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Abstract
In previous edge detection schemes based on the spin-orbit interaction of light, the direction and intensity of the edge-enhanced images are influenced by the incident polarization state. In this study, we develop an edge detection strategy that is insensitive to changes in both the incident polarization and the incident angle. The output intensity and transfer function remain entirely impervious to changes in incident polarization, being explicitly formulated as functions of the incident angle, specifically in terms of $\cot ^2{\theta _{i}}$ and $\cot {\theta _{i}}$, respectively. This behavior is attributed to the opposing nature of the polarization components $\tilde E_r^{H-H}$ and $\tilde E_r^{V-V}$ in the x-direction after undergoing mapping through the Glan polarizer, while the sum of polarization components $\tilde E_r^{H-V}$ and $\tilde E_r^{V-H}$ in the y-direction can be simplified to terms independent of incident polarization. Furthermore, we propose a metasurface design to achieve the required optical properties in order to realize the derived edge detection scheme.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Computing is applied in diverse field globally [1,2], yet traditional electronic systems, despite thier historical dominance in technology, grapple with challenges like sluggish processing speeds, high energy consumption, and costly components [3–5]. Recently, attention has shifted towards the field of optical computing [6–9], encompassing spatial differentiation, integration, and convolution. In comparison to conventional electronic computing, optical computing offers advantages such as large bandwidth, fast response speed, low power consumption, and low interference [10–14], due to the advantages that of photons.
Optical differentiation computing, capable of handling gradients and variations in the light field, is often employed for extracting crucial geometric features from images (i.e. edge detection) [15–28]. This reduces the computational load in image processing, leading to enhanced operational efficiency and precision. Recently, due to its ease of setup (requiring only a simple optical interface) and the capability to achieve two-dimensional imaging [29,30], edge detection based on the spin-orbit interaction of light stands out [31–35]. For example, Zhu et al. utilized an ordinary glass interface to achieve edge information extraction through the photonic spin Hall effect (PSHE) [16], Further, making the in-plane shift equal to the out-of-plane shift enables the realization of two-dimensional edge detection [35]. However, current research in edge detection is often associated with the polarization of incident light [36,37]. Variations in the incident polarization can impact the edge information, potentially limiting its practical application in complex environments.
In this paper, we propose a polarization-independent edge detection scheme based on the spin-orbit interaction of light, provide analytical results and propose a metasurface capable of achieving the required optical properties. Changes in incident polarization do not influence the edge information, including the output intensity and transfer function. Similarly, alterations in the incident angle do not affect the direction of edge imaging but exhibit a cotangent dependent function relationship with output intensity and transfer function. This scheme exhibits exceptional robustness in incident polarization, and holding the promise of extending edge detection based on the spin-orbit interaction of light to complex real-world environments.
2. Theoretical analysis
2.1 Theoretical calculation of spatial spectral transfer function
We consider a light beam reflected through a medium-medium interface. The $xy$ plane of the experimental Cartesian coordinate system is parallel to medium-medium interface. We use coordinate systems ($x_{i}$, $y_{i}$, $z_{i}$) and $(x_{r}$, $y_{r}$, $z_{r}$) to represent incident and reflected beams, respectively, where the $z_{i}$ and $z_{r}$ axes are parallel to the propagation direction of the central wave vector of the light beam. The angular spectrum of the incident Gaussian beam can be expressed as follows [31]:
Here, $R_{m}$ is the transfer matrix, $\beta _{i}$ is the polarization angle of the incident beam, which is set as the angle between the incident light beam and $x_{i}$ axis, and $\varphi$ denotes the phase difference along the $x_{i}$ and $y_{i}$ axes, where $\varphi =0,\pi /2$ correspond to linearly polarized and circularly polarized light, respectively. Please note that the cross-Fresnel coefficients of anisotropic materials are not considered here, only isotropic cases are taken into account. According to the Taylor series expansion, the Fresnel coefficients $r_p$ and $r_s$ can be expanded as follows:
According to Eq. (2), we can get the evolution of parallel and vertical components as follows:
In the expression above, the values of in-plane spin separation $\Delta {x}$ and transverse spin separation $\Delta {y}$ are given as follows:
By introducing approximation $1+\Delta {x}k_{rx}+\Delta {y}k_{ry}\approx \exp (\Delta {x}k_{rx}+\Delta {y}k_{ry})$, we can get
In order to eliminate the overlapping part in the reflected beam, we use a Glan laser polarizer with the polarization axis at an angle of $\beta _r = \arctan \left ( {\frac {{r_s }}{{r_p }}\tan {\beta _i } } \right ) + \frac {\pi }{2}$ to the $x$ axis. Finally, we can get the final output field ${\tilde E}_{out}({x},{y})$, which can be approximately written as the spatial full differentiation of the input ${\widetilde E_i }$:
2.2 Analytical study of incident-polarization-independent output intensity distribution and spatial transfer function
A close examination of Eqs. (8)–(10) shows that the two field profiles appear symmetrical in $\Delta {x}$ and $\Delta {y}$ $( {r_p}{ r_s}{\tan \beta _i } \Delta _{H} \to {r_p}{ r_s}{\tan \beta _i } \Delta _{V}$ and ${{{r}}_{p}^{2}} {\delta _{r}^{H}} \to {{{r}}_{p}^{2}} ,{e}^{ 2 i \varphi } \tan ^{2} \beta _i {r}_{s}^{2} {\delta _{r}^{V}} \to {e}^{2 i \varphi } \tan ^{2} \beta _i {r}_{s}^{2} )$. Therefore, if $r_{p}=r_{s}$, the final output field at this time can be modified as follows:
The modified spatial transfer function can be obtained as
Referring to Eqs. (14), (15), under the condition $r_{p}=r_{s}$, the polarization angle $\beta _i$ of the incident beam has been simplified, while retaining the phase difference $\varphi$ between the $x_i$ and $y_i$ component. Consequently, by maintaining the phase difference unchanged, both the output light intensity and transfer function remain unaffected by the polarization angle. It is evident that the in-plane spin separation consistently equals 0. Additionally, the transverse spin separation remains unaffected by the incident polarization angle but does depend on the incident angle $\theta _{i}$ and wave vector $k_{0}$ when keeping the phase difference constant. This further indicates that the output light intensity distribution $\tilde {E}_{out}({x},{y})$ and the transfer function $H ({x},{y})$ are independent of the incident polarization angle when the incident beam is only polarized light of the same phase difference; they are solely dependent on the input light intensity distribution, wave vector and incident angle, when neglecting the Fresnel coefficients. It should be noted that in this work, we primarily focus on linearly polarized light. This choice is made because linearly polarized, circularly polarized, and elliptically polarized light exhibit similar polarization-independent properties.
3. Results and discussion
3.1 Polarization-independent edge detection
In Figs. 1(a), we present the theoretical intensity distribution of the incident light field and the resulting output light field under varying incident linear polarizations while maintaining the condition $r_p=r_s$. To discern the trend, we select incidence polarization angles $\beta _i$ of $20^\circ$, $50^\circ$, and $80^\circ$ with intervals of 30 degrees. The incident angle is fixed at $\theta _i=45^\circ$ for all cases. It is crucial to emphasize that the Fresnel coefficients are treated as constant terms. From Figs. 1(a), it is evident that the incident light field follows a Gaussian distribution, and the output light field exhibits transverse spin separation in the $k_y$ direction. Remarkably, this transverse spin separation remains constant even when the incident polarization is set to $20^\circ$, $50^\circ$, or $80^\circ$. This observation underscores the occurrence of the PSHE, leading to the transverse splitting of circularly polarized light. It is clear to see that the PSHE is independent of the incident polarization, emphasizing its polarization insensitivity. To illustrate the spatial differentiation, we also examine the theoretical spatial spectral transfer function $H({x},{y})$ under the condition $r_{p}=r_{s}$, as shown in Figs. 1(b). At $k_y/k_{0}=0$ (indicated by the white dashed line), the transfer function consistently maintains a minimum value of 0, while also exhibiting a symmetrical distribution along highlighting that variations in the incident polarization do not modify the transfer function. This proves the potential for spatial differentiation and edge detection in the $k_y$ direction, independent of incident polarization.
Fig. 1. The output light intensity distribution (a) and transfer function (b) under different incident polarizations $\beta _i$ with $20^o$, $50^o$, and $80^o$, respectively. The inserted images in (a) depict the light field distribution of the incident light. The incident angle is $\theta _{i}=45^o$ throughout.
To investigate the relationship between the output light field, the transfer function and the incident angle, we plot the output light field and the transfer function at different incident angles in Figs. 2 and 3. Here, we select incidence polarization angles $\theta _i$ of $45^\circ$, $60^\circ$, and $75^\circ$ with intervals of 15 degrees. All data is normalized with regards to $\theta _i = 45^\circ$ while the incident polarization is fixed at $\beta _{i}=50^o$. From Fig. 2, it can be found that as the incident angle increases, the output light field consistently displays splitting along the $k_y$ direction, while the peak intensity gradually decreases. Furthermore, it shows a positive correlation with $\cot ^2 \theta _i$. In Fig. 3, the numerical values of the transfer function maintain symmetry along $k_y/k_0=0$ (indicated by the white dashed line), and its peak exhibits a relationship with the incident angle represented by $\cot \theta _i$. This indicates that changing the incident angle does not alter the direction of spatial differentiation and edge detection, it only affects the magnitude of the output light intensity.
Fig. 2. The output light intensity distribution (a) are under different incident angle $\theta _i$ with $45^o, 60^o$, and $75^o$, respectively. The corresponding two-dimensional output light intensity distribution (b) is presented at the position $k_x=0$. The incident polarization is $\beta _{i}=50^o$ throughout.
Fig. 3. The transfer function (a) under different incident angles $\theta _i$ with $45^o, 60^o$, and $75^o$, respectively. The corresponding two-dimensional transfer function (b) is presented at the position $k_x=0$. The incident polarization is $\beta _{i}=50^o$ throughout.
In consideration of the potential experimental implementation of the edge detection scheme, we outline the experimental setup shown in Fig. 4(a), and conduct theoretical simulations of the polarization-independent edge detection scheme shown in Fig. 4(b). A laser is employed to generate linearly polarized light, while a half-wave plate (HWP) is utilized for controlling the intensity of the light. The object, serving as the edge imaging target, is positioned at the focal plane of Lens1. Lens1, Lens2, Lens3, and Lens4 are configured to form two sets of 4f systems. The quarter-wave plate (QWP) within the GLP1 polarization prism is utilized to manipulate the polarization state of incident light. A medium is incorporated to induce the photonic spin Hall effect. The second GLP2 polarization prism is employed to select the desired polarization state, and a CCD is utilized for image detection. The CCD should be positioned at the focal plane of Lens4. Here, the incident polarization and angles in Figs. 4(b) are set to $\beta _{i}=20^o$ and $\theta _i=45^o$, $\beta _{i}=50^o$ and $\theta _i=45^o$, $\beta _{i}=50^o$ and $\theta _i=60^o$, respectively. From Figs. 4(b), it can be observed that the imaging direction remain unaffected by the polarization and angle of incidence. However, the variations in the incidence angle lead to changes in the final output light intensity. This aligns with the earlier predictions regarding the output light intensity and transfer function.
Fig. 4. The experimental setup (a) and theoretical prediction (b) for edge detection with spin–orbit interaction of light. The light source is a He–Ne laser (wavelength $\lambda = 632.8$ nm); HWP, half-wave plate; GLP, Glan laser polarizer; medium, the Fresnel coefficient $r_{p}=r_{s}$; CCD, charge-coupled device. Lens1 and Lens2 form the first 4f system, the object (resolution target) is placed at the front focal plane of Lens1. Lens3 and Lens4 constitute the second 4f system, and the CCD is placed at the rear focal plane of Lens4. (b) Edge detection is simulated under incident angles $\theta _i$ and $\beta _i$ with a value of ($45^o$, $20^o$), ($45^o$, $50^o$), and ($60^o$, $50^o$), respectively.
3.2 Analytical results for polarization-independent
To analyze the factors contributing to incident polarization independence, we decompose the reflected angular spectrum into four polarization components, considering cross-polarization. Eq. (11) can be reformulated as [30]:
The terms $\tilde {E}_{r}^{H-H}$ and $\tilde {E}_{r}^{H-V}$ represent the reflected wavepacket components with horizontal (H) and vertical (V) polarizations, respectively, when the incident light is H polarized and reflects at a medium interface. Similarly, $\tilde {E}_{r}^{V-V}$ and $\tilde {E}_{r}^{V-H}$ denote the reflected wavepacket components when the incident light is V polarized. These four components can be defined as follows:
Upon careful analysis of Eqs. (16)–(21), we observe that when the Fresnel coefficients $r_p = r_s$, the angle of the Glan polarizer $\beta _r$ satisfies the incident polarization angle $\beta _i$ as follows: $\beta _r = \beta _i + \frac {\pi }{2}$. Additionally, the polarization component $\tilde E_r^{H - H}$ undergoes a mapping through the Glan polarizer opposite to the mapping of the polarization component $\tilde E_r^{V - V}$ (i.e. $\tilde E_r^{H - H} \cos \beta _r + \tilde E_r^{V - V} \sin \beta _r = 0$). This implies that the component in the x-direction is always 0. For the y-direction, considering that $\cos \beta _i \sin \beta _r - \cos \beta _r \sin \beta _i$ is always equal to 1, its component can be simplified to $e^{i \varphi } \frac {{2{r_{p,s} }\cot \theta _i}}{{k_0}}$. Therefore, we believe that the reason for polarization independence is that, after undergoing the mapping by the Glan polarizer, the components $\tilde E_r^{H - H}$ and $\tilde E_r^{V - V}$ in the x-direction are completely opposite, while the sum of components of $\tilde E_r^{H - V}$ and $\tilde E_r^{V - H}$ in the y-direction can be simplified to terms independent of incident polarization.
Although we have discussed a one-dimensional polarization-independent edge detection scheme in isotropic interfaces in this work, we believe this approach can be extended to anisotropic interfaces. When the cross Fresnel coefficients satisfy $r_{sp}=-r_{ps}$ and $r_{pp}=r_{ss}$ in anisotropic interfaces [39], the cross-polarization component in the x-direction becomes zero, and the polarization component in the y-direction simplifies the polarization term. This results in a one-dimensional polarization-independent edge detection scheme. Additionally, anisotropic interfaces may also exhibit mapping relationships similar to Eqs. (14), (15), allowing for the elimination of polarization-related terms and satisfying the transfer function $H(k_{rx},k_{ry})=$ $\Delta {x} k_{rx}+i\Delta {y}k_{ry}$ [40], where $\Delta {x}=\Delta {y}$. This facilitates a two-dimensional polarization-independent edge imaging scheme.
3.3 Proposal of metasurface realization satisfying $r_{p}=r_{s}$
In order to achieve equal Fresnel coefficients $r_{p}=r_{s}$, we design a metasurface, as shown in Fig. 5. It is made up of a two-dimensional arrangement of subwavelength sized structures to manipulate optical properties of the reflected electromagnetic waves. Considering sidewall slope angle of the meta-atom to be 5 degrees due to etching processes, we optimize the design of a rectangular prism shape [41]. We choose the target wavelength to be 632.8 nm. The meta-atoms are made up of hydrogenated amorphous silicon [42], which has low loss in visible region, on top of a glass substrate. To optimize the geometry of the meta-atom to provide the required optical properties of $r_{p}=r_{s}$, we employ particle swarm optimization (PSO) [43,44] and choose five parameters: periodicity along the $p_x$ and $p_y$ axes, the length $a$ and the wight $b$ of top rectangular face and the height $h$ as shown in Fig. 5. PSO iterates through variations in geometric parameters in an effort to identify a configuration that minimizes the fitness function. Simulations are performed using the Finite element method-based simulation software, COMSOL Multiphysics. The Fresnel reflection coefficients for s-polarization and p-polarization of the unit cell structure are extracted from the simulation system and the fitness function, which is defined as $(r_p-r_s)/(r_s+r_p)$, are calculated as figure of merit.
The simulation is repeated 200 times and converges to an optimized design as shown in Fig. 6(a). The optimized geometric parameters are determined as follows: $p_x = 289.67$ nm, $p_y = 251.44$ nm, $h = 200$ nm, $a = 24.29$ nm, and $b = 240$ nm. Commercial electron-beam lithography (EBL) in combination with reactive ion etching (RIE) processes can be employed for the fabrication of the metasurface. This technique is well-established in the field and offers precise control over the nanostructures essential for the functionality of metasurface [45]. We can anticipate that by incorporating the metasurface, the edge detection image can be obtained by substituting the metasurface for the medium section in the experimental setup depicted in Fig. 4. Figures 6(b)-(c) illustrate that the amplitude and phase values for both s-polarization and p-polarization are remarkably close in the incident angle $45^o$, indicating that the required output field and spatial transfer function can be successfully obtained. It should be noted that the metasurface we designed satisfy $r_p = r_s$ only at $\theta _i=45^o$. To better observe the influence of the incident angle on the output light intensity and transfer function, we opted not to utilize the metasurface data for simulation of edge detection. Instead, we envisaged an ideal scenario where $r_p = r_s$ holds at every incident angle.
Fig. 6. The iterative data (a) for optimizing the metasurface. Amplitude (b) and phase (c) of the optimized metasurface.
In addition, to ensure that the Fresnel coefficients $r_{p}=r_{s}$ holds across any angles, two other practical schemes are introduced. The first satisfies this condition at the isotropy-anisotropic interface, where $\varepsilon _{2x} \varepsilon _{2y} = \varepsilon _{2x} \varepsilon _{2z} = \varepsilon _1^2$ [46], with $\varepsilon _1$ denoting the dielectric constant of the isotropic material, and $\varepsilon _{2x}, \varepsilon _{2y}$, and $\varepsilon _{2z}$ representing the dielectric constants of the anisotropic material in different directions. Moreover, the dielectric-metal-dielectric layer system [47], formed by the integration of waveguide effects and plasma phenomena, offers a second method aimed at restricting the transmission of both s-polarization and p-polarization.
4. Conclusion
In summary, we have revealed a polarization-independent edge detection approach based on the spin-orbit interaction of light, and revealed its origins. When the Fresnel coefficients $r_p=r_s$, the in-plane spin separation $\Delta {x}$ along the $k_x$ direction remains constantly at 0, while the transverse spin separation $\Delta {y}$ along the $k_y$ direction becomes polarization independent. This results in both the output light intensity and transfer function being insensitive to incident polarization, exhibiting functional dependencies of $\cot ^2{\theta _{i}}$ and $\cot {\theta _{i}}$ with respect to the incident angle. The origin of this phenomenon can be attributed to the fact that, after undergoing mapping through the Glan polarizer, the polarization component $\tilde {E}_r^{H - H}$ opposite to that of the polarization component $\tilde {E}_r^{V - V}$. Additionally, the summation of components $\tilde {E}_r^{H - V}$ and $\tilde {E}_r^{V - H}$ in the $y$-direction can be simplified to terms independent of incident polarization. Furthermore, we designed a metasurface that achieves the required Fresnel coefficients over a large range of incident angles, proving the feasibility of the system. This polarization-independent edge detection scheme exhibits exceptional robustness to incident polarization, promising to broaden the practical applications of edge detection based on the spin-orbit interaction of light.
Funding
Institute of Information and Communications Technology Planning and Evaluation (IITP) (2019-0-01906); Chung Mong-Koo Foundation (NRF-2022R1A6A3A13066251); China Scholarship Council (202308410546); Ministry of Science and ICT, South Korea (NRF-2019R1A2C3003129, NRF-2019R1A5A8080290, NRF-2022M3C1A3081312, NRF-2022R1C1C2004662); National Natural Science Foundation of China (12074100).
Acknowledgments
P. T. acknowledges the China Scholarship Council (Grant No. 202308410546). Y. K. acknowledges the Hyundai Motor Chung Mong-Koo fellowships and the NRF Ph.D. fellowship (Grant No. NRF-2022R1A6A3A13066251) funded by the Ministry of Education of the Korean government. T. B. acknowledges the NRF grant (NRF-2022R1A6A1A03052954) funded by the Ministry of Education (MOE) of the Korean government, the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korean government (MSIT) (No.2019-0-01906, Artificial Intelligence Graduate School Program (POSTECH)), and the POSTECH PIURI fellowship.
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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