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Abstract
Optical differential operation and image edge detection have shown some unique advantages, including faster operating speeds and lower power consumption, compared to a digital image processor. Here, we examine the origin of optical differentiator in the photonic spin Hall effect. We find that the optical differentiator in x direction is related to the beam divergence when the incident angle of the central wave vector meets Brewster's, while the optical differentiator in y direction is related to the cross polarization component. By eliminating the H-H or V-V polarization component, the H-V or V-H polarization component induced the spin-dependent splitting in the photonic spin Hall effect, which can be used for the one-dimensional optical differentiator in an arbitrary direction. Furthermore, we investigate the photonic spin Hall effect and double-peak profile near Brewster angle and predict that a perfect two-dimensional optical differentiator will become a reality.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The photonic spin Hall effect (PSHE), manifesting itself as spin-dependent splitting of light, is considered as a result of the spin-orbit interaction of light when the spectrum components experience different rotations in order to satisfy the transversality in the reflected or refracted interface [1–6]. As a fundamental physical effect in light-matter interaction, the PSHE plays an important role in different physical systems, such as high-energy physics [7,8], plasmonics [9,10], metamaterials [11–13], and Weyl semimetal [14–16]. Specifically, the PSHE has been employed for precision metrology benefiting from its high sensitivity to changes in physical parameters of the optical interface. The physical effects or parameters at liquid molecules [17,18], nanostructures [19–21], and two-dimensional atomic crystals [22,23] have been precise measurement. More recently, the image processing of edge detection in PSHE has been reported and attracted tremendous attentions because of its unique advantages including faster operating speeds and lower power consumption compared to digital image processor [24,25]. The spin-splitting in PSHE has been used to control and manipulate the optical differentiator in image edge detection, which is generally on subwavelength scales and can be significantly enhanced by the so-called quantum weak measurement technique.
Optical edge detection is to extract meaningful information and preserve important geometric features with obvious brightness changes in images, which can greatly reduce the amount of data and significantly improve the efficiency and accuracy of system [26,27]. The spatial differentiator is a core device in optical edge detection and enables realtime and continuous processing of edge detection from an entire image. In recent years, many methods for analog spatial differentiation in optical edge detection have been investigated, for example, deliberately designed layered structure [28], surface plasmon [29,30], the grating nanostruces [31,32], and optical metamaterials or metasurfaces [33–35]. However, layered structure and surface plasmon depend on complex layered structures or critical plasmonic coupling condition and can only achieve partial differentiation. The fabrication processes of Bragg grating and metamaterials are very complicated or expensive, and also limit the resolution of edge detection. The PSHE can be a good substitution of spatial differentiation and overcome those disadvantages above. Unfortunately, the two-dimensional optical differentiator in PSHE has not been achieved, the physical mechanism and adjustable resolution of the optical differentiator still require more research.
In this work, we examine the origin of optical differentiator in PSHE. As shown in Fig. 1, we find that the cross polarization components, when a H or V linearly polarized light reflects at the air-glass interface, act as the optical differentiator in y direction, which can carry the edge information of the image in optical path. As the H and V-induced cross polarization components have opposite differential direction, we can theoretically realize a one-dimensional optical differentiator in arbitrary direction by modulating the incident polarization state and the incident angle. However, a part of edge information will inevitably lost along with eliminating the influence of H or V polarization component. It is interesting that the beam divergence, when the incident angle of the central wave vector meets Brewster's law, plays the role of optical differentiator in x direction. By introducing a phase factor $\pi /2$ between the optical differentiators in x and y directions and matching the polarization angle $\alpha $ and incident angle ${\theta _i}$, the phase singularity will appear in arbitrary direction along the coordinate origin and lead to a perfect two-dimensional optical differentiator.
Fig. 1. Schematic diagram of the optical differentiators in photonic spin Hall effect. The H and V represent the incident wavepacket with H and V polarization, respectively. The H-H and H-V denote the reflected wavepacket with H and V polarization when the H polarization reflects at an air-glass interface, while the V-V and V-H denote the reflected wavepacket when the V polarization reflects at an air-glass interface. The H-H component acts as optical differentiator in x direction near the Brewster angle. The H-V and V-H components play the role of optical differentiator in y direction with opposite sign. ${\theta _i}$ and ${\theta _r}$ are the incident and reflected angles.
2. Theoretical analysis
We first theoretically analyze a light beam reflection at the air-glass interface. The $x\textrm{y}$-plane of the laboratory Cartesian frame ($x,y,z$) is parallel to the air-glass interface. We use the coordinate frames (${x_i},{y_i},{z_i}$) and (${x_r},{y_r},{z_r}$) to denote incident and reflected beams, respectively. The ${z_{i,r}}$ axis attaches to the direction of the central wave vector. We assume that the wavepacket with $|\textrm{H}\rangle $ or $|\textrm{V}\rangle $ polarization reflects at the air-glass interface. The corresponding individual wave-vector components of $|\textrm{H}({\textrm{k}_{i,r}})\rangle $ and $|\textrm{V}({\textrm{k}_{i,r}})\rangle $ can be expressed by $|\textrm{P}({\textrm{k}_{i,r}})\rangle $ and $|\textrm{S}({\textrm{k}_{i,r}})\rangle $ [20]
In order to examine the actual significance of the four polarization components in image processing of edge detection, we need obtain the complex amplitude ${E_i}$ and ${E_r}$ in position space. By making use of Taylor series expansion based on the arbitrary angular spectrum component ${k_x}$, ${r_{p,s}}$ can be expanded as
The expansion of the Fresnel coefficient induces the Goos-Hänchen effect [41,42] and double-peak profile near Brewster angle [43]. We consider the incident beam with a Gaussian distribution and its angular spectrum can be written as
Here, the ${r_0} = {k_i}w_0^2/2$ represents the Rayleigh length. Similarly, the four polarization components of the reflected filed in position space are given by
Here, $\rho = \frac{{\partial {r_p}}}{{\partial {\theta _i}}}$ and $\chi = \frac{{\partial {r_s}}}{{\partial {\theta _i}}}$. The incident and reflected fields are expressed in their own coordinate systems, respectively. As shown in Fig. 2, the spin-splitting in PSHE is actually the superposition of the cross-polarization component. Specifically, the incident H polarization, reflected at an air-glass interface, induces a H polarization component (H-H) and its cross polarization component (H-V). Similarly, the incident V polarization results in its V polarization component (V-V) and H polarization component (V-H). The power-weight of H-V (V-H) is much smaller than the H-H (V-V), leading to the extremely small weighted shifts in PSHE. The only exception is in the Brewster angle where the intensity distributions are balanced between the H-H and H-V, resulting in a huge double-peak profile as shown in Fig. 3. By eliminating the influence of H-H or V-V component, the spin-dependent splitting in PSHE can be enhanced and observed.
Fig. 2. The intensity distributions of the four polarization components in position space when the incident angle ${\theta _i}$ is far away from the Brewster angle ${\theta _B}$, where the refractive index of the glass is 1.515, the value of the ${\theta _B} = {56.57^\circ }$, the wavelength $\lambda = 632nm$. (a) and (b) represent the intensity distributions of the H-H and H-V, respectively. (d) and (e) represent the intensity distributions of the V-V and V-H. (c) and (f) show the cross-polarization intensity ratio in the case of H and V input polarization, respectively.
Fig. 3. The intensity distributions of the four polarization components of the reflected filed in position space when the incident angle ${\theta _i}$ is at the Brewster angle ${\theta _B}$. (a) and (b) represent the intensity distributions of the H-H and H-V, respectively. (d) and (e) denote the intensity distributions of the V-V and V-H. (c) and (f) show the total intensity distributions of the reflected filed in the case of H and V input polarization, respectively.
3. Results and discussion
In this section, we will use the theoretical analysis to study the origin of optical differentiators in x and y directions. Our study will mainly focus on how to realize the one-dimensional optical differential operation in arbitrary direction. The previous researches show that the Goos-Hänchen shifts and spin-splitting in PSHE enabled optical differential operation and image edge detection [24,25,42]. Therefore, the properties of optical differentiators in x and y directions on far away from the Brewster angle or at the Brewster angle are taken into account. In addition, we also briefly discuss and predict that a perfect two-dimensional optical differentiator will become a reality by introducing a phase factor in y direction.
Now, we consider the optical differentiators in x and y directions. As the $\rho $ and $\chi $ are the crucial small quantity, compared to the ${r_p}$ and ${r_s}$ when the incident angle is far away from the Brewster angle, we can simplify Eqs. (13), (14), and (15) as follows:
The intensity distributions of four polarization components given above are shown in Fig. 2. It is apparent that the H-V and V-H components act as the optical differentiator in y direction with the opposite differential direction, while the H-H and V-V components do not work in x direction. It is worth to note here that the cross-polarization H-V (V-H) will be completely covered up by the H-H (V-V) component because of its low power-weight as shown in Fig. 2(c) and (f). When a target image is processed by this differentiator in y direction, the output edge image of the target in y direction will be leaved out by eliminating the H-H (V-V) component. Meanwhile, the $E_r^{H - V}(x,y,z)$ and $E_r^{V - H}(x,y,z)$ are approximately proportional to the first-order spatial differentiation of the input ${E_i}(x,y,z)$ as the $\rho $ and $\chi $ are much smaller than the ${r_p}$ and ${r_s}$.
When the incident angle is at the Brewster angle with ${r_p}\textrm{ = }0$, the H-V component is no longer a small quantity compared to the H-H component, we can simplify Eq. (13) as follows:
Equation (19) shows that the H-H component plays the role of optical differentiator in x direction when the incident angle ${\theta _i} = {\theta _B}$. As shown in Fig. 3, the optical differentiator in x direction is due to the beam divergence when the incident angle of the central wave vector meets Brewster's law, resulting in that the intensity distribution of the H-H component approximately linearly increases along with the intensity distribution of the input H polarization increased. Similarly, there exists a hidden optical differentiator in the V-V component with the factor $\frac{{i\chi }}{{{k_i}}}\frac{{\partial {E_i}(x,y,z)}}{{\partial x}}$, completely covered by the factor $\frac{{{r_s}({r_0} + iz)}}{{{k_i}x}}\frac{{\partial {E_i}(x,y,z)}}{{\partial x}}$. It is interesting that the process of achieving the one-dimension edge detection at any desirable direction is just through leaving out the hidden optical differentiator in x direction.
To examine the spatial differentiation effect, the position space transfer functions of the optical differentiators in x and y directions near the Brewster angle are shown in Fig. 4. The spatial transform between the incident and reflected electric fields in position space is determined by a spatial transfer function ${H_{p,s}}(x,y) = E_r^{H - H,H - V}(x,y)/{E_i}(x,y)$ with $z = 250mm$ [44]. Figures 4(a) and (c) show that the ${H_p}(x,y)$ is strictly linear distribution along with x axis while the ${H_s}(x,y)$ is approximately linear distribution along with y axis, which leads to a tiny distortion of the edge image information in y direction. Figures 4(b) and (d) indicate that the optical differentiation effect is due to the phase gradient near the coordinate origin.
Fig. 4. The position space transfer function of the optical differentiator in x and $\textrm{y}$ directions near the Brewster angle. (a) and (c) denote the intensity distributions of the position space transfer function in x and y directions, respectively. (b) and (d) show the phase distributions of the position space transfer function in x and y directions.
As we all know, the polarized light carries some meaningful image information or features of an object after illuminating it. When the image information is processed by an optical differentiator, the left- and right-handed photons with the opposite spin angular momentum carry the same image information with a tiny shift at the image plane. The overlap of image plane can be eliminated by a Glan laser polarizer, leaving out only the edge information available for detection. The differential direction of optical differentiator and tiny shift at the image plane determine how sharp the edge can be resolved.
We next consider the one-dimensional optical differentiator in arbitrary direction. To understand this spatial differentiation effect, we employ the input photons with arbitrary linear polarization state
$\alpha $ is the incident polarization angle. After reflection, the polarization states in position space evolve asAccording to the analysis above, we need leave out the optical differentiators in x and y directions. In order to eliminate the factor $\frac{{{r_p}({r_0} + iz)\cos \alpha }}{{{k_i}x}}$ and $\frac{{{r_s}({r_0} + iz)\sin \alpha }}{{{k_i}x}}$ in H-H and V-V components, we use a Glan laser polarizer whose polarization axis is set to be a $\gamma $ angle with the x axis. When the $\gamma = \beta + \frac{\pi }{2}$ with $\beta = \arctan (\frac{{{r_s}\sin \alpha }}{{{r_p}\cos \alpha }})$, the final field in the whole differentiator system evolve as
Equation (25) shows the mathematical relationship between the output wave function and the input wave function in the whole differentiator system. The differential direction can be modulated by the incident polarization angle and the incident angle. The phase distribution of transfer function in the whole differentiator system, $H(x,y) = {E_{out}}(x,y)/{E_i}(x,y)$, is shown in Fig. 5. (a) illustrates how to leave out the optical differentiator we desired. (e), (b), (c) and (d) reveal that there exists distinct spatial differentiation rotation at arbitrary incident angle, which appears a clockwise rotation as the polarization angle increased. (e)-(f) display that the differential direction will rotate clockwise along with the incident angle increased and reverse near the Brewster angle. In general, the edge information of an image parallel to the spatial differential direction will be detected, the edge information of an image perpendicular to the spatial differential direction will disappear. In order to achieve high resolution in the image processing of edge detection, the spatial differential direction must be parallel to the edge information as much as possible. The same resolution can be achieved by rotating the object with a desired direction, which is verified in Ref. [42].
Fig. 5. Schematic of the phase distributions of the position space transfer function in the final field changing with the polarization angle $\alpha $ and the incident angle . (a) represents that the factor $\frac{{{r_p}({r_0} + iz)\cos \alpha }}{{{k_i}x}}$ and $\frac{{{r_s}({r_0} + iz)\sin \alpha }}{{{k_i}x}}$ in H-H and V-V components are eliminated. (e), (b), (c) and (d) are related to the phase distributions with the ${\theta _i} = {50^\circ }$ and $\alpha = {10^\circ },{20^\circ },{30^\circ },{40^\circ }$, respectively. (e)-(h) denote the ${\theta _i}$ phase distributions with the $\alpha = {10^\circ }$ and ${\theta _i} = {50^\circ },{55^\circ },{57^\circ },{60^\circ }$.
In addition, we also discuss the case of a perfect two-dimensional optical differentiator briefly. We have know that the differentiator in PSHE can perform differential operations in x and y directions simultaneously, the differential operation in x direction is related to the phase gradient near the x axis and the differential operation in y axis is due to the phase gradient near the y axis. As the phase of the differentiators in x and y directions remain synchronized, the one-dimensional optical differentiator in arbitrary direction is the combined direction of these two fundamental vectors. Hence, if we introduce a phase factor $\pi /2$ between x and y directions, the phase gradient will appear in radial direction along the coordinate origin and lead to a two-dimensional optical differentiator. The final field in the whole differentiator system can be written as follows:
The phase distribution of transfer function in the two-dimensional optical differentiator is shown in Fig. 6. The radial phase gradient along the coordinate origin can be adjusted by $\alpha $ and ${\theta _i}$. In order to realise the perfect two-dimensional optical differentiator near the coordinate origin, the phase gradient near the x and $iy$ axis need to be consistent. Namely, the phase gradient in arbitrary direction along the coordinate origin must be equal, otherwise the one will be disturbed by the other. As illustrated in Fig. 6, (a)-(c) show that the perfect two-dimensional optical differentiator will appear near $\alpha = {60^\circ }$ when ${\theta _i} = {\theta _B}$, (d)-(f) reveal that the optimum polarization angle $\alpha $ will increase along with ${\theta _i}$ increased.
Fig. 6. Schematic of the phase distributions of transfer function in the two-dimensional optical differentiator changing with the polarization angle $\alpha .$ (a)-(c) denote that the phase distributions change with the ${\theta _i} = {56.57^\circ }$ and $\alpha = {40^\circ },{60^\circ },{80^\circ }$, respectively. (d)-(f) represent that the phase distributions change with the ${\theta _i} = {60^\circ }$ and $\alpha = {40^\circ },{60^\circ },{80^\circ }$.
By balancing the power weight of optical differentiators between x and $iy$ axis, we can seek out the perfect two-dimensional optical differentiator. As shown in Fig. 7, the perfect two-dimensional optical differentiator will appear for $\alpha = {67^\circ }$, ${\theta _i} = {56.57^\circ }$ or $\alpha = {67.7^\circ }$, ${\theta _i} = {60^\circ }$. In fact, there exist three problems that restrict the realization of the two-dimensional optical differentiator: eliminating the interference factor $\frac{{{r_p}({r_0} + iz)\cos \alpha }}{{{k_i}x}}$ and $\frac{{{r_s}({r_0} + iz)\sin \alpha }}{{{k_i}x}}$ in H-H and V-V components, introducing a phase factor $\pi /2$ in the optical differentiators between x and y directions, keeping power weight of the optical differentiators in x and y directions to be consistent. The incident elliptically polarized state, incident angle and Glan laser polarizer can exactly solve those problems.
Fig. 7. Schematic of the intensity distributions of transfer function in the two-dimensional optical differentiator changing with the polarization angle $\alpha $. (a)-(d) denote that the intensity distributions change with the ${\theta _i} = {56.57^\circ }$ and $\alpha = {40^\circ },{60^\circ },{67^\circ },{80^\circ }$, respectively. (e)-(f) represent that the intensity distributions change with the ${\theta _i} = {60^\circ }$ and $\alpha = {40^\circ },{60^\circ },{67.7^\circ },{80^\circ }$.
4. Conclusion
In conclusion, we have revealed the origin of optical differentiator in PSHE. We find that the H-V and V-H components in reflected filed act as the optical differentiator in y direction with the opposite differential direction, while the H-H component plays the role of optical differentiator in x direction when the incident angle ${\theta _i} = {\theta _B}$ with ${r_p} = 0$. Similarly, a hidden x direction optical differentiator in the V-V component will appear when the ${r_s} = 0$. As its low power-weight in the whole reflected field, those optical differentiator will be completely covered up by the interference factor $\frac{{{r_p}({r_0} + iz)}}{{{k_i}x}}$ and $\frac{{{r_s}({r_0} + iz)}}{{{k_i}x}}$. The one-dimensional optical differentiator in arbitrary direction can be achieved by eliminating the interference factor $\frac{{{r_p}({r_0} + iz)\cos \alpha }}{{{k_i}x}}$ and $\frac{{{r_s}({r_0} + iz)\sin \alpha }}{{{k_i}x}}$ in H-H and V-V components. Meanwhile, introducing a phase factor $\pi /2$ in the optical differentiators between x and y directions and keeping them consistent, we can extend the one-dimensional optical differentiator to two-dimensional optical differentiator. The investigations of optical differentiators in PSHE may provide insights into the fundamental properties of optical edge detection, and find important applications in image processing, high-contrast microscopy, and augmented reality.
Funding
Education Department of Hunan Province (19A388).
Disclosures
The authors declare no conflicts of interest.
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