1. Introduction

Optical differential operation is a technique that utilises optical tools and devices to perform analog computational processing and transmission of image information, which has garnered widespread attention in image processing due to the advantages such as massively parallel processing, ultra-high speed and markerless [1–4]. In particular, one- and two-dimensional (1D-2D) optical differential operations based on the photonic spin Hall effect (PSHE) have come to the forefront due to the flexible manipulation [5–11]. The PSHE refers to the spin splitting when a linearly polarized light is transmitted at a non-uniform medium interface, which has been widely proposed in optical research [12–14]. The PSHE induces the transverse shift in the y-direction, which is closely related to spin-orbit interaction of light, and the transverse shift is the particular case of Imbert-Fedorov shift [15,16]. Meanwhile, there is the longitudinal in-plane spin separation of light (IPSSL) that is observed experimentally, which originates from the tiny in-plane spread indirection of the wave vector [17]. On this basis, 1D differentiators provide a clear and intuitive approach for capturing information in a single direction, while 2D differentiators are more comprehensive for image processing, and both are significant for edge detection. The researchers have successfully implemented optical differential operations via the PSHE on materials including air-glass interfaces [5–7] and high-efficiency metasurfaces [8–10]. In addition, the optical differential operation based on the Goos-Hänchen effect have also been studied and applied [18]. In recent years, there has also been an increased interest in enhancing the resolution of optical differential operation via the PSHE [19–22]. Tunable resolution of the optical differential operation can be involved by modulating the shifts through changing the physical parameters of the material. However, due to the material constraints, current differential operations rarely enable fast switching between 1D-2D edge detections, with few degrees of freedom and low contrast. To improve the performance of differential operation, we require a scheme to realize optical differential operation with tunable dimension and contrast.

The Weyl semimetal (WSM) is a novel topological material when time-reversal or inversion symmetry is broken, where a pair of Weyl points (WPs) with opposite handedness exists in bulk Brillouin zone [23–26]. The type-I WSM including TaAs and TaP families have been proved by observing the Fermi arcs with separated WPs [27–29], and have attracted widespread attention in topological optics due to its unique properties induced by the WPs, such as anomalous Hall effect [23,30] and negative magneto-resistivity [31,32]. Recently, the spin-dependent beam shifts can be enhanced by adjusting the physical parameters of the WSM, including lattice spacing [33], the tilt angle of the Weyl cone [34,35] and WPs separation [36], which provides a possibility to modulate the optical differential operation. Compared to the materials such as conventional glass and metasurfaces that currently employed in optical differential operations, the WSM offers abundant degrees of freedom that can enhance the beam shifts by changing its internal properties. Therefore, the optical differential operation based on the PSHE in the WSM has significant potential to switch the dimensions and modulate the contrast of the optical differential operation.

This paper constructs an optical differential operation via the PSHE in a WSM, which enables 1D-2D edge detection and high contrast. The incident and polarization angles can scale the in-plane and transverse shifts to switch between 1D-2D differential operations. With the efficient optical transmission characteristics of the WSM, the shifts can be enhanced and the contrast of the differential operation can be changed by regulating the physical parameters of the WSM. The high contrast is also verified by comparing the differential operation with the glass. In this case, we design an efficient differential operation with switchable dimension and high contrast, which may develop important applications in optical image processing.

2. Theoretical analysis

A structure with three layers consisting of air, the WSM TaAs and glass is constructed, as shown in Fig. 1(a). The layer of WSM TaAs with lattice spacing a = 3.44 Å, thickness $d = 50$ nm and fermi velocity ${v_F} = {10^6}{\rm {m}} \cdot {{\rm {s}}^{ - 1}}$ is laid on the BK7 glass with refractive index $n = 1.5$ [27,36]. Here, two Weyl points in the Brillouin zone are separated by the wave vector $\pm b$, and $b = {b_0}{{ \times 2\pi } \mathord {\left / {\vphantom {{ \times 2\pi } a}} \right.} a}$ indicates the degree of separation of the Weyl points, where $b_0$ indicates the coefficient of the WPs separation. After reflecting on the WSM-glass substrate, the Gaussian beam splits into left and right circularly polarized components, which are spatially shifted in both in-plane and transverse directions. $\left ( {X ,Y ,Z } \right )$ is the experimental coordinate system, where the z-axis is mutually perpendicular to the WSM interface, and $\left ( {x_i ,y_i ,z_i } \right )$ is the coordinate system of incidence.

 

Fig. 1. Schematic diagram of second-order differential operation in a WSM. (a) The light beam is reflected from the air to the WSM-glass interface and splits into left-circularly and right-circularly polarized components, which produces spatial shifts $\Delta _x$ and $\Delta _y$, and the objective image "$\theta$" gets the 2D edge imaging through the optical differential operation. Here, left- and right-circularly polarized components is represented by red and blue curves. (b) A pair of Weyl points with opposite chirality of "+" and "-" is represented by blue and orange balls, with the outer arrow as the source and the inner arrow as the sink of Berry curvature. The pseudo-magnetic field ${B_{{\rm {e}}l}}$ is induced by the WPs separation $b$, and the WPs interact with the generated pseudo-magnetic field when the WPs are separated.

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The WSM is a novel topological semimetal characterized by Weyl fermions as quasi-particles, which exists only in the presence of either time-reversal or inversion symmetry breaking. Although we consider a WSM with broken time-reversal symmetry, we can introduce a slight break in inversion symmetry through a momentum-independent spin-orbit interaction [24]. The tiny breaking of inversion symmetry results in charge imbalance between the WPs, leading to the separation of WPs in momentum space. The spatial variation of WPs induces a pseudo-magnetic field ${B_{{\rm {e}}l}}$ interacting with fermions of opposite chirality, as illustrated in Fig. 1(b). Unlike an externally applied magnetic field, the pseudo-magnetic field can attract carriers of opposite chirality. During transport, ${B_{{\rm {e}}l}}$ in the chiral anomaly enhances the filter-induced longitudinal photoconductivity [37,38]. It is known that changing the optical conductivity will directly influence the PSHE, so that regulating the position of WPs will affect the PSHE, thereby providing a theoretical foundation for tuning optical differential operations.

This incident light is selected as the the single wavelength of the Gaussian beam generated by the He-Ne Laser, and its wavelength is $\lambda = 632.8$ nm. The incident angular spectrum can be expressed as [36]:

The linearly polarized light is converted into elliptically polarized light after passing through the quarter-wave plate, and then the beam is reflected through the WSM-substrate interface. The angular spectrum after reflection can be written as $\tilde E_r = M_R \left \langle {QWP} \right |\tilde E_{in}$. Here, $\left \langle {QWP} \right |$ is the Jones matrix equation for the quarter-wave plate whose optical axis is $0^\circ$ with x direction:

After the beam is reflected on this three-layer structure, the angular spectrum of the reflected light field can be written as [5]:

Here, ${e_ + }$ and ${e_ - }$ indicates the left and right rotation of the beam of light, $\kappa _1^ \pm = {r_{pp}}{\chi _{pp}} \pm i{r_{ps}}{\chi _{pp}}$, $\kappa _2^ \pm = {r_{ss}}{\chi _{ss}} \pm i{r_{ps}}{\chi _{ps}}$ and ${\chi _{pp}} = \partial \ln {r_{pp}}/\left ( {{k_0} \partial {\theta _i}} \right )$, ${\chi _{ps}} = \partial \ln {r_{ps}}/\left ( {{k_0}\partial \theta _i} \right )$, ${\chi _{ss}} = \partial \ln {r_{ss}}/\left ( {{k_0}\partial \theta _i} \right )$, and $\delta = \left ( {{r_{pp}} + {r_{ss}}} \right ){\rm {cot}}{\theta _i} / {k_0}$ is set here. It is worth noting that the coefficients in the formula are set to ${F_L} = \left ( {{r_{pp}} - i{r_{ps}}} \right ){\rm {cos}}\alpha {\rm {\ +\ }}\left ( {{r_{ss}} + i{r_{ps}}} \right ){\rm {sin}}\alpha$ and ${F_R} = \left ( {{r_{pp}} + i{r_{ps}}} \right ){\rm {cos}}\alpha + i\left ( {{r_{ps}} + i{r_{ss}}} \right ){\rm {sin}}\alpha$, which are modulo equivalent.

There is an overlap between the left and right circular polarization components, which are recombined into a linear polarization. In order to eliminate the overlap and retain edge information, we introduce another Glan Laser polarizer that is cross polarized with the first one. The output optical field in momentum space can be represented by a simplification of Euler’s theorem as:

Here, $\Delta _x$ and $\Delta _y$ are in-plane shift of the IPSSL and transverse shift of the PSHE in momentum space [15–17], and these shifts have been well-studied in different systems [42–48]. Meanwhile, they are also the important basis for the optical differential operation.

The reflected field in position space can be obtained by the Fourier transform, which can be written as:

A uniform 2D isotropic edge is obtained when $\Delta _x$ and $\Delta _y$ reach equilibrium, as shown in Fig. 1(a). When $\Delta _x$ vastly surpasses $\Delta _y$, the in-plane shift dominates and 1D x-direction differential imaging is obtained. Conversely, 1D y-direction optical differential operation is realized.

The transfer function is a 2D differential expression for the $x$ and $y$ directions, which reflects the performance of a differential operation. Within the diffraction limit, the larger the slope of the transfer function is, the higher the contrast of imaging can be obtained [49,50]. According to the expression of the spatial transfer function, it can be written as:

Combined with Eqs. (4)–(8), (11) and (12), we can deduce that there are abundant degrees of freedom to modulate the optical differential operation in WSM, such as thickness, WPs separation, lattice spacing. Importantly, WPs separation is a particular parameter to WSM and offers unique properties in beam shifts, such as anomalous Hall effect [23,30] and negative magneto-resistivity [31,32]. On this basis, the optical differential operation can be modulated by adjusting the WPs separation.

3. Results and discussions

We know that the magnitudes of the in-plane and transverse shifts determine the dimensionality of the optical differential operation, so the enhancement of the beam shifts is the key to explore. In this section, we investigate the in-plane and transverse shifts for different incident conditions, analyse the spatial spectral transfer functions and deduce the necessary conditions for 1D-2D differential operations. Subsequently, numerical simulations are carried out to verify the switchable dimension of the differential operation.

Firstly, we explore the in-plane and transverse shifts with polarization angle for different incident angles. The spatial spectral transfer function of the differential operation can theoretically verify that the operation enables dimension switching. By regulating the incident and polarization angles, the transfer function changes abruptly in one direction when the in-plane shift $\Delta _x$ is significantly larger or much smaller than the transverse shift $\Delta _y$, which means that the differential operation enables the 1D edge detection. As $\Delta _x$ and $\Delta _y$ are balanced, namely ${\Delta _x} = {\Delta _y}$, the edge information obtained in the x and y directions is equal after filtering out the recombined linearly polarization, which means the 2D edge detection can be realized. Variation of the in-plane and transverse shifts with the polarization angle at the incident angles of 30$^\circ$, 50$^\circ$ and 80$^\circ$ are shown in Fig. 2(a). In this figure, the 1D x- and y-direction differential operations can be accomplished in the case of point I ($\alpha =50^\circ$ and $\theta =80^\circ$) and point II ($\alpha =90^\circ$ and $\theta =30^\circ$). The spatial transfer functions are respectively linear distribution in the x and y wavevector regions as shown in Figs. 2(b) and 2(c), and their phases jump along the zero-value axis that are obtained from Figs. 2(e) and 2(f). The condition for uniform 2D edge detection is satisfied at point III ($\alpha =66.5^\circ$ and $\theta =50^\circ$), and it has a uniformly tapered transfer function with a minimum value of 0 in Fig. 2(d), and there is a helical phase in Fig. 2(g). These findings set the stage for differential operation to switch dimension and tune the contrast.

 

Fig. 2. Theoretical switching of 1D-2D differential operations. (a) Variation of in-plane shift $\Delta _x$ and transverse shift $\Delta _y$ with the polarization angle $\alpha$ at the incident angle $\theta = 30^\circ$, $50^\circ$ and $80^\circ$, where the $\Delta _x$ and $\Delta _y$ are represented by solid and dashed lines, respectively. (b) Transfer function |H| and (e) the phase distribution Arg[H] in the 1D x-direction of point I. (c) |H| and (f) Arg[H] in the 1D y-direction of point II. (d) |H| and (g) Arg[H] in the uniform 2D of point III.

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Next, we numerically simulated the intensity distributions of the incident light field, the 1D-2D output light fields and the edge detection of the target image through the differential operations. Figure 3(a) shows the intensity distribution of the light field generated by the Gaussian beam, and Figs. 3(b)-(d) show the intensity distributions of the output light fields in 1D x-direction, 1D y-direction and 2D detection, respectively. Note that the parameters are selected corresponding to the points points I, II, and III in Fig. 2(a). It can be observed that the left and right circularly polarized components of the light beam are spatially shifted in opposite directions, which is an important basis for optical edge detection. Subsequently, by performing a Fourier transform on the input image "$\theta$" in Fig. 3(e), and applying a transfer function and performing an inverse Fourier transform, the output images are generated as shown in Figs. 3(f)-(h). It is found that the dimension and direction of the optical differential operation can be realized by adjusting the incident angle and polarization angle, and the feasibility and correctness of the differential operation are verified by the output results.

 

Fig. 3. Numerical simulation of switching between 1D-2D differential imaging. (a)-(d) Intensity distributions of the incident and 1D-2D output light fields of the Gaussian beam. (e)-(h) Input image and its output results of edge detection.

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In edge imaging, researchers have achieved a large number of results, which show that the steeper the slope of the transfer function, the more pronounced the mutation detected in the image, and the higher the contrast of the differential operation. Combining Eq. (14), the intensity coefficient can be extracted from the transfer function expressed as:

From the above formula, it is clear that the magnitude of the beam shifts affects the modulus of the transfer function, which represents the contrast of optical differential operation is closely related to the shifts. Due to the abundant degrees of freedom and highly sensitive properties of the WSM, we will further change the physical parameters of the WSM to modulate the shifts and achieve high contrast of the optical differential operation.

Subsequently, we explore the ratio of the intensity coefficients to compare the contrast of the differential operation in the WSM with that in the glass [5]. It can be inferred from the theoretical analysis that the optical differential operation can be regulated by changing the physical parameters of the WSM. Since the WPs separation is a unique internal parameter in the WSM, we investigate the variation of incident angle and WPs separation on the optical differential operation. Figure 4(a) shows the intensity coefficient ratio of the WSM and glass as a function of the incident angle and the x-direction wave vector at the WPs separation ${b_0}=0.05$. It is worth noting that, considering the comparability of the two differential operations, we will choose the polarisation 67.1$^\circ$ from the differential operation in the glass. As shown in the figure, the intensity coefficient of the operation in the WSM is improved by two orders of magnitude compared to that in the glass at $\theta =26.5^\circ$. The effect of the WPs separation on the differential coefficients in the x-direction is shown in Fig. 4(b) by setting the incident angle as 26.5$^\circ$. Notably, in the wavevector region of the momentum space, the steeper slope of the intensity coefficients denotes a more pronounced mutation in the image detection, and results in the higher contrast of the optical differential operation. It is obvious from the figure that the contrast of the optical differential operation can be modulated by changing the internal parameters of the WSM, and improved by two orders of magnitudes compared to that in conventional glass. This phenomenon is due to the fact that the WPs separation effects the longitudinal conductivity and the beam shifts, and further modulate the optical differential operation. It is worth mentioning that the optical differential operation can be achieved in inhomogeneity and anisotropy materials including Q-plates due to the enhancement of the beam shifts near the Brewster angle or normal incidence [42,43]. However, the optical differential operation in the WSM can be modulated by changing the internal parameters such as WPs separation and lattice spacing instead of being limited at these special angles. WSM has a lot of application value due to its advantages of flexible modulation and abundant degrees of freedom compare to the other materials, which is the important basis for tuning the optical differential operation.

 

Fig. 4. Theoretical effects of the incident angle and the WPs separation on the contrst of differential operations. Ratio of the differential intensity coefficients of the WSM and glass as functions of x-direction wave vector component and (a) the incident angle at $b_0$=0.05 and (b) the WPs separation at $\theta$ =26.5$^\circ$. Here, the other parameters are chosen in agreement with the optical differential operation of the glass.

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Finally, to verify the high contrast of the optical differential operation in the WSM, we perform the differential operation on the feature images with the glass and WSM, respectively. Figure 5(a) shows the edge imaging when the differential operation in glass is applied to edge detection, and Figs. 5(b) and 5(c) represent the edge imaging through the differential operation in the WSM at $b_0 = 0.2$ and $b_0 = 0.05$. The result of image edge detection in the glass is considerably darker than in the WSM, and the edge imaging in WSM at $b_0 = 0.05$ obviously brighter. The horizontal light intensity distributions shown in Figs. 5(d)-(f) are normalised by the maximum light intensity of the differential operation in WSM at $b_0 = 0.05$, and it can be seen that the light intensity obtained by the differential operation is higher in WSM and the image edge features are more prominent than in glass. It is convincing that the beam shifts of the beam transport model in the WSM is more significant compared to conventional glass. This is because the fact that the WSM can impact on the spin-orbit interaction of light and the physical parameters can improve the optical conductivity, which is an essential factor for the high contrast of the differential operation. The optical differential operation in the WSM provides greater potential for edge imaging due to the advantages of high contrast.

 

Fig. 5. Numerical simulation of the differential operation in the glass and the WSM at $\theta = 26.5^\circ$. (a) shows the output images through the differential operation in the glass. (b) and (c) present the edge imaging through the differential operation in the WSM at $b_0 = 0.2$ and $b_0 = 0.05$, respectively. (d)-(f) correspond to the horizontal intensity distributions at the green dashed line in Figs. 5(a)-(c), respectively.

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4. Conclusion

In conclusion, this paper constructs an optical differential operation based on the PSHE in a WSM, enables dimension switching and high contrast. The model of beam transport is established and the differential results are numerically simulated. It is found that the incident and polarization angles affect the balance between the in-plane and transverse shifts, and 1D differential operation can be achieved when one of these shifts is considerably superior. Interestingly, the regulation process has a unique position where the in-plane and transverse shifts reach a balance, and the differential operation switches from 1D to 2D detection. Due to the effect of the WSM, the spin-orbit interaction of light can be efficiently regulated by physical parameters such as WPs separation, and a substantial enhancement of contrast can be achieved compared to conversational glass. It is noteworthy that the study of optical differentiators may find important applications in smart driving and high-contrast microscopy.