1. Introduction

Optical spatial differentiation is a method of all optical image processing that can achieve high-speed, low energy consumption, large bandwidth, and large-scale parallel operations [1–7]. Spatial differentiation can be used for extracting image edge information, which has broad application prospects in fields such as microscopic imaging, image enhancement, and pattern recognition [8–11]. The traditional 4F spatial optical filtering system is widely used in spatial optical analog computing, which mainly consists of Fourier transform lenses and spatial frequency filters. This system utilizes the Fourier transform effect of the lens and loads various filters with specific functions on the Fourier plane to process the spectral information of the incident image. It can obtain corresponding frequency distribution of output light field by eliminating the low frequency information and extracting higher-order information of an image, thereby enhancing the edge contour of object. Such an edge detection technique has important applications in the fields of image processing and geometric feature information extraction in computer vision [12–14].

The traditional 4F system uses large-sized optical components for information processing, which is not compatible with modern miniaturized and integrated signal processing devices. Combining optical analog computing with artificial microstructure materials provides a possibility for further integration and miniaturization of spatial optical analog processors [1,15,16]. The corresponding theoretical design of optical differential combinators with the use of metamaterials was proposed for the foundation of image processing [1]. Then, the feasibility of the 4F system method and the Green’s function method was verified by experiments [4,5]. Subsequently, spatial optical analog computing devices such as integrators [3], differentiators [17,18], and equation solvers were implemented [19]. Recently, based on the photonic spin Hall effect (SHE) in reflection or refraction on the surface between two media [20,21], a differentiator was proposed to realize optical image processing [15]. Inducing the photonic SHE acts as a filter on the Fourier plane, an intensity function of eliminating intermediate low frequency information and retaining marginal high frequency information can be achieved [22,23]. Similarly, based on the spin-orbit coupling of light in a dielectric metasurface, a kind of optical analog computing for broadband edge image has also been experimentally demonstrated [16].

With the rapid development of optical metamaterials in analog spatial differential imaging, the photonic SHE has become an effective method for image edge detection based on an optical differential operation on light field distribution [15,20,24,25], which is different from traditional lens and filter combinations. Among them, the study of photonic SHE in two-dimensional atomic crystals (2DAC) [26] has attracted much attention [27–31]. As an important example, there are two theoretical models to study the optical properties of 2DAC [32], and several experimental validations have been made through different research methods [33–39]. Based on the models, some critical parameters such as number of graphene layers can be accurately measured. However, these methods are usually subject to the factors of complexity and high cost of the experimental equipments, as well as the decrease of precision and sensitivity as the layers increase. Here, we have not verified which model is closer to the theoretical truth, but only provided an alternative method to characterize the number of graphene layers for reference based on these two models.

In this work, our goal is to achieve a spatial differentiation in the reflection of light on the surface of a graphene stacking. We first establish the paraxial model to describe the light-matter interaction in the multi-layered structures. To accomplish this goal, we solve Maxwell’s equations at the boundary condition to obtain the Fresnel coefficients. The theoretical models with zero-thickness and slab features are established by the relation between reflection and transmission wave functions. Then, an one-dimensional optical differential operation in graphene stacking is theoretically studied for different models. In order to clearly describe the influence of the use of graphene on the optical differentiation, we study the reflection field on the surface for different layers of graphene when the reflection coefficient $r_p$ is near zero. Finally, in order to further clarify the optical spatial differential operation, we discuss the amplitude and phase of the spatial spectral transfer function on graphene surface with the variation of incident angle. To visually show the graphene differential operation, an image edge detection is demonstrated, and the graphene stacking play the role of a band-pass filter during the operation process. Due to the sensitivity between the detection resolution and the number of graphene layers, our research result may provide an alternative method to characterize the layer number of 2DAC.

2. Optical model of two-dimensional atomic crystals

To obtain a clear physical picture for describing the reflection and refraction of a light beam on the surface of 2DAC, we establish an optical transmission model to describe its vector field structure. As an example, let’s assume that a graphene film is placed at a glass substrate, as shown in Fig. 1. In the theoretical analysis and derivation process, we only considered a simple stack of monolayer graphene, and did not consider the interaction between graphene layers. That is, we study the physical model of a multi-layered structures that consists of air, graphene stacking, and a substrate. The structure consists of three homogeneous transparent medium $m_1$, $m_g$, $m_2$, whose refractive indices are $n_{(m_1, m_g, m_2)}$, respectively. $d_{m_g}$ is the effective thickness of graphene [40]. When a Gaussian beam passes through a square target image and enters the graphene stacking surface, the graphene will function as a spatial differentiator of the light field during this transmission process, performing an optical spatial differential operation to achieve square edge imaging, as shown in Fig. 1(a). We show that there are two theoretical models to describe this differential process. As the number of layers increases, the high-frequency edge information components will pass through the filter, thus realizing a layer-sensitive imaging edge detection. The corresponding transfer function becomes a linear form, which is necessary for the spatial differential operation. It also turns out that the sensitivity of the graphene layer number can be used to modulate the two-dimensional optical differential operation, which enables controllable an asymmetric imaging edge detection.

 

Fig. 1. (a) Schematic representation of image edge detection of sharp-edged object based on a graphene film interface. (b) The structure consists of three homogeneous transparent medium $m_1$ (Air layer), $m_g$ (Graphene film layer), and $m_2$ (Silica glass substrate layer). Here, $d_{m_g}$ is the effective thickness of graphene, and $m_g$ marks the graphene stacking layer. Also, $E_{i}$ and $E_{r}$ show the incident and reflected electric fields, respectively.

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2.1 Fresnel coefficients for zero-thickness model

Graphene is supposed as an isotropic homogeneous medium, namely, a uniaxial crystal whose optical axis along the graphene peel axis [32]. For the zero-thickness model, atomically thick graphene is considered to be an infinitely film, whose surface charge interacts with the incident light. So the properties are characterized by optical conductivity [36]. The thin layer graphene boundary characterized by surface conductivity and surface susceptibility without the concept of effective thickness. That is, the effective thickness is $d_{m_g}=0$ and the refractive index of graphene $n_{m_g}=0$. Such a model considers the angular frequency of the incident wave function, and the phases of the reflected and transmitted wave functions are related to the material parameters and incidence angle. So that it can well simulate the phase evolution in the process of optical transmission.

We next solve Maxwell’s equations at the interface through the integral form of modified Ampere’s law and Faraday electric field induction law. Consider the boundary condition for the electromagnetic field is continuous at the interface to obtain the Fresnel coefficients for the zero-thickness model [32]. Since the graphene is regarded as an infinite plane with no thickness, the Fresnel coefficients are determined only by the surface conductivity and electric susceptibility. Suppose that the electric and magnetic fields in different medium $m_1$ and $m_2$ are respectively expressed as $\vec {\mathbf {E}}_{(m_1,m_2)}$ and $\vec {\mathbf {H}}_{(m_1,m_2)}$. At the interface of the multi-layered structure, the total current density generated by the electric field is

When an electric field $\vec {\mathbf {E}}$ is applied, a macroscopic dipole moment is produced (Except when the electric field is perpendicular to the graphene surface). According to Ohm’s law in electrostatic field, the current density $\vec {\mathbf {J}}_{\alpha }=\sigma _{m_g}\vec {\mathbf {E}}$ is related to the surface conductivity $\sigma _{m_g}$. The variation of polarization strength with time will lead to the variation of current density $\vec {\mathbf {J}}_{\beta }=\partial \vec {\mathbf {P}}/\partial t$, where the density of polarization $\vec {\mathbf {P}}=\varepsilon _0 \chi _{m_g} \vec {\mathbf {E}}$ can be defined. $\varepsilon _0$ and $\chi _{m_g}$ is the vacuum permittivity and the surface electric susceptibility of graphene, respectively. Therefore, the boundary conditions at the medium interface can be expressed as

2.2 Fresnel coefficients for slab model

For the slab model case, the graphene film in the structure is usually regarded as a uniform plate with an effective thickness, and its dielectric constant is a complex constant in connection with the refractive index of graphene (In the experiment, the refractive index of graphene is generally $n_{m_g}=3.0+1.149i$ at the wavelength $632.8\, nm$ [45]). Considering the thickness of an atom as the equivalent thickness of graphene, some experimental studies have been performed with the use of uniform equivalent refractive index, and their experimental and theoretical results are in good a agreement [32,36,38]. Here, we assume that the effective thickness of graphene is only related to the number of graphene layers, and is independent of other parameters. The reflection and refraction of light on the surface of graphene is regarded as the reflection and refraction of two different dielectric interfaces, and its Fresnel coefficients are determined by the dielectric constant $\varepsilon _{m_g}$ and the effective thickness $d_{m_g}$. Here, the refractive index of graphene $n_{m_g}$ between two media ($m_1$ and $m_2$) need to be considered. When a light beam is incident on this multi-layer structure, the propagation of wave vectors $k_{(i,m_g,t)}$ in the medium $m_1$, the graphene film $m_g$, and the medium $m_2$ should be taken into account.

Due to the extremely thin feature of graphene samples, it is usually necessary to study the optical properties of graphene with a substrate. Therefore, we consider the propagation mode of graphene with multi-layer structure, as shown in Fig. 1(b). When the light beam undergoes multiple reflections and refractions in the multi-layer media, this multi-layer structure can be simplified into multiple reflection interfaces. The light beam passes through whole system of the multi-layer structure can be described by a 2$\times$2 transmission matrix $T_{m_1\rightarrow m_g\rightarrow m_2}$, this matrix of the light field transmission sequentially through the medium $m_1$, $m_g$ and $m_2$ is denoted by

Here, the transmission matrix of light field across the interface of different medium is described as

Considering the light goes from $m-1$ layer medium to $m$ layer medium, $T_{(p,s)}^{m-1,m}$ and $R_{(p,s)}^{m-1,m}$ are defined as the refraction and reflection coefficients for $H$ or $V$ polarization, respectively.

The transmission matrix of light field is related to the effective thickness of the graphene film:

Note that the expression for the effective thickness of graphene $d_{m_g}=m_g d$ needs to be taken into account ($m_g$ is the number of layers, and the thickness of a single atomic layer is $d=0.34 nm$ [8]). The wave vector in the graphene film becomes $k_{m_g}=n_{m_g} k_0$, $\theta _{m_g}$ is the refraction angle on the graphene film, and $\cos \theta _{m_g}=\sqrt {1-(n_{m_1}\sin \theta _i/n_{m_g})^{2}}$. According to the transmission matrix $T_{m_1\rightarrow m_g\rightarrow m_2}$ for the whole structure system, we can easily obtain the following Fresnel reflection coefficients by the relation $r_{(p,s)}(\varepsilon _{m_g}, d_{m_g})=T_{21}/T_{11}$,

The reflection coefficient of different interfaces in the above formula Eq. (8) is expressed as

Because of the refraction law, the cosine of refraction angle of the medium $m_2$ is $\cos \theta _{m_2}=\sqrt {1-(n_{m_1}\sin \theta _i/n_{m_2})^{2}}$. The above equations Eqs. (8) and (9) show that the Fresnel reflection coefficients of the multi-layer structure are determined by the refractive index $n_{m_g}$ (The graphene permittivity $\varepsilon _{m_g}/\varepsilon _0=n_{m_g}^{2}$) and the effective thickness of graphene $d_{m_g}$. Meanwhile, it is also affected by the refractive indices of medium (Here, $n_{m_1}=1$ and $n_{m_2}=1.515$) and the incident angle $\theta _i$. It should be noted that the slab model does not take into account the interaction between the graphene layers during the derivation, nor did it consider the light absorption properties of the dielectric material. The slab model describes the phase change of light by considering it passes through the graphene with thickness. Since the refractive index of graphene $n_{m_g}$ is complex constant, the absorption of light through the graphene arises. As can be seen from Eq. (8), during the interaction between the light and graphene, the optical path increases with the increase of the thickness of graphene, thus leading to the phase change.

Due to the atomic-scale thickness of graphene, it is often necessary to study the graphene supported on the substrate. The final results for two models need to be considered for the existence of light absorption. Especially, when a lossy substrate is used (The refractive index is complex), there will be the light absorption on the substrate. Two models have to be taken into account the boundary conditions between the interfaces, and the physical model is established by the relation between electric field and magnetic field. The behavior of the reflection and transmission light wave can be obtained, which is related to the graphene layer number and optical properties [32,38].

Because of the thermal motion of the molecules, the surface of graphene is not always smooth, and the height of the folds may be even greater than the thickness of monolayer graphene. These show that the thickness of graphene can not accurately describe the optical properties of graphene. The graphene smoothness correlates with its optical properties because the thickness of graphene affects the optical path difference in the slab model. The zero-thickness model does not depend on the thickness of the graphene. Consider the polarization in matter changes with time an electric current is generated, i.e., $\vec {\mathbf {J}}_{\beta }=\partial \vec {\mathbf {P}}/\partial t=i\omega \mathbf {P}$. Therefore, this generated current is directly related to the angular frequency, as can be seen from Eq. (3). In this work, Our aim here is to provide a more comprehensive guidance for the optical analog imaging experiments with two graphene models in the future.

Figure 2 shows the variation law of the Fresnel coefficient $r_{p}$ with incident angle for two different models. According to the formula Eqs. (4) and (8), the Fresnel coefficients can be expressed in this form $r_{(p,s)}=\left |r_{(p,s)}\right | e^{i\phi }=Re[r_{(p,s)}]+i Im[r_{(p,s)}]$. Here, the focus of its change rule is on the incident near the Brewster and the pseudo-Brewster angle [46]. In the Figs. 2(a1) and (a2), we can find that the variation pattern of the real part of the Fresnel reflection coefficient $Re[r_{p}]$ is almost the same with different incident angles for the two models. With the increase of the number of graphene layers, the pseudo-Brewster angle ($\theta _{pB}>\theta _{B}$) also increases. It is attributed to the linear increase of the surface conductivity $\sigma _{m_g}$ with the number of graphene layers $m_g$ for the zero-thickness model, and for the slab model, the reason for the change is the effective thickness of graphene $d_{m_g}$.

 

Fig. 2. Schematic diagram of the variation law of Fresnel coefficient $r_{p}$ with incident angle for two different models. (a1)-(c1) In the zero-thickness model case, the change rule of the real part $\mathbf {Re[r_{p}]}$, the imaginary part $\mathbf {Im[r_{p}]}$, and the amplitude $\mathbf {Abs[r_{p}]}$ of Fresnel reflection coefficients for $H$ polarization are shown, respectively. (a2)-(c2) The change law of the Fresnel’s coefficients are shown for the slab model case. Here, $m_g$ represents the number of graphene layers.

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The imaginary part of the Fresnel coefficient $Im[r_{p}]$ becomes smaller and smaller as the incident angle increases. Moreover, when the number of graphene layers is zero (Only existence of the glass substrate, $m_g=0$), there is no imaginary part ($Im[r_{p}]=0$). This is related to the cosine of the incident angle. As the number of layers increases, an increase in the imaginary part is attributed to the curvilinear increase of the surface susceptibility $\chi _{m_g}$ for the zero-thickness model, and that the reason for this change is caused by the effective thickness for the slab model. The value of imaginary part is antisymmetric about the zero axis. And its absolute value for the slab model is nearly twice size for the zero-thickness model, as shown in Figs. 2(b1) and (b2). Therefore, it is evident from Figs. 2(c1) and (c2), the amplitude $Abs[r_{p}]=\left |r_{p}\right |$ have a turning point in Brewster angle and pseudo-Brewster angle. With the increase of pseudo-Brewster angle with different layers, the amplitude value is more and more far from zero. Moreover, by comparing the Figs. 2(c1) and (c2), since the absolute value of the imaginary part of the slab model is greater than the zero-thickness model, the amplitude of the slab model at the pseudo-Brewster angle have a larger non-zero value for the same number of layers, and the corresponding pseudo-Brewster angle will increase when the number of layers increases.

3. Optical spatial differential operation

We next establish the paraxial model to describe the vector field for optical spatial differential operation. We reveal how the cross polarization component of the reflected beam evolves, and how the incident angle affects its intensity distribution. Therefore, we need to establish the electric field relationship on the surface of the 2DAC [27]. Considering an incident Gaussian beam with $H$ or $V$ polarization, we assume that the incident field in wave vector space is described by $\tilde {\mathbf {E}}_i=(w_{0}/\sqrt {2\pi })\exp [-w^{2}_{0}(k_{x_i}^{2}+k_{y_i}^{2})/4]$.

It is known that a linear polarization can be decomposed into $H$ and $V$ components. The reflected angular spectrum are related to incident angular spectrum at the interface by means of the relation below

In the usual paraxial approximation, various components of the angular spectrum correspond to different Fresnel coefficients. We use a Taylor series expansion based on the arbitrary angular spectrum component in the locating $k_{x_i}=0$. $r_p$ and $r_s$ can be expanded in the first-order approximation, respectively:

Next we consider incident field with linear polarization, the reflected angular spectrum for the two polarized components is expressed as:

Here, our main theoretical result are exhibited in Eqs. (12) and (13). When the incident field is $H$ polarization, a $V$ polarization component can be induced in the reflected field. In other words, the reflection field has not only $H$ polarization component but also $V$ polarization component. To clearly demonstrate such an interesting phenomenon, here, we choose the incident field with the $H$ polarization for different incidence angles.

Reflection of incident field at interface of anisotropic medium can realize a spatial differentiation operation, which can be used to edge detection of a object image [7]. The cross-polarization effect originates from the spin-orbit coupling of light [47]. When the light field is selected by two orthogonal polarizers, the differential operation of the spatial distribution of light field is realized [22]. Specific steps are as follows: as an example, we consider an incident Gaussian beam with $H$ polarization, and the light field propagates to the focal position for edge imaging. The reflected field in Eqs. (12) and (13) can be obtained. Then, selecting a $H$ or $V$ polarization analyzer, the output light field is evolved as

Hence, the angular spectrum of output field can be written as the spatial differentiation of input field. Therefore, an expression of the optical spatial differentiation of the output field based on the 2DAC is established with different polarizations ($H$ or $V$ polarization), so that imaging edge detection can be realized simultaneously in the $x$ and $y$ directions.

In order to clearly describe the use of graphene layers to regulate the one-dimensional optical differential operation, we study the reflected field on the surface of different layers of graphene along with $r_p\approx 0$ near the pseudo-Brewster angle in Fig. 3. The reflected field at the graphene-glass interface is obtained by inverse Fourier transformation. Let us first consider an incident $H$ polarization Gaussian field ($\mathbf {E}^{H}_{in}$), and then output select the same $H$ polarization analyzer. The output field is evolved as shown in Fig. 3. For both the zero-thickness and slab models, symmetrical distribution of reflected field under different incident pseudo-Brewster angle corresponding to different graphene layers are shown in the Figs. 3(b1)-(d1) and (b2)-(d2). By comparing the Brewster effect of the two models, it is found that the zero field intensity region in the middle of the symmetric light spot decreases with the increase of the number of layers, especially for the slab model. Besides, Figs. 3(e1) and (e2) show the intensity distribution of reflected field along the $x$-direction. As the number of graphene layers increases, it can be seen that the Brewster effect changes more significantly in the slab model, and the middle zero field strength is moving further and further away from the zero point.

 

Fig. 3. The light intensity distribution of reflected field changes under different incident Brewster and pseudo-Brewster angle corresponding to different graphene layers. (a1) represents an incident Gaussian field with horizontal polarization $\mathbf {E}^{H}_{in}$. (a2) shows the distribution of reflected field intensity $\mathbf {E}^{H}_{out}$ under Brewster angle incidence $\theta _{B}=56.57^{\circ }$ (Zero layers of graphene). (b1)-(d1) For the zero-thickness model, symmetrical distribution of reflected field under different graphene layers with the incident different pseudo-Brewster angle $\theta _{pB}=57.15^{\circ },57.70^{\circ }$, and $58.23^{\circ }$, respectively. (b2)-(d2) For the slab model, symmetrical distribution of reflected field under different incident pseudo-Brewster angle $\theta _{pB}=57.15^{\circ },57.71^{\circ }$, and $58.25^{\circ }$, respectively.

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According to formula Eq. (14), it can be found that the light field is related to $r_p$ and its first partial derivative. Only when $r_p$ is satisfied to approximate zero, one-dimensional optical differential operation in $x$ direction can be better performed. Also because the amplitude of the slab model $Abs[r_{p}]$ at the pseudo-Brewster angle exist a non-zero value larger than the zero-thickness model for the same number of layers, as shown in Figs. 2(c1) and (c2). Therefore, the comparison of the Brewster effect between two models show that the slab model is more sensitive to the number of graphene layers, as shown in Fig. 3. As well as, the zero-thickness model is more suitable for the one-dimensional optical differential operation by modulating zero-value $r_p$, so as to achieve better image edge detection.

Next, we have a comparative study the process of reflected field on the surface of different layers of graphene with the incident Brewster angle $\theta _{B}=56.57^{\circ }$. Figures 4(b1)-(d1) and (b2)-(d2) show the asymmetric distribution of the reflected field with the change of graphene layer number for the zero-thickness and slab models, respectively. Among them, we show the asymmetric distribution of the reflected field. As the number of layers increases, it is found that the central minimum point of the asymmetric distribution moves towards the side of $x$-axis $x<0$, while the light field intensity quickly redistributes the energy to the other side of $x>0$. This phenomenon can also be seen from one-dimensional distribution of the reflected field corresponding to different graphene layers ($m_g=1,2,3$), and it contains the symmetric distribution of the reflected field ($m_g=0$) as a contrast, as shown in Figs. 4(e1) and (e2). Therefore, by modulating zero-value $r_p$ and its first partial derivative on the surface of graphene at the incident Brewster angle, one-dimensional optical differential operation of the light field is carried out, which can better realize asymmetric imaging edge detection. For the case of different incident pseudo-Brewster angle, the sensitivity of the graphene layer number is used to modulate the one-dimensional optical differential operation, which can also achieve controllable asymmetric imaging edge detection.

 

Fig. 4. For zero-thickness and slab model, the change of reflection field varies with the number of graphene layers under the Brewster angle incidence ($\theta _{B}=56.57^{\circ }$). Here, $m_g=0,1,2$, and $3$ indicate different layers of graphene.

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To realize the optical differential operation, according to the formula Eqs. (14) and (15), the spatial spectral transfer function is simplified as

The transfer function is related to $r_p$ first partial derivative for formula Eq. (16) in $x$ directions, and it is also affected by the zero value of $r_p$. The formula Eq. (17) shows that one-dimensional optical differential operation can be realized via the Fresnel coefficient on the surface of different layers of graphene with varying incident angle.

In order to verify the optical spatial differential operation, we theoretically study the amplitude and the phase of spatial spectral transfer function on the surface of different layers of graphene for corresponding pseudo-Brewster angle and Brewster angle, as shown in Fig. 5. It is shown that the optical spatial differential results from the Fresnel coefficient $r_p$ of the incident Brewster and pseudo-Brewster angle. According to Eq. (16), in the wave vector space, the amplitude form of transfer function are typical for the V-shaped distribution in the Fig. 5(a1)-(a2) for both the zero-thickness and slab models, and the minimum value of transfer function is zero at wave vector $k_{rx}=0$. That is due to the fact that $r_p$ undergoes a sharp change near the incident Brewster angle, thus providing a large partial derivative. At this time, the transfer function will take the linear form (in the case of $m_g=0,\theta _{B}=56.57^{\circ }$). Comparing different layers of graphene (In the case of $m_g=1,2,3,4,5$, corresponding pseudo-Brewster angle $\theta _{pB}$), the amplitude of transfer function has shown the arc-shaped V distribution near the wave vector $k_{rx}=0$. As the number of graphene layers increases, the minimum value of amplitude gets far away from zero due to the influence of other spectral components.

 

Fig. 5. (a1)-(d1) and (a2)-(d2) show the variation pattern of the spatial spectral transfer function and corresponding phase with the number of graphene layers for the zero-thickness and slab models, respectively. (a1, b1) and (a2, b2) are the change of the transfer function and corresponding phase at pseudo-Brewster angle for different layers. (c1, d1) and (c2, d2) are the result at the same Brewster’s angle $\theta _{B}=56.57^{\circ }$ for different layers.

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In particular, the variation pattern of the slab model is more obvious. It also proves that the zero-thickness is more suitable for one-dimensional optical differential operation by modulating zero-value $r_p$. Here, the graphene stacking have been used as band-pass filter of spatial differention, a wider range of frequency components near the wave vector $k_{rx}=0$ will pass through the spatial filter as the number of layers increases. For the spectral form of V-like distribution mentioned above, the zero-thickness model will be more suitable for high-pass filtering. Therefore, as the number of layers increases, the high-frequency edge information components will pass through the filter, thus realizing layer-sensitive imaging edge detection. According to the amplitude distribution, Fig. 5(b1)-(b2) illustrate that the phase variation of the spatial spectral transfer function exists a phase $\pi$ transition [In the case of $\theta _{B}=56.57^{\circ }$]. With an increase in the incidence angle (For different pseudo-Brewster angles), the phase of the transfer function shows a gradient along the wave vector, moreover, undergoing a sharp reversal change near the wave vector $k_{rx}=0$, with a phase $\pi /2$ transition. Here, for the slab model, the phase occurs a positive and negative transition as shown in Fig. 5(b2), due to the imaginary part of the Fresnel coefficient [Fig. 2(b1)-(b2)].

Next, we comparative study the amplitude and phase change of spatial spectral transfer function for different layers of graphene with the incident angle $\theta _{i}=\theta _{B}=56.57^{\circ }$, as shown in Figs. 5(c1)-(c2) and (d1)-(d2). As the number of graphene layers increases, the symmetric distribution of the amplitude gradually shifts towards the wave vector $k_{rx}<0$. That is, for a band-pass filter like the graphene stacking, the symmetry central frequency is away from the wave vector $k_{rx}=0$, and the bandwidth is expanding. This situation for the slab model is more clearly shown from the values in Fig. 5(c1)-(c2). Based on its amplitude distribution, the phase variation of the transfer function shows a gradient change near the symmetry central frequency, and this gradient changes more slowly as the number of layers increases. Meanwhile, the phase $\pi /2$ transition is shifted to zero phase. The transfer function varies linearly when the wave vector is far from the symmetry central frequency, and this linear change is more pronounced for the wave vector $k_{rx}>0$, as shown in Fig. 5(c1)-(c2). The phase change of transfer function trend slowly approaches to the zero phase for the wave vector $k_{rx}>0$, as shown in Fig. 5(d1)-(d2). Therefore, the transfer function becomes a linear form, which is necessary for the spatial differential operation. It also turns out that the sensitivity of the graphene layer number can be used to regulate the one-dimensional optical differential operation, which enables controllable asymmetric imaging edge detection.

To verify the effectiveness of the optical differentiation operations, we take the input square pattern as an example to study the edge imaging results, as shown in Fig. 6. We present the output imaging results obtained by applying a first-order differentiator to a input square target image. The input image undergoes Fourier convolution modeling, meaning that the output image is generated by a procedure of performing Fourier transform, applying the spatial spectral transfer function, and then performing Fourier inverse transform. In addition, during the preprocessing stage, the original image was smoothed using a 7$\times$7 Gaussian kernel [48,49]. Figures 6(a) and (b) are two- and one-dimensional bright field images of the initial square object, respectively. As a contrast, the edge imaging of zero layer graphene is shown in Figs. 6(c) and (d). The edge detection with clear resolution is realized by one-dimensional differential operation in $x$ direction. Due to the Brewster effect, the graphene as a band-pass filter makes the internal light field no intensity. The square target can be differentiated by using the graphene, and the image resolution is still high with the increase of the number of layers, which shows that using graphene as a differential operator is effective in the Figs. 6(e1)-(e3) and (g1)-(g3). For the zero-thickness model, as the number of layers increases, the graphene acts as a band-pass filter makes the light field still have small intensity values as shown in Fig. 6(f1)-(f3). Compared with the zero-thickness model, the intensity of internal light field of the slab model is larger in the Fig. 6(h1)-(h3). The Brewster effect of reflected field induced by the graphene stacking, whose non-zero intensity can be used as a weak background light for imaging edge detection. This means that for more layers of graphene, the internal light intensity becomes background noise in Fig. 6(g1)-(g3). Therefore, few-layer graphene based on the zero-thickness model is more suitable for imaging edge detection, and this also validates the conclusion of Fig. 3.

 

Fig. 6. (a) shows two-dimensional brightfield image of an initial object, and (b) correspondingly displays one-dimensional intensity distribution. (c) and (d) display an one-dimensional edge imaging of the square target and the intensity profiles along red line (zero layer of graphene, $m_g=0$). (e1)-(e3) and (g1)-(g3) Imaging edge detection can be realized in $x$ directions with the number of graphene layers ($m_g=1,2,3$) for the zero-thickness model and the slab model, respectively. The corresponding one-dimensional intensity profiles are shown in (f1)-(f3) and (h1)-(h3).

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We study the reflected field and the spatial spectral transfer function with different layers of graphene. The results show that the slab model is more sensitive to the number of graphene layers since it is more scientific for light passing through large thickness. When the graphene layers are less than two, the layered three-dimensional (3D) system will reduce to the two-dimensional (2D) one, and the zero-thickness model is suitable for one-dimensional optical differential operation. In addition, in the slab model, since it is the bulk material domain, it is necessary to apply the complex optical conductivity $\sigma _{b}$, and it is related to the complex dielectric function, the vacuum permittivity, and the angular frequency. For the really 2D case, its optical conductivity $\sigma$ is considered as the product of the corresponding bulk quantities and the effective thickness $d$ (equal to the interlayer spacing), i.e., $\sigma =d\times \sigma _{b}$ [50,51]. Therefore, the slab model will reduce to the 2D model when d is small enough (2D case). This has been showed by some studies, in which the results for the expression of a layered 3D system changes directly to a 2D model by calculating the limit of phase shift in the thin film [52].

4. Conclusion

We establish the optical model to describe the vector field at a multi-layered structures that consists of air, graphene stacking, and a substrate. The comparison of the Brewster effect between zero-thickness model and slab model is shown that the slab model is more sensitive to the number of graphene layers. As well as, the zero-thickness model is more suitable for one-dimensional optical differential operation. This is due to the change law of Fresnel coefficient. Specifically, the imaginary part of Fresnel coefficient is antisymmetric about the zero axis, and its absolute value for the slab model is nearly twice size for the zero-thickness model. Meanwhile, the amplitude of Fresnel coefficient have a turning point in different Brewster and pseudo-Brewster angles, and the amplitude of the slab model at the pseudo-Brewster angle has a larger non-zero value. In addition, we have a comparative study the process of reflection field on the surface of different layers of graphene with the incident Brewster angle. As the number of layers increases, it is found that the central minimum point of the asymmetric distribution moves towards the side of $x$-axis $x<0$, while the light field intensity quickly redistributes the energy to the other side of $x>0$. Therefore, by modulating zero-value $r_p$ and its first partial derivative on the surface of graphene at the different incident Brewster and pseudo-Brewster angle, the sensitivity of the graphene layer number is used to modulate the one-dimensional optical differential operation. Next, we theoretically study the spatial spectral transfer function with different layers of graphene. As the number of layers increases, the high-frequency edge information components will pass through the filter, thus realizing layer-sensitive imaging edge detection.