1. Introduction

In recent years, optical analog computing has become an important topic for its faster running speed and higher parallelism [1,2]. Compared with digital computing [3,4], analog computing is designed based on the fluctuation and interference principles of light, without cost of analog-to-digital conversion, and low power consumption, thus it meets the requirements for real-time processing of information in many fields. In the field of optical analog computing, mathematical operations such as differential operation [5], integral operation [6–8], convolution [9] and Fourier transform [10] have been realized. Among them, the differential operation has attracted extensive attention for its significant applications in artificial intelligence such as image processing and other fields [11–14]. Importantly, various methods of optical differential operation based on the 4f system [5], Bragg grating structure [15,16], Multi-layered plate structure [17], dipole scattering [18], photonic crystals [19] and metasurface [20–23] have been successively proposed. However, we note that the process of designing a differentiator is extremely complex whether it is micro-integrated structure or other micro-nanostructure [12]. Here, differentiators with a simple and thin structure have been developed based on Brewster effect [24,25], Goos-Hänchen effect [11], photonic SHE [26–29] and other methods. This kind of differentiator has a simpler structure, without complex metasurfaces or filter-based Fourier optics, and it can enable differential operations through reflection or refraction at the optical interface.

Recently, graphene has been extensively studied based on its unique electronic and optical properties [30,31]. It is a single layer structure of carbon atoms with a hexagonal honeycomb lattice. In particular, the conductivity can be flexibly modulated by varying the Fermi energy such as chemical doping and electrostatic field biasing [32,33]. Photonic SHE based devices have been designed based on graphene materials in previous reports [34–40]. When a Gaussian beam [41] of linearly polarized light is reflected by an optical interface containing a graphene layer, photons with opposite spin angular momentum (left and right circularly polarized) are separated from each other in the transverse direction perpendicular to the incident plane due to the refractive index gradient acting as the role of the applied electric field, resulting in a corresponding transverse shift of the reflected beam in the direction perpendicular to the plane of incidence [42,43]. Although the transverse shift is tiny, it is extremely sensitive to the relevant physical parameters of the optical model. Since graphene is a single layer carbon atomic structure and it is sensitive to the external environment. Small external perturbations can cause significant changes in the optical properties of graphene. We can easily adjust the optical properties of graphene accurately by varying the external electric field without changing the structural parameters of the model to realize the real-time tunable differential operation. In addition, the optical properties of graphene are better tunable in the telecommunication wavelength than in the visible wavelength and a variety of graphene based photonic devices have been theoretically proposed for operation in the telecommunication wavelength [44]. In fact, based on the above advantages, it seems feasible to design a tuneable differentiator for real time edge imaging switching at a telecommunication wavelength.

In this paper, we propose a tunable optical differential operation based on isotropic graphene at 1550nm. Here, a four-layered structure model consisting of prism-graphene-air gap-substrate is proposed. Gaussian light is reflected through the model, the photonic SHE will occur during the reflection process, and the reflected beam obtains transverse shifts. Moreover, we prove through the theory that Gaussian light can realize differential operation after being reflected by the four-layered structure. Also, this paper theoretically predicts that by adjusting the incident angle of the beam, the Fermi energy of the graphene, the thickness of the air gap and other related parameters near the Brewster angle, the transfer function required for tunable optical differential can be achieved by varying the Fresnel reflection coefficient and the transverse spin shift. Finally, we theoretically predict that fast switching of image edge resolution can be realized by electric field modulation and get a conclusion.

2. Theoretical analysis

As indicated in Figure 1, we consider a four-layered reflective structure consisting of prism-graphene-air gap-substrate. In this four-layered structure, the prism and the substrate have a refractive index of 1.5, the air gap has a refractive index of 1, and the optical conductivity of the monolayer graphene can be expressed as $\sigma$.

 

Fig. 1. Schematic diagram of the wave reflection at a four-layered structure of prism-graphene-air gap-substrate. The incident beam is incident along the $z_{i}$ axis, and after passing through the four-layered nanostructure, it is reflected along the $z_{r}$ axis. $x_{i}$, $y_{i}$, $z_{i}$ and $x_{r}$, $y_{r}$, $z_{r}$ represent incident and reflected beams coordinates, respectively.

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When an incident beam of linearly polarized light is reflected through the four-layered structure with an angle $\theta _{i}$, the photonic spin Hall effect occurs. Then, the corresponding reflected beam would be split into left-handed and right-handed circularly polarized spin components. $x_{i}$, $y_{i}$, $z_{i}$ and $x_{r}$, $y_{r}$ and $z_{r}$ represent the coordinates of the incident and the reflected electric fields, respectively, where the $z_{i}$ and $z_{r}$ axes are parallel to the propagation direction of the central wave vector of the light beam, respectively. Here, we consider the incident light beam is a linearly polarized Gaussian beam and its angular spectrum can be written as

$M_{R}$ represents the reflection matrix, the angular spectrum of reflected beam can be expressed as

Here, the boundary conditions $k_{rx}=-k_{ix}$ and $k_{ry}=k_{rx}$, $k_{0}$ is the wave vector in vacuum. $\widetilde {E}_i^H$ and $\widetilde {E}_i^V$ denote the angular spectrum of the incident light beam along the $x_{i}$ and $y_{i}$ axes, respectively. $\widetilde {E}_r^H$ and $\widetilde {E}_r^V$ denote the angular spectrum of the reflected beam along the $x_{r}$ and $y_{r}$ axes, respectively, where $\widetilde {E}_r^H=\frac {1}{\sqrt {2}}\left (\widetilde {E}_{r+}+\widetilde {E}_{r-}\right )$, $\widetilde {E}_r^V=\frac {1}{\sqrt {2}} i\left (\widetilde {E}_{r-}-\widetilde {E}_{r+}\right )$. Here, $\mathrm H$ and $\mathrm V$ respectively represent horizontal and vertical polarizations. The positive and negative signs denote left and right circularly polarized spin components, respectively.

In this paper, we assume that a beam of light is incident with a horizontal polarization state. Since the vertical polarization state of the incident beam is insensitive to the graphene layer, it is not discussed too much in our calculation. Correspondingly, the vertical polarization state can be analyzed in the similar way. According to Eq. (3), we can obtain the expressions of the reflected angular spectrum

When the reflected photons pass through a Glan laser polarizer whose polarization axis is orthogonal to the polarization of the center wave vector. Therefore, the final output electric field going through the whole differentiator system can be rewritten as

As shown in Figs. 2(a) and (b), we respectively plot theoretical diagrams of the intensity distribution of the incident light field and reflected light field. We can see that when a beam of Gaussian light incident on the four-layered structure with a graphene layer, the reflected light field has a transverse spin separation in the vertical $y$ direction. When the photon is reflected, due to the existence of spin-orbit coupling, the spin angular momentum of the photon is converted into orbital angular momentum, resulting in the transverse spin separation of the beam, it also represents the photonic spin Hall effect after beam reflection. Figs. 2(c) and (d) represent intensity distributions of the incident and reflected light fields in the $y$ direction at $x = 0$ after normalization, respectively. We can see that the reflected beam exhibits a first-order Hermite Gaussian distribution with zero amplitude at $y = 0$, which corresponds to a first-order differentiation in the y direction of the incident light field.

 

Fig. 2. Spatial differentiation demonstration for a Gaussian light illumination. (a) and (b) respectively represent theoretical intensity profiles of the incident and reflected beams. (c) and (d) represent the theoretical intensity of incident and reflected beams in the $y$ direction with $x$ = 0, respectively.

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Then we also theoretically derive the spatial spectral transfer function, the spatial transform between the incident and reflected electric fields is determined by a spatial spectral transfer function $H(k_x,k_y)=E_\text {out}(k_x,k_y)/E_\text {in}(k_x,k_y)$, where $E(k_x,k_y)$ is the Fourier transform of $E(x,y)$. Finally, the spatial spectral transfer function can be simplified as

In Figs. 3(a), (b) and (c) all represent the spatial spectral transfer function of the four-layered structure containing the graphene layer. It can be seen that at the special point $k_{y}/k_{0} = 0$, the minimum of the spatial spectral transfer function is 0, and it is a linear distribution as well. For the spatial spectral transfer function has a phase transition at the special point $k_{y}/k_{0} = 0$, as shown in Fig. 3(d), this also proves that the differentiator we design can achieve spatial differentiation in the y direction. In Fig. 3(c), we also add the spatial spectral transfer functions with the incident angles of $30^{\circ }$ and $35^{\circ }$. It can be seen that when the incident angles are selected as $30^{\circ }$, $33,61^{\circ }$ and $35^{\circ }$ at $k_{x}/k_{0} = 0$, the slope of the spatial transfer function is different for different incident angles. Therefore, the incident angle has a certain influence on the result of spatial differentiation.

 

Fig. 3. The spatial spectral transfer function of the four-layered structure of prism, graphene, air gap and substrate. (a) and (b) represent the spatial spectral transfer function when the incident Angle is selected at $33.61^{\circ }$. (c) represents the spatial transfer functions when $k_{x}/k_{0} = 0$ and the incident angles are set to $30^{\circ }$, $33.61^{\circ }$ and $35^{\circ }$, respectively. (d) shows the phase distribution of the spatial spectral transfer function.

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Next, we will discuss and analyze the transverse spin shift after the incident beam is reflected. At any given plane $z_{r}$ = const. The transverse spin shift of the field centroid compared to the geometrical-optics prediction is given by

In Eq. (9), $\delta _{\pm }$ respectively represent the transverse spin shift of the left and right circular polarization components,then we can get

In which $r_{p}$ and $r_{s}$ denote Fresnel reflection coefficients for parallel and perpendicular polarizations, respectively, where $r_{p,s}=\mid r_{p,s}\mid \text {exp}(i\varphi _{p,s})$. By making use of a Taylor series expansion based on the arbitrary spectrum component, $r_{p}$ and $r_{s}$ can be defined as

From Eqs. (10) and (11), it can be seen that the Fresnel reflection coefficient is the key parameter for the transverse spin shifts, they can be expressed as

Here, $n\in \{p,s\}$, m ,l = 1,2,3,4 (1,2,3,4 represent prism, graphene, air gap and substrate) and $k_{m z}=k_{0}\sqrt {\varepsilon _{m}-\varepsilon _{1}(\sin \theta _{i})^{2}}$. $r_n^{12}$, $r_n^{23}$ and $r_n^{34}$ represent the Fresnel reflection coefficients of the first interface (prism-graphene), the second interface (graphene-air gap) and the third interface (air gap-substrate), respectively. $d_{2}$ and $d_{3}$ represent the thickness of the graphene layer and the air gap, respectively.

Here, $\varepsilon _m$ is the permittivity of the m-th layer medium. In which the relative equivalent permittivity of the graphene layer can be expressed as

In Eq. (18), t = 0.34 nm is the thickness of the graphene layer, and $\varepsilon _0=8.854\times {10}^{-12}~\mathrm F{/ \mathrm m}$ is the permittivity of vacuum. In particular, the optical conductivity of graphene $\sigma$ can be expressed as

Therefore, we plan to influence the optical properties and Fresnel reflection coefficient of graphene by tuning its Fermi energy. For a monolayer graphene, we can easily adjust the Fermi energy of graphene through external electrostatic field biasing, so as to realize the modulation of the photonic spin Hall effect in the above four-layered structure. In addition, we also explore the effects of the incident angle of the beam and the thickness of the air gap on the modulation of the photonic SHE, and apply tunable optical differential operation to target edge detection.

3. Results and discussions

In this section, we will further discuss and analyze the influence of graphene on the photonic spin Hall effect. We can easily see that it changes with the graphene conductivity through the Fresnel reflex coefficient. Therefore, the photonic spin Hall effect is also adjusted when we try to change the graphene Fermi energy. As shown in Figs. 4(a) and (b), we can see that at a wavelength of 1550 nm, both the electrical conductivity and permittivity of graphene vary with the Fermi energy. In particular, when the Fermi energy is at 0.4 eV, the conductivity and permittivity will change greatly, and the spin shift will also change greatly at this point. As shown in Fig. 4(c), when we change the incident angle of the beam to the vicinity of the Brewster angle ($32^{\circ }$-$35^{\circ }$), it can be seen that the spin shift is very sensitive to the incident angle when the beam is reflected near the Brewster angle due to the presence of the monolayer graphene. In the part before 0.4 eV, the spin shift has basically no effect on the Fermi energy of graphene, and the maximum spin shift is nearly invariable that remains nearby 15$\lambda$. The incidence angle characteristic of the Brewster angle is $33.55^{\circ }$, which does not alter. However, when the Fermi energy increases above 0.4 eV, it can be clearly seen that the spin shift changes sharply and the incident angle exhibiting Brewster angle characteristics is $33.69^{\circ }$. Obviously, we can see that when the Fermi energy in Fig. 4(a) is 0.4 eV, the real part of the graphene conductivity has a step down to 0 and the imaginary part has a minimum value, meanwhile the relative permittivity and Fresnel reflection coefficient of graphene change, which causes a drastic change of the spin shift and the position of the Brewster angle at such Fermi energy. When the Fermi energy is increased to 0.49 eV, we discover that the spin shift starts to decrease slowly after a drastic change. The Fermi energy is at 0.49 eV, what we care about is that the imaginary part of the graphene conductivity in Fig. 4(a) gradually increases from negative to positive. And the Fermi energy increases above 0.49 eV, the spin shift will gradually decrease with the increase of Fermi energy, which we can clearly see from Fig. 4(c). Therefore, we can conclude that the position of the Brewster angle is controlled by the real part of graphene conductivity, while the value of the spin shift is mainly controlled by the imaginary part of graphene conductivity from the above two important features. In Fig. 4(d), we plot the Fermi energies versus spin shifts for incident angles of $33.31^{\circ }$, $33.61^{\circ }$, and $33.91^{\circ }$, respectively. We can see that at $33.31^{\circ }$ and $33.91^{\circ }$, the spin shift drops sharply nearby the Fermi energy of 0.4 eV and 0.49 eV. At $33.61^{\circ }$, when the Fermi energy is 0.4 eV, the spin shift changes from $-y$ direction to $+y$ direction, which also corresponds to the change of Brewster angle in Fig. 4(c). Meanwhile, in the part above 0.49 eV, the spin shifts under the three incident angles gradually decrease with the increase of Fermi energy.

 

Fig. 4. (a) and (b) respectively represent the optical conductivity and relative permittivity of the monolayer graphene as a function of Fermi energies (0-1.0 eV) when the incident wavelength is 1550 nm. (c) shows the change of spin shifts with Fermi energies and incident angles ($32^{\circ }$-$35^{\circ }$) when the beam is incident in the horizontal polarization state. (d) shows when the incidence angles are set to be $33.31^{\circ }$, $33.61^{\circ }$, and $33.91^{\circ }$, tuning of the spin shift of the horizontal polarization state versus the Fermi energies.

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In addition, we also discuss the correlation between the spin shift and the thickness of the air gap, as shown in Fig. 5(a). When we set the beam waist $w_0=30\mathrm {\lambda }$, we can see that the spin shift exhibits periodic oscillations with increasing the thickness of the air gap, which is caused by the Fabry-Perot resonance in the four-layered nanostructure. Furthermore, in Fig. 5(a) we intercept four different air gap thicknesses of 650 nm, 1350 nm, 2050 nm and 2750 nm, this corresponds to the four spin shifts of Fig. 5(b). It can be seen that when the thickness is 650 nm (or 2050 nm), the spin shift can reach a maximum of 15$\lambda$, which is half of the beam waist. At 1350 nm (or 2750 nm), the maximum spin shift is only about 4$\lambda$. We also found that changing the thickness of the air gap does not have a large effect on the Brewster angle, which is at $33.55^{\circ }$. From the above analysis, we know that a giant spin shift can be achieved by optimizing the thickness of the air gap near the Brewster angle, which also provides greater flexibility for modulating the photonic spin Hall effect.

 

Fig. 5. (a) shows the spin shift varies with thicknesses of air-gap(0-3000 nm) and incident angles($30^{\circ }$-$37^{\circ }$). (b) represents the spin shift diagram under four different air gap thicknesses of the optimal thickness 650 nm, 1350 nm, the optimal thickness 2050 nm, 2750 nm.

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Finally, the optical differential operation plays a very significant role in the ultra-fast image processing of image edge detection. In order to verify the edge detection ability of the differential operation, we theoretically predict the edge of the target under different graphene Fermi energies at a telecommunication wavelength. Figure 6 exhibits the edge image of the measured targets "1" and "0" under different Fermi energies. Here, the incident beam with a horizontal polarization state is applied, and we fix that the incident angle of the Gaussian beam at $33.61^{\circ }$ and the thickness of the air gap is 650 nm. Figs. 6(a) and (i) represent the measured target images, while Figs. 6(b)-(d) and (j)-(l) represent the edge detection images at different graphene Fermi energies. Notably, the differentiation is along the y direction, and the edge perpendicular to the y direction is the most obvious. When we adjust the Fermi energy of the monolayer graphene from 0.4 eV to 0.42 eV, it is obvious that the edge thickness corresponding to the measured target becomes thicker in the y direction with the increase of spin shift. It can be clearly seen from Figs. 6(e)-(h) and (m)-(p), which represent the horizontal intensity distribution of the blue dotted lines in Figs. 6(a)-(d) and (i)-(l), respectively. Therefore, we can easily adjust the edge image resolution by modulating the Fermi energy of graphene without changing the model structural parameters. In addition, with the adjustment of some sensitive parameters, such as the incident angle of the beam or the thickness of the air gap, we can obtain tunable differentiators with different sensitivities.

 

Fig. 6. Theoretical diagram of target edge detection images under different graphene Fermi energies at $33.61^{\circ }$. (a) and (i) represent the target image of the number "1" and "0" for testing. (b)-(d) and (j)-(l) represent corresponding edge detection images at different Fermi energies, respectively. (e)-(h) and (m)-(p) correspond to the horizontal intensity profiles of the blue dotted line in (a)-(d) and (i)-(l), respectively.

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4. Conclusion

In summary, we theoretically analyze the tunable optical differentiator based on the four-layered structure of prism-graphene-air gap-substrate by using the principle of tunable optoelectronic properties of graphene at a telecommunication wavelength. Here, we apply the photonic SHE in the proposed nanostructure to the spatial differential operation in the y direction. Then, we mainly discuss the relationship between the transverse spin shift and the Fermi energy of graphene, the beam incident angle, and the air gap thickness under the incident state of horizontally polarized light near the Brewster angle. By optimizing these parameters to obtain the maximum spin shift, precise control of the transverse shift can be achieved. Importantly, adjusting the Fermi energy of graphene electrically could control the magnitude of the transverse shift to modulate the resolution of the resulting edge image. Finally, it may lead to potential opportunities for future tunable spin optoelectronic devices, and open up more applications in edge detection, object recognition, and biomedical imaging.