. X 9, 021032 (2019).
[Crossref]

T. W. Hughes, I. A. Williamson, M. Minkov, and S. Fan, “Wave physics as an analog recurrent neural network,” Sci. Adv. 5, eaay6946 (2019).
[Crossref]

N. M. Estakhri, B. Edwards, and N. Engheta, “Inverse-designed metastructures that solve equations,” Science 363, 1333–1338 (2019).
[Crossref]

F. Zangeneh-Nejad and R. Fleury, “Topological analog signal processing,” Nat. Commun. 10, 2058 (2019).
[Crossref]

2018 (6)

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659–670 (2018).
[Crossref]

X. Lin, Y. Rivenson, N. T. Yardimci, M. Veli, Y. Luo, M. Jarrahi, and A. Ozcan, “All-optical machine learning using diffractive deep neural networks,” Science 361, 1004–1008 (2018).
[Crossref]

C. Guo, M. Xiao, M. Minkov, Y. Shi, and S. Fan, “Photonic crystal slab Laplace operator for image differentiation,” Optica 5, 251–256 (2018).
[Crossref]

H. Kwon, D. Sounas, A. Cordaro, A. Polman, and A. Alù, “Nonlocal metasurfaces for optical signal processing,” Phys. Rev. Lett. 121, 173004 (2018).
[Crossref]

Y. Hwang, T. J. Davis, J. Lin, and X.-C. Yuan, “Plasmonic circuit for second-order spatial differentiation at the subwavelength scale,” Opt. Express 26, 7368–7375 (2018).
[Crossref]

D. A. Bykov, L. L. Doskolovich, A. A. Morozov, V. V. Podlipnov, E. A. Bezus, P. Verma, and V. A. Soifer, “First-order optical spatial differentiator based on a guided-mode resonant grating,” Opt. Express 26, 10997–11006 (2018).
[Crossref]

2017 (3)

W. Wu, W. Jiang, J. Yang, S. Gong, and Y. Ma, “Multilayered analog optical differentiating device: performance analysis on structural parameters,” Opt. Lett. 42, 5270–5273 (2017).
[Crossref]

T. Zhu, Y. Zhou, Y. Lou, H. Ye, M. Qiu, Z. Ruan, and S. Fan, “Plasmonic computing of spatial differentiation,” Nat. Commun. 8, 15391 (2017).
[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

2016 (3)

H. Takesue, O. Tadanaga, Y. Haribara, K. Enbutsu, T. Sonobe, P. L. McMahon, K. Inoue, S. Tamate, A. Marandi, T. Honjo, K. Igarashi, H. Takenouchi, S. Utsunomiya, K.-I. Kawarabayashi, T. Umeki, K. Aihara, and T. Inagaki, “A coherent Ising machine for 2000-node optimization problems,” Science 354, 603–606 (2016).
[Crossref]

R. Athale and D. Psaltis, “Optical computing: past and future,” Opt. Photon. News 27(6), 32–39 (2016).
[Crossref]

A. Youssefi, F. Zangeneh-Nejad, S. Abdollahramezani, and A. Khavasi, “Analog computing by Brewster effect,” Opt. Lett. 41, 3467–3470 (2016).
[Crossref]

2015 (2)

D. R. Solli and B. Jalali, “Analog optical computing,” Nat. Photonics 9, 704–706 (2015).
[Crossref]

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
[Crossref]

2014 (2)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–164 (2014).
[Crossref]

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8, 937–942 (2014).
[Crossref]

2011 (1)

D. R. Jackson, P. Burghignoli, G. Lovat, F. Capolino, J. Chen, D. R. Wilton, and A. A. Oliner, “The fundamental physics of directive beaming at microwave and optical frequencies and the role of leaky waves,” Proc. IEEE 99, 1780–1805 (2011).
[Crossref]

2010 (2)

A. Neto, “UWB, non dispersive radiation from the planarly fed leaky lens antenna—part 1: theory and design,” IEEE Trans. Antennas Propag. 58, 2238–2247 (2010).
[Crossref]

A. Neto, S. Monni, and F. Nennie, “UWB, non dispersive radiation from the planarly fed leaky lens antenna–part II: demonstrators and measurements,” IEEE Trans. Antennas Propag. 58, 2248–2258 (2010).
[Crossref]

1993 (1)

D. F. Filipovic, S. S. Gearhart, and G. M. Rebeiz, “Double-slot antennas on extended hemispherical and elliptical silicon dielectric lenses,” IEEE Trans. Microwave Theory Tech. 41, 1738–1749 (1993).
[Crossref]

Abdollahramezani, S.

Aihara, K.

H. Takesue, O. Tadanaga, Y. Haribara, K. Enbutsu, T. Sonobe, P. L. McMahon, K. Inoue, S. Tamate, A. Marandi, T. Honjo, K. Igarashi, H. Takenouchi, S. Utsunomiya, K.-I. Kawarabayashi, T. Umeki, K. Aihara, and T. Inagaki, “A coherent Ising machine for 2000-node optimization problems,” Science 354, 603–606 (2016).
[Crossref]

Alù, A.

A. Cordaro, H. Kwon, D. Sounas, A. F. Koenderink, A. Alù, and A. Polman, “High-index dielectric metasurfaces performing mathematical operations,” Nano Lett. 19, 8418–8423 (2019).
[Crossref]

H. Kwon, D. Sounas, A. Cordaro, A. Polman, and A. Alù, “Nonlocal metasurfaces for optical signal processing,” Phys. Rev. Lett. 121, 173004 (2018).
[Crossref]

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–164 (2014).
[Crossref]

Antoniou, A.

A. Antoniou, Digital Filters: Analysis, Design, and Signal Processing Applications (McGraw-Hill Education, 2018).

Athale, R.

R. Athale and D. Psaltis, “Optical computing: past and future,” Opt. Photon. News 27(6), 32–39 (2016).
[Crossref]

Baehr-Jones, T.

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005).

Bernstein, L.

R. Hamerly, L. Bernstein, A. Sludds, M. Soljačić, and D. Englund, “Large-scale optical neural networks based on photoelectric multiplication,” Phys. Rev. X 9, 021032 (2019).
[Crossref]

Bezus, E. A.

Bozhevolnyi, S. I.

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Analog computing using reflective plasmonic metasurfaces,” Nano Lett. 15, 791–797 (2015).
[Crossref]

Burghignoli, P.

D. R. Jackson, P. Burghignoli, G. Lovat, F. Capolino, J. Chen, D. R. Wilton, and A. A. Oliner, “The fundamental physics of directive beaming at microwave and optical frequencies and the role of leaky waves,” Proc. IEEE 99, 1780–1805 (2011).
[Crossref]

Byer, R. L.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8, 937–942 (2014).
[Crossref]

Bykov, D. A.

Capolino, F.

D. R. Jackson, P. Burghignoli, G. Lovat, F. Capolino, J. Chen, D. R. Wilton, and A. A. Oliner, “The fundamental physics of directive beaming at microwave and optical frequencies and the role of leaky waves,” Proc. IEEE 99, 1780–1805 (2011).
[Crossref]

Castaldi, G.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–164 (2014).
[Crossref]

Chen, C.-F.

J. Zhou, H. Qian, C.-F. Chen, J. Zhao, G. Li, Q. Wu, H. Luo, S. Wen, and Z. Liu, “Optical edge detection based on high-efficiency dielectric metasurface,” Proc. Natl. Acad. Sci. USA 116, 11137–11140 (2019).
[Crossref]

Chen, J.

D. R. Jackson, P. Burghignoli, G. Lovat, F. Capolino, J. Chen, D. R. Wilton, and A. A. Oliner, “The fundamental physics of directive beaming at microwave and optical frequencies and the role of leaky waves,” Proc. IEEE 99, 1780–1805 (2011).
[Crossref]

Chen, R.

Y. Zhou, W. Wu, R. Chen, W. Chen, R. Chen, and Y. Ma, “Analog optical spatial differentiators based on dielectric metasurfaces,” Adv. Opt. Mater. 8, 1901523 (2020).
[Crossref]

Y. Zhou, W. Wu, R. Chen, W. Chen, R. Chen, and Y. Ma, “Analog optical spatial differentiators based on dielectric metasurfaces,” Adv. Opt. Mater. 8, 1901523 (2020).
[Crossref]

Chen, W.

Y. Zhou, W. Wu, R. Chen, W. Chen, R. Chen, and Y. Ma, “Analog optical spatial differentiators based on dielectric metasurfaces,” Adv. Opt. Mater. 8, 1901523 (2020).
[Crossref]

Cordaro, A.

A. Cordaro, H. Kwon, D. Sounas, A. F. Koenderink, A. Alù, and A. Polman, “High-index dielectric metasurfaces performing mathematical operations,” Nano Lett. 19, 8418–8423 (2019).
[Crossref]

H. Kwon, D. Sounas, A. Cordaro, A. Polman, and A. Alù, “Nonlocal metasurfaces for optical signal processing,” Phys. Rev. Lett. 121, 173004 (2018).
[Crossref]

Davis, T. J.

T. J. Davis, F. Eftekhari, D. E. Gómez, and A. Roberts, “Metasurfaces with asymmetric optical transfer functions for optical signal processing,” Phys. Rev. Lett. 123, 013901 (2019).
[Crossref]

Y. Hwang, T. J. Davis, J. Lin, and X.-C. Yuan, “Plasmonic circuit for second-order spatial differentiation at the subwavelength scale,” Opt. Express 26, 7368–7375 (2018).
[Crossref]

Doskolovich, L. L.

Edwards, B.

N. M. Estakhri, B. Edwards, and N. Engheta, “Inverse-designed metastructures that solve equations,” Science 363, 1333–1338 (2019).
[Crossref]

Eftekhari, F.

T. J. Davis, F. Eftekhari, D. E. Gómez, and A. Roberts, “Metasurfaces with asymmetric optical transfer functions for optical signal processing,” Phys. Rev. Lett. 123, 013901 (2019).
[Crossref]

Enbutsu, K.

H. Takesue, O. Tadanaga, Y. Haribara, K. Enbutsu, T. Sonobe, P. L. McMahon, K. Inoue, S. Tamate, A. Marandi, T. Honjo, K. Igarashi, H. Takenouchi, S. Utsunomiya, K.-I. Kawarabayashi, T. Umeki, K. Aihara, and T. Inagaki, “A coherent Ising machine for 2000-node optimization problems,” Science 354, 603–606 (2016).
[Crossref]

Engheta, N.

N. M. Estakhri, B. Edwards, and N. Engheta, “Inverse-designed metastructures that solve equations,” Science 363, 1333–1338 (2019).
[Crossref]

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–164 (2014).
[Crossref]

Englund, D.

R. Hamerly, L. Bernstein, A. Sludds, M. Soljačić, and D. Englund, “Large-scale optical neural networks based on photoelectric multiplication,” Phys. Rev. X 9, 021032 (2019).
[Crossref]

Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
[Crossref]

Estakhri, N. M.

N. M. Estakhri, B. Edwards, and N. Engheta, “Inverse-designed metastructures that solve equations,” Science 363, 1333–1338 (2019).
[Crossref]

Fan, S.

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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Discrete space optical signal processing. (a) Illustration of the concept. An array of optical antennas is used to sample an impinging optical field in space. Then, the desired operation is performed on the sampled field through a network with a discrete number of input/output ports. (b) Realistic implementation of the proposed concept. The top apertures are used to sample an incoming wave profile. The sampled signals are supplied to a waveguide structure that realizes the desired mathematical operation. The composition of the waveguide may be complex, depending on the operation. Photodetectors collect the output waves at bottom openings. The lenses are used to increase the effective area of the apertures and enhance the efficiency of the system.
Fig. 2.
Fig. 2. Design of the sampling antenna. (a) Optical fields are sampled in space through apertures on a metallic wall. The apertures are supplemented with a leaky-wave air cavity and a lens to enhance their effective area and consequently maximize the amount of collected power. (b) Radiation pattern of the lens-aperture system. (c) Electric field for excitation with a plane wave with incident angle $\theta ={ 0^ \circ}$ (left) and $\theta ={ 3^ \circ}$ (right). The dimensions of the structures have the following values: $L = 22\lambda$, ${h_1} = 0.038\lambda$, ${h_2} = 0.45\lambda$, $W = \lambda /(2\sin {3^ \circ})$, where $\lambda$ is the wavelength in free space.
Fig. 3.
Fig. 3. Discrete port network for implementing a difference operation. (a) The network is based on a parallel-plate waveguide with periodic arrays of input/output channels at opposite walls. The collected signals from the antenna array are injected to the network through the top channels, while the processed signal is retrieved through the bottom ones. The difference operation is achieved by displacing the output channels compared to the input ones by an appropriately selected distance $d$. The obstacles in the vicinity of the T-junctions between the channels and the waveguide are added for matching purposes. The dimensions of the structure are as follows: $d = 0.477W$, ${w_{\rm{g}}} = 0.18\lambda$, $r = \frac{{{w_{\rm{g}}}}}{2} - 0.02{\lambda _{\rm{g}}}$, and ${w_{\rm{o}}} = 0.018{\lambda _{\rm{g}}}$, where ${\lambda _{\rm{g}}}$ is the guided wavelength inside the core waveguide. $W$ is the same as for the sampling array in Fig. 2. (b) Power transmission coefficient through the waveguide versus the incidence angle $\theta$ and the frequency. The center frequency is ${f_0} = 1.1{f_{\rm{c}}}$, where ${f_{\rm{c}}}$ is the cutoff frequency of the core waveguide.
Fig. 4.
Fig. 4. Response of the full structure (antenna array and ensuing waveguide). (a) Amplitude and phase for the field transfer function, defined as the ratio of the output and incident electric fields. The results are obtained at the design frequency ${f_0}$. (b) Power transmission coefficient defined as the ratio between the output power at the channels at the bottom of the waveguide and the incident power over a period of the antenna array. The results are presented versus the frequency and the illumination angle.
Fig. 5.
Fig. 5. Response of the structure under illumination with a generic optical field. Three cases for the length of the input image are analyzed, $500\lambda$, $700\lambda$, and $900\lambda$, with the smallest length yielding an angular spectrum equal to the one of the structure. The results are derived for five different frequencies spanning the entire bandwidth of the structure. In all cases, the response is very close to the ideal difference response.

Equations (9)

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T a r r a y = 1 k W D ( θ ) cos θ ,
D ( θ ) = { k W cos θ , θ max < θ < θ max 0 , o t h e r w i s e .
y n = k = M N h k x n k ,
y n 1 2 x n + 1 e j β ( W d ) + 1 2 x n e j β W ,
y n e j β ( W d ) ( x n + 1 x n ) ,
{ 1 1 2 cos [ β ( W 2 d ) ] e j β W } y n 1 4 e j β W ( y n + 1 + y n 1 ) = j e j β W { x n sin [ β ( W d ) ] + x n + 1 sin ( β d ) } .
T w g ( k t , β ) = | y 0 | 2 | x 0 | 2 = | e j β W 2 cos [ k t W β ( W 2 d ) 2 ] e j β W 2 cos [ k t W + β ( W 2 d ) 2 ] | 2 | Δ ( k t , β ) | 2 ,
Δ ( k t , β ) = 1 1 2 { cos ( k t W ) + cos [ β ( W 2 d ) ] } e j β W .
T w g ( k t , β ) T w g ( k t , β 0 ) + cos 2 ( k t W 2 ) sin 2 ( β 0 W 2 ) | Δ ( k t , β 0 ) | 2 [ Δ β ( W 2 d ) ] 2 ,

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