All-optical image edge detection based on the two-dimensional photonic spin Hall effect in anisotropic metamaterial

Jin Zhang; Su Zhou; Xin Dai; Mian Huang; Xiaoyan Yu

doi:10.1364/OE.476492

1. Introduction

Photonic spin Hall effect (SHE) is characterized by the separation of photons with opposite spin angular momentum in the direction perpendicular to the incidence plane. Due to the spin-orbit interaction of photons, such a fundamental effect generally occurs on light reflection and refraction at planar interfaces. With the deepening of research, the photonic SHE has been studied in various optical systems [1–4], such as air-glass systems [5–8], different material systems [9–14], two-dimensional (2D) metamaterials [15–20], optical detection systems [21–24]. In general, the photonic SHE at interfaces is a weak effect of light-matter interaction, and its spin-dependent splitting is only on the order of wavelengths, which greatly hinders its potential applications. Therefore, extensive investigations of the modulation and enhancement of the photonic SHE in different ways have been a hot issue in recent years [25–31]. Although the photonic SHE is a tiny effect, it is extremely sensitive to the physical parameters of the optical structures, in which symmetry breaking in anisotropic metamaterial structures plays a crucial role in this process [32–36].

Recently, the photonic SHE has become an effective method for image edge detection. Such an optical method to implement analog signal processing is based on the optical differential operation that refers to the mathematical operation of the light field distribution [37–41]. The process is intrinsically due to the spin-dependent separation of photons induced by the photonic SHE. The essential features of optical images are mainly related to the local changes in the distribution of intensity, phase and polarization [42]. Therefore, it is widely used in artificial intelligence, microscopic imaging, quantum imaging and other fields. Depending on the photonic SHE, the image edge detection can be realized by an air-glass interface, optical metasurfaces. and surface plasmon [43–46]. Most of these existing detecting methods are limited to one-dimensional (1D) optical detection, high cost, or a lack of adjustable factors. Optical edge detection is mainly based on the differential operation of the spin splitting of the reflected light field caused by the photonic SHE.

Therefore, by designing the periodic symmetry breaking in the anisotropic medium, changing the certain filling factor, the optical axis angle, and the incident angle of light, the anisotropic metamaterial provides opportunities to restudy the photonic SHE at the unique optical interface. The spin-dependent displacement induced by the spin-orbit interaction of light at the anisotropic metamaterial interface may become adjustable or even two dimensional. That is, the longitudinal displacement will take place except for the transverse displacement. As a result, we can adjust the specific parameters to modulate the longitudinal and transverse displacements to develop or extend applications of the photonic SHE. Then, we demonstrate that the tunable photonic SHE in anisotropic metamaterial will open up an opportunity to improve optical edge detection. And finally, by loading the linear polarization selection on the Fourier plane to eliminate the image information of the overlapped part of left and right circular polarizations light, the transverse and longitudinal displacements can be modulated, and the image edge can be flexibly realized by the extraction of contours.

2. General theoretical model

The schematic sketch of the particular structure anisotropic metamaterial is shown in Fig. 1, consisting of a periodic uniaxial crystals and a cover made substrate, the entire structure appears as an anisotropic medium [47–49]. Figure 1(a1) shows the propagation of a linearly polarized Gaussian beam along an anisotropic medium interface, the reflected beam will split into left- and right-handed circularly polarized light, and the reflection surface is parallel to the incident surface. Due to the interaction of spin-orbit angular momentum, the spin-dependent displacements occur in the $x$-direction and the $y$-direction, and the magnitude of the displacements are related to the change of filling factor, incident angle, and optical axis angle. Here $y{'}$-axis represents the rotation axis of the optical axis of the medium, and the rotation angle with $y$-axis is the optical axis angle $\phi$. Figure 1(a2) is the 2D representation of periodic uniaxial crystals, the period and the groove width of the grating are p and g, the thickness of the medium layer and substrate show d and t, respectively. We theoretically studied the effects of incident angle, optical axis angle and filling factor on the photonic SHE. Only the filling factor is related to the material structure, and the filling factor is equal to the ratio of the groove width and the period of the grating. Therefore, the other specific parameters of the anisotropic metamaterial are not considered at present, these considerations will not influence the integrity of the content and framework of the paper.

Fig. 1. (a1) Schematic representation of the wave reflection at an anisotropic medium interface. On the reflecting surface, the giant transverse and in-plane displacements occur different values. Where y${'}$-axis represents the rotation axis of the optical axis of the medium, and the rotation angle with y-axis is the optical axis angle $\phi$. (a2) A two-dimensional representation of periodic uniaxial crystals, where the period and the groove width of the grating are p and g, the thickness of the medium layer and substrate show d and t, respectively. (b1) and (b2) Effective dielectric constant with the relation between the filling factor and refractive index of medium.

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When a plane wave is incident on the anisotropic medium, two transmitted plane waves (ordinary and extraordinary modes) will take place. From the Maxwell’s equations, one can deduce the boundary conditions in anisotropic medium. In the uniaxial crystals structures, the effective ordinary and extraordinary refractive indices are [47,48]

(1)$$n^{2}_{o}=\epsilon_{o}=f n^{2}_{1}+(1-f)n^{2}_{2},$$

(2)$$n^{2}_{e}=\epsilon_{e}=[f/n^{2}_{1}+(1-f)/n^{2}_{2}]^{{-}1},$$

for an anisotropic medium structure made of two materials with refractive indices $n_{1}$ and $n_{2}$, and the filling factor $f$=g/p. From these expressions above, we have $n_{o}>n_{e}$. The corresponding effective dielectric constant are shown in Fig. 1(b1) and Fig. 1(b2), here the refractive indices of vacuum and medium are taken as $n_{1}=1$ and $n_{2}$ (from 1 to 10), respectively. It can be seen that the effective dielectric constant $\epsilon _{o}$ and $\epsilon _{e}$ decrease with the increase of the filling factor $f$ (from 0 to 0.8), respectively. The closer the refractive indices $n_{1}$ and $n_{2}$ of two materials, the less the effective dielectric constant. In addition, when the filling factor $f$ is small, the effective dielectric constant $\epsilon _{o}$ and $\epsilon _{e}$ have a rapid increase because of increasing of the refractive index $n_{2}$; as the filling factor $f$ get larger, with the increase of the medium refractive index, the ordinary dielectric constant $\epsilon _{o}$ increases, but the variation of the extraordinary dielectric constant $\epsilon _{e}$ is not obvious. The Maxwell’s equations can be written as

(3)$$\begin{array}{r}\nabla\times\mathrm E={-}\frac{\partial \mathrm B}{\partial t},\qquad \nabla\times\mathrm H=\frac{\partial \mathrm D}{\partial t},\\ \mathrm B=\mu_{0}\mu\mathrm E,\qquad \mathrm D=\epsilon_{0}\epsilon\mathrm H, \end{array}$$

where $\epsilon$ is the dielectric tensor, $\epsilon _{0}$ is the permittivity of vacuum. Consider a plane wave $\mathrm E\cdot \exp (ik_{ix}x+ik_{iy}y+ik_{iz}z-i \omega t)$, we get the differential equation

(4)$$\nabla\times\mu^{{-}1}\cdot\nabla\times\mathrm E-k^{2}_{0}\epsilon\cdot\mathrm E=0,$$

then the Hamiltonian is

(5)$$\nabla\times\to \begin{bmatrix} 0 & -k_{z} & k_{y}\\ k_{z} & 0 & -k_{x}\\ -k_{y} & k_{x} & 0 \end{bmatrix},$$

where $k_{0}=2\pi /\lambda$ is the vacuum wave vector. Here, we assume that the incident plane wave propagate along the $xoz$ plane ($k_{y}=0$), and the optical axis is located at $xoy$ plane. We can get

(6)$$\begin{aligned}k^{o}_{z}&=\sqrt{\epsilon_{o}k^{2}_{0}-k^{2}_{ix}},\\ k^{e}_{z}&=\sqrt{\epsilon_{e}k^{2}_{0}-(\epsilon_{o}+\alpha^{2} \Delta\epsilon)k^{2}_{ix}/\epsilon_{o}}, \end{aligned}$$

where the subscripts $o$ and $e$ denote ordinary and extraordinary, and $k$ is the normal wave vector of the surface. Here, the anisotropy is defined as $\Delta \epsilon =\epsilon _{e}-\epsilon _{o}$, the eigenvector of electric field can be obtained by solving the Null space of the Matrix. Consequently, we get

(7)$$\begin{aligned}\mathrm E^{o}_{t}&=N_{o}({-}k^{o}_{z},k^{o}_{z}\alpha/\beta,k_{ix}),\\ \mathrm E^{e}_{t}&=N_{e}[-(\epsilon_{o}k^{2}_{0}-k^{2}_{ix})/k_{ix},-\beta\epsilon_{o}k^{2}_{0}/\alpha k_{ix},k^{e}_{z}], \end{aligned}$$

where $N$ is a normalization factor, $\alpha =\cos \phi$ and $\beta =\sin \phi$ stand for the direction cosines and sine of the optical axis $\phi$, respectively.

(8)$$\begin{aligned}N^{o}&=1/\sqrt{{k^{o}_{z}}^{2}+\beta^{2}k_{ix}^{2}},\\ N^{e}&=1/\sqrt{\alpha^{2}{k^{o}_{z}}^{4}+\beta^{2}\epsilon^{2}_{o}k^{4}_{0}+\alpha^{2}k^{2}_{ix}{k^{e}_{z}}^{2}}. \end{aligned}$$

To calculate the reflection and the transmission of an incident wave plane-polarized in an arbitrary direction, we decompose the input field and the transmission field into its $s$ (vertical) and $p$ (horizontal) polarization components, where $E_{s}$ is perpendicular to the incident plane (the $zx$ plane) and $E_{p}$ lies in the plane of incidence. We firstly consider the $s$ polarization. The $z$-dependence of the electric field components are

(9)$$\begin{array}{r} Incident:(0,1,0)\exp(ik_{ix}x+ik_{iy}y+ik_{iz}z-i \omega t),\\ Reflected:(r_{sp}\cos\theta_{i},r_{ss},r_{sp}\sin\theta_{i})\exp(ik_{ix}x+ik_{iy}y-ik_{iz}z-i\omega t),\\ Transmitted:t^{o}_{s}(E^{o}_{tx},E^{o}_{ty},E^{o}_{tz})\exp(ik^{o}_{x}x+ik^{o}_{y}y+ik^{o}_{z}z-i \omega t)\\+(t^{e}_{s}(E^{e}_{tx},E^{e}_{ty},E^{e}_{tz}))\exp(ik^{e}_{x}x+ik^{e}_{y}y+ik^{e}_{z}z-i \omega t), \end{array}$$

where $r_{ss},r_{sp},t^{o}_{s},t^{e}_{s}$ are the reflection and the transmission amplitudes of incident wave; $\theta _{i}$ is the angle of incidence; $k^{o}$ and $k^{e}$ is vector normal components of the ordinary and the extraordinary modes; and $\mathrm E^{o}$ and $\mathrm E^{e}$ denote the electric field vectors. Based on the boundary conditions and Faraday Law, there exists the following equations at the reflected plane.

(10)$$\begin{aligned}1+r_{ss}&=t^{o}_{s}E^{o}_{ty}+t^{e}_{s}E^{e}_{ty},\\ k_{iz}(1-r_{ss})&=k^{o}_{z}t^{o}_{s}E^{o}_{ty}+k^{e}_{z}t^{e}_{s}E^{e}_{ty},\\ r_{sp}\cos\theta_{i}&=t^{o}_{s}E^{o}_{tx}+t^{e}_{s}E^{e}_{tx},\\ -r_{sp}(k_{iz}\cos\theta_{i}+k_{ix}\sin\theta_{i})&=k^{o}_{z}t^{o}_{s}E^{o}_{tx}+k^{e}_{z}t^{e}_{s}E^{e}_{tx}-k_{ix}(t^{o}_{s}E^{o}_{tz}+t^{e}_{s}E^{e}_{tz}), \end{aligned}$$

the above equation can be solved

(11)$$\begin{aligned}r_{ss}&=[(k_{iz}-k^{e}_{z})AE^{e}_{ty}-(k_{iz}-k^{o}_{z})BE^{o}_{ty}]/C,\\ r_{sp}&=2k_{iz}(AE^{e}_{tx}-BE^{o}_{tx})/(C\cos\theta_{i}), \end{aligned}$$

where

(12)$$\begin{aligned}A&=[(k_{iz}+k^{o}_{z})\cos\theta_{i}+k_{ix}\sin\theta_{i}]E^{o}_{tx}-k_{ix}\cos\theta_{i}E^{o}_{tz},\\ B&=[(k_{iz}+k^{e}_{z})\cos\theta_{i}+k_{ix}\sin\theta_{i}]E^{e}_{tx}-k_{ix}\cos\theta_{i}E^{e}_{tz},\\ C&=A(k_{iz}+k^{e}_{z})E^{e}_{ty}-B(k_{iz}+k^{o}_{z})E^{o}_{ty}. \end{aligned}$$

Then, we will consider the $p$ polarization. The $z$-dependence of the electric field components are

(13)$$\begin{array}{r} Incident:(\cos\theta_{i},0,-\sin\theta_{i})\exp(ik_{ix}x+ik_{iy}y+ik_{iz}z-i \omega t),\\ Reflected:(r_{pp}\cos\theta_{i},r_{ps},r_{pp}\sin\theta_{i})\exp(ik_{ix}x+ik_{iy}y-ik_{iz}z-i \omega t),\\ Transmitted:t^{o}_{p}(E^{o}_{tx},E^{o}_{ty},E^{o}_{tz})\exp(ik^{o}_{x}x+ik^{o}_{y}y+ik^{o}_{z}z-i \omega t)\\+(t^{e}_{p}(E^{e}_{tx},E^{e}_{ty},E^{e}_{tz}))\exp(ik^{e}_{x}x+ik^{e}_{y}y+ik^{e}_{z}z-i \omega t), \end{array}$$

where $r_{pp},r_{ps},t^{o}_{p},t^{e}_{p}$ are the reflection and the transmission amplitudes of incident wave. Based on the boundary conditions and Faraday Law, they are satisfied the following equations at the reflected plane,

(14)$$\begin{aligned}r_{ps}&=t^{o}_{p}E^{o}_{ty}+t^{e}_{p}E^{e}_{ty},\\ -k_{iz}r_{ps}&=k^{o}_{z}t^{o}_{p}E^{o}_{ty}+k^{e}_{z}t^{e}_{p}E^{e}_{ty},\\ (1+r_{pp})\cos\theta_{i}&=t^{o}_{p}E^{o}_{tx}+t^{e}_{p}E^{e}_{tx},\\ (1-r_{pp})k_{iz}/\cos\theta_{i}&=k^{o}_{z}t^{o}_{p}E^{o}_{tx}+k^{e}_{z}t^{e}_{p}E^{e}_{tx}-k_{ix}(t^{o}_{p}E^{o}_{tz}+t^{e}_{p}E^{e}_{tz}), \end{aligned}$$

similarly, we will get

(15)$$\begin{aligned}r_{pp}&=2k_{iz}[(k_{iz}+k^{e}_{z})E^{o}_{tx}E^{e}_{ty}-(k_{iz}+k^{o}_{z})E^{o}_{ty}E^{e}_{tx}]/C-1,\\ r_{ps}&=2k_{iz}(k^{e}_{z}-k^{o}_{z})E^{o}_{ty}E^{e}_{ty}/C. \end{aligned}$$

It is worth noting that $r_{ps}=r_{sp}$, after the formula is solved. The Fresnel reflection coefficients corresponding to different polarizations are obtained by establishing a light field transmission model on the surface of an anisotropic medium [50]. This is the first step to obtain the quantitative relationship of photonic SHE at the interface of different medium.

3. Photonic spin Hall effect

The photonic SHE is induced by spin-dependent geometric phases, which correspond to spin-orbit interactions [3]. When the beam is incident on a non-uniform anisotropic medium, a spatially varying geometric phase (only related to the change of the polarization state) will be generated, which can cause the transverse and in-plane spin-dependent splitting of the beam. Therefore, the modulation of the photonic SHE is essentially the polarization control. The photonic SHE and Imbert-Fedorov shift are derived from spin-orbit coupling [1]. In order to satisfy the angular momentum conservation in the $z$-component, the reflected and transmitted light of a finite beam width at the interface must generate an additional transverse orbital angular momentum, and the longitudinal and transverse displacements are derived from the transverse linear momentum conservation, and these displacements are spin-dependent. Moreover, Goos-Hänchen shift is not the result of spin-orbit interaction. It is generally the longitudinal movement of the whole beam after the incident horizontal or vertical polarized light, which is a kind of spin-independent displacement.

In this section, we discuss the specific expressions of the spatial displacements caused by the photonic SHE in the anisotropic medium, and find out the related parameters to achieve the effectively controlling the magnitude of the shifts. The reflected field can be written as

(16)$$E^{p,s}_{rx,ry,rz}={M}_{R}E^{p,s}_{ix,iy,iz},$$

(17)$$\begin{aligned} {M}_{R}= \left[ \begin{array}{cc} r_{pp}-k_{ry}(r_{ps}-r_{sp})\cot\theta_{i}/k_{0} & r_{ps}+k_{ry}(r_{pp}+r_{ss})\cot\theta_{i}/k_{0}\\ r_{sp}-k_{ry}(r_{pp}+r_{ss})\cot\theta_{i}/k_{0} & r_{ss}-k_{ry}(r_{ps}-r_{sp})\cot\theta_{i}/k_{0} \end{array}\right]. \end{aligned}$$

Here, the boundary condition $k_{ry}=k_{iy}$ is applied. The fresnel reflection coefficients have been expanded to the first order in Taylor expansion. In the spin basis set, the polarization of $e^{H}_{r}$ and $e^{V}_{r}$ can be decomposed into two orthogonal spin components

(18)$$e^{H}_{r}=\frac{1}{\sqrt{2}}(e^{+}_{r}+e^{-}_{r}),$$

(19)$$e^{V}_{r}=\frac{1}{\sqrt{2}}i(e^{-}_{r}-e^{+}_{r}),$$

where $e^{+}$ and $e^{-}$ denote the left and right circular polarization components, respectively. We consider an incident Gaussian beam with horizontal or vertical polarizations, and the wave function in momentum space possess the following expression

(20)$$\widetilde{\xi}_{i}=\frac{w_{0}}{\sqrt{2\pi}}\exp\left[-\frac{w^{2}_{0}(k_{ix}^{2}+k_{iy}^{2})}{4}\right],$$

where $w_{0}$ is the width of wave function. The electric field can be simplified by the Talor expansion and the Euler Equation as

(21)$$\begin{aligned} \widetilde{E}_r^{H}&=\frac{r_{pp}-i r_{sp}}{\sqrt{2}}\exp(ik_{ry}\delta_{y}^H+k_{ry}\Delta_{y}^H)\exp(ik_{rx}\delta_{x}^H+k_{rx}\Delta_{x}^H)\widetilde{\xi}_{r}e^{+}\\ &+\frac{r_{pp}+ir_{sp}}{\sqrt{2}}\exp({-}ik_{ry}\delta_{y}^H+k_{ry}\Delta_{y}^H)\exp({-}ik_{rx}\delta_{x}^H+k_{rx}\Delta_{x}^H)\widetilde{\xi}_{r}e^{-},\end{aligned}$$

(22)$$\begin{aligned} \widetilde{E}_r^{V}&=\frac{r_{ps}-i r_{ss}}{\sqrt{2}}\exp(ik_{ry}\delta_{y}^V+k_{ry}\Delta_{y}^V)\exp(ik_{rx}\delta_{x}^V+k_{rx}\Delta_{x}^V)\widetilde{\xi}_{r}e^{+}\\ &+\frac{r_{ps}+ir_{ss}}{\sqrt{2}}\exp({-}ik_{ry}\delta_{y}^V+k_{ry}\Delta_{y}^V)\exp({-}ik_{rx}\delta_{x}^V+k_{rx}\Delta_{x}^V)\widetilde{\xi}_{r}e^{-},\end{aligned}$$

where $\delta _{x,y}^{H,V}$ and $\Delta _{x,y}^{H,V}$ are the spatial shifts and the angular shifts, respectively. For isotropic 2D atomic crystal [16], the fresnel reflection coefficients $r_{ps}=-r_{sp}=0$ is attributed to the crossed conductivity $\sigma _{sp}=\sigma _{ps}=0$, we then get angular shifts $\Delta _{x,y}^{H,V}=0$. The in-plane spin Hall shifts of wave-packet at initial position ($z_r=0$) are given by

(23)$$\langle{\Delta x_{r\pm}^{H,V}}\rangle=\frac{\langle E_{r\pm}^{H,V}|\partial_{k_{rx}}|E_{r\pm}^{H,V}\rangle}{\langle E_{r\pm}^{H,V}|E_{r\pm}^{H,V}\rangle}.$$

Substituting Eqs. (21) and (22) into Eq. (23), we obtain the in-plane displacements of two spin components

(24)$$\langle \Delta x^{H}_{r\pm}\rangle={\mp} \mathrm{Re}\bigg[\frac{r_{pp}}{k_{0}(r^{2}_{pp}+r^{2}_{sp})}\frac{\partial r_{sp}}{\partial \theta_{i}}-\frac{r_{sp}}{k_{0}(r^{2}_{pp}+r^{2}_{sp})}\frac{\partial r_{pp}}{\partial \theta_{i}}\bigg],$$

(25)$$\langle \Delta x^{V}_{r\pm}\rangle={\mp} \mathrm{Re}\bigg[\frac{r_{ps}}{k_{0}(r^{2}_{ss}+r^{2}_{ps})}\frac{\partial r_{ss}}{\partial \theta_{i}}-\frac{r_{ss}}{k_{0}(r^{2}_{ss}+r^{2}_{ps})}\frac{\partial r_{ps}}{\partial \theta_{i}}\bigg].$$

The transverse displacements of two spin components can be calculated by

(26)$$\langle{\Delta y_{r\pm}^{H,V}}\rangle=\frac{\langle E_{r\pm}^{H,V}|\partial_{k_{ry}}|E_{r\pm}^{H,V}\rangle}{\langle E_{r\pm}^{H,V}|E_{r\pm}^{H,V}\rangle}.$$

Substituting Eqs. (21) and (22) into Eq. (26), respectively. We get the transverse displacements for two spin components are obtained as

(27)$$\langle \Delta y^{H}_{r\pm}\rangle={\mp} \mathrm{Re}\bigg[\frac{(r_{pp}+r_{ss})r_{pp}}{k_{0}(r^{2}_{pp}+r^{2}_{sp})} \cot\theta_{i}-\frac{(r_{ps}-r_{sp})r_{sp}}{k_{0}(r^{2}_{pp}+r^{2}_{sp})}\cot\theta_{i}\bigg],$$

(28)$$\langle \Delta y^{V}_{r\pm}\rangle={\mp} \mathrm{Re}\bigg[\frac{(r_{pp}+r_{ss})r_{ss}}{k_{0}(r^{2}_{ss}+r^{2}_{ps})} \cot\theta_{i}+\frac{(r_{ps}-r_{sp})r_{ps}}{k_{0}(r^{2}_{ss}+r^{2}_{ps})}\cot\theta_{i}\bigg].$$

It can be seen from the displacement formula Eqs. (24), (25), (27) and (28) that both the in-plane displacements and the transverse displacements are related to the Fresnel coefficient, and the spin-dependent displacements are also dependant on the incident angle.

To verify these relationships, we theoretically calculate the relationship between the displacements and filling factor, incident angle and optical axis angle. As an example, we consider an incident Gaussian beam with vertical polarization, the wavelength of He-Ne laser source is $\lambda =632.8nm$. Some parameters in the literature [48] are used in the calculation, the refractive index of silicon is $n_{2}=3.5$, and the groove width and the period of the grating are the wavelength order (the filling factor is equal to their ratio). It is interesting to note that the in-plane displacement increase with the increases of the incident angles, and the overall displacements become larger with the increase of the filling factor. Meanwhile, the displacements are antisymmetric about $90^{\circ }$ optical axis angle as shown in Figs. 2(a)-(c) [$\theta _{i}=30^{\circ }; 45^{\circ }; 60^{\circ }$]. This makes it possible to achieve large in-plane splitting by constructing large filling factors. At the same time, as shown in Figs. 2(b)-(f) [$\theta _{i}=30^{\circ }; 45^{\circ }; 60^{\circ }$], the transverse displacements decrease with the increase of the incident angle, and the overall absolute displacements become larger with the increase of the filling factor. The displacements are symmetrical about the $90^{\circ }$ optical axis angle, and when the filling factor approaches $f=1$, the displacement have a maximum value. This makes it possible to achieve large spin-dependent splitting by fabricating large filling factors of anisotropic materials.

Fig. 2. The in-plane and transverse displacements of the incident vertical polarizations with different incident angle are plotted. (a)-(c) represent the in-plane displacement, (d)-(f) denote the transverse displacement. Here, each column corresponds to different incident angles $\theta _{i}=30^{\circ }$ [(a) and (d)], $45^{\circ }$ [(b) and (e)], $60^{\circ }$ [(c) and (f)], and the displacements are measured in nanometers.

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In order to illustrate the contrast between the in-plane displacement and the transverse displacement caused by the photonic SHE in more detail, we still take the spatial shift obtained by vertically polarized incidence as an example. Comparing the parameters in Fig. 2, the one-dimensional relationship between the spatial shift and the optical axis angle are described in Fig. 3, when the incident angles are $30^{\circ }$, $45^{\circ }$, $50^{\circ }$, $55^{\circ }$, $60^{\circ }$, and $70^{\circ }$, respectively. With the increase of the incident angle, the in-plane displacements show a gradually increasing trend, and they become larger with the increase of the filling factor, as shown in Figs. 3(a)-(f). When the optical axis angle is $90^{\circ }$ (that is, the optical axis is in the incident plane), it is obvious that the in-plane displacements are zero. When the optical axis angle are on both sides of $90^{\circ }$, there are an extreme point in the in-plane displacements, and they are about zero-point antisymmetric.

Fig. 3. The in-plane and transverse displacements with different incident angle and the filling factor ($f=0.1; 0.5; 0.8$) are plotted. (a)-(f) The solid line indicates the displacement in the $x$-direction, and the dashed line indicates the displacement in the $y$-direction, under the incident angle $\theta _{i}=30^{\circ },45^{\circ },50^{\circ },55^{\circ },60^{\circ },70^{\circ }$, respectively.

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Meanwhile, comparing with the in-plane displacement, it is found that the overall transverse displacements decrease as the incident angle increase. When the optical axis angle is $90^{\circ }$, it is obvious that the transverse displacements have a maximum point. Here, an interesting phenomenon is that the transverse displacements increase with the decrease of the filling factor, when the optical axis angle are away from $90^{\circ }$. However, when the optical axis angle is around $90^{\circ }$, as the incident angle increases to $50^{\circ }$, the transverse displacements increase as the filling factor increase, as shown in Figs. 3(a)-(c). Then, when the incident angle reaches $60^{\circ }$, the transverse displacement will increase with the decrease of the filling factor as shown in Figs. 3(e)-(f). While the transverse displacements change around the incident angle of $55^{\circ }$ are shown in Fig. 3(d), without the above regular changes. This makes it possible to accuracy control of the transverse displacements with different incident angle.

Through comparison, different incident angles, filling factors and optical axis angles were designed to obtain different variation laws of the in-plane and transverse displacements. It was found that the size of the spatial displacement can be adjusted by setting the specific parameters, so as to realize the 2D edge detection imaging in specific direction. Then, the next section will realize how to use the spatial displacement caused by the photonic SHE to realize the manipulation of 2D edge imaging.

4. Optical spatial differentiation

The photonic SHE reflected by the incident beam at the interface of the anisotropic medium can realize the spatial differential operation, which can be used to detect the edge of the object image [39]. The photonic SHE originates from the spin-orbit coupling of light. When the light field is selected by two orthogonal polarizers, the left-handed and right-handed components of the spatial displacement are subtracted and cancelled, and the differential operation of the spatial shift of the light field is realized. Specific steps are as follows: As an example, we still consider an incident Gaussian beam with vertical polarization. The beam propagates to the focal position for edge imaging, so the electric field Eqs. (21) and (22) ignore angular shifts $\Delta _{x,y}^{H,V}$, the reflected electric field can be obtained as

(29)$$\widetilde{E}_r^{V}=\frac{r_{ps}-i r_{ss}}{\sqrt{2}}\exp[i(k_{ry}\delta_{y}^V+k_{rx}\delta_{x}^V)]\widetilde{\xi}_{i}e^{+}+\frac{r_{ps}+ir_{ss}}{\sqrt{2}}\exp[{-}i(k_{ry}\delta_{y}^V+k_{rx} \delta_{x}^V)]\widetilde{\xi}_{i}e^{-},$$

here, the spatial shifts are expressed as follows

(30)$$\delta_{y}^V=\frac{(r_{pp}+r_{ss})r_{ss}}{k_{0}(r^{2}_{ss}+r^{2}_{ps})}\cot\theta_{i}+\frac{(r_{ps}-r_{sp})r_{ps}}{k_{0}(r^{2}_{ss}+r^{2}_{ps})} \cot\theta_{i},$$

(31)$$\delta_{x}^V=\frac{r_{ps}}{k_{0}(r^{2}_{ss}+r^{2}_{ps})}\frac{\partial r_{ss}}{\partial \theta_{i}}-\frac{r_{ss}}{k_{0}(r^{2}_{ss}+r^{2}_{ps})}\frac{\partial r_{ps}}{\partial \theta_{i}}.$$

Loading a horizontal polarization post-selection, the output field is evolved as

(32)$$\widetilde{E}_{out}^{V}=\frac{r_{ps}-i r_{ss}}{2}\exp[i(k_{ry}\delta_{y}^V+k_{rx}\delta_{x}^V)]\widetilde{E}_{in}^{V}+\frac{r_{ps}+ir_{ss}}{2}\exp[{-}i(k_{ry}\delta_{y}^V+k_{rx}\delta_{x}^V)] \widetilde{E}_{in}^{V},$$

because of $r_{ss}\gg r_{ps}$, the output field is rewritten as

(33)$$\widetilde{E}_{out}^{V}\approx\frac{-i r_{ss}}{2}\left\{\exp[i(k_{ry}\delta_{y}^V+k_{rx}\delta_{x}^V)]-\exp[{-}i(k_{ry}\delta_{y}^V+k_{rx}\delta_{x}^V)]\right\}\widetilde{E}_{in}^{V},$$

by introducing Taylor expansion in the first-order approximation $\exp [ix]\approx 1+ix$. The spatial spectral transfer function is simplified as

(34)$$H=\frac{\widetilde{E}_{out}^{V}}{\widetilde{E}_{in}^{V}}\approx r_{ss}(k_{ry}\delta_{y}^V+k_{rx}\delta_{x}^V).$$

Hence, the output field can be written as the spatial differentiation of the input field

(35)$$\widetilde{E}_{out}^{V}\approx r_{ss}(\delta_{y}^V \frac{\partial \widetilde{E}_{in}^{V}}{\partial y}+\delta_{x}^V \frac{\partial \widetilde{E}_{in}^{V}}{\partial x}),$$

where the spatial shifts are the in-plane displacements in the $x$-direction and the transverse displacements in the $y$-direction mentioned in the previous section. Therefore, an expression of the optical spatial differentiation of the output light field based on the 2D photonic SHE is established as the formula in Eq. (35), so that edge detection imaging can be realized simultaneously in the $x$ and $y$ directions.

According to the quantitative analysis in the above section, we can choose the incident angles $\theta _{i}$ and the optical axis angles $\phi$ corresponding to Fig. 3(e), the purpose of this was to realize that the in-plane displacements are equal to the transverse displacements for a certain filling factor and the corresponding incident angle. To demonstrate the optical spatial differentiation, Fig. 4 shows amplitude and phase change of the spatial spectral transfer function in Eq. (34) for different incident and optical axis angles, the matching angles for each column: $\theta _{i}=90^{\circ }$, $\phi =60^{\circ }$ [Figs. 4(b1) and (b2)]; $\theta _{i}=60^{\circ }$, $\phi =99.3^{\circ }$ [Figs. 4(c1) and (c2)]; $\theta _{i}=60^{\circ }$, $\phi =27.7^{\circ }$ [Figs. 4(d1) and (d2)]; $\theta _{i}=60^{\circ }$, $\phi =90^{\circ }$ [Figs. 4(e1) and (e2)], respectively. Figures 4(b1)-(e1) show that amplitude change of the spectral transfer function coincides with theoretical prediction, which have the minimum at $k_{rx}=0$ [Fig. 4(b1)], $k_{rx}=-k_{ry}$ [Fig. 4(c1)], $k_{rx}=k_{ry}$ [Fig. 4(d1)], $k_{ry}=0$ [Fig. 4(f1)], respectively. In the wave vector $K$ space, The amplitude of the spatial spectral transfer function presents a V-shaped distribution, and satisfies a strict linear distribution along the zero value axis. Note that the output field caused by the photonic SHE is a superposition of left- and right-handed circular polarization fields. Then, by the polarization post-selection of linear polarization, the spatial spectral transfer function is equivalent to a linearly polarized light intensity filter. These light intensity of the centre part is eliminated, meanwhile, the left- and right-handed circular polarization light intensity parts of the edge are retained for optical differential operation, so it presents a V-shaped $K$ space distribution. Corresponding to the amplitude distribution, Figs. 4(b2)-(e2) exhibit that phase change of the spectral transfer function coincides with theoretical prediction, which have the minimum at $k_{rx}=0$ [Fig. 4(b2)], $k_{rx}=-k_{ry}$ [Fig. 4(c2)], $k_{rx}=k_{ry}$ [Fig. 4(d2)], $k_{ry}=0$ [Fig. 4(f2)], respectively. The phase of the spatial spectral transfer function exist a gradient along the zero value axis, and a phase $\pi$ transition occurs, presenting an opposite symmetrical phase.

Fig. 4. The spatial spectral transfer function corresponding to the image edge detection. (b1)-(e1) show the amplitude change of the spatial spectral transfer function for different incident angles $\theta _{i}$ and optical axis angles $\phi$. (b2)-(e2) The corresponding phase gradient change is displayed. Here, the matching angles of each column: $\theta _{i}=90^{\circ }$, $\phi =60^{\circ }$ [(b1) and (b2)]; $\theta _{i}=60^{\circ }$, $\phi =99.3^{\circ }$ [(c1) and (c2)]; $\theta _{i}=60^{\circ }$, $\phi =27.7^{\circ }$ [(d1) and (d2)]; $\theta _{i}=60^{\circ }$, $\phi =90^{\circ }$ [(e1) and (e2)], respectively.

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The 2D differential operation of edge imaging can be realized based on the 2D photonic SHE, so as to realize the image edge detection of the object in any direction. For example, Fig. 5 show that the use of characteristic images to verify the realization of edge detection based on different in-plane and transverse displacements, and note that these displacements corresponding to Fig. 3(e). Figures 5(a1) and (a2) show the bright field distribution for different specific images. Corresponding to the amplitude and phase change of spatial spectral transfer function in four column Figs. 4(b)-(e), we select the parameter conditions of the incident angle and the optical axis angle in Fig. 4.

Fig. 5. The adjustable edge detection are displayed for different specific images, and note that the in-plane and transverse displacements corresponding to Fig. 3(e). (a1) and (a2) show brightfield images of the input image. (b1)-(e1) and (b2)-(e2) The edge detection of a simple letter diagram and a specific logo are displayed, respectively. Corresponding to the parameters in Fig. 4, (b1) and (b2) show 1D edge imaging along $x$ direction ($\theta _{i}=90^{\circ }$, $\phi =60^{\circ }$), the transverse displacement in the $y$ direction is zero (according to the formula Eq. (30)); (c1) and (c2) represent 2D edge imaging ($\theta _{i}=60^{\circ }$, $\phi =99.3^{\circ }$), the in-plane displacement is approximately equal to the transverse displacement; (d1) and (d2) denote 2D edge imaging ($\theta _{i}=60^{\circ }$, $\phi =27.7^{\circ }$), the absolute value of the in-plane displacement and the transverse displacement are approximately equal; (e1) and (e2) indicate 1D edge imaging along the $y$ direction ($\theta _{i}=60^{\circ }$, $\phi =90^{\circ }$), the in-plane displacement in the $x$ direction is zero.

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Selecting the incident angle $\theta _{i}=90^{\circ }$ and the optical axis angle $\phi =60^{\circ }$, the in-plane displacement in the $x$ direction is $\delta _{x}=107.945nm$, the transverse displacement in the $y$ direction is zero (According to the formula Eq. (30), the transverse displacements is related to $\cot \theta _{i}=0$), and 1D edge detection along the $x$ direction are realized, as shown in Figs. 5(b1) and (b2). Selecting the incident angle $\theta _{i}=60^{\circ }$ and the optical axis angle $\phi =99.3^{\circ }$. The in-plane displacement $\delta _{x}=-54.366nm$ is approximately equal to the transverse displacement $\delta _{y}=-54.387nm$, at this point, 2D edge detection in second and fourth quadrants of the Cartesian coordinate axis can be realized, as shown in Figs. 5(c1) and (c2). Selecting the incident angle $\theta _{i}=60^{\circ }$ and the optical axis angle $\phi =27.7^{\circ }$, The absolute value of the in-plane displacement $\delta _{x}=32.945nm$ and the transverse displacement $\delta _{y}=-32.946nm$ are approximately equal, thereby realizing 2D edge detection in the first and third quadrants of the coordinate axis, as shown in Figs. 5(d1) and (d2). Selecting the incident angle $\theta _{i}=60^{\circ }$ and the optical axis angle $\phi =90^{\circ }$, the in-plane displacement in the $x$ direction is zero, the transverse displacements in the $y$ direction is $\delta _{y}=-64.513nm$, and 1D edge detection along the $y$ direction is realized, as shown in Figs. 5(f1) and (f2).

Figure 5(a1) is the bright field distribution of a simple letter diagram, 1D and 2D edge imaging resolution performance is comparable, there are no complex detail performance, as shows in Figs. 5(b1)-(e1). By contrast, Fig. 5(a2) is the bright field distribution of a specific logo with a complex detail design, it is found that 2D edge imaging have excellent performance compared with 1D edge imaging in the central detail imaging, as shows in Figs. 5(b2)-(e2). These studies have shown that 2D imaging has good application value in image definition and discrimination.

We propose a 2D image edge detection method based on photonic SHE in anisotropic metamaterials. The 2D photonic SHE can be controlled by the filling factor of the crystal structure, the optical axis angle, and the incident angle, this intrinsic effect can be used to realize a tunable edge imaging. These effects can be modulated by adjusting the relevant parameters. Therefore, it can provide some theoretical proposal for the future experimental verification. Based on text parameters, we can also fabricate the anisotropic material with periodic variation by laser etching and other methods [49,51], and the experimental setup for realizing the above theoretical results can be similar to this one in [37]. These already existing experimental systems may provide a practical and applicable method for the two-dimensional image edge detection based on photonic SHE. As a well-known example with structural materials for edge detection, the dielectric metasurface is fabricated on optical glass by laser direct writing technique, which can be integrated on traditional optical components [45]. This structural material has high optical efficiency for edge detection. By designing the periodic of the structural material, the imaging resolution can be modulated and the highest imaging resolution close to the diffraction limit can obtained. In particular, the optical differential operation based on photonic SHE is independent of optical frequency, which can achieve broadband edge detection [38]. Similarly, in our work the imaging resolution can be modulated by the induced transverse and in-plane displacements in photonic SHE.

5. Conclusion

In conclusion, we applied the photonic SHE in an anisotropic metamaterial to the spatial differential operation to realize edge detection. It is found that precise control of spatial displacements can be achieved by designing the filling factor of a specific anisotropic medium, properly rotating the optical axis of the medium, and changing the incident angle of light field. The overall in-plane displacements increase and the overall transverse displacements decrease with the increase of incident angle. The in-plane and transverse displacements are zero-point antisymmetric and maximum symmetric about the $90^{\circ }$ optical axis angle, respectively. Meanwhile, the in-plane displacements increase with the increases of filling factor, this makes it possible to achieve large in-plane splitting by making large filling factors. Here, an opposite phenomenon is that the transverse displacement increases with the decrease of filling factor, when the optical axis angle is away from $90^{\circ }$. Interestingly, when the optical axis angle is $90^{\circ }$, the transverse displacements have a maximum value, and the large spin-dependent splitting can be achieved by adjusting small incidence angles. Then, the adjustable one-dimensional edge detection is realized. Finally, by changing the optical axis of anisotropic metamaterial, the transverse and in-plane displacements are equivalent for a certain filling factor and the corresponding incident angle, 2D edge detection is realized in specific direction by setting the specific parameters of anisotropic metamaterial structures, and the imaging resolution can be modulated by the magnitude of transverse and in-plane displacements. Therefore, The modulation of the photonic SHE and the edge detection relationship are explored, it can provide some theoretical proposal for the future experimental verification, and the effects in the interaction between light and micro-nano structures are revealed.

Funding

National Natural Science Foundation of China (42164009); The Training Program for High-level Innovation Youth Teachers of Guiyang (GYCQJ[2018]36); The Special Funding of Guiyang Science and Technology Bureau and Guiyang University (GYU-KY-[2021]).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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All-optical image edge detection based on the two-dimensional photonic spin Hall effect in anisotropic metamaterial

Abstract

1. Introduction

2. General theoretical model

3. Photonic spin Hall effect

4. Optical spatial differentiation

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (35)

Optics Express