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Abstract
In recent years, optical analog computing has experienced rapid development, among which optical differential operation has attracted great attention. Here, based on the unique optical properties of graphene, we propose an electrically tunable optical spatial differentiation by introducing a graphene layer at a quartz substrate. It is found that the output light field is sensitive to the graphene layer near the Brewster angle for small polarization output at the graphene-quartz substrate interface and can be modulated by changing the Fermi energy of graphene. In this case, the result of the optical differential operation can be dynamically regulated. Almost strict one-dimensional differential operations in different directions and almost perfect two-dimensional differential operations can be achieved. In addition, two-dimensional edge detection with different degrees of distortion in different directions can also be realized when applied to image processing. This new modulation method may provide more possibilities for tunable image edge detection and provide a potential way for developing more versatile optical simulators in the future.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, optical analog computing has attracted significant attention due to its advantages of ultra-high speed, real-time, high parallel processing, and low power consumption [1–3]. It can perform a variety of simple tasks, such as differentiation, integration, and convolution [4,5]. Among them, optical differentiation, as a basic mathematical operation, can extract the edge information of objects with obvious brightness changes when it is applied to image processing. Image edge detection has important application value in automatic driving, microscopic imaging, and target recognition [6–11]. Therefore, the optical differential operation provides the possibility for applications that require real-time processing of target images. At present, a variety of optical differentiators capable of image edge detection have been widely reported, such as optical differentiators based on surface plasmons [12,13], photonic crystals [14], grating nanostructures [15,16], metasurface [17–20] and photonic spin Hall effect [21–24]. In addition, some concepts from the field of topological photonics are applied to optical computation to realize differential operations [25].
The photonic spin Hall effect (SHE), which is thought to be the result of spin-orbit interactions, is a tiny effect, but it is highly sensitive to changes in system parameters. Therefore, it plays a key role in many fields [26–28] and has been studied in various optical structures [29–35]. Optical differentiation based on photonic SHE has become an effective and accurate method for image edge detection. This process is essentially due to the spin-dependent separation of photons caused by photonic SHE. Here, the spin-dependent displacements include in-plane spin separation and transverse spin separation. In image processing, it can control the direction and resolution of one-dimensional edge detection of target objects. At present, it has been reported that beam displacement can be adjusted through various means, such as changing the incident angle of the light beam, modifying the optical axis of anisotropic metamaterials, and altering the thickness of the reflected layer [22,23,36,37]. The differentiators based on these methods make it possible to realize tunable edge imaging. However, these approaches often require modifying the device structure or adjusting the incident light beam. Therefore, is there an easy way to implement tunable differentiation?
In this work, we propose a mechanism to achieve electrically tunable optical differential operation by utilizing the unique optical properties of graphene. Graphene is a two-dimensional carbon nanomaterial with a hexagonal honeycomb lattice composed of carbon atoms with $sp^{2}$ hybridization orbitals [38,39]. Its optical parameters can be dynamically adjusted by changing Fermi energy through electrostatic doping [40–42], which directly provides efficient real-time control of reflected waves. In the past, the introduction of graphene layers in various structures to research beam displacement has been widely reported. For example, considering the reflection of a light beam at a graphene-substrate system, the spin-dependent shift in the photonic SHE, Imbert-Fedorov shift, or Goos-Hänchen shift are investigated [31,41,43,44]. Based on the reflection of the light beam at gain-loss media embedded with monolayer graphene, a feasible method is also theoretically proposed to amplify and control photonic SHE in wide range of incident angles [45]. Here, we consider light beams reflected at the graphene-quartz substrate, where graphene is treated as a zero-thickness model. By eliminating the linear polarization in the center wave vector of the reflected beam, the system can perform optical differential operations. First, we theoretically derived the relationships between the graphene conductivity and the transfer function, as well as the output light field. On this basis, we analyze the influence of Fermi energy on the output light field by combining the coefficients in the transfer function. It is found that the output light field is sensitive to the change in the Fermi energy of graphene near the Brewster angle when incident polarization is very small. In this case, the result of the optical differential operation can be dynamically adjusted. Approximate or almost strict one-dimensional or two-dimensional optical differential operations can be achieved. Further numerical simulation results confirm the theoretical suggestion. This new modulation method may provide more possibilities for tunable image edge detection and provide a potential way for developing more versatile optical simulators in the future.
2. Theoretical framework
Firstly, we consider a Gaussian wave packet with monochromatic frequency $\omega$ impinging from air to the graphene-quartz substrate interface as shown in Fig. 1(a). The $z$ axis of the Cartesian coordinate system ($x$, $y$, $z$) is perpendicular to the interface, and ($x_{i}$, $y_{i}$, $z_{i}$) and ($x_{r}$, $y_{r}$, $z_{r}$) represent the coordinate systems of the incident and reflected beams, respectively. The $z_{i}$ ($z_{r}$) axis is parallel to the incident (reflected) beam propagation direction, and the electric field components of the incident (reflected) wave along the $x_{i}$ ($x_{r}$) and $y_{i}$ ($y_{r}$) directions correspond to the $p$ and $s$ polarizations, respectively. According to boundary conditions, the amplitudes of incidence, reflection, and refraction satisfy the following equations:
Here, $s$ and $p$ represent vertical and parallel polarization states respectively, $r_{pp}=E_{r}^{p}/E_{i}^{p}$, $r_{ss}=E_{r}^{s}/E_{i}^{s}$, $r_{ps}=E_{r}^{p}/E_{i}^{s}$ and $r_{sp}=E_{r}^{s}/E_{i}^{p}$. $\sigma _{ss}$, $\sigma _{pp}$, $\sigma _{sp}$ ($\sigma _{ps}$) denote the transverse, longitudinal, and crossing conductivities, respectively. $k_i$ and $k_t$ are the incident and refracted wave vectors, respectively. $k_{iz}$ and $k_{tz}$ are the wave vectors along the $z$ axis in the air and quartz substrate, respectively. $Z_{1}$ and $Z_{2}$ denote air and quartz substrate impedances, respectively. If we restrict isotropy for graphene, there are $r_{ps}=-r_{sp}=0$ and $\sigma _{ps}=\sigma _{sp}=0$. $r_{pp}$ and $r_{ss}$ are represented by $r_{p}$ and $r_{s}$, respectively. Therefore, the Fresnel reflection coefficients of $p$ and $s$ waves are given by
Here, the first and second terms on the right-hand side of the equation are contributed by intraband scattering and interband transition, respectively. $\rm {step}()$ represents the step function, $E_{f}$ is the Fermi energy of graphene, $\hbar$ denotes the reduced Planck function, and $e$ is the unit charge. $\tau ={\mu }E_{f}/ev_{f}^{2}$ represents the relaxation rate, which is related to the mobility $\mu =10^{4}$ $\rm {cm^{2}Vs^{-1}}$ and Fermi velocity $v_{f}= 10^{6}$ $\rm {m/s}$. The values of these parameters were selected as in Ref. [41]. It is worth noting that the total conductivity of graphene is no longer real, so the Fresnel reflection coefficients $r_{p}=|r_p|\exp (i\varphi _p)$ and $r_{s}=|r_s|\exp (i\varphi _s)$ are also complex with different phases after introducing a graphene layer at a quartz substrate.
Fig. 1. (a) Diagram of wave reflection at the graphene-quartz substrate interface in Cartesian frame. The $z$ axis of the Cartesian coordinate system is perpendicular to the base surface, and the graphene is uniformly placed on top of the quartz substrate at $z=0$. ($x_{i}$, $y_{i}$, $z_{i}$) and ($x_{r}$, $y_{r}$, $z_{r}$) represent coordinates of the incident and reflected beams, respectively. Transverse and in-plane splitting occurs for the light beam at the interface. (b) The front view of the proposed reflection structure. $\theta _{i}$ and $\theta _{r}$ represent incident and reflected angles, respectively. The relative dielectric constants of air and quartz substrate are $\varepsilon _{1}=1$ and $\varepsilon _{2}=2.15$, respectively. (c) The variation of the intensity distribution of the output light field with Fermi energy is described.
Next, we consider an incidence of Gaussian wave packet with a wavelength of 1550 $\rm {nm}$, where all wavevector components have the same polarization $\boldsymbol {u_{in}}=(u_{in}^{x}, u_{in}^y)^{T}$. The electric field of the incident wave can be written as $\boldsymbol {\tilde {E}_{in}}(k_{ix},k_{iy})=\boldsymbol {u_{in}}\tilde {E}_{in}(k_{ix},k_{iy})$. Here, $\tilde {E}_{in}(k_{ix},k_{iy})=w_{0}/\sqrt {2\pi }\exp {[-w_{0}^{2}(k_{ix}^{2}+k_{iy}^{2})}/4]$, where $w_{0}$ represents the beam waist. After reflection through the graphene-quartz substrate interface, the reflected field can be expressed as $\boldsymbol {\tilde {E}_{r}}(k_{rx},k_{ry})=\boldsymbol {R}(k_{rx},k_{ry})\cdot \boldsymbol {\tilde {E}_{in}}(k_{ix},k_{iy})$, where $\boldsymbol {R}(k_{rx},k_{ry})$ represents the reflection matrix and is two dimensional. It can be expressed as
In general, the reflected waves with different wavevectors can have different polarizations. The output polarization $\boldsymbol {u_{out}}=(u_{out}^{x}, u_{out}^y)^{T}$ can be selected through a linear polarizer to obtain the final output electric field, which can be expressed as $\boldsymbol {\tilde {E}_{out}}(k_{rx},k_{ry})=\boldsymbol {u_{out}}\tilde {E}_{out}(k_{rx},k_{ry})$. The whole process can be described by $\tilde {E}_{out}(k_{rx},k_{ry})=t(k_{rx},k_{ry})\tilde{E} _{in}(k_{ix},k_{iy})$. It is worth noting that $t(k_{rx},k_{ry})$ is used to describe the spatial transformation from incident field to output field, which is known as the spatial spectral transfer function. It can be calculated by the formula $t(k_{rx},k_{ry})=\boldsymbol {u_{out}}^{\dagger }\boldsymbol {R}(k_{rx},k_{ry})\boldsymbol {u_{in}}$, and we can get
Here, $\eta =r_{p}u_{in}^{x}{u_{out}^{x}}^*+r_{s}u_{in}^{y}{u_{out}^{y}}^*$, $\xi _{1}=-\frac {1}{k_{0}}({\rho }u_{in}^{x}{u_{out}^{x}}^*+{\chi }u_{in}^{y}{u_{out}^{y}}^*)k_{rx}$, and $\xi _{2}=\frac {1}{k_{0}}(r_{p}+r_{s})\cot {\theta _{i}}(u_{in}^{y}{u_{out}^{x}}^*-u_{in}^{x}{u_{out}^{y}}^*)k_{ry}$, where $\rho =\partial {r_{p}}/\partial {\theta _{i}}$, and $\chi =\partial {r_{s}}/\partial {\theta _{i}}$. In the above equation, the boundary conditions $k_{ix}=-k_{rx}$ and $k_{iy}=k_{ry}$ are applied, and the Fresnel reflection coefficients $r_p$ and $r_s$ are developed by a first-order Taylor expansion. It is worth noting that the coefficients $\xi _{1}$ and $\xi _{2}$ here are related to in-plane splitting and transverse splitting in the photonic SHE, respectively. By the way, the expression $E_{in, out}(x,y)$ for the input and output fields in the position space are obtained through the spatial Fourier transform, and they describe the scalar field of the field distribution in the input or output plane, respectively.
In order to realize the spatial differential operation, we expect the transfer function to satisfy the condition $t(k_{rx}=0,k_{ry}=0)=0$. It can be concluded from Eq. (9) that the purpose can be achieved only by making $\eta =0$. The above equation is called the cross-polarization condition. In this work, we can choose the input and output polarizations appropriately to satisfy this condition. According to the above analysis, the output polarization selected by the linear polarizer can be expressed specifically as $\boldsymbol {u_{out}}=(cos{\alpha }, sin{\alpha })^T$, where $\alpha$ is the angle between the polarization axis and the $x_r$ axis, as shown in Fig. 1(a). Here, the input polarization can be described by the output polarization, as shown in the following equation
The final output electric field in position space is expressed by the following formula
It can be seen that $E_{out}$ appears as a differential form of the input electric field. The coefficients $\xi _1$ and $\xi _2$ in the transfer function together determine the final differential effect. According to previous studies, when $\xi _{1}/\xi _{2}=\pm {i}$, the transfer function has a rotation invariant magnitude [25]. At this point, the anisotropic differential operation can be realized based on the graphene-quartz substrate interface, so perfect two-dimensional image edge detection can also be realized. This can be achieved by adjusting the output polarization angle (the incident polarization also changes), the incidence angle, and the Fermi energy of the graphene. Based on this, some interesting situations may arise when the Fermi energy of graphene is controlled simply by adjusting the grid voltage so that $\xi _{1}$ and $\xi _{2}$ become other values.
3. Results and discussion
Due to the unique optical properties of graphene, the spatial displacement of a light beam after reflection at the graphene-quartz substrate interface may be sensitive to changes in certain physical parameters. This may have an impact on optical differentiation operations and result in different edge images for target objects during edge detection. It can be seen from the theoretical part that the coefficients $\xi _1$ and $\xi _2$ in the transfer function are related to the in-plane splitting and transverse splitting in the photonic SHE, respectively. Here, $\xi _1$ and $\xi _2$ determine the differentiators in the $x$ and $y$ directions, respectively. It can be seen from Eqs. (11) and (12) that the Fresnel reflection coefficients have a significant influence on it. As shown in Fig. 2, the amplitude and phase of Fresnel reflection coefficients are plotted. It is found that the values of $|r_p|$ and $|r_s|$ have a jump when the Fermi energy is about 0.401 eV. The phase $\varphi _p$ has special changes at $E_f\rightarrow {0.401}\,\rm {eV}$ or $E_f=0.481\,\rm {eV}$, while the phase $\varphi _s$ has a $2\pi$ transition at $E_f=0.481\,\rm {eV}$. It can also be clearly seen from Fig. 2(a) that the value of $|r_p|$ varies significantly near the Brewster angle, so the corresponding coefficients $\xi _1$ and $\xi _2$ should also change significantly. Here, the value of Brewster angle is basically stable in the range of $56.3^{\circ }$ to $56.4^{\circ }$ at $E_f<0.4\,\rm {eV}$, and in the range of $55.7^{\circ }$ to $55.9^{\circ }$ at $E_f>0.401\,\rm {eV}$. At the Fermi energy of 0.4 eV to 0.401 eV, the Brewster angle does not exceed the above-mentioned range, but there is a jump at a specific Fermi energy.
Fig. 2. (a) and (b) The amplitude of the Fresnel reflection coefficient at the graphene-quartz substrate interface. (c) and (d) The phase of the Fresnel reflection coefficient at the graphene-quartz substrate interface.
The Fresnel reflection coefficients are no longer real due to introduce the graphene layer, so the coefficients $\xi _1$ and $\xi _2$ are complex, and the values of the coefficients together determine the differential effect of the differentiator designed based on the graphene-quartz substrate interface. As shown in Fig. 3, we plot the real and imaginary parts of $\xi _1$ and $\xi _2$ near the Brewster angle in order to analyze their relationship to the Fermi energy of graphene and output polarizations. Figures 3(a)–3(d) reveal $\textrm{Re}[\xi _1]$, $\textrm{Im}[\xi _1]$, $\textrm{Re}[\xi _2]$ and $\textrm{Im}[\xi _2]$ at the incident angle $\theta _{i}=55.6^\circ$, respectively. Figures 3(e)–3(h) show $\textrm{Re}[\xi _1]$, $\textrm{Im}[\xi _1]$, $\textrm{Re}[\xi _2]$ and $\textrm{Im}[\xi _2]$ at the incident angle $\theta _{i}=55.8^\circ$, respectively. It can be seen that their changes are relatively obvious at small polarization outputs, and a jump occurs when the Fermi energy of graphene is close to 0.401 eV. When the incident angle $\theta _{i}=55.8^\circ$, the real and imaginary parts of the coefficients increase dramatically at a certain position between the Fermi energy of 1.5 eV to 1.55 eV. Therefore, the range of variation for $\xi _1$ or $\xi _2$ is larger in this case. For small polarization outputs, these particular changes with Fermi energy may result in tunable edge imaging.
Fig. 3. The coefficients $\xi _1$ and $\xi _2$ in transfer function at different incident angles are plotted. (a)–(d) Real and imaginary parts of the coefficients when the incident angle is $55.6^\circ$. (e)–(h) Real and imaginary parts of the coefficients when the incident angle is $55.8^\circ$.
Based on the previous analysis, we choose smaller output polarizations and plot the variation in the coefficients $\xi _1$ and $\xi _2$ with the Fermi energy of graphene, to conveniently analyze the influence of the Fermi energy on the differential effect. Figure 4(a) shows the coefficients change with Fermi energy at $\theta _{i}=55.6^{\circ }$ and $\alpha =0.34^{\circ }$, Fig. 4(b) shows the ratio of coefficients change with Fermi energy on the condition. It can be clearly seen that the changes of $\xi _1$ and $\xi _2$ are relatively gentle with $E_f<0.4\,\textrm{eV}$ or $E_f>0.8\,\textrm{eV}$. When the Fermi energy is between 0.4 eV and 0.8 eV, $\textrm{Re}[\xi _1]$ rapidly increased to the maximum value after a sharp decrease, then increased after a slight decrease, and finally decreased gradually. The absolute value of $\textrm{Re}[\xi _2]$ increases slightly and then gradually decreases to the minimum value, then gradually increases, and finally slowly decreases. $\textrm{Im}[\xi _1]$ and $\textrm{Im}[\xi _2]$ have a similar trend with the change of Fermi energy, but the change range of $\textrm{Im}[\xi _1]$ is larger. With the change of Fermi energy, the values of the coefficients have different changing trends. Therefore, there is no doubt that the tunable optical differential operation can be realized by changing the Fermi level under the condition that other parameters are fixed.
Fig. 4. (a) The variation of the real and imaginary parts of the coefficients $\xi _1$ and $\xi _2$ with Fermi energy. The horizontal coordinates corresponding to the black dotted lines are $E_f=0.1\,\textrm{eV}$ and $E_f=0.481\,\textrm{eV}$ (from left to right), respectively. (b) The variation of the real and imaginary parts of the ratio of the coefficients $\xi _1$ and $\xi _2$ with Fermi energy. The red dots (from left to again) correspond to Fermi energy of 0.1 eV, 0.403 eV, 0.481 eV, and 1.9 eV, respectively. (c)-(f) The distribution of the output light field at different Fermi energy. The Fermi energy selected corresponds in turn to the positions of the red dots on the green line in (b) (from left to right). (c1)-(f1) The spatial spectral transfer function at different Fermi energy. Here, the output polarization is $0.34^\circ$ and the incident angle is $55.6^\circ$.
To demonstrate the tunability of the differentiator based on the graphene-quartz substrate design, we selected different Fermi energy to plot the theoretical intensity distribution of the output light field, as shown in Figs. 4(c)–4(f). The parameters selected in the intensity distribution diagram correspond, in turn, to those at the red dot on the green line in Fig. 4(b). At Fermi energy of 0.1 eV and 0.481 eV, Figs. 4(c) and 4(e) show that the output light splits essentially in a single direction. From the position of the black dotted line in Fig. 4(a), it can be seen that $\textrm{Im}[\xi _1]$ and $\textrm{Im}[\xi _2]$ are close to 0 at this time, which is negligible compared with the real part. Then the phase of two coefficients in the transfer function can be approximately synchronous, and thus almost strict one-dimensional differential operations in the corresponding direction can be implemented. It is worth mentioning that the direction of one-dimensional differential operation is also different with different Fermi energy of graphene. Figures 4(c1) and 4(e1) reveal that the spatial spectral transfer functions present an almost strictly linear distribution in the direction of spot splitting [corresponding to Figs. 4(c) and 4(e)], which further proves that the almost strictly one-dimensional differential operation can be achieved, and the differential direction can be adjusted by changing the Fermi energy. However, both the real and imaginary parts of $\xi _1$ and $\xi _2$ play an important role in differentials at $E_f=0.403\,\textrm{eV}$. At this time, the phase of the differentiator in the $x$ and $y$ directions are not synchronized, resulting in a tendency of vortex distribution of the output light field intensity, and the weight close to the $y$ direction is large, as shown in Fig. 4(d). Therefore, the differentiator designed based on this condition can achieve approximate two-dimensional differential operations, but there is a small distortion in the direction perpendicular to the larger weight. Figure 4(d1) shows that the transfer function does not have a rotational invariant magnitude in the radial direction, but is approximately linear in the radial direction of coordinate origin, which also indicates that the differentiator under this condition can not achieve strict two-dimensional differential operation, and there is a slight distortion in the corresponding direction. When the Fermi energy is 1.9 eV, Fig. 4(f) shows that the beam splits approximately in the $y$ direction, and Fig. 4(f1) shows that the corresponding transfer function is approximately linear distribution in the direction slightly off the $y$ direction. At this time, there should be a large amount of distortion in the direction slightly off the $x$ direction when the differentiator performs image edge detection. From the above analysis, by adjusting the Fermi energy of graphene, it can be seen that almost strict or approximated one-dimensional differential operations and approximate two-dimensional differential operations can be achieved at $\theta _i=55.6^{\circ }$ and $\alpha =0.34^{\circ }$.
For other output polarization and incident angles, other interesting results can occur by adjusting the Fermi energy of graphene. Figures 5(a) and 5(b) show that the changes of $\xi _1$, $\xi _2$, and their ratios with Fermi energy at $\theta _i=55.8^{\circ }$ and $\alpha =0.84^{\circ }$. When the Fermi energy is 0.1 eV or 0.481 eV, Figs. 5(c) and 5(f) show that the output light is also basically split in a single direction. As previously analyzed, in this case, $\textrm{Im}[\xi _1]$ and $\textrm{Im}[\xi _2]$ are very close to 0 and negligible, and the differentiator can realize almost strict one-dimensional differential operations. When the Fermi energy of graphene is 0.401 eV (1.338 eV), it can be seen from the black dotted line in the locally enlarged diagram (main diagram) in Fig. 5(a) that $\textrm{Re}[\xi _1]$ and $\textrm{Im}[\xi _2]$ are close to 0, while $\textrm{Im}[\xi _1]$ and $\textrm{Re}[\xi _2]$ are approximately equal. Corresponding to Fig. 5(b), $\textrm{Im}[\xi _1/\xi _2]=-0.988\,(1)$ and $\textrm{Re}[\xi _1/\xi _2]=-0.003\,(-0.001)$ can be obtained, in which case $\xi _1/\xi _2=-i\,(i)$ can be approximated, so the two-dimensional differential operation can be achieved. Figures. 5(d) and 5(g) present an almost perfect vortex for the intensity distribution of the output light field, which also confirms the previous sentence. For the Fermi energy is 0.404 eV (1.96 eV), the intensity distribution of the output light field is similar to Fig. 4(d), showing an approximate vortex distribution. The weight of the light field intensity distribution is greater in the direction of slight deviation from the $x$($y$) direction, and there is a tiny distortion in the direction of slight deviation from the $y$($x$) direction when the differentiator performs image edge detection. In addition, we also plot the corresponding transfer functions and phase distribution, as shown in Fig. 6. Here, the parameters selected in Figs. 6(a)–6(f) also correspond to the red dots on the green line in Fig. 5(b) in sequence (from left to right). The results of Figs. 6(a) and 6(d) are similar to those of Figs. 4(c1) and 4(e1), and the transfer function is almost strictly linear distribution in the direction of light spot splitting. Figures 6(b) and 6(e) reveal that the transfer functions show almost linear distribution along the radial direction from the coordinate origin with an almost rotationally invariant magnitude, which further proves that almost perfect two-dimensional differential operations can be achieved. By the way, the phase distribution of the transfer function Corresponding to Figs. 6(b1) and 6(e1) has a helical phase. The transfer functions shown in Figs. 6(c) and 6(f) are similar to that shown in Fig. 4(d1), also revealing a slight distortion in the corresponding direction for edge detection. In general, through the above series of analyses, it can be seen that the differentiator designed based on the graphene-quartz substrate interface can achieve different differential results at different Fermi energy. In particular, under certain conditions, almost perfect two-dimensional differential operations can also be achieved.
Fig. 5. (a) The variations of the real and imaginary parts of the coefficients $\xi _1$ and $\xi _2$ with Fermi energy. The horizontal coordinates of the locally enlarged diagram are 0.4 eV to 0.402 eV. The black dotted line in the main diagram corresponds to the Fermi energy of 0.401 eV, and the black dotted line in the locally enlarged diagram corresponds to the Fermi energy of 1.338 eV. (b) The variations of the real and imaginary parts of $\xi _1/\xi _2$ with Fermi energy. The Fermi energy at the red dots (from left to back) is 0.1 eV, 0.401 ev, 0.404 eV, 0.481 eV, 1.338 eV, and 1.96 eV, respectively. (c)-(h) The distribution of the output light field at different Fermi energy. The Fermi energy selected corresponds in turn to the positions of the red dots on the green line in (b) (from left to right). Here, the output polarization is $0.84^\circ$ and the incident angle is $55.8^\circ$.
Fig. 6. (a)-(f) The spatial spectral transfer when different Fermi energy is 0.1 eV, 0.401 eV, 0.404 eV, 0.481 eV, 1.338 eV, and 1.96 eV, respectively. (b1) and (e1) The phase distribution of transfer function at Fermi energy of $E_f=0.401\,\textrm{eV}$ and $E_f=1.338\,\textrm{eV}$, respectively. Here, $\theta _i=55.8^{\circ }$ and $\alpha =0.84^{\circ }$.
Fig. 7. Numerical demonstration of tunable edge detection for different specific images. (a) and (f) The bright-field images with different input images. (b)–(e) Edge detection image for the simple letter “EDGE” at $\theta_i=55.5^{\circ}$ and $\alpha=0.34^{\circ}$. The parameters correspond to Figs. 4(c)–4(e), the Fermi energy of graphene is 0.1 eV, 0.403 eV, 0.481 eV, and 1.9 eV, respectively. (g)-(j) Edge detection image for the smiley face pattern at $\theta_i=55.8^{\circ}$ and $\alpha=0.84^{\circ}$. The parameters correspond to Figs. 6(d), 6(e), 6(g), and 6(h), respectively.
In order to verify the realization of edge detection based on different Fermi energy, the edge images under different conditions are obtained through theoretical simulation by using a simple letter diagram and a smiley face pattern as input images, respectively, as shown in Fig. 7. Figures 7(a) and 7(f) show the bright field images at different input images. Figures 7(b)–7(e) display the edge images of the simple letter diagram, where the parameter settings correspond to Figs. 4(c)–4(f) one by one. Obviously, by changing the Fermi energy of graphene, the differentiator can realize not only one-dimensional edge detection in different directions [Figs. 7(b) and 7(d)], but also two-dimensional edge detection with different degrees of distortion in different directions [Figs. 7(c) and 7(e)]. Figures 7(g)–7(j) show edge detection diagrams of smiley face pattern, where parameter settings correspond to Figs. 5(d), 5(e), 5(g) and 5(h) in turn. The two-dimensional edge detection images [Figs. 7(h) and 7(j)] have a slight distortion in the direction of a slight deviation from the $y$ or $x$ direction, while Figs. 7(g) and 7(i) reveal almost perfect two-dimensional edge detection images, respectively. These results correspond to Fig. 4 and Fig. 5, respectively. It should be noted that the differentiator with parameters corresponding to Figs. 5(c) and 5(f) can get one-dimensional edge detection images in the corresponding direction, even though we do not give the simulation results here. In summary, tunable image edge detection can be achieved by changing the Fermi energy of graphene.
We propose a tunable differential operation based on reflections from the graphene-quartz substrate interface. The beam displacement, which affects the result of the final differential operation, can be modulated by changing the incident polarization and incident angle of the light beam, and the Fermi energy of graphene. Based on a simple optical interface, direction-adjustable one-dimensional image edge detection can be realized by changing incident polarization [36], and adjustable resolution edge image can be achieved by changing incident angle [46]. This has been proven experimentally and theoretically. In our work, the Fermi energy of graphene directly affects the value of the transfer function, and the coefficients in the transfer function are related to the in-plane and transverse displacements in the photonic SHE, respectively. Therefore, we have focused on changing the Fermi energy of graphene by adjusting the grid voltage to control the light field, which makes the implementation of the tunable optical differential operation based on a simple method, rather than by changing the device structure [22,37] (the optical axis of the anisotropic metamaterial and the thickness of the reflector) or the incident beam (incident angle) [23] to achieve. This provides the possibility for the development of photonic devices with more tuning functions in the future. It is worth mentioning that different detection effects can be achieved by adjusting related parameters when edge detection is carried out on the target. Therefore, our method may provide some theoretical suggestions for future experimental verification.
4. Conclusion
In this paper, considering light beam reflection at the graphene-quartz substrate interface, we theoretically propose a mechanism for electrically tunable optical differential operation using the unique optical properties of the graphene layer. Here, we choose the linear output polarization, and the incident polarization is given by satisfying the corresponding cross-polarization conditions. It is found that the coefficient in the transfer function has some interesting changes near the Brewster angle when the output of small polarization is selected. At this point, by adjusting the gate voltage to change the Fermi energy, we reveal that the differentiator can not only achieve almost strict one-dimensional differential operations in different directions but also achieve almost perfect two-dimensional differential operations. Moreover, two-dimensional image edge detection with different degrees of distortion in different directions can also be realized. It is worth mentioning that we can achieve almost perfect two-dimensional optical differential operations without using total internal reflection or operating at Brewster angle because the Fresnel reflection coefficients are complex. Compared with previous studies, our proposed method for implementing the tunable optical differential operation is no longer limited to adjusting the incident polarization, incident angle, and thickness of the reflected layer. This new modulation method may provide more possibilities for adjustable image edge detection and provide a potential path toward developing more versatile optical simulators in the future.
Funding
National SKA Program of China (2022SKA0130100, 2022SKA0130104); National Natural Science Foundation of China (12273008); Natural Science and Technology Foundation of Guizhou Province ([2023]024); the Foundation of Guizhou Provincial Education Department (KY (2020) 003).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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