Enhanced optical spatial differential operations via strong spin-orbit interactions in an anisotropic epsilon-near-zero slab

Several works on optical higher-order differential operations have recently attracted attention, particularly in image processing for edge detection. However, the inefficient differential operation leads to barriers to practical applications. Here, we report an anisotropic epsilon-near-zero slab to significantly enhance the transmission efficiency of second-order differentiators and discuss the Berry phase mechanism of this optical calculation process. Through a rigorous full-wave analysis of the process, we find that the conversion efficiency of differential operation depends on the spin-orbit interactions. Our scheme can strengthen the spin-orbit interaction by introducing anisotropy, which significantly enhances the transmission efficiency. We finally give transfer functions to reveal how to improve the efficiency and compare the quadratic coefficient among different systems. This highly efficient differentiation operation may develop significant applications in fast, compatible, and power-efficient ultrathin devices for data processing and biological imaging.

Figure 1

Schematic diagram of the anisotropic epsilon-near-zero slab as a second-order differentiator operates at normal incidence mode. $|L\rangle$ and $|R\rangle$ are the left and right circular polarizations, respectively.

Figure 2

Illustration of a finite-width Gaussian beam vertically incident at the interface in momentum space; $\vartheta$ is the propagation angle of the arbitrary wave components in the spherical coordinate system, $k_r=\sqrt{{k_x^2+k_y^2}^{\phantom{\int }}}$ is the corresponding transverse wave vector, $cos\varphi =k_x/k_r$, $sin\varphi =k_y/k_r$, $\mathrm{\Delta }k=2\pi /w_0$ is transverse half width.

Figure 3

Geometric representation of Berry phase. A Pancharatnam-Berry phase is gained as a left-handed CP plane wave experiences a spin-reversal propagation, which is half of the solid angle of the shaded area enclosed by two paths ($k$ and $k^{\prime }$) connecting the north and south poles on the Poincaré sphere.

Figure 4

The transmission coefficients and efficiency for (a) glass slab with $\varepsilon =2.25$, (b) isotropic ENZ slab with $\varepsilon =0.01$, and (c) anisotropic ENZ slab with ${\varepsilon }_x={\varepsilon }_y=1,{\varepsilon }_z=0.01$. We set an incident beam with $\lambda =1$, incident vector $k_0=2\pi /\lambda$, and the thickness of slab $h=2\lambda$.

Figure 5

The amplitude distribution of transfer function of anisotropic ENZ with free momentum plane for (a) $h=2\lambda$ and (b) $h=4\lambda$. (c) magnitude and (d) phase of transfer function for anisotropic ENZ with a finite-width Gaussian beam incidence. The waist width is set as $w_0=15\lambda$, so that transfer function can provide good conversion efficiency and uniform additional phase; the other parameters are specified in Fig. 3.

Figure 6

The quadratic coefficient $Q$ of the transfer function magnitude in various parameters. (a) Normalized quadratic coefficient of the transfer function magnitude for anisotropic ENZ slab (${\varepsilon }_x={\varepsilon }_y=1$, ${\varepsilon }_z=0.01$; red dashed line), anisotropic ENZ slab with loss (${\varepsilon }_z=0.01+0.05i$; green solid line), and isotropic slab ($\mathit{\varepsilon }=0.01$; blue dotted line) as a function of transverse wave vector $k_r/k_0$ ($h=2\lambda$). (b) The quadratic coefficient with different thickness $h/\lambda$ of those three slabs under the beam width as $k_r/k_0=0.02$. (c) The influence of anisotropic item (${\varepsilon }_z$) on the quadratic term coefficients with three different thickness. $Q_0$ is defined as the quadratic coefficient of uniform media $\varepsilon =2.25$, $h=2\lambda$. The other parameters are specified in Fig. 4.

Figure 7

Numerical demonstration of second-order edge detection of the entire image. (a) is the grayscaling representation of incident images. (b) is intensity images transmitted through the ENZ differentiator with certain parameters. (c) and (d) correspond to the horizontal intensity distributions of images (a) and (b), respectively, at the green dotted line. The horizontal and vertical coordinates represent the position and light intensity, respectively. The other parameters are specified in Fig. 4.