Laplace metasurfaces for optical analog computing based on quasi-bound states in the continuum

Danping Pan; Lei Wan; Min Ouyang; Wei Zhang; Alexander A. Potapov; Weiping Liu; Zixian Liang; Tianhua Feng; Zhaohui Li

doi:10.1364/PRJ.426827

Abstract

Laplace operation, the isotropic second-order differentiation, on spatial functions is an essential mathematical calculation in most physical equations and signal processing. Realizing the Laplace operation in a manner of optical analog computing has recently attracted attention, but a compact device with a high spatial resolution is still elusive. Here, we introduce a Laplace metasurface that can perform the Laplace operation for incident light-field patterns. By exciting the quasi-bound state in the continuum, an optical transfer function for nearly perfect isotropic second-order differentiation has been obtained with a spatial resolution of wavelength scale. Such a Laplace metasurface has been numerically validated with both 1D and 2D spatial functions, and the results agree well with that of the ideal Laplace operation. In addition, the edge detection of a concerned object in an image has been demonstrated with the Laplace metasurface. Our results pave the way to the applications of metasurfaces in optical analog computing and image processing.

1. INTRODUCTION

Optical analog computing has attracted intensive attention recently because it holds the promise to tackle the issues of low speed, high power consumption, and complexity in traditional digital signal processing with electric solutions, especially for the scenarios that require real-time and high throughput [1 –3]. As an efficient platform for realizing optical analog computing, artificially engineered photonic structures have been proposed and demonstrated to perform spatial differentiation, integration, convolution, even equation solving, and so on [4 –9]. Of particular importance, differentiation is one of the most fundamental mathematical operations, and its realization has gained much research interest. Various principles have been exploited, including metasurface [10 –16], Brewster effect [17,18], and surface plasmon polariton [19 –22], among others [23 –28]. By employing the optical analog differentiation on spatial functions, edge detection and further processing for images can be conducted in an efficient manner [29 –33]. Therefore, optical analog computing can facilitate the development of image-processing technology together with electronic platforms.

Among differentiation operators, the Laplacian is a basic and important one that performs the isotropic second-order differentiation. It plays a necessary role in image processing for applications from medical imaging to object detection. It is also an essential operator in various fields of physical science and engineering, for example, wave phenomena of optics and vibration, diffusion process of carriers in semiconductors [34]. The realization of optical analog Laplace operator may offer a powerful tool for solving complex and large-scale mathematical problems. Nevertheless, few works have been reported because it is a challenge to realize the same second-order differentiation for every direction even in a 2D plane. Recently, a multilayer phase-shifted Bragg grating has been proposed [35]. The multiple reflections at different interfaces facilitate the optical transfer function (OTF) as a quadratic function of an in-plane wave vector, thus normally reflecting an incident Gaussian beam into a Laguerre–Gaussian beam. In order to minimize the bulky size of the devices, a photonic crystal slab system has been proposed [36]. The realization of Laplace operation relies on the excitation of a guided resonance by carefully tuning the photonic crystal size parameters and the distance between the photonic crystal slab and the uniform slab. A nearly perfect Laplacian and excellent edge detection of an image have been demonstrated, but the separated-slab system presents challenges for practical fabrication and applications. Meanwhile, it works only for a tiny incident angle and leads to a small numerical aperture (NA) of about 0.01, introducing rigorous requirements on the light path and hampering the improvement of spatial resolution to the wavelength scale.

In this paper, we introduce a single-layer dielectric metasurface that can perform Laplace operation nearly perfectly, the simple design of which can also release the restrictions of fabrications and applications. Most importantly, the maximum incident angle has been widely expanded, and the NA can approach 0.14, benefiting the practical setup and leading to a spatial resolution about 4 times the working wavelength. The proposed Laplace metasurface relies on the excitation of bound state in the continuum (BIC), which has been demonstrated with exotic optical properties [37 –42]. The highly symmetric mode profile enables a nearly isotropic OTF of Laplace operation. We have demonstrated that such an OTF can not only produce the second-order differentiation results for a 1D spatial Gaussian function, but also output the correct results for a 2D Bessel function. In addition, we have also shown that the Laplace metasurface can be employed to realize edge detection of objects of concern in an image. The proposed Laplace metasurface can be tuned to work at different wavelengths in a transmission mode, thus benefiting the applications of optical computing, medical diagnostics, machine vision, and so on.

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2. PRINCIPLE AND DESIGN

The optical analog computing of Laplace operation for a 2D spatial function can be expressed in the Cartesian coordinate as $\nabla^{2} E (x, y) = \partial^{2} E (x, y) / \partial x^{2} + \partial^{2} E (x, y) / \partial y^{2},$ (1)where $\nabla^{2}$ is the Laplace operator and $E (x, y)$ is the electric field profile of the input optical signal. The input light field can be written as the combination of plane waves with different wave vectors, $E (x, y) = \iint \tilde{E} (k_{x}, k_{y}) e^{i (k_{x} x + k_{y} y)} d k_{x} d k_{y},$ (2)where $\tilde{E} (k_{x}, k_{y})$ is the weight of plane waves and $k_{x}$ and $k_{y}$ are the wave vectors along the two orthogonal axes. Consequently, the Laplace operation on a light field is $\nabla^{2} E (x, y) = - k_{∥}^{2} E (x, y),$ (3)where $k_{∥}^{2} = k_{x}^{2} + k_{y}^{2}$ and is the in-plane wave vector. It means that the OTF for the Laplace metasurface should be $T (k_{∥}) = - k_{∥}^{2}$ , which is a quadratic function of the in-plane wave vector of the input light field. Because $k_{∥} = k \sin θ$ , where $k$ is the incident wave vector and $θ$ is the incident angle, the transmission of the Laplace metasurface should be zero for normal incidence, while it should be increased when the incident angle is increased.

In order to satisfy the requirements of the OTF and the transmission properties, we have designed a novel Laplace metasurface, which is shown in Fig. 1(a). When passing through the Laplace metasurface, the input light field will be automatically processed, and the results of Laplace operation can be obtained just by recording the output light field in a simple and instantaneous way. The Laplace metasurface consists of a square lattice of a modified Si brick on a glass substrate, which is shown in Fig. 1(b). The high symmetry of square lattice and the defects at the four faces of Si brick ensure an isotropic OTF for in-plane directions. We have calculated the transmittance spectra of the Laplace metasurface at different incident angles and shown them in Fig. 1(c). It can be found that a resonance exhibits at the wavelength of approximately 740 nm when the light field is incident obliquely. The transmittance increases as the incident angle is increased. As a result, it provides the necessary condition to realize the Laplace operation.

Figure 1.(a) Illustration of a dielectric metasurface transforming an input 2D spatial function to another function as a Laplace operator; (b) unit cell of the dielectric metasurface. Left, 3D view of the unit cell. It consists of a Si brick (light blue color) with a thickness of $h = 163 nm$ and a glass substrate (gray color). Right, top view of the unit cell. The period is $a = 331 nm$ , and the width of Si brick is $d = 251 nm$ . There are four square voids with a width of $s = 33 nm$ at the centers of all edges. (c) Transmission coefficient spectra of the Laplace metasurface at different incident angles along $x$ direction for a $p$ wave.

To unambiguously explain the dependence of the transmittance versus the incident angle for the Laplace metasurface, we have calculated the dispersion bands around 740 nm of the proposed dielectric metasurface, which are shown in Fig. 2(a). To capture the essential mechanism, we considered a lossless Si metasurface with a refractive index of 3.73 that was fitted from the experimental results [43]. Because the differentiation for the Laplace operator is isotropic, we consider both $Γ X$ and $Γ M$ directions of the irreducible Brillouin zone. It can be found that the middle band highlighted with red color is nearly the same along both directions at around the $Γ$ point, promising the isotropic property. The corresponding quality factors of the five bands are shown in Fig. 2(b). For the quasi-BIC band highlighted with red color, the quality factor approaches infinity at the $Γ$ point, indicating a BIC that cannot be accessed by the plane waves outside. Away from the $Γ$ point, the BIC becomes a quasi-BIC, and it can partially couple with the incident plane wave, as the corresponding quality factor is finite. Meanwhile, other bands feature lower quality factors, and they can also couple with the incident wave. The BIC at the $Γ$ point is protected by symmetry. This can be found by examining the mode patterns, which are shown in Fig. 2(c). For the normal component of the electric field of the mode shown at the top panel, it is even under the operation of 180° rotation around the normal direction ( $C_{2}$ ) of the metasurface, while it is odd for the plane waves. In fact, the symmetry of the mode profile is necessary for achieving isotropic OTF. Furthermore, it can be found that the normal component dominates the electric field, as shown at the bottom panel of Fig. 2(c). Therefore, the quasi-BIC can be accessed by the $p$ -polarization plane waves with an incident angle. In addition, we have calculated the transmittance spectra as functions of incident angle along both $Γ X$ and $Γ M$ directions, which are shown in Figs. 2(d) and 2(e). It can be found that at the target wavelength of 740 nm, the quasi-BIC is excited gradually, with an increased transmittance as the incident angle increases. The resonating wavelength of the quasi-BIC shifts little, but the spectral bandwidth is extended, showing agreement with the calculated dispersion band structure and $Q$ factors presented in Figs. 2(a) and 2(b).

Figure 2.(a) Dispersion bands around 740 nm of the proposed dielectric metasurface Laplace operator in both $Γ X$ and $Γ M$ directions. The band of interest highlighted with red corresponds to the quasi-BIC. The inset shows the irreducible Brillouin zone. (b) Corresponding Q factors of the five bands in (a). The red color corresponds to the quasi-BIC band. (c) $E_{z}$ field distribution of the BIC at the $Γ$ point. Top panel, top view at the central plane of the unit cell; bottom panel, cross-sectional view at the central plane of the unit cell. The yellow dashed lines highlight the profile of the structure, while the green arrows indicate the electric field. The red and blue colors represent the positive and negative values, respectively. (d) and (e) Transmittance spectra as functions of the incident angle for the $p$ polarization in the $Γ X$ and $Γ M$ directions, respectively.

3. RESULTS AND DISCUSSION

It is important to examine whether the transmittance is a quadratic function of the in-plane wave vector. Without loss of generality, we show the transmittance amplitude versus the incidence angle in Fig. 3(a) along the typical directions of $Γ X$ and $Γ M$ . One can notice that these two curves nearly overlap each other, intimating the possibility of achieving an isotropic OTF. To show the quadratic behavior of the OTF, we fitted the OTF in the $Γ X$ direction, and a quadratic function $T (k_{x}) = α k_{x}^{2}$ with $α = 26.45 / k^{2}$ can be obtained, which is also plotted in Fig. 3(a) and considered as the ideal case. It is obvious that the transmittance curves for the Laplace metasurface are very close to the ideal case with a parabolic line shape within the incidence angle up to 8°, which is improved by nearly 1 order of magnitude compared to previous works and leads to an NA of about 0.14. Such a large NA can help to maintain more information on the incident light field and thus achieve a high spatial resolution approaching the wavelength scale. In addition, we have shown the corresponding phases in Fig. 3(b), and there is only a little difference between the results of Laplace metasurface and the ideal one. To further demonstrate the isotropic OTF, we have studied the transmittance of the Laplace metasurface for arbitrary in-plane wave vectors and the 2D OTF for an incident $p$ wave, which is shown in Fig. 3(c). The equal-transmittance contours are with a nearly circular shape, indicating good isotropic property. Moreover, the transmittance phase plotted in Fig. 3(d) shows that the Laplace metasurface can provide nearly the same phase for any in-plane wave vector, making the OTF greatly satisfy the definition of the Laplace operation. We have also studied the OTFs for the $s$ wave and polarization conversion, which are significantly smaller than that of the $p$ wave (see details in Appendix A). Because of the small values of the transmission for $s$ wave, it is hard to realize the Laplace operation for linearly polarized waves. Nevertheless, the OTF for unpolarized waves is similar to the case of the $p$ wave as an average result of both polarizations. In this case, the transmission drops a little compared to the case of $p$ -polarization. To increase the overall transmission efficiency, ensuring high symmetry of the structure might be a solution. Therefore, the OTF of the designed metasurface makes it possible to act as a practical Laplace operator for unpolarized waves. In addition, the isotropic OTF can be maintained even when the incidence angle is extended to 20°, at which the transmittance amplitude can reach about 0.76 and the phase is still close to $- π$ (see details in Appendix B). As the excitation of quasi-BIC is general, similar OTFs can be obtained for various wavelengths. For example, we have designed another Laplace metasurface working at the wavelength of 1550 nm (see details in Appendix C). Therefore, the flexibility of the Laplace metasurface can be applied in various scenarios.

Figure 3.(a) Transmittance amplitudes of the Laplace metasurface as functions of the incidence angle along both $Γ X$ (red solid lines) and $Γ M$ (red dotted lines) directions for a $p$ wave at the wavelength of 740 nm. The ideal results (black dashed lines) are fitted from those along the $Γ X$ . (b) Corresponding phases of the Laplace metasurface and the ideal case. A reference plane is purposefully set to make the value as $- π$ at normal incidence. (c) 2D transmittance amplitudes as functions of incidence angle. The equal-transmittance contours (black lines) resemble well-defined circles, indicating good isotropic properties. (d) Corresponding 2D transmittance phases of the Laplace metasurface.

In order to verify whether the designed metasurface can output correct results of Laplace operation, we first investigate the second-order differentiation on a 1D spatial function, which is shown in Fig. 4(a). This spatial function consists of three Gaussian envelopes and can be regarded as the electric field profile of an input light beam. To calculate the output function, we first calculated the Fourier spectrum $\tilde{E} (k_{x}, k_{y})$ of the input function. We then applied the OTF to the Fourier spectrum. Finally, we conducted the inverse Fourier transform, and the output function can be expressed as $E_{o} (x, y) = F^{- 1} [T (k_{x}, k_{y}) \tilde{E} (k_{x}, k_{y})]$ . With this processing procedure, we have studied both cases of the ideal Laplacian OTF and the metasurface, which are shown in Figs. 4(b) and 4(c), respectively. It can be found that the output results from the metasurface faithfully show the spatial profile of the ideal one, indicating that the metasurface can correctly conduct the second-order differentiation. The tiny difference between the results of the ideal case and that of the metasurface can be attributed to the small deviation of the OTF of the metasurface from the ideal one. Furthermore, both OTFs in the $Γ X$ and $Γ M$ directions have been considered, and the results overlap each other very well, which can be clearly observed in Fig. 4(c). In addition, we have set the spatial resolution of $3.6 λ$ for the input function, which is also nearly the maximum spatial resolution of the proposed metasurface [16]. Consequently, the metasurface can work at a resolution of wavelength-scale well, which is improved nearly 1 order of magnitude compared to previous work [36]. Therefore, the designed metasurface can indeed realize the second-order differentiation for a 1D spatial function.

Figure 4.(a) 1D spatial function with three Gaussian envelopes as the input light field; (b) output results after the operation of ideal second-order differentiation; (c) output results from the Laplace metasurface. Both OTFs along the $Γ X$ (solid line) and $Γ M$ (symbols) directions were considered. Here the pixel size of input is $3.6 λ$ .

The Laplace operation not only includes the second-order differentiation but also requires that the differentiation is isotropic. In order to validate the Laplace operation of the designed metasurface, we have further investigated a 2D spatial function $E (r, φ) = J_{2} (r) \cos 2 φ$ , where $J_{2} (r)$ is a Bessel function of the first kind. Such a function is a possible solution of the Laplace equation in the polar coordinate. Furthermore, the results of the Laplace operation on this function can be obtained analytically (see details in Appendix D). Therefore, it is an appropriate example for validating the performance of the Laplace metasurface. Here we assumed that this spatial function was loaded on the electric field of an unpolarized light; the corresponding light-intensity profile is shown in Fig. 5(a). In the case of ideal Laplace operation, the output results can be derived analytically; its light-intensity profile is shown in Fig. 5(b). It should be noted that although the intensity distributions of both light fields are very similar, the phases between them are totally different (see details in Appendix E). For the Laplace metasurface, we have applied the similar processing procedure mentioned above to obtain the corresponding results, which are shown in Fig. 5(c). It is obvious that the result from metasurface is consistent with that of the ideal case, showing the effectiveness of Laplace operation with a metasurface. To further explain the accurate operation of the Laplace metasurface, we show the light-field intensity along the radial direction with both $φ = 0$ and $φ = π$ in Figs. 5(d)–5(f), which correspond to the signals of input, output of the Laplace operating of the ideal case, and metasurface, respectively. The profiles are actually the cross-sectional views of the light-field patterns along the $x$ direction. One can find that the results of the ideal operation and Laplace metasurface have, indeed, a similar profile. In addition, we have also shown the results with $φ$ varied from 0 to $2 π$ and $r = 576 λ$ in Figs. 5(g)–5(i). According to the analytical expression of the input function and its Laplacian results, the corresponding output should behave as $\cos^{2} 2 φ$ , which is indeed correctly shown in the figures. Therefore, the proposed metasurface has demonstrated a valid Laplace operation.

Figure 5.(a) Input 2D spatial function for Laplace operation; (b) analytical solution of the input function; (c) output from the Laplace metasurface; (d)–(f) radial profiles of the corresponding functions along $x$ direction at $y = 1800 λ$ in (a)–(c), respectively; (g)–(i) angular profiles of the corresponding functions at $r = 576 λ$ in (a)–(c), respectively. Here all the pixel sizes are set to be $3.6 λ$ .

One of the applications of Laplace operation is the edge detection of targets of concern in an image. Here, we also demonstrated that the proposed Laplace metasurface can be employed to recognize a traffic sign, which is critical for automatic driving. A general traffic sign was selected and is shown in Fig. 6(a). To extract the essential information, we first converted the false-color image to a gray-scale one, shown in Fig. 6(b). Such a gray-scale image can be considered as an intensity-modulated unpolarized light field that is impinged on the Laplace metasurface. From the results after ideal Laplace operation that are shown in Fig. 6(c), we can notice that the edges of both outer circles and the arrow can be clearly identified. Therefore, the necessary information can be safely retained, while the redundant one has been discarded, accelerating the information processing in the first step. We have also shown the output results from the Laplace metasurface in Fig. 6(d), where it can be found that similar edge detection can also be successfully realized. The good agreement between the results implies again that the designed metasurface can perform a Laplace operation with high quality. It should be noted that a monochromatic light beam is necessary for achieving excellent performance due to the narrow bandwidth of the BIC. In addition, it is also important to ensure the symmetry and the fabrication precision of the structure. Otherwise, the Laplace operation and edge detection will degrade. The edge detection for images can also be achieved at the wavelength of 1550 nm (see details in Appendix F).

Figure 6.(a) False-color image of a traffic sign; (b) corresponding gray-scale image as the input; (c) and (d) output image from the ideal Laplace operation and the Laplace metasurface corresponding to (b), respectively.

4. CONCLUSION

In summary, we have proposed and numerically demonstrated a Laplace metasurface that can perform the isotropic second-order spatial differentiation. The angle-resolved excitation of a BIC of the Laplace metasurface provides an OTF that is close to that for ideal Laplace operation at the wavelength of 740 nm. The maximum incident angle has been widely extended to about 8° compared to previous works, improving the spatial resolution. The mode pattern with high symmetry in the Laplace metasurface ensures the isotropic property of the Laplace operation. The Laplace operations of 1D Gaussian function and 2D spatial functions with the proposed Laplace metasurface have been successfully demonstrated, which agree well with that of the ideal Laplace operation. Furthermore, we have demonstrated that the Laplace metasurface can also be employed to realize the edge detection with high quality for images, which would find important applications not only in the fields of optical analog computing, but also in the fields of object recognition.

Acknowledgment

Acknowledgment. T. Feng acknowledges financial support of the Double Hundred Talents Program of Jinan University (DHT2020096).

APPENDIX A: OTFs FOR THE S WAVE AND POLARIZATION CONVERSION

s

Figure 7.(a) OTF of the metasurface under the $s$ -wave illumination; (b) corresponding transmittance phases. A reference plane is purposefully set to make the phase value at the normal incidence as $- π$ . (c) and (d) OTFs for the cases of polarization conversion with the $p$ - and $s$ -wave incidences, respectively.

APPENDIX B: ISOTROPY OF THE OTF

- π

Figure 8.(a) 2D transmittance amplitudes as functions of incidence angle under the $p$ wave illumination at 740 nm. The equal-transmittance contours from inner to outer indicate the transmittance from 0.1 to 0.7 at a step of 0.1. (b) Corresponding phases. A reference plane is purposefully set to make the phase value at the normal incidence as $- π$ .

APPENDIX C: LAPLACE METASURFACES AT THE WAVELENGTH OF 1550??nm

a = 743 nm

Figure 9.(a) 2D transmittance of a Laplace metasurface at the wavelength of 1550 nm under the $p$ -wave illumination; (b) corresponding 2D transmittance under the $s$ -wave illumination; (c) transmittance of the $s$ wave under a $p$ -wave incidence; (d) transmittance of the $p$ wave under an $s$ -wave incidence.

APPENDIX D: ANALYTICAL RESULTS OF THE LAPLACE OPERATION ON THE 2D BESSEL FUNCTION

E (r, φ) = J_{2} (r) \cos 2 φ .

APPENDIX E: RESULTS OF THE INPUT AND OUTPUT ELECTRIC FIELD DISTRIBUTIONS

- π

Figure 10.(a) Real part of the electric field of the input light field; (b) analytical solution of the Laplace operation on the input; (c) output from the Laplace metasurface; (d)–(f) radial profiles of the corresponding functions along the $x$ direction at $y = 1800 λ$ in (a)–(c), respectively.

APPENDIX F: EDGE DETECTION OF IMAGES AT THE WAVELENGTH OF 1550??nm

As the principle of Laplace metasurface is general, we demonstrated the edge detection of an image at the wavelength of 1550 nm with the results presented in Fig. 9 . We have employed a typical QR code as the input 2D image, since the QR codes are now important in our daily life, and the edge detection for them plays a critical role in locating the QR code region. Therefore, it is meaningful to perform high-speed edge detection on QR codes. The QR code we have chosen is shown in Fig. 11 (a), which carries the information of a Chinese character meaning “light.” Through the processing procedure discussed in this context, we can obtain the results from the ideal Laplace operation and that from the Laplace metasurface, which are shown in Figs. 11 (b) and 11 (c), respectively. It can be seen that the edges of all elements have been successfully extracted. Furthermore, the light-intensity output from metasurface is very similar to the ideal one, implying the high quality of edge detection for images with the Laplace metasurface.

Figure 11.(a) Input image consisting of a QR code; (b) output image of the ideal Laplace operation; (c) output from the Laplace meatsurface. All images are the light-intensity profile; the pixel sizes are set as $2.88 λ$ .

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