1. Introduction

Traditional analog computers mainly process information with the forms of mechanical, electronic, or hybrid computers [1, 2], but their development has been severely limited by their essential defects, such as relatively large size and slow response. The form of all-optical information processing can theoretically break through these restrictions, but optical transistors and digital logic circuits still have many problems [3]; consequently, they cannot be applied to all-optical information processing systems. Also the traditional lens-based optical signal processors depend on accumulated phase delay during light propagation to manipulate the impinging light [4], which causes these devices to have a large volume, diffraction limit [5], and other intrinsic drawbacks. Therefore, an optical information processor that can overcome the above shortcomings and can be used to authentically realize all-optical information-processing systems should be developed [6].

In 2014, Silva et al. introduced the concept of “computational metamaterials”, theoretically analyzed their physical properties, and designed a conceptual metasurface and a realistic three-layered thin metasurface that could be used to perform mathematical operations in the infrared band [7]. Metasurface is a two-dimensional metamaterial with subwavelength thickness in the propagation direction, which provides an arbitrary abrupt change in phase or amplitude for the impinging light by adjusting the size, shape, and composition of the constituent structures [8]. Due to their relatively simple preparation process and low loss, metasurfaces have attracted a large number of researchers [9–21]. All-optical information processors, which are based on computational metasurfaces, are several orders of magnitude thinner than conventional lens-based optical signal processors, thereby providing the possibility for realizing direct, miniaturized, integratable, ultra-fast all-optical information processing systems. Motivated by the work of Silva et al. [7], researchers have proposed other metasurfaces that can perform mathematical operations in the near-infrared or infrared band [22–24]. However, for most of these computational metasurfaces, their structures are periodic, requiring precise regulation of the size, shape, or composition of each nanostructure unit. Consequently, the preparation of these metasurface samples is difficult. Thus, most of these metasurfaces are studied only through simulations, and only a few are fabricated using the complex etching technology, which is costly and difficult to use widely. So far, in visible light, there are few reports about the computational metasurfaces, because the structures of the periodic metasurface are smaller and its preparation is more difficult.

In order to break through the bottleneck of preparing the metasurface that operates in visible light. In this study, an easy-to-prepare, low-cost, reflective metasurface with quasi-periodic silver dendritic structures [25] is presented by considering the metasurface preparation process proposed by our group [26, 27]. The proposed metasurface consists of silver dendritic structures atop a fused silica spacer and silver film substrate. Supporting both electric and magnetic resonance in visible light, as well as multiple reflections within the dielectric layer, enables us to fully manipulate not only the amplitude but also the phase profile of the reflected co-polarized light spatially by varying the shapes of silver dendritic structures. The differential operation property of the metasurface is proved through simulation in red, yellow, and green bands. Silver dendritic metasurface samples are prepared by adopting the electrochemical deposition technology. Experimental results indicate that the metasurfaces can perform differential operation in the green and red bands. The differential processors based on the metasurfaces can be used for real-time edge detection and image contrast enhancement; thus, they can be applied in some practical scenarios, such as enhancing the detection of hidden objects or lumps in the body [28].

2. Design of the silver dendritic metasurface

2.1 Design principle

Firstly, let us review the general ideal of designing a computing metasurface. For a linear space-invariant system, the relationship between arbitrary input function $f\left(x,y\right)$and corresponding output function $g\left(x,y\right)$can be described by the following linear convolution [7]:

Corresponding to the schematic shown in Fig. 1, $f\left(x,y\right)$represents the incident linearly polarized light $E_{in}\left(x,y\right)$, and $g\left(x,y\right)$ represents the reflected co-polarized light $E_{re}\left(x,y\right)$. $H\left(k_x,k_y\right)$ is a transfer function related to the desired mathematical operation and is equal to the position-dependent reflection coefficient $r\left(x,y\right)$ of the metasurface. The real-space coordinates $\left(x,y\right)$at the metasurface represent $\left(k_x,k_y\right)$. Accordingly, Eq. (2) can be expressed as

 

Fig. 1 Schematic of performing a mathematical operation for a reflective metasurface. The system consists of a Fourier transforming (FT) block and a reflective metasurface with position-dependent reflection coefficient$r\left(x,y\right)$.

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According to Eq. (4), in order to perform first-order differentiation of $E_{in}\left(x\right)$, $r\left(x\right)$ must mimic the form of the operator $\partial /\partial x$ in the Fourier space. Since the operator $\partial /\partial x$ is transformed into $i{k}_x$in the Fourier space, the reflection amplitude of the first-order differential metasurface will be

2.2 Design of the silver dendritic metasurface

After a considerable number of simulations, as shown in Figs. 2(a) and 2(b) in which m represents the scaling factor of a dendritic structure (inset in Fig. 2(a)) in the x direction with the origin as the scaling center and n represents the scaling factor of the dendritic structure in the y direction with the origin as the scaling center, we determine that changing the shape of the silver dendritic structure in the resonant band can allow an arbitrary abrupt change in the amplitude and phase of the reflected co-polarized light. The inset in Fig. 2(a) is the initial dendritic structure with rod length l = 75.6 nm, rod width w₁ = 48 nm, thinner rod width w₂ = 36 nm, the angle between inside rods θ₁ = 120°, and the angle between outside rods θ₂ = 36°. In this study, a metasurface composed of eleven different shapes of silver dendritic structure units is designed to realize Eq. (5), as shown in Fig. 2(c). The unit consists of three layers with fixed side length W = 250 nm, silver dendritic structure thickness t₁ = 30 nm, silicon dioxide (SiO₂) spacer thickness t_s = 30 nm, and silver substrate thickness t₂ = 100 nm (such a thick silver substrate can reflect all the incident light), as shown in Fig. 2(d). The permittivity of SiO₂ is described by the constant ${\text{ε}}_{{\text{SiO}}_2}$ = 2.1, and the permittivity of silver is set to the realistic Drude model value [31].

 

Fig. 2 Design of the silver dendritic metasurface for first-order differential operation and the corresponding position-dependent reflection coefficient. The maps of the reflection coefficient (a) amplitude and (b) phase as function of the shape of the dendritic structure. (c) Metasurface composed of eleven different shapes of silver dendritic structure units. The length of the metasurface is L = 11W = 2750 nm; (d) A three-dimensional view of a silver dendritic structure unit shown in the dashed box in Fig. (c). The parameters are t₁ = 30 nm, t_s = 30 nm, t₂ = 100 nm, and W = 250 nm, and the incident x-polarized light propagates normally to the metasurface along the –z-axe direction; (e) Comparison of the theoretical (dashed) and simulated (solid) results of the position-dependent reflection coefficient for the first-order differentiation functionality at a wavelength of 586 nm. The black and red lines represent the amplitude and phase of the reflection coefficient, respectively.

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3. Simulation and experimental verification

3.1 Simulation

The simulations are conducted using the commercial finite element software Comsol Multiphysics, and the Floquet periodic boundary condition is used in the directions of the x- and y-axes. When the incident x-polarized light with a wavelength of 586 nm propagates normally to the metasurface along the –z-axe direction, as shown in Figs. 2(c) and 2(d), the position-dependent reflection coefficient is obtained, as shown in Fig. 2(e). Compared with the desired reflection coefficient, the maximum deviation of the amplitude is approximately 0.3, except for parts close to the metasurface boundaries, and the maximum deviation of the phase is approximately 60°, except for parts near the center and the boundaries of the metasurface. Although the deviations are larger than those in the published paper [24], these results are generally consistent with the desired reflection coefficient. In this way, we prove that the silver dendritic metasurface has the ability to perform first-order differential operation in the yellow band.

Next, we try to change the operating wavelength of the metasurface by uniformly scaling its transverse dimension (i.e., the size in the xoy plane). Figures 3(a) and 3(b) show the metasurfaces obtained by uniformly enlarging the transverse dimension of the metasurface shown in Fig. 2(a) to 1.22 times and reducing the transverse dimension of the metasurface shown in Fig. 2(a) to 0.7 times, respectively. Their theoretical and simulation results at wavelengths of 625 and 540 nm are shown in Figs. 3(c) and 3(d), respectively. Compared with the desired reflection coefficient, except for parts close to the boundaries of the metasurfaces, the maximum deviations of reflection amplitude are approximately 0.45 and 0.28, respectively, which are larger than that in the published paper [24]; except for parts near the center and the boundaries of the metasurfaces, the maximum deviations of the reflection phase are approximately 32° and 29°, respectively, which are similar to that in the published paper [24]. These results are generally consistent with the desired reflection coefficients, so the silver dendritic metasurfaces have the ability to perform first-order differential operation in the red and green bands.

 

Fig. 3 Silver dendritic metasurfaces obtained by uniformly scaling the transverse dimension of the metasurface shown in Fig. 2(a) and the corresponding position-dependent reflection coefficients. The transverse dimension of the metasurface shown in Fig. 2(a) is uniformly enlarged to (a) 1.22 times and reduced to (b) 0.7 times, respectively. The corresponding position-dependent reflection coefficients are achieved at wavelengths of (c) 625 nm and (d) 540 nm.

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Therefore, we suggest and demonstrate, for the first time, that a silver dendritic metasurface can perform the first-order differential operation in the red, yellow, and green bands. We denote that, as the size of the metasurface becomes larger, the working wavelength shows red shift, and the deviation between simulated results and desired results gets larger. It can be seen that, in the case of shorter operating wavelengths, such as green and yellow bands, the simulated results are in good agreement with the desired results and very similar to the simulated results reported in the literature [7, 22–24, 29]; in the case of longer operating wavelengths, such as red band, the deviation between simulated results and desired results increases, which arises from the larger difference between morphology of the dendritic units and that of other periodic structures [23, 24]. Figures 2(e), 3(c), and 3(d) indicate that, as the transverse dimension of the metasurface decreases, the corresponding maximum amplitude of the reflection coefficient decreases. Because the units in the metasurface become smaller, the loss gradually increases, thereby resulting in weakened reflected light. These results also reveal that the operating wavelength of the metasurface will change when its transverse dimension is scaled. Its differential operation functionality can thus be readily extended to the infrared and telecommunication wavelengths.

3.2 Preparation by the electrochemical deposition method

According to the abovementioned design, we adopt our proposed bottom-up electrochemical deposition method [32, 33] to prepare the silver dendritic metasurface samples. The preparation process is shown in Fig. 4(a). The sample is composed of a single-layer silver dendritic metasurface sample and a silver film sample, which are all prepared by using the electrochemical deposition method.

 

Fig. 4 Preparation and characterization of the silver dendritic metasurface samples. (a) Preparation process diagram of the single-layer silver dendritic metasurface sample and the silver film sample, which are subsequently assembled into a silver dendritic metasurface sample; (b) SEM of the single-layer silver dendritic metasurface sample; (c) Transmission spectra of the single-layer silver dendritic metasurface samples; (d) Optical photograph of the silver film sample; (e) Reflectance spectrum of the silver film sample.

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The area of the prepared single-layer silver dendritic metasurface sample is 13 mm × 10 mm. Figure 4(b) presents the scanning electron microscopy (SEM) image of the sample. On the whole, the dendritic structure units are uniformly distributed onto the surface of an ITO glass substrate, and the diameter of the unit is approximately 200 nm to 300 nm. Hence, there are approximately 10⁸ dendritic units in the prepared metasurface sample. Also we can see that there are various dendritic structures in the sample, and their distribution is quasi-periodic. These dendritic structures must contain the dendritic structures designed in the simulation. Therefore, the sample can be used to verify the differential operation functionality of the designed metasurface. Figure 4(c) shows the transmission spectra of the single-layer silver dendritic metasurface samples S₁ (green curve) and S₂ (red curve), which have higher transmission resonance peaks at 528 and 620 nm, respectively, except for the intrinsic absorption peak of silver at approximately 400 nm. The two samples are thus resonant in the green and red bands, that is, they can represent the metasurfaces shown in Figs. 3(b) and 3(a) respectively. The resonant wavelengths of the samples can be changed by adjusting the deposition conditions, including deposition voltage, deposition time, experimental temperature, and electrolyte concentration [33].

The general process of preparing the same area silver film sample is similar to the preparation of single-layer silver dendritic metasurface samples, except the AgNO₃ electrolyte is different, and the deposition time is changed to 20 s. The AgNO₃ solution with a mass fraction of 4% is prepared by adding 0.08 g AgNO₃ to 2 ml ultrapure water and then dripping triethanolamine (analytical grade) into the solution until it becomes colorless and transparent. The required AgNO₃ electrolyte (prepared as it needs to be used) for preparing the silver film sample is obtained. Figure 4(d) shows the optical photograph of the prepared silver film sample, which is dense and has high brightness. Figure 4(e) shows that when the wavelengths of the incident light are 532 and 632 nm, the reflectance values of the silver film are 87% and 88%, respectively, which basically meet the requirement to reflect all the incident light.

We then obtain the silver dendritic metasurface samples S₃ and S₄ for the final measurement by placing the silver film sample close to samples S₁ and S₂, respectively, as shown in Fig. 4(a). Therefore, the whole silver dendritic metasurface sample is actually composed of the top of the antioxidant layer of polyvinyl alcohol (PVA) film, the layer of silver dendritic structures, ITO conductive glass, silver film, and the bottom of ITO conductive glass. It is worth noting that the entire preparation process of the metasurface is easy and mature. Unlike the metasurfaces fabricated by using the expensive etching technology, this metasurface is affordable, thereby making it suitable for widespread application.

3.3 Reflectivity measurement

The measurement principle of the first-order differential operation property of the silver dendritic metasurface sample is shown in Fig. 5(a). The x-polarized light beam from the laser passes through the filter, and its strength is adjusted to be appropriate. The spot diameter is enlarged when the beam travels through the beam expander. The spot diameter of the expanded beam is regulated to a proper size by the circular diaphragm. The beam vertically irradiates the sample after passing through the beam splitter (BS). The light reflected from the sample is reflected to the mirror by the BS. The light reflected from the mirror is then detected by the probe of the fiber optic spectrometer. During the test, the intensity of the reflected light spot at each position is obtained by moving the probe along the x direction and through the spot center. The wavelengths of the lasers used for testing samples S₃ and S₄ are 532 and 632 nm, respectively. The spot diameter of the incident light passing through the BS is 3 mm.

 

Fig. 5 Measurement of the first-order differential operation property of the silver dendritic metasurface samples. (a) Schematic of the experimental setup for testing the intensity of the light reflected from the samples; Normalized co-polarized reflection intensity (i.e., reflectivity) distributions of the silver dendritic metasurface samples (b) S₃ and (c) S₄ along the x-coordinate.

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The intensity distributions of the light reflected from the silver dendritic metasurface samples and the silver film sample along the x-coordinate are measured using the experimental setup shown in Fig. 5(a). The reflection intensity distributions of the samples S₃ and S₄ are normalized to that of the silver film sample. Since the area of the sample is much larger than that of the incident spot, the region of the sample that is actually tested is the region covered by the spot. Therefore, the range of the region in the x direction is −1.5 mm to 1.5 mm. For the silver dendritic metasurface samples S₃ and S₄, the distributions of the normalized reflection intensity (i.e., the reflectivity) in the range are shown in Figs. 5(b) and 5(c) respectively. The reflectivity curves of the samples S₃ and S₄ are basically parabolic, which are generally consistent with the desired reflectivity. However, near the center of the tested region in the samples, the reflectivity is larger than the simulated and the desired reflectivity. Compared with that in the published work [23], the general trend of the reflectivity curve is basically the same, but the reflectivity near the sample center is significantly large. This is partly because the size of the prepared silver dendritic structure deviates from the design, and the prepared silver film does not reflect all the incident light. And we know that the structure of the periodic metasurface is homogeneous, which is analogous to the single crystal in crystallography. The metasurface prepared in this study contains various dendritic structures, which is analogous to the polycrystal. It is well known that the structures of the single crystal and the polycrystal can be obtained by X-ray diffraction, but the diffraction pattern of the polycrystal is resulted from superposition of the diffractions produced by many crystal structures, which is a statistical effect. Therefore, the reflectivity distributions of the dendritic metasurfaces are actually statistical results, and their performance is consequently reduced compared with that of the periodic metasurface.

4. Conclusion

First-order differential metasurfaces are proposed and designed using silver dendritic structures at red, yellow, and green wavelengths. Simulation results show that the position-dependent reflection coefficients agree well with the desired reflection coefficients. The silver dendritic metasurface samples are prepared by using the bottom-up electrochemical deposition method, which is easy, mature, and low cost. Therefore, the samples can be prepared in a large area. Their reflectivity distributions are measured at green and red wavelengths respectively. The experimental results are mainly consistent with the simulated and theoretical prediction. Thus, the silver dendritic metasurface that can perform the mathematical operation in visible light provides the potential for miniaturized, integratable all-optical information processing systems. Its ability to perform differential operation provides it with a great practical value in real-time ultra-fast edge detection, image contrast enhancement, hidden object detection, and other applications.