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Abstract
Various optical differential computing devices have been designed, which have advantages of high speed and low power consumption compared with traditional digital computing. In this paper, considering the reflection of a light beam through a three-layer structure composed of glass, metal and air, we propose a designable optical differential operation based on surface plasmon resonance (SPR). When the SPR is excited under certain conditions, the spin-dependent splitting in the photonic spin Hall effect (SHE) changes dramatically. We first prove theoretically that this three-layer structure can realize one-dimensional optical differential operation. By discussing the transverse beam displacement under different conditions, it is found that the designable differential operation with high sensitivity can be realized by slightly adjusting the incident angle and the thickness of metal film. We design the differentiator which can obtain the image of measured target edge in real time and get different edge effects at different times. This will provide more possible applications for autonomous driving and target recognition.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of information technology, the devices that can realize information transmission and processing with large bandwidth, high speed, real time and low power consumption are becoming more and more popular. Compared with the traditional electronic devices for information processing [1], the optical signal as a carrier for information processing has been widely concerned because of its advantages of large bandwidth, fast response speed, low power consumption and little interference [2,3]. The optical computing can be divided into digital computing [4,5] and analog computing [6,7]. However, while providing high speed and reliable operation, the digital signal processor has disadvantages of high power consumption, huge costs on analog-to-digital converters and sharp performance degradation at high frequencies. Therefore, the optical analog computers are more reasonable and economical.
It is well known that optical analog computers can perform some specific and simple tasks, such as spatial differentiation, integration and convolution. The optical differentiator has attracted much attention because of its great role in edge detection and image processing [8–12]. As a result, various differential operation have emerged. For example, some concepts from topological photonics are applied to optical calculations to realize broadband isotropic two-dimensional differentiation [13], and spatial differentiation can also be achieved at a single metal-dielectric interface [14]. In addition, there are some differential operations based on photonic crystals [15], grating nanostructures [16,17] and metasurface [18–21]. However, the process of making a differentiator using metasurface or some specific nanostructures is extremely complex and the production cost is also high. Therefore, it is more desirable to use some simple devices to make the differentiator. When a light beam is reflected through an optical interface, the corresponding beam displacement occurs. For this reason, the differential operation devices based on SHE of light [22–30], Goos-H$\ddot {a}$nchen effect [31–33] and Brewster effect [34,35] were designed. In the past, based on the SPR effect, the Goos-H$\ddot {a}$nchen shift and Imbert-Fedorov shift during light beam reflection [36–40] have been extensively studied, as well as the spin-dependent shift [41–43]. In fact, based on the SPR, it seems feasible to design a differentiator which can not only realize differential operation and edge image processing, but also realize sensitive adjustment of edge thickness. This may provide good prospects for the development of optical image processing in the future.
In this paper, we propose a designable optical differential operation based on surface plasmon resonance. It is assumed that a Gaussian beam is reflected through a three-layer structure composed of glass, metal and air, the spin Hall effect of light will occur during the reflection process. It is worth noting that the SPR occurs when the incident angle approaches resonant angle. At this time, the p-wave Fresnel reflection coefficient will drop sharply, while the s-wave Fresnel reflection coefficient basically does not change. This would lead to a dramatic change for spin-dependent shift in the photonic SHE. Here, the sharp decrease of p-wave reflection coefficient is attributed to the generation of vanishing wave, and the resonance between the vanishing wave and the plasma wave occurs when the vanishing wave meets the plasma wave in the metal film. In our work, firstly, it is theoretically proved that the spatial differentiation operation can be realized after a light beam is totally internally reflected through the three-layer structure. Secondly, by discussing changes of the Fresnel reflection coefficient and transverse beam displacement at different incident angles and thicknesses of metal film, we theoretically predict that the highly sensitive designable differential operation can be realized by slightly adjusting the incident angle and the thickness of metal film. Finally, we get a conclusion.
2. Theoretical analysis
As shown in Fig. 1, we consider a light beam is reflected through a three-layer structure composed of glass, metal and air. Here, the metal is chosen as the Au. The corresponding refractive indexes of the three media are $n_{1}=1.515$, $n_{2}=\sqrt {-10.4+1.4i}$ and $n_{3}=1$, respectively. The $xy$ plane of the experimental Cartesian coordinate system is parallel to both the glass-metal interface and the metal-air 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 Gaussian distribution of the incident light beam can be written as:
Fig. 1. Schematic diagram of the wave reflection at a three-layer structure composed of glass, metal and air, by considering surface plasmon resonance. $x_{i}$, $y_{i}$, $z_{i}$ and $x_{r}$, $y_{r}$, $z_{r}$ denote the coordinates of incident and reflected beams, respectively. $\delta _{+}$ and $\delta _{-}$ represent the transverse beam displacements for left- and right-handed circularly polarized components. $x$, $y$, $z$ denote the Cartesian coordinates of laboratory. $\theta _{i}$ and $\theta _{r}$ are incident and reflected angles.
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 $\alpha$ to the $x$ axis, where $\alpha = \beta _{r}+\frac {\pi }{2}$. After introducing the Jones matrix of the Glan laser polarizer, we can obtain
As shown in Fig. 2(a) and (b), we plot the theoretical intensity distribution of the incident light field and the output light field. The incident light field has a Gaussian distribution and the output light field has a transverse spin separation in the $y$ direction. It is proved that the SHE occurs after the light beam is reflected through three-layer structure, leading to the transverse shift of left- and right-handed circular polarization components in opposite directions. Figure 2(c) and (d) are the intensity distributions of the input and output light fields in the $y$ direction when $x=0$. Obviously, this is consistent with Fig. 2(a) and (b). To illustrate the spatial differentiation, we also measure the spatial spectral transfer function $H(k_{x},k_{y})=E_{out}(k_{x},k_{y})/E_{in}(k_{x},k_{y})$, which is represented by the spatial transformation between the incident and reflected electric fields. We can get $H(k_{x},k_{y}) =\chi \sin {(\Delta {y}k_{y})}$, further according to $|\Delta {y}k_{y}|\ll 1$ , the transfer function can approximate to spatial spectrum
Fig. 3(a) and 3(c) are the theoretical spatial spectral transfer functions, it can be seen that there is a minimum value at $k_{y}/k_{0}=0$. As shown in Fig. 3(b), we find a phase transition at $k_{y}/k_{0}=0$ for the spatial spectral transfer function. In order to clearly observe spatial differentiation, we plot the spatial spectral transfer function when the incident angle is chosen as $30^{\circ }$, $44.16^{\circ }$ and $60^{\circ }$ at $k_{x}/k_{0}=0$, as shown in Fig. 3(d). It is obvious that the transfer function has a minimum of 0 at $k_{y}/k_{0}=0$, and a linear distribution at other positions. Therefore, this further demonstrates that the differentiator which we design can achieve spatial differentiation in the $y$ direction. At the same time, the different slope of the spatial spectral transfer function at different incident angles indicates that the incident angle has a certain influence on the spatial differentiation.Fig. 2. Demonstration of spatial differentiation under Gaussian light irradiation. (a) and (b) Theoretical intensity profiles of incident and reflected beams. (c) and (d) Theoretical intensity of incident and reflected beams in $y$ direction with $x=0$.
Fig. 3. The spatial spectral transfer function on a three-layer structure composed of glass, metal and air. (a) and (c) Theoretical spatial spectral transfer function with the incident angle is chosen as $44.16^{\circ }$. (b) The phase distribution for the spatial spectral transfer function. (d) Theoretical spatial spectral transfer function ($k_{x}/k_{0}=0$) when the incident angle is set to be $30^{\circ }$, $44.16^{\circ }$ and $60^{\circ }$, respectively. Here, the thickness of metal film is chosen as 48 nm.
3. Results and discussion
We have proved theoretically that the spatial differential operation can be realized after the total reflection of light beam through the three-layer structure composed of glass, metal and air. Therefore, we design the differentiator which can also play a role in image processing and edge detection. Due to the presence of metal film, the interesting SPR effect occurs under certain conditions. This will greatly change the value of transverse beam displacement, thus affecting the effect of edge detection. Next, we will discuss and analyze them one by one. According to Eq. (3) and (4), we can obtain the parallel and vertical components of reflected field as follows:
It can be seen that the spin-dependent shift after reflection of H and V polarization components is related to Fresnel reflection coefficient. The Fresnel reflection coefficient of s- and p-waves with three-layer structure can be expressed as:
In order to study the transverse beam displacement (the value of $\Delta {y}$ is affected by Fresnel reflection coefficient) based on three layers structure, we first observe the Fresnel reflection coefficient with the change of the incident angle or the thickness of metal film. As shown in Fig. 4(a), $r_{s}$ changes with the incident angle or the thickness of metal film in two basic trends: one is gradually increasing and finally becoming stable, the other is always basically keeping a stable value. On the contrary, as shown in Fig. 4(b), when the thickness of metal film is fixed at 50 nm, it is found that the $r_{p}$ decreases sharply and then rises rapidly near the incident angle of $44.16^{\circ }$, which will have a great influence on spin-dependent shift. Here, we find a good resonant angle $\theta _{R}=44.16^{\circ }$ where the SPR is excited. However, when the metal thickness is chosen as other values, the obvious resonant effect can not be found. As shown in Fig. 4(c), only when the incident angle is fixed at $45^{\circ }$ has the p-wave reflection coefficient an obvious trend of decreasing and then increasing. However, obvious resonant effect can not be found at other incident angles. As shown in Fig. 4(d), we fixed the incident angle at $44.16^{\circ }$ and found an optimal resonant thickness of about 48 nm.
Fig. 4. The effect of the thickness of metal film and incident angle on Fresnel reflection coefficient. (a) and (b) Fresnel reflection coefficient changes with incident angle when the thickness of metal film is chosen as 1 nm, 10 nm, 50 nm and 90 nm, respectively. (c) Fresnel reflection coefficient varies with the metal thickness when the incident angle is set to be $30^{\circ }$, $45^{\circ }$ and $60^{\circ }$, respectively. (d) Fresnel reflection coefficient changes with the thickness of metal film when the incident angle is $44.16^{\circ }$.
According to Eq. (13) and (14), $Re(r_{s}/r_{p})$ and $Re(r_{p}/r_{s})$ determine the spin-dependent shift of the H and V polarization components, respectively. From the theoretical analysis, we know that the transverse beam displacement $\Delta {y}$ is related to $\delta _{r}^{H}$ and $\delta _{r}^{V}$ (the real parts of $\delta _{r}^{H}$ and $\delta _{r}^{V}$ are closely related to $\delta _{\pm }^{H}$ and $\delta _{\pm }^{V}$, respectively). Therefore, we need to study the impact of the thickness of metal film and incident angle on them. It can be clearly seen from Fig. 5(a) and (b) that $Re(r_{s}/r_{p})$ changes sharply around $d=48$ nm or $\theta _{i}=44.19^{\circ }$ when the incident angle is fixed at $\theta _{R}=44.16$ or the thickness of metal film is fixed at 48 nm, respectively, and has a large negative value at the corresponding position, due to the SPR is excited. The above results can also be clearly seen from Fig. 5(c). This means that the spin-dependent shift $\delta _{\pm }^{H}$ of the H polarization component will be greatly enhanced at $d=48$ nm or $\theta _{i}=44.16^{\circ }$. On the contrary, as is shown in Fig. 5(d), $Re(r_{p}/r_{s})$ cannot achieve large negative or positive values in any case, so that $\delta _{\pm }^{V}$ does not have the same huge enhancement as $\delta _{\pm }^{H}$. Therefore, the SPR effect has a great influence on the spin-dependent shift of the H polarization component. As shown in Fig. 6, it is obvious that the edge of the measured target becomes thicker in the $y$ direction with the increase of transverse beam displacement $\Delta {y}$ (Experimental suggestion: It is indispensable to deposit different thickness metal film samples on glass. In order to achieve a slight change for incident angle in the experiment, a high-precision turntable with an angle setting can be used to place and fix the glass deposited with the metal film. It is hoped that these suggestions will be useful for experimental operations). In order to enable the differentiator, which is designed based on surface plasmon resonance, to achieve sensitive adjustment for measured target edge, we expect that $\Delta {y}$ will have similar changes to $\delta _{\pm }^{H}$ near the resonant angle or the optimal thickness. Therefore, we next performed the analysis from the fixing the optimal metal thickness $d=48$ nm and fixing the relatively good resonant angle $\theta _{i}= 44.16^{\circ }$, respectively.
Fig. 5. The effects of thickness of metal film and incident angle on $Re(r_{s}/r_{p})$ and $Re(r_{p}/r_{s})$. (a) $Re(r_{s}/r_{p})$ changes with incident angle when the thickness of metal film is set to be 1 nm, 10 nm, 50 nm and 90 nm, respectively. (b) $Re(r_{s}/r_{p})$ changes with metal thickness when the incident angle is selected as $30^{\circ }$, $44.16^{\circ }$ and $60^{\circ }$, respectively. (c) and (d) $Re(r_{s}/r_{p})$ or $Re(r_{p}/r_{s})$ varies with the thickness of metal film and incident angle. Here, $44.16^{\circ }$ is a good resonance angle, and 48 nm is an optimal thickness.
Fig. 6. Edge image of the measured target under different beam transverse displacements predicted by theory. (a) and (e) The original image of the measured target. (b) - (d) and (f) - (h) Edge image of the measured target “a” in the $y$ direction when the transverse beam displacement $\Delta {y}$ is chosen as 1346 nm, 3372 nm and 6065 nm, respectively. (i) - (k) Correspond to the horizontal intensity distributions of images (f) - (h), respectively, at the white dotted line. Here, $\beta _{i}=0^{\circ }$ and $d=48$ nm. The incident angle is chosen as $44.13^{\circ }$, $44.17^{\circ }$ and $44.19^{\circ }$, respectively.
When the fixed incident angle is $44.16^{\circ }$, as shown in Fig. 7(a), the spin-dependent shift of H polarization is greatly enhanced near $d$=48nm. As the transverse beam displacement $\Delta {y}$ is also related to incident polarization $\beta _{i}$, we select different incident polarization angles to observe the changes of the transverse beam displacement $\Delta {y}$ in Fig. 7(b). When the incident polarization is chosen as $0^{\circ }$, the changes of $\Delta {y}$ and $\delta _{+}^{H}$ are basically the same. At this time, the edge of the target detection object can be sensitively adjusted by slightly changing the thickness of metal film. However, with the increase of the incident polarization angle, $\Delta {y}$ cannot get a large value, and the change is gentle, by this time the sensitivity of the differentiator may be low. When the thickness of the fixed metal film is 48 nm, as shown in Fig. 7(c), the spin-dependent displacement of H polarization is greatly enhanced around the resonant angle $44.19^{\circ }$ as predicted. Similarly, as shown in Fig. 7(d), when the incident polarization is $0^{\circ }$, the change of $\Delta {y}$ is basically the same as $\delta _{+}^{H}$. By slightly adjusting the incident angle near the resonant angle $\theta _{i}=44.19^{\circ }$, the edge of the target detection object can be sensitively adjusted. With the increase of the incident polarization angle, The change of $\Delta {y}$ near the resonant angle is no longer as dramatic as that at the incident polarization angle of $0^{\circ }$, and the peak gradually decreases. However, we can also adjust the thickness of the measured target edge by adjusting the value of the incident angle, but the sensitivity is reduced. In particular, the designable differentiators with different sensitivities can be obtained when we select different incident polarization angles.
Fig. 7. The spin-dependent displacement of H polarization and the transverse beam displacement $\Delta {y}$ vary with the thickness of metal film or incident angle. (a) The spin-dependent displacement of H polarization varies with incident angle at the optimal thickness $d=48$ nm. (b) The transverse beam displacement changes with incident angle when the incident polarization angle is set to be $0^{\circ }$, $10^{\circ }$, $20^{\circ }$ and $30^{\circ }$, respectively. (c) The spin-dependent displacement of H polarization varies with thickness of the metal film at the incident angle $\theta _{R}=44.16^{\circ }$. (d) The transverse beam displacement varies with thickness of metal film when the incident polarization angle is selected as $0^{\circ }$, $10^{\circ }$, $20^{\circ }$ and $30^{\circ }$, respectively.
4. Conclusion
In summary, based on the three-layer structure composed of glass, air and metal, we propose a designable optical differentiator based on surface plasmon resonance. Firstly, we prove theoretically that we design the device which can perform one-dimensional differential operation. Secondly, we analyze the variation of Fresnel reflection coefficient, so as to find a relatively good resonant angle and an optimal thickness of metal film. It is found that the transverse beam displacement is greatly enhanced when the surface plasmon resonance is excited. Therefore, we can achieve designable differential operation by slightly adjusting the incident angle or changing the thickness of metal film, which get the image with different edge thickness of the measured target. It is worth noting that we only need to adjust slightly near the resonant angle or optimal thickness, so this differentiator will likely have very high sensitivity. In particular, we can obtain designable optical differentiators with different sensitivities by changing the incident polarization when the metal film is fixed to the optimal thickness. Finally, the designable differentiator with high sensitivity and changeable sensitivity can be applied to object edge detection and realize edge tunability. This will provide more possible applications in autonomous driving and target detection.
Funding
National Natural Science Foundation of China (U1731238, 12273008); The Foundation of Science and Technology Program of Guizhou Province ([2016]4008, [2017]5726-37, [2018]5769-02); The Foundation of Department of Education of Guizhou Province (KY (2020) 003).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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