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Abstract
To achieve optical differential operation based on the cross-polarization effect at the optical interface, one just needs an optical interface composed of two uniform media with different refractive indices. When certain conditions are satisfied, the reflection co-efficient of the light field at the interface conforms to the form of the spatial spectrum transfer function required by the spatial differentiation, the spatial analog operation can be achieved with a single interface. In this paper, based on the optical differentiation of Brewster effect, we propose a tunable optical differentiation based on the cross-polarization effect at the optical interface. We theoretically derive the tunable optical differentiation and then conduct an experiment to demonstrate theoretical results. It is found that the differentiator can achieve the tunable optical differentiation by adjusting the polarization of output beam. While getting the clear edge of the object, we can also observe the imaging of the middle part to different degrees, which realizes the multi-degree of freedom imaging for the measured target. This provides a potential way to develop devices more suitable for microscopic imaging and target detection.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As the fast development of information storage, processing and transmission, there is an increasing requirement for the internal performance of information processing. While the development of digital analog computer is gradually becoming mature, it also has some obvious disadvantages, such as high power consumption and slow running speed. In order to solve the problems faced by digital computers, the optical analog computer has attracted extensive attention and studies, due to its advantages of superspeed, large bandwidth and low loss [1–3].
In recent years, the field of optical analog computer based on optical differential computing devices has made great progress. Many kinds of methods of optical analog signal operation have been proposed successively. In principle, the existing optical analog differential operation is divided into four categories: the integrated 4F system [4,5], optimizing the spatial point spread function [6,7], spatial pattern coupling interference [8–12] and reflection or refraction at the optical interface [13–19]. The optical differential operation based on the integrated 4F system consists of two Fourier transformation and a metasurface. The design of metasurface [20] is complicated and the production cost is high. Therefore, it is difficult to fabricate experimental devices. The size of the optical differential device based on the optimized spatial point spread function is smaller than that of the 4F system. But it requires the design of complex multi-layer metamaterials and the production cost is high, thus it is difficult to meet the needs of the rapid development of modern information technology. It is not necessary for the spatial differential operation [21–23] based on spatial pattern coupling interference to design complex multi-layer metamaterials. However, the devices designed based on this method usually have limited spatial bandwidth due to their dependence on resonance. Compared with the other three optical differential operation device, the optical differential operation device based on the reflection or refraction at the optical interface require neither resonant structures nor complex metasurface or multilayer metamaterials. It can be proved that optical differential operation can be realized with a single optical interface. Typical examples are the optical differential operation devices based on Brewster angle [24] and that on photonic spin Hall effect [25,26]. Both devices are capable of edge detection. On this basis, it is also meaningful to study the realization of tunable optical differentiation operation with a single optical interface.
In monochromatic polarized light, the feasibility of two-dimensional optical differentiation [27] and image edge detection based on the Brewster effect has previously been experimentally demonstrated. In this paper, aiming at the optical differential operation device based on the Brewster effect [28–31], we propose a tunable optical differential operation based on the cross-polarization effect at the optical interface. The cross-polarization effect is an interesting phenomenon [32–34]. It can be expressed as when the light beam is reflected at the dielectric interface, the polarization component orthogonal to the polarization direction of the incident light is found in the reflected light. The differentiator is made up of two polarizers, a quarter-wave plate and a prism. The Gaussian beams through a quarter-wave plate becomes elliptical polarized light, and then incident at the Brewster angle and reflect off the air-glass interface. Due to the spin-orbit interaction of light [35], the unique properties of Brewster effect and cross-polarization effect, we get a clear vortex beam output through corresponding adjustment. When the sample is placed, a clear edge detection of the two-dimensional image is obtained. In our work, one can get the output images of different visibility mainly by slight change of the polarization of the output beam. In fact, we can not only achieve two-dimensional optical edge detection, but also observe the object more completely through regulation. Therefore, this differentiator that we designed can achieve multi-degree of freedom imaging.
2. Theoretical analysis
In order to achieve tunable optical differential operation, the schematic diagram is shown in Fig. 1(a). we input a complete “Object”, its complete two-dimensional edge can be detected, and a complete “Object” edge image is obtained. On this basis, we slightly change the polarization state of the output beam, then we will get different output image. Theoretically, if we input a Gaussian beam as shown in Fig. 1(b) or a complete image of an object as shown in Fig. 1(e), the optical differentiator will output a standard vortex light as shown in Fig. 1(d) or a clear and complete edge image of an object as shown in Fig. 1(g). Based on this, we slightly change the polarization state of the output beam, and then we will get the intensity distribution diagram of the output light field as shown in Fig. 1(c) or the image of the object as shown in Fig. 1(f). As can be seen Fig. 1(c), it is no longer a standard vortex. In addition, in Fig. 1(f), one can clearly see the image of the object, but its edge becomes unclear. If we continue to slightly change the polarization state of the output beam, the intensity distribution of the output light field will tend to become the input Gaussian beam. At the same time, except the edge of the output image , the other parts are gradually appearing. Finally, it will tend to become the complete image of the input object. Therefore, one can design the tunable optical differentiator which can realize multi-degree of freedom imaging of the object.
Fig. 1. (a) Schematic diagram of tunable optical differential operation based on the cross-polarization effect at the optical interface. The red ellipses indicate that the incident light is elliptically polarized, and the red arrows indicate that the polarization axis of related polarizers, when this polarization axis is $0^{\circ }$ with the $x$ axis, a complete image of the object can be input to get a clear edge image. (b) The theoretical intensity profiles of the incident beam. (c) Theoretical intensity profiles of the output light field when the polarization of the output beam is slightly changed. (d) The intensity profiles of the output standard vortex light. (e) The complete image of the input object. (f) The output image when the polarization of the output beam is slightly changed. (g) The clear edge of the output object.
Next, we will carry out the theoretical derivation of the paper. Firstly, we consider a Gaussian beam incident along $z$ axis, and the angular spectrum of the incident Gaussian beam in the momentum space is expressed as
Then, the light beam passes through a quarter-wave plate (QWP) with the optical axis at an angle of $0^{\circ }$ to the $x$ axis , so the Jones matrix for the QWP can be specified by the following expression:
Next, the beam is incident at Brewster angle $\theta _{i}$, and reflected at the air-glass interface. The reflection matrix can be written as [24]
Here, $k_{0}$ is the wave vector in vacuum, and its values is $\frac {2\pi }{\lambda }$. $r _{p}$ and $r _{s}$ are the Fresnel reflection coefficients of $p$ and $s$ waves, respectively. Their values are related to the incident angle. The expressions can be described as
Therefore, the angular spectrum of the beam reflected through the air-glass interface is obtained as
Here, $\tilde {\boldsymbol {E}}_{r}=\tilde {E}_{r}^{H}\boldsymbol {e}_{H}+\tilde {E}_{r}^{V}\boldsymbol {e}_{V}$, $\tilde {E}_{r}^{H}$ is the component of $\tilde {\boldsymbol {E}}_{r}$ on the $x$ direction , and $\tilde {E}_{r}^{V}$ is the component of $\tilde {\boldsymbol {E}}_{r}$ on the $y$ direction . Hence, the components of the reflection angle spectrum are as follows
In the paraxial approximation, according to the Taylor series expansion, $r_{p}$ and $r_{s}$ can be expanded as follows:
Next, we substitute them into Eqs. (8) and (9) to obtain
Here, $\delta _{H}=\frac {(r_{p}+r_{s})\cot {\theta _{i}}}{k_{0}r_{p}}$, $\delta _{V}=\frac {(r_{p}+r_{s})\cot {\theta _{i}}}{k_{0}r_{s}}$, $\triangle _{H}=\frac {\partial {\ln {r_{p}}}}{k_{0}\partial {\theta _{i}}}$, and $\triangle _{V}=\frac {\partial {\ln {r_{s}}}}{k_{0}\partial {\theta _{i}}}$. We substitute $\boldsymbol {e}_{H}=\frac {1}{\sqrt {2}}(\boldsymbol {e}_{-}+\boldsymbol {e}_{+})$ , $\boldsymbol {e}_{V}=\frac {i}{\sqrt {2}}(\boldsymbol {e}_{-}-\boldsymbol {e}_{+})$, Eqs. (12) and (13) into $\tilde {\boldsymbol {E}}_{r}=\tilde {E}_{r}^{H}\boldsymbol {e}_{H}+\tilde {E}_{r}^{V}\boldsymbol {e}_{V}$, so the reflected field in the momentum space can be expressed as
In the expression above, the values of $x$ and $y$ are set as follows:
Here, we introduce the approximation: $1+\Delta {x}k_{x}+\Delta {y}k_{y}\approx \exp {(\Delta {x}k_{x}+\Delta {y}k_{y})}$, then substituting $\boldsymbol {e}_{+}=\frac {1}{\sqrt {2}}(\boldsymbol {e}_{H}+i \boldsymbol {e}_{v})$ and $\boldsymbol {e}_{-}=\frac {1}{\sqrt {2}}(\boldsymbol {e}_{H}-i \boldsymbol {e}_{V})$ into Eq. (14), we get the reflected field in the momentum space:
With Fourier transform of the reflected light field in the momentum space, as follows:
The reflected light field for the position space can be written as
Then, the reflected wave passes through a Glan laser polarizer (GLP2) with the polarization axis at an angle of $\beta$ to the $x$ axis. If $\Delta x$ and $\Delta y$ are small enough, and the reflected field passes through GLP2 with the polarization angle $\beta$ chosen as $0^{\circ }$, the output field in the whole differentiator system can be acquired as
Therefore, the output field $E_{out}(x,y)$ is the two-dimensional optical differential of the incident field $E_{in}(x,y)$, which can detect the two-dimensional edge of the target effectively in real time.
On this basis, we fine-tune the polarization angle $\beta$ of GLP2. And the value of $\beta$ is negative when we adjust the polarization axis to the left, on the contrary, the value of $\beta$ is positive. At this point, the final electrical field through the whole differentiator system can be obtained as
This expression is no longer in differential form, at this time, the standard edge image cannot be obtained by this system. With the change of the polarization angle of GLP2, the intensity distribution of the output light field also changes.
As shown in Fig. 2, the theoretical intensity profiles of output light field changes with the polarization angle when the polarization angle of GLP2 is adjusted to the left or right. Figures 2(a)–2(d) are the theoretical intensity profiles of the output light field when the polarization angle of GLP2 is adjusted to the left at $0^{\circ }$, $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively. Figures 2(e)–2(h) are the theoretical intensity profiles of the output light field when the polarization angle of GLP2 is adjusted to the right at $0^{\circ }$, $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively. As can be seen from Fig. 2, with the increase of the polarization angle of GLP2, the intensity distribution of the output light field tends to change from the standard vortex light to the Gaussian light, no matter it is fine-tuned to the left or the right. Figures 2(a) and 2(f) are standard vortex beams, and the final electric field obtained by the corresponding device is shown in Eq. (20). Hence, the corresponding experimental device can realize standard two-dimensional differential operation, that is, it can realize real-time edge detection of the target (This has also been proved in Ref. [24]). However, when the polarization angle of GLP2 is fine-tuned to the left or right, the output light field is no longer a standard vortex light. At this point, the final electric field obtained by the corresponding device is shown in Eq. (21). So we think that the strict two-dimensional differential operation cannot be realized, but real-time imaging of the detected target object can still be carried out. As can be seen from Fig. 2, as the polarization angle of GLP2 changes, the output image obtained by the corresponding device will be different. Because the output beam still exists vortex, the object image obtained by the device should also highlight the edge to varying degrees. In particular, when the output beam changes from vortex light to Gaussian light, we get a complete image of the object. Therefore, our method can realize multi-degree of freedom imaging of the object.
Fig. 2. Theoretical intensity profiles of the reflected light field with left or right fine-tuning of the polarization angle $\beta$ of GLP2. (a) and (e) are the theoretical intensity profiles of the output light field when the polarization angle $\beta =0^{\circ }$. (b)-(d) are the theoretical intensity profiles of the output light field when the polarization angle of GLP2 is adjusted to the left at $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively. (f) - (h) These are the theoretical intensity profiles of the output light field when the polarization angle of GLP2 is adjusted to the right at $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively.
As shown in Fig. 3, when $y=0$, the output intensity in the $x$ direction changes with the polarization angle of GLP2. Figures. 3(a)–3(d) are the output intensity in the $x$ direction ($y=0$) when the polarization angle of GLP2 is adjusted to the left at $0^{\circ }$, $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively. Figures 3(e)–3(h) are the output intensity in the $x$ direction ($y=0$) when the polarization angle of GLP2 is adjusted to the right at $0^{\circ }$, $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively. It can be clearly observed that when the polarization angle of GLP2 is fine-tuned to the left, the right peak of the output intensity in the $x$ direction ($y=0$) gradually decreases, while the left peak gradually increases. When the adjusted angle is close to a certain value, the right wave peak will be reduced to zero, leaving only the left peak. Similarly, by fine-tuning the polarization angle of GLP2 to the right, the light peak of the output intensity in the $x$ direction ($y=0$) gradually decreases, while the right peak gradually increases. When the adjusted angle is close to a certain value, the left peak decreases to zero, leaving only the right peak. Specifically, the output beam tends to change from a vortex beam to a Gaussian beam, which is mutually verified with the results in Fig. 2. Therefore, in the process of GLP2 polarization angle change, we can achieve multi-degree of freedom imaging of the measured target.
Fig. 3. The influence of the output intensity in the $x$ direction ($y=0$) by adjust the polarization angle $\beta$ of GLP2. (a) and (e) These are the output intensity in the $x$ direction ($y=0$) of the polarization angle $\beta$ = $0^{\circ }$. (b) - (d) These are the intensity in the $x$ direction ($y=0$) when the polarization angle of GLP2 is adjusted to the left at $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively. (f) - (h) These are the output intensity in the $x$ direction ($y=0$) when the polarization angle of GLP2 is adjusted to the right at $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively.
3. Experimental demonstration
According to theoretical analysis, when the polarization angle $\beta$ of GLP2 is $0 ^{\circ }$, the output field is vortex light, which is the two-dimensional optical differential of the incident light field. Based on Eq. (21), we slightly change the polarization angle $\beta$ of GLP2 to the left or right. The changed process of the output beam from vortex beam to Gaussian beam can be clearly observed. In order to prove the imaging characteristics of the tunable optical differential operation based on the cross-polarization effect at the optical interface, we designed experiments to verify that the output image of the object varies with the polarization of output beam.
The experimental device is shown in Fig. 4, we take the USAF-1951 resolution target as the object, and select the vertical ripple and the horizontal stripe to conduct the experiment. The He-Ne laser with a wavelength of 532nm is used for irradiation. HWP is used to adjust the intensity of the incident beam to make the experimental results easier to observe. GLP1 is used to adjust the polarization state of the incident beam. In the experiment, we set the polarization angle of GLP1 as $67.1^{\circ }$. The measurement target is placed on the front focal plane of L1, enabling it to be transmitted to the air-glass interface, and L1 and L2 constitute the first 4F system. QWP is a quarter-wave plate which changes the beam from circular to elliptically polarized. The prism with a refractive index of 1.5 is used to act as an air-glass interface, and the incident beam is reflected through the prism after incident at the Brewster angle. GLP2 is used to adjust the polarization state of the outgoing beam to the desired position, and it is a key instrument in our experiment. Then L3 and L4 constitute the second 4F system, and the optical path between them is the sum of their focal lengths. Finally, the CCD is placed on the rear focal plane of the L4 to receive the sharpest image and preserve the record.
Fig. 4. Experimental setup: The light source is a He-Ne laser with a wavelength of 532nm; HWP, half-wave plate; L, lens; GLP, Glan laser polarizer; QWP, quarter-wave plate, the inclined surface of the prism acts as the air-glass reflection interface; CCD, receiving the output images; L1 and L2 form the first 4f system, the object is placed at the front focal plane of L1. L3 and L4 constitute the second 4f system, and the CCD is placed at the rear focal plane of L4. The focal lengths of L1, L2, L3 and L4 are 25mm, 75mm, 200mm and 100mm, respectively. The inset shows the USAF-1951 resolution target.
Firstly, we rotate the polarization axis angle of GLP2 to darken the light field in the middle of the target image, so that CCD can receive a clear two-dimensional edge detection image. Then, we fine-tune the polarization angle of GLP2 based on this. In this experiment, we used the Glan laser polarizer with a scale of $0.04^{\circ }$ for each small grid. Therefore, we rotate the polarization angle of GLP2 one grid at a time to the left or right (we rotate to the third grid, the position when the polarization angle of GLP2 was $0.12^{\circ }$), and save the images received on CCD respectively. Vertical ripples and horizontal stripes are selected as measured objects for the experiment, thus the experimental results are shown in Fig. 5. Figures 5(a), 5(a1), 5(e) and 5(e1) are two-dimensional edge detection image of the measured target. Figures. 5(b)–5(d) and 5(f)–5(h) are the output image of the measured target when the polarization angle of GLP2 is adjusted to the left at $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively. Figures 5(b1)–5(d1) and 5(f1)–5(h1) are the output image of the measured target when the polarization angle of GLP2 is adjusted to the right at $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively.
Fig. 5. Output images of different measured targets changing with polarization angles of GLP2. (a), (a1), (e) and (e1) These are two-dimensional edge detection image of the measured target. (b) - (d) and (f) - (h) These are the output image of the measured target when the polarization angle of GLP2 is adjusted to the left at $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively. (b1) - (d1) and (f1) - (h1) These are the output image of the measured target when the polarization angle of GLP2 is adjusted to the right at $0.04 ^{\circ }$, $0.08 ^{\circ }$ and $0.12 ^{\circ }$, respectively.
Obviously, on the basis of obtaining a clear two-dimensional edge detection image, we adjusted the polarization angle of GLP2 to the left or right. As the polarization angle of GLP2 increases, the middle of the output image changes from dark to bright, and gradually tends to the original complete image of the input object. In this process, we get the image of multi-degrees of freedom of the object, which is mutually verified with our theoretical analysis. Therefore, based on the two-dimensional differentiation achieved by the Brewster effect, we propose the tunable optical differentiation can achieve multi-degree of freedom imaging. It provides more imaging options for microscopic imaging and high contrast imaging.
4. Conclusion
In this paper, on the basis of the two-dimensional optical differential operation based on the Brewster effect, we propose a tunable optical differential operation based on the cross-polarization effect at the optical interface. Theoretically, we derive the final output light field after the beam passes through the differentiator. The form of two-dimensional differential is obtained when the output polarization angle is $0^{\circ }$, so the theoretical output diagram is vortex light. It is found that the output light field changes from vortex light to Gaussian light when the polarization of output beam changes. In the process, we think that the corresponding device can image the object with different degrees of freedom. In the experiment, we build a tunable differentiator which is mainly composed of two polarizers, a prism and a quarter-wave plate, and select the corresponding target image. It is found that we fine-tune the output polarization state when the output object presents a clear edge, with the increasing of the polarization angle, the output image has the characteristics of bright edge and gradually bright middle. Therefore, we can adjust the output polarization state according to our needs. While getting the clear edge of the object, we can also observe the imaging of the middle part to different degrees, which realizes the multi-degree of freedom imaging of the measured target. In fact, we propose the tunable optical differential operation which can also achieve real-time imaging, low power consumption and high efficiency optical analog calculation. It should be noted that our scheme may be able to observe the measured target more comprehensively, which provides a potential way for the research and development of simple and flexible microscopic imaging.
Funding
National Natural Science Foundation of China; Science and Technology Program of Guizhou Province; Department of Education of Guizhou Province.
Disclosures
The authors declare no conflicts of interest.
Data availability
No date were generated or analyzed in the presented research.
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