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Abstract
Full polarization imaging plays an important role in remote sensing to distinguish artificial objects from the natural environment, recognizing objects in shadows and sun glint suppression. In this paper, we propose a broadband full Stokes channeled modulated polarization imaging system based on a Mach-Zehnder-grating interferometer (MZGI) with advantages such as compact size, low cost, snapshot ability, and high optical efficiency. It uses gratings to compensate for the dispersion of the carried frequency when inputting broadband light to generate interference fringes. Two detectors are assembled to the output plane to acquire the interference fringes. Each image obtained by the detectors can be individually demodulated into different Stokes parameters individually. When the two groups are combined together, the full Stokes parameters are obtained. The simulation and optical efficiency analysis demonstrate that the interference fringes can obtain the full polarization information simultaneously with high optical efficiency in broadband wavelengths.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Polarization information is sensitive to the surface features, shape, and roughness of detected objects. It is important for remote sensing to highlight the difference between interesting objects and the background. Stokes polarization imaging is a special case of general polarimetry that is used to map the polarization state across a scene of interest. The Stokes vector is a matrix with the form [1]:
In the CMPI system, the Stokes parameters are modulated on different carried frequencies (CF) through interference. The interference fringe image appearing on the detector contains the Stokes polarization information. It is evident in the frequency domain using the Fourier transform, where each parameter remains in its own channel. By means of demodulation, the Stokes parameters are restored to provide the polarization information contained in the target’s two-dimensional image. CMPI was first proposed by K. Oka in 2003 [17]. The key to CMPI is the polarization modulation module that divides the input light to generate shearing distance and phase differences. When the beams combine in the detector plane, interference generates CF to realize the modulation process. The shearing distance generator is the core device in the polarization modulation module. It is classified as three categories: birefringent crystals [18–22], interferometer [23,24], and polarization gratings [25]. CMPI based on birefringent crystals is mainly a Savart plate. In 2016, we analyzed the factors that impact the width of the spectral imaging band and calculated the criterion of the CMPI system [26]. We later calculated the error in assembling the half wave plate [27]. Although the birefringent crystal CMPI system has a competitive advantage as one optical path, it cannot use broadband input light, which significantly limits the application domain. CMPI based on polarization gratings is sensitive to the temperature and high cost. A Sagnac interferometer is used in CMPI, but it is difficult to design with a large aperture. To adapt the system for remote sensing, such as a larger aperture, lower temperature sensitivity, low cost, full polarization, broadband imaging, and compact configuration, we proposed a novel CMPI structure using a Mach-Zehnder-grating interferometer (MZGI-CMPI). First, we introduced a principle for a dispersive shearing distance generator based on the MZGI. We then demonstrated full polarization imaging. In the third section, we provided the structure’s test results. Then we discussed the demodulation, optical efficiency, and errors in the simulation results.
2. Mach-Zehnder-grating interferometer (MZGI) acts as dispersive shearing distance generator
The MZGI divides the input light into two beams with a dispersive shearing distance shown as Fig. 2. It contains two mirrors (${M_1}$ and ${M_2}$), two polarization beam splitters, and four transmission blazed gratings. The incident light is divided into two beams after vertically entering the polarization beam splitter.
Taking the reflected light as an example, it is diffracted by grating ${G_1}$ into the first order with a diffraction angle $\theta $ with the form:
where $\lambda $ is the wavelength and d is the period of the gratings. When d is larger than 30 um, the diffraction angle is simplified approximately as $\theta \approx \lambda /d$. When the beam is transmitted by the following grating ${G_2}$, the diffraction angle is removed as parallel to the optical axis. The shearing distance generated by gratings is expressed as: where ${l_1}$ means the distance between ${G_1}$ and ${G_2}$. In the similar way, the shearing distance ${\textrm{D}_2}$ generated by gratings ${G_3}$ and ${G_4}$ can be expressed as ${\textrm{D}_2} \approx {l_2}\lambda /d$, where ${l_2}$ means the distance between ${G_3}$ and ${G_4}$. The total shearing distance equals:Equation (4) clearly demonstrates that the shearing distance is dispersive with the wavelength when the period and distance between the gratings are decided.
3. Demonstration of the full polarization imaging system
We use extra pupils to obtain the full polarization information. A new design that utilizes two detectors is provided in Fig. 3. It includes a fore-collimation module, a common beam splitter, four mirrors (${M_1}$, ${M_2}$, ${M_3}$, and ${M_4}$), two polarization beam splitters ($\textrm{PB}{\textrm{S}_1}$ and $\textrm{PB}{\textrm{S}_2}$), four transmitted diffraction gratings (${G_1}$, ${G_2}$, ${G_3}$,and ${G_4}$), two polarizer oriented at ${45^ \circ }$, two imaging lenses, and two detectors ($\textrm{Detecto}{\textrm{r}_1}$ and $\textrm{Detecto}{\textrm{r}_2}$) acting as focal plane arrays (FPAs).
The input light transmits through the collimation module and is splited into a transmitted ray (${E_t}$) and a reflected ray (${E_r}$) with Jones matrixes as:
Then the ${E_r}$ is reflected into the quart wave plate (QWP) oriented at ${45^ \circ }$, and becomes the new form:
The parallel and perpendicular components of ${E_r}$ are split into ${E_{rs}}$ and ${E_{rp}}$ by the first polarization beam splitter ($\textrm{PB}{\textrm{S}_1}$).
The ${E_{rs}}$ and ${E_{tp}}$ are diffracted by ${G_1}$ into the first order. When these dispersed rays are incident on ${G_2}$, the diffraction angle induced previously to be ${G_1}$ is removed. The rays are parallel to the optical axis and offset by a distance equal to:
Then the rays enter $\textrm{PB}{\textrm{S}_2}$ after the reflection generated by the mirror ${M_4}$. The ${E_{rs}}$ is reflected continuously and the ${E_{tp}}$ transmits $\textrm{PB}{\textrm{S}_2}$ with the electronic field in the form:
The ${E_{rp}}$ and ${E_{ts}}$ are diffracted, offer a dispersive shearing distance, and enter $\textrm{PB}{\textrm{S}_2}$. The ${E_{rp}}$ will transmit through $\textrm{PB}{\textrm{S}_2}$ to meet the ${E_{rs}}$. The rays have the following form:
The two rays have the same vibration direction after going through the polarizer oriented at ${45^ \circ }$. The FPA of $\textrm{Detecto}{\textrm{r}_1}$ acquires the interference fringes with intensity:
Simplifying Eq. (10) and substituting Eq. (1) into it, and we can get the new form:
Substituting Eq. (11) into Eq. (12), it can be re-expressed as:
Thus, the partial Stokes parameters (${S_0}$, ${S_1}$, and ${S_2}$) are amplitude modulated onto the interference fringes. Taking the transmitted light into consideration, the ${E_t}$ light is divided into two beams after $\textrm{PB}{\textrm{S}_1}$ as ${E_{ts}}$ and ${E_{tp}}$ with the Jones matrixes
Similarly, the intensity acquired by $\textrm{Detecto}{\textrm{r}_2}$ equals:
Expand Eq. (15) and get:
Its phase factors equal:
Then the intensity can be expressed as:
Taking the diffractions efficiency of the gratings varying with the wavelength into consideration, the Stokes parameter ${S_j}(j = 0,1,2,3.)$ we acquired has a relationship with the objects’ true Stokes parameter $S_j^{\prime}(j = 0,1,2,3.)$ like that:
4. Test results of CMPI based on MZGI
Simulation are used to verify this system’s operating principles. simulation plays an important role in the principle verification process. Simulating ideal input light to test the principle is necessary to optimize the system and design appropriate experiments. We used a simulation to rapidly verify this system. MATLAB software is commonly used to simulate the intensity and sample process. The platform was Windows 10 with Intel Core i5-7500@3.4 GHz and 8.00 GB RAM. The software was MATLAB R2014a. To more comprehensively cover the detection cases, three kinds of input light were used: ${0^ \circ }$ linear polarized light, ${45^ \circ }$ linear polarized light and right circular polarized light. The imaging scale was $L \times L = 256 \times 256$, the pixel space equaled $4.65\mu m$, $f = 25mm$, the input light band was $0.174\mu m$ from $0.526\mu m$ to $0.700\mu m$, and the commercial elements’ capabilities were considered. The polarization state of input light was divided into four areas with different polarization states. In the upper left there was a circular area with the Stokes vector ${\left[ {\begin{array}{cccc} 1&0&1&0 \end{array}} \right]^T}$, in the bottom left there was a circular area with the Stokes vector ${\left[ {\begin{array}{cccc} 1&1&0&0 \end{array}} \right]^T}$, in the upper right there was a circular area with the Stokes vector ${\left[ {\begin{array}{cccc} 1&0&0&1 \end{array}} \right]^T}$ and the rest area was set to be unpolarized with the Stokes vector ${\left[ {\begin{array}{cccc} 1&0&0&0 \end{array}} \right]^T}$. The simulation results are shown as Fig. 4.
Fig. 4. The interference polarization image (a) $\textrm{CC}{\textrm{D}_1}$ detects ${S_0}$, ${S_2}$, and ${S_1}$. (b) $\textrm{CC}{\textrm{D}_2}$ detects ${S_0}$, ${S_2}$, and ${S_3}$.
In the Fig. 4, the fringes in area A and C indicate the ability to detect ${45^ \circ }$ linear polarized light, the fringes in area B indicate the ability to detect ${0^ \circ }$ linear polarized light and the fringes in area D indicate the ability to detect right circular polarized light.
Taking Fourier transform on the images, we can get the distribution of channels in the frequency domain:
The Stokes parameters are recovered by filtering the channels and conducting the inverse Fourier transform on the amplitude such that:
To calculate the phase factors in Eq. (21), introducing two reference beams is important [28], where the phase factors can be measured for a known Stokes vector over a uniformly illuminated scene. Reference data are obtained by a polarized orientation at ${0^ \circ }$ to isolate $\textrm{exp(j4}\pi l{z_i}/df)$ and a polarized orientation at ${0^ \circ }$ to isolate $\textrm{exp(j4}\pi l{x_i}/df)$. The Stokes parameters are expressed as:
The RMS error between the demodulated results and the simulation input is calculated by:
Further, we provide the preliminary experimental setup shown as Fig. 6. One $25.4mm \times 25.4mm$ beam splitter was used as the BS in the first light-splitter module. Then one precision achromatic QWP oriented at ${45^ \circ }$ was used in the reflected light path. Two 25.4 mm wire grid polarizing beam splitter cubes were used as the PBSs in the setup. Four mirrors were 25 mm diameter 1/10th wave optical flats. The four transmission gratings were ${\textrm{G}_1}{\sim }{\textrm{G}_4}$, 35 grooves/mm. Two imaging lenses of 25 mm focal length and two CMOSs (Allied Vision GT1660) were used to take the interference images.
The pictures taken by two CMOSs were cut to $256 \times 256$ and shown as Fig. 7(a) and Fig. 7(b). The fringes carried the polarization information. The demodulation results are shown as Fig. 7(c) and Fig. 7(d). It shows that the input light is partial polarized, and the degree of polarization is 43.2%. The setup can show the ability to acquire the fringes carrying polarization information.
Fig. 7. The results of the setup. (a) the fringe picture by COMS1, (b) the fringe picture by CMOS2, (c) the demodulation of ${S_0}$, and (d) the distribution of degree of polarization
5. Discussion
5. Optical efficiency
The main reason for incomplete utilization of incident light is that the transmittances of some elements vary with the wavelength, especially polarization beam splitters and diffraction gratings. We assume that the input light is unpolarized and the intensity is ${I_{in}}$. Then it is split into ${R_1}{I_{in}}$ and ${T_1}{I_{in}}$ after the beam splitter (BS) where ${R_1}$ and ${T_1}$ are the reflectivity and transmittance of the BS, respectively. The next elements are mirrors and QWP that have transmittances over 95%. Then the beams will transmit through the $\textrm{PB}{\textrm{S}_\textrm{1}}$ and become four rays with intensities ${R_1}{R_2}{I_{in}}$, ${R_1}{T_2}{I_{in}}$, ${T_1}{R_2}{I_{in}}$, and ${T_1}{T_2}{I_{in}}$, where ${R_2}$ and ${T_2}$ are the reflectivity and transmittance of $\textrm{PB}{\textrm{S}_1}$, respectively. The diffracted gratings weaken the intensity further with the forms ${R_1}{R_2}D{E^2}{I_{in}}$, ${R_1}{T_2}D{E^2}{I_{in}}$, ${T_1}{R_2}D{E^2}{I_{in}}$, and ${T_1}{T_2}D{E^2}{I_{in}}$. The following $\textrm{PB}{\textrm{S}_2}$ has the same effect on $\textrm{PB}{\textrm{S}_1}$ and the beams’ intensity become ${R_1}R_2^2D{E^2}{I_{in}}$, ${R_1}T_2^2D{E^2}{I_{in}}$, ${T_1}R_2^2D{E^2}{I_{in}}$, and ${T_1}T_2^2D{E^2}{I_{in}}$. The analysis will abstract the half intensity and whole intensity, and the FAP has an efficiency that equals:
For ${R_1}(\lambda )+ {T_1}(\lambda )\approx 1$ in the BS, Eq. (24) can be simplified as:
The relations between parameters ${T_1}(\lambda )$, ${R_1}(\lambda )$, $D{E_1}(\lambda )$ and wavelength are depicted in Fig. 8(a). The total efficiency $\eta (\lambda )$ is shown in Fig. 8(b). Although the device can broaden the incident spectrum domain into white light, the elements we chose will impact the experimental results. The spectral domain $0.\textrm{ }174\mu m$ has an optical efficiency over 30%.
Fig. 8. The optical efficiency based on commercial elements [29,30]. (a) The performance of commercial element. (b) The optical efficiency.
5.2 Difference between CMPI based on MZGI and DoAm
The novel CMPI system based on the Mach-Zehnder interferometer has two detectors, which is similar to the DoAm. However, there is an intrinsic difference between them as shown in Fig. 9. The DoAm needs at least 4 cameras to acquire the full Stokes polarization information. Figure 9 uses one Stokes parameter ${S_0}$ as an example. ${S_0}$ needs the ${0^ \circ }$ polarized light plus the ${90^ \circ }$ polarized light. If the two detectors’ view fields are not identical, the result error will be non-negligible. This problem can be solved by optical engineering matching or image processing software. But it is a challenge for the researchers. CMPI based on MZGI needs only two detectors, and the matching is not as strict as the DoAm. Each picture acquired by the detectors can be demodulated to the partial Stokes polarization information. If the detectors’ view fields do not properly match, the pictures are independent, and the calculated Stokes polarization information is not effected by the matching. The full Stokes parameters acquired by two detectors are shown in Fig. 9.
5.3 Difference between CMPI based on MZGI and conventional CMPI
Compared with the conventional CMPI system, the system based on MZGI has an extra detector to acquire the full Stokes parameters. The ability to obtain full Stokes polarization information is so important to remote sensing that CMPI based on MZGI cannot abandon the two detector design. Adding an extra detector increases the assembly’s complexity. But CMPI based on MZGI has some advantages over conventional CMPI.
First, obtaining a sufficiently large aperture is challenging for the other systems. Using CMPI based on the Sagnac interferometer as an example, the hypotenuse was 57.2 mm in a prior study [23]. This is because the gratings in the compensation part should be set at a distance so that the input and exit pupils are limited by the design. In CMPI based on MZGI, the compensation part has two group gratings. The distance between the two gratings depends on the size of the entire construction so that the pupils can be designed to match the remote sensing requirements.
Second, in the conventional modulated polarization imaging system, there are 7 channels limited to one rectangle [18]. On the one hand, the channels’ crosstalk will increase the recovered program’s noise. On the other hand, if the object’s details are very rich and appear as high frequency information in the frequency domain, filtering the channels will damage the information’s integrity, leading to a lack of detail. In the new system, there are only 3 channels in one image, including one DC component and a pair of conjugated channels. As a result, the channel crosstalk will decrease and more information will be retained.
The detection accuracies also differ. In conventional CMPI, four beams generate interference fringes on the detector, and there are seven channels in the frequency domain. The channels exist as the sinc function. Then the channels’ crosstalk will influence the demodulation results. But the crosstalk can be reduced in CMPI based on MZGI because there are only three channels in the frequency domain.
Figure 10 shows a uniform distributed light detection situation with the Stokes vector ${\left[ {\begin{array}{cccc} 1&{0.75}&{0.5}&{0.433} \end{array}} \right]^T}$. In conventional CMPI, the interference fringes are more complicated than those of CMPI based on MZGI shown as the Fig. 10. In the frequency domain, the value of each point equals to the sum of all channels’ contribution. Channels would intertwine with each other. Seven channels’ crosstalk would be more complicated than three channels’ crosstalk. The enlarged picture of region A shows a clear crosstalk generated by conventional CMPI. But in the same region B and C generated by CMPI based on MZGI, crosstalk is smaller. For the real target imaging, high frequency information would be all around the frequency domain leading to much more crosstalk. The more channels there are, the value distribution becomes more complex. So the crosstalk can be reduced when the number of channels decreases. Although the crosstalk is still present but the demodulation accuracy can be improved.
Fig. 10. Comparison of two kinds of CMPI for detecting elliptically polarization. (a): The interference fringes and frequency domain of conventional CMPI. (b): The interference fringes and frequency domain of CMPI based on MZGI.
6. Conclusion
In conclusion, we provided a novel CMPI configuration that takes advantage of the Mach-Zehnder interferometer’s two entrance pupils and two exit pupils. It is possible to assemble two detectors to acquire the full polarization information. The simulated results showed the reliability of the novel configuration. We provided the preliminary experimental setup to show the ability to acquire the fringes carrying polarization information. The new CMPI based on MZGI has a wider imaging spectrum range of $0.174\mu m$ that is limited by commercial elements. It has competitive advantages of being compact, inexpensive, and stable, with a high optical efficiency and larger aperture.
Funding
National Natural Science Foundation of China (61890961); Natural Science Foundation of Shaanxi Province (2018JM6008); Scientific Research Plan Projects of Shaanxi Education Department (19JK0856).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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