Optical analog computing has attracted intensive attention recently in application scenarios that call for high computing speed. (1−3) Optical analog computing devices based on conventional optical components face limitations such as complex systems, alignment errors, and bulky sizes. On-chip metasurfaces based on optical waveguides can simplify the system, while lithography technology makes in-plane alignment easier than free-space alignment. Such nanotechnology-enabled metasurfaces have the potential to miniaturize optical analog computing devices to the micron or even nanoscale. (4,5) Unlike conventional metasurfaces, which can only manipulate light in free space, (6) on-chip metasurfaces can not only control the in-plane waveform for light focusing and mode conversion (7,8) but also precisely control the out-of-plane light for holography. (9,10) Additionally, the fabrication process of metasurfaces on the SOI platform is compatible with CMOS technology, enabling large-scale manufacturing at low costs. (11)

In 2014, Silva et al. reported the first metasurface-enabled optical analog computing based on the fundamental principle of optical Fourier transform (OFT) and Green’s function. (12) Although the OFT method can perform a large number of mathematical operations, free-space alignment in multilayer metasurfaces is challenging. (13) Though the Green’s function method-based optical computing metasurface avoids the problem of alignment, its range of operations is restricted by symmetrical distribution. In order to solve the problem of alignment of metasurfaces in the OFT-based optical computing system, Wang et al. demonstrated a spatial-domain differentiator by employing in-plane metasurfaces based on optical waveguides. (14) This approach applied in mathematical operations such as differentiation, (14) integration, (15) convolution, (16) addition, and equation solution; (17) however, due to the influence of the vignetting effect, (18) the accuracy of their computational results still needs to be improved.

As a fundamental mathematical operation in signal processing, the Hilbert transform can be widely applied in various fields such as biomedicine, (19,20) vibration analysis, (21) image processing, (22−24) and spectral analysis. (25,26) In the frequency domain, optical Hilbert transformers (OHTs) based on fiber-optic Mach–Zehnder interferometers and fiber Bragg gratings have been reported. (27,28) However, these structures face limitations such as environmental temperature, coupling loss, and mechanical stability. Compared to fiber-based OHTs, integrated OHTs based on Bragg gratings, (29) Mach–Zehnder interferometers and microring resonators (30) can significantly reduce the impact of some of these challenges with compact sizes and low costs. In the spatial domain, OHTs based on OFT achieve light manipulation by inserting a phase plate at the common focal point of two lenses. (31) Nevertheless, issues such as large volumes and alignment errors exist, which restrict the application of the reported spatial OHTs.

The implementation of an integrated spatial-domain Hilbert transformer (ISHT) using in-plane metasurfaces, to the best of our knowledge, has not been reported. However, two problems remain in implementing the Hilbert transform operation with in-plane metasurfaces based on OFT: (1) The high spatial frequency components of the input signal are lost by the 4f system due to the lens’s finite relative aperture, resulting in vignetting effects. (2) The large differences in geometric parameters between adjacent meta-atoms no longer satisfy the periodic approximation, resulting in performance reduction.

As illustrated in Figure 1, the proposed ISHT consists of three cascaded in-plane metasurfaces on a slab waveguide according to the optical 4f system in OFT. (18) Meta I and III are identical in-plane metalenses designed to perform optical spatial Fourier transforms in nearly real-time, with a focal length of f. If d = f, the Fourier transform of input signal is formed at the focal plane of Meta I. The focal plane behind Meta I is known as the Fourier plane. The convolution result of the input signal and Meta II is displayed at the output plane, which is the focal plane of Meta III.

The designed metasurfaces are slot waveguide arrays along the y-axis with a period (p) of 500 nm. The slot waveguides are defined on an SOI platform with a 3 μm-thick substrate and a 220 nm-thick device layer. The cladding layer is 2.2 μm-thick silicon dioxide. The slot-waveguide-based meta-atoms are numerically simulated by the three-dimensional finite-difference time-domain (FDTD) method. The incident light is set to the fundamental TE mode at a wavelength of 1550 nm. A more detailed discussion on optical properties of slot waveguides can be found in our previous work. (33) The transmission of the in-plane metasurfaces can be manipulated by adjusting the width w, leading to amplitude modulation. The phase of the in-plane wave can be controlled by adjusting the length l, thereby achieving wavefront control of the in-plane wave.

Etched slots of the metalens with varying geometrical characteristics arranged along the y-axis follow eq 1, which defines the phase shift distribution of the transmitted wave for in-plane wave focusing (14): where n_eff is the effective refractive index of the guided light confined in the silicon slab waveguide.

Figure 2a depicts the simulated performance of metalenses with a D of 15 μm at a working wavelength λ of 1550 nm. The target f varies from 5 to 30 μm at intervals of 5 μm. The simulated focal lengths fluctuate around the target focal lengths of the designed in-plane metalens. The deviation between the target f and the simulated f varies from −2.7 to 39.4% as the target f decreases, which arises from weakening of the periodic approximation condition. This is also the reason why the focusing efficiency (FE) of the in-plane metalens dramatically decreases from 74.3 to 28.4% as the target f reduces from 30 to 5 μm. Here, the FE is defined as the ratio of the power passing through a rectangular aperture on the focal plane, with a width three times that of the spot size (FWHM) and a height of 0.5 μm, to the total power incident on the metalens.

From Figure 2a, it is observed that the focusing performance of the metalens decreases as the relative aperture (f/D) declines. However, to mitigate the vignetting effect, a metalens with a larger relative aperture, while still maintaining adequate focusing performance, should be selected. Therefore, the focal length of the metalens is set to 15 μm, with a focusing efficiency of 58.7%. To correct the focus offset of the device and improve the FE, an inverse design is employed to optimize the device. Based on the selected metalens, this method improves the performance of the metalens by fine-tuning the geometric parameters (w and l) of the slot waveguides. The figure of merit (FOM) composed of the absolute value of the focus offset, the FE, and the FWHM as following: where f_sim represents the simulated focal length. Based on the simulation results, f, FE, and FWHM are critical parameters of the metalens. To ensure that the relative aperture of the metalens is 1, the weight of the focal length offset, as the most important variable in the FOM, is set to 0.5. Simultaneously, the weights of the FE and FWHM are set as 0.3 and 0.2 to enhance the focusing performance, respectively.

The normalized intensity distribution of the optimized metalens along the propagation direction at a wavelength of 1550 nm is shown in Figure 2b. The focus offset of the designed in-plane metalens has decreased by approximately 80%, with the simulated f falling from 16.18 to 15.26 μm. The normalized intensity distribution on the focal plane is depicted in Figure 2c, where the spot size has been reduced from 0.60 to 0.53 μm. The simulated results for the optimized in-plane metalens performance in the 1450–1650 nm wavelength range are displayed in Figure 3d. Compared to that in Figure 2a, the FE has been improved by 6.8% through optimization. Its transmission and focal spot size remain nearly unchanged within the 1450–1650 nm wavelength range, while the focusing efficiency improves slightly from 60.2 to 65.5%, and the simulated f decreases from 16.67 to 13.80 μm as the wavelength increases. This indicates that the optimized in-plane metalens has a broad bandwidth of 200 nm and can achieve the efficient focusing of light beams. Such a metalens with a relative aperture of 1 reduces vignetting effects and ensures high transmission and FE, making it a reliable Fourier transform component for our Hilbert transformer.

The integrated optical 4f system is formed by placing the optimal in-plane metalenses at x = −15 μm and x = 15 μm. A rectangular silicon waveguide with a width W along the y-axis is on the left of Meta I for forming signal input. Due to the rectangular waveguide used as the input waveguide in our device, only Gaussian input signals exist and transmit within it. By changing the width W of the input waveguide to alter the input signal, Figure 3a shows the light intensity distribution in the x–y plane. The Fourier transform of the input signal is formed at the Fourier plane (x = 0), and the input signal is reconstructed on the output plane at x = 30 μm.

The coefficient of determination (COD) is used to quantify the degree of fit between the output value of this system and the theoretical value. The coefficient of determination can be represented by the symbol R², and it is defined by the formula R² = 1 – SS_res/SS_tot, where SS_res represents the residual sum of squares, and SS_tot denotes the total sum of squares. A greater goodness of fit between two variables is indicated by a value that is closer to 1.

The simulated transmission and the calculated COD for various W values of the designed optical 4f system are displayed in Figure 3b. When W is below or equal to 5 μm, the COD increases quickly as the waveguide width increases due to the mitigation of vignetting effects. When the W is larger than 5 μm, the COD gradually drops as the waveguide width increases. Transmission follows the same trend as the COD and peaks at W = 4 μm. It can be inferred that the optimum working range of W is 4–10 μm, where transmission greater than 85% and a COD above 0.96 can be obtained at a wavelength of 1550 nm.

The signal recovery function of the integrated optical 4f system at various wavelengths is compared in Figure 3c, where W is set to 4 μm. According to calculations, the COD at 1450, 1550, and 1650 nm is 0.94, 0.97, and 0.99, respectively. This suggests that the system has accurate signal recovery functions and offers a reliable on-chip optical convolution system for the ISHT. It is noteworthy that, as Figure 3d illustrates, the simulated transmission of this system in the 1450–1650 nm wavelength range is greater than 91%. Since the input waveguide width of this system is 4 μm, the high-frequency components of the input signal are less than those of smaller W, and the scattering in the edge area of the in-plane metalens has very little effect on the in-plane wave.

To realize the Hilbert transform, Meta II for modulating the phase and amplitude of the in-plane wave should be defined as (18) where j is the imaginary unit. From eq 3, it can be observed that the Hilbert transform introduces a ±π/2 phase shift to negative and positive spatial frequencies. According to OFT, reducing the distance d between the input plane and Meta I decreases the vignetting effect but introduces a certain phase deviation in the Fourier transform of the input signal at the same time. Thus, in Meta II, a phase modification factor is required to be included: (18)

The phase distribution along the y axis of the Meta II is depicted in Figure 4a based on eq 4. The phase and transmission distribution of the selected slot waveguides are indicated by the red squares and blue dots in Figure 4a, respectively. The average transmission is 98.43%, and the maximum phase difference between the selected and target phase shifts is 0.02 rad. These selected slot waveguides are etched along the y axis at x = 0. The proposed ISHT is made up of Meta II and the integrated optical 4f system composed of Meta I and III with a footprint of 45 μm × 15 μm. Then, the proposed ISHT is numerically simulated by the FDTD. Figure 4b shows the TE field distribution of the designed ISHT, and the inset is the magnified TE field distribution for Meta II. When the light travels through Meta II, there is a phase difference of roughly π between the positive and negative y axes. According to the convolution theorem, by modifying the transmission and phase distribution of META II, various operations and functions can be achieved. The essence of this device is to modulate the input wave in the Fourier space. Although the function of a single metasurface-based computing device is fixed, multiplexing optical computing can be achieved by utilizing metasurface components with different computing functions in combination.

Since the periodic approximation is no longer satisfied, optimization by inverse design is utilized to enhance performance. The COD between the theoretical values and the simulation results at a wavelength of 1550 nm is used as the FOM. The intensity distributions before and after optimization of the designed ISHT are compared with the theoretical values shown in Figure 4c. In Figure 4d, the COD between the simulation results and the theoretical values is improved from 0.9 to 0.99 at a wavelength of 1550 nm, and it is greater than 0.9 in the wavelength range of 1550–1630 nm.

Figure 4e indicates the impact of fabrication errors in the slot waveguide width on the COD. At a wavelength of 1550 nm, when the fabrication error ranges from −10 to +10%, the COD between the simulated and theoretical results for the proposed Hilbert transformer remains above 0.93. It conveys that manufacturing errors within ±10% have a limited impact on the calculation results of the proposed Hilbert transformer.

In order to further verify the performance of the device, the input is set as three parallel rectangular optical waveguides with the simulation results shown in Figure 4f. The difference is observed between the simulated and the theoretical values. It is attributed to that the period of the meta unit cell is relatively larger compared to the input waveguide width, resulting in insufficient spatial sampling and processing of the input signal.

The designed in-plane metalens and ISHT are fabricated on an SOI platform using a CMOS-compatible process. The fabricated devices are shown in Figure 5. As shown in Figure 5a, light enters from the grating coupler and then widens to 15 μm by a taper; finally, it focuses on the focal plane through the in-plane metalens. Here, a waveguide array with a width of 0.5 μm and a spacing of 0.5 μm is used to sample the output-focused light. To obtain a higher resolution sampling accuracy, two sets of waveguide arrays arranged in cross patterns are placed on the same focal plane, as shown in Figure 5b. The same output-sampling arrays are also used in the fabricated ISHT.

The devices are characterized using the measurement setup utilized in our previous work. (34,35) Figure 6a,b shows a comparison of the measured and simulated results of two in-plane metalenses before and after optimization, with Δx = 15 μm. The blue square represents the experimental data (Exp), the blue dashed line represents the fitted data obtained by Gaussian fitting of the experimental data (Exp-fit), and the red solid line shows the FDTD simulation data (Sim). The experimental data at y = ± 1 μm are about 0.3, deviating from the simulated data, possibly due to fabrication errors. According to the fitted data, the FWHM is about 1.59 μm. The fitted light distribution of the in-plane metalens optimized by inverse design shows an obvious narrowing trend, and the normalized transmission at y = ± 1 μm is close to 0, which is in good agreement with the simulation data. According to the experimental fitting data, its FWHM is about 0.94 μm. Compared to Figure 6a,b, the spot size of the metalens atΔx = 15 μm is reduced by 40.9% through inverse design optimization.

Figure 6c shows the comparison of the normalized intensity distributions of experiments and simulations at different wavelengths along the x-direction (y = 0). The simulated results show that f of the metalens slightly decreases as the wavelength increases. Limited by the optimal working wavelength range of the grating couplers and the operating wavelength range of the laser used in the experiment, the device is characterized in the wavelength range of 1530–1568 nm. Compared to the simulation, the fitting curve shows the measured f is slightly longer than the simulated one. The fabricated in-plane metalens with a relative aperture of 1 exhibits the on-chip light beam focusing function on the experiment over a broad working bandwidth, with a focus depth range of approximately 2.5 μm. It provides a real-time Fourier transformer for on-chip optical computing applications based on OFT.

The normalized intensity distributions of the two input signals corresponding to Figure 5c are displayed in Figure 6d. As shown in Figure 6e, the intensity distributions of the simulated results of the ISHT agree well with the theoretical values at a wavelength of 1550 nm, and the calculated COD between the theoretical and experimental values is approximately 0.90. This indicates that the manufactured device is able to perform a precise Hilbert transform of the input signal. The experimental and simulated normalized intensities at different output planes are shown in Figure 6f. The measured results of the fabricated ISHT are consistent with the simulated results in different output planes, proving that light propagates and computes according to the expected path in the fabricated device.