A complete full Stokes imaging spectropolarimeter is proposed. Four separate polarized spectra are fed into the Sagnac Fourier transform spectrometer without slit using different angle combinations of the polarized elements. The four polarized spectra are separated without spatial aliasing. And the system has a good performance to resist the instrument noise due to its high light throughput. The mathematical model for the approach is derived and an optimization of the retardance is discussed. For acquiring the four spectra simultaneously, an improved robust polarization modulator using aperture division is outlined. Then the system is discussed in detail including the imaging principle and spectral resolution. Lastly, two proven experiments are carried out and the experimental results in visible light are outlined.
© 2013 Optical Society of America
Spectrum implies the materials of the target, while the polarization information tells us the surface features, shape, shading and roughness of the target . Imaging spectropolarimeter provides us spatial, spectral and polarization information together and improves the ability to effectively recognize the target. It has become a well-recognized technique and has been used in many scientific applications, such as environment monitor  and biomedical diagnosis . A number of productive strategies have been developed and tested in the last two decades for the goal of retrieving the spectropolarimetric information with full Stokes vector, high spectral resolution using robust structure with fast test [4–7].
Fourier transform imaging spectropolarimeter without slit has the advantages of high radiation throughput and high space resolution due to the absence of the exit slit. And it is a viable alternative to dispersive instrument in visible and infrared spectral range [8–11].
Several methods are usually used to modulate the Stokes vector into the Fourier transform imaging spectrometer. One is using the KD*P crystal to modulate the linear Stokes parameters into a folded Michelson Fourier transform spectrometer, which is utilized to measure the linear solar polarization spectrum . Another method is working with the spectropolarimetric technique implemented by K. Oka , in which the full Stokes parameters are obtained simultaneously and separated in the interferogram taken from the Fourier transform spectrometer [14, 15]. When a narrow-band spectrum is measured by this way, there may be aliasing among the fringes of different channels. A complicated method should be utilized to revise the retrieve spectrum . Besides, the method using the variable-retardance to modulate polarized spectra onto a temporally varying fringe pattern is also attractive. We can obtain a high resolution spectrum in this way. A complex device must be used to make sure that the change of the retardance is accuracy and the detection time is increased [17, 18].
In this paper, we present a complete Fourier transform imaging spectropolarimeter. Four polarized spectra are launched into the Fourier transform spectrometer by changing the angles of the polarized elements instead of changing the retardance. It can be used in different system architectures such as time division structure. An improved polarization modulator based on our prime work with micro-lens array is designed to create the four polarized spectra at once. The Sagnac Fourier transform spectrometer without slit is used to generate the fringe pattern.
We will first describe the theoretical model behind the spectropolarimetric technique and analyze the optimization of the retardance. In Sect. 3 the system architecture is outlined including an improved modulator and the Sagnac Fourier transform spectrometer. The complete system is discussed in Sect. 4 including the analysis of the image system and recovery spectrum. And Sect. 5 shows the reconstruction strategy of the spectropolarimetric images and experimental results of two experiments with error analysis, followed by a conclusion in Sect. 6.
2. Theoretical model
2.1 Spectropolarimetric measurements
For acquiring polarized spectra without spatial aliasing, we generate four polarized spectra using simply polarization elements. A retarder and a linear polarizer in broad-band spectrum are installed before the Fourier transform spectrometer as shown in Fig. 1.Unlike traditional imaging polarization system which is usually used for monochromatic light, the presented system is utilized to measure the polarization information in various wavenumber. The input polychromatic light from the scene is launched into the retarder first and then passes through the polarizer.is the fast axis angle of the retarder with the reference direction x axis, andis the transmission angle of the polarizer with the reference direction x axis.
The Stokes vector of the emergent light from the linear polarizer can be expressed by the Muller calculation
is the unknown input Stokes vector of the scene.is the total intensity of light.denotes preference for linear 0° over 90°,for linear 45° over 135°, andfor circular right over circular left polarization states. The quantitiesandare the Muller matrices of the linear polarizer and retarder, respectively.is the retardance of the retarder in various wavenumber. Because the camera responds to light intensity regardless of the polarization state, only the first parameterofcan be measured. When the polarized spectrum passed through the Fourier transform spectrometer, fringe pattern is created whose intensity is shown as
When the retardanceis known, we can construct an equation set of the input Stokes parameters in various wavenumber by choosing at least four polarized interferometric fringes with different combinations of19]. The equation set can be expressed by matrix
The four Stokes parameters can be expressed by calculating the inverse matrix of
The four polarized spectra are separated without spatial aliasing, so both broad-band spectrum and narrow-band spectrum can be detected. Besides, the spectral resolution is improved compared to the channeled Fourier transform spectropolarimeter. The four polarized spectraof the scene can be generated in different ways, such as time division and amplitude division. And there is no extra device to control the retardance of the retarder.
2.2 Optimization of the retardance
Matrixis the analysis matrix of the polarization system. Amrit, et al. have proved that the error of the Stokes parameters is affected by the condition number of matrix . When the system relative error is small, the estimated errorof the Stokes parameters is shown asEq. (7) is small. Three types of condition number (,and) are usually used to optimize the system error and they are equivalence . Regular angle combinations of the retarder and polarizer (0°, 0°), (0°, 45°), (45°, 0°), (45°, 45°) are used in the paper which can make the simultaneous polarization modulator simply as explained in the next section. The value ofwithincreasing from 0° to 180° is shown in Fig. 2(a). The retarder with retardance from 70-deg to 160-deg is recommended where the measurement error can be resistant effectively. Because the condition number changes slowly in the retardance range as shown in Fig. 2(b). And the maximum condition number is about two times of 5.864 which is the minimum condition number in the polarization system using a polarizer and a quarter waveplate .
3. System architecture
3.1 Polarization modulator using aperture division
In this paper, we generate the four spectra by the simultaneous polarization modulator (SPM) based on aperture division, which is usually used in 3-D light field imaging  and multi-spectral imaging . It has been proposed in our primer work . The SPM consists of a polarization array, an objective lens L1 and a micro-lens array as shown in Fig. 3(a). The polarization array is composed of four polarization channels as shown in Fig. 3(c). Benefiting from the new angles combination, an improved structure of the polarization array can be used, where four rectangular polarization elements are applied instead of eight ones, as shown in Fig. 3(b). The structure of the new polarization array is simply and the adjustment is easier.
Light from a point on the scene is first modulated by the polarization array and splits into four paths with different polarization states. Then the light is launched into the objective lens L1 and focuses on a lenslet of the micro-lens array which is located in the image plane of L1. The lenslet separates the four beams based on direction and produces four images of the point on its focal plane, and the images contain four polarization states
A pinhole array can take place of the micro-lens array. However, the radiation throughput of the system will reduce. Compared to other methods to generate the four polarization spectra, the SPM has several advantages as shown followed
- (a) The four polarization spectra are fed into the system simultaneously and there is no spatial aliasing between them.
- (b) There is no movable element in the SPM, so many estimation errors are reduced which can rise from vibrations and thermal drift.
- (c) The four polarization spectra pass through the same imaging lens and are captured by a camera, making that the four images of an object point have near identical optical aberration compared to other aperture division system .
3.2 Sagnac Fourier transform spectrometer
A Sagnac Fourier transform spectrometer without slit is incorporated into the SPM and used to generate the polarized interferometric images, as shown in Fig. 4..
The focal plane of the micro-lens array is coincided with the front focal plane of the collimating lens L2. The light from the polarized image is first collimated by the lens L2, then travels the interferometer by means of a beam-splitter BS and two reflectors M1 and M2, and lastly imaged on the focal plane of the lens L3.
The Sagnac interferometer, which plays a role of lateral shearing beam splitter, is the heart of the spectrometer. It is used to separate the input beam into two coherent interfering beams and provide phase-delay between them, so that the OPD changes with varying the angleof the entering ray with respect to the normal of the lateral shearing splitter. It produces a fringe pattern of equal thickness that is localized at infinity and the intensity of the fringe pattern is shown as
The accuracy of the recovery spectrum is affected by the signal-to-noise-ratio (SNR) of the Fourier transform spectrometer, which is expressed as
When the spectral content and maximum optical difference path are fixed, the SNR of the Fourier transform spectrometer is invariable if the photonics noise is only taken into account . However when the instrumental noiseis taken into account, the SNR of the spectrometer increases with the increasing of the signal intensity. Benefiting from the cancelling of the slit, the Sagnac Fourier transform spectrometer without slit owns the advantage of high light throughput, which is about two orders of magnitude larger than filtering spectrometer and the Fourier transform spectrometer with slit . So it has a higher SNR compared to the Fourier transform spectrometer with slit and has a good performance to resist the noise of the instrument. Compared to the birefringent Fourier transform imaging spectrometer [27, 28] and Michelson Fourier transform imaging spectrometer , the Sagnac interferometer is easily adjustable and has a high resistance to vibration benefitting from the common path structure of the interferometer
4. System analysis
4.1 Imaging system
The optical layout of the complete system is shown in Fig. 5.The imaging progress to capture the polarization images with fringe pattern is outlined and discussed.
The scene is first imaged by the SPM to generate the polarized image which contains four different polarization states separated in space on the front focal plane of L2. Compared to the traditional imaging system, the image captured by the SPM contains not only the two-dimension space information but also the polarimetric information modulated in the direction vector of the light. The scene is first imaged on the micro-lens array by the objective lens L1 to detect the two-dimension space information, which is akin to traditional imaging progress. Each lenslet of the micro-lens array represents a point on the scene. So the space resolution of the scene is given by
Then the light is imaged by the micro-lens array to separate the four polarization states. The light beam of a point is divided by the polarization array, and each lenslet forms a conjugate image of the polarization array on its focal plane where the light from the four polarization channels is absolutely separated in space. The image generated by the SPM is shown in Fig. 6(a), where four images of a point are separated in space and we can obtain the four polarized images of the scene by de-interlace, as shown in Fig. 6(b).
For extracting clearly polarization images from the captured image, the four image points created by a lenslet should be uniform distribution in space. So the polarization array should be the aperture stop of L1. Otherwise the radiation throughput of the four channels will change with different field of view because light from some channel will be kept out by the aperture diaphragm in a wide field of view as shown in Fig. 6(c). Besides, the image points created by the micro-lens array should be separated absolutely. So the relative apertures of L1 and lenslet array should satisfyFig. 6(d).
The lens L2 and L3 compose the second imaging system where the light from the polarized image is collimated by L2 and passes through the Sagnac interferometer, then imaged by L3 on its focal plane. The Sagnac interferometer can be folded to a plate and used to create a fringe pattern superposed on the polarized image. The field of view of the complete imaging system is expressed by
4.2 Spectrum analysis
The spectrum is retrieved by Fourier transform of the fringe pattern. So the spectral resolution is limited by the maximum OPD. A Large maximum OPD brings us a high resolution spectrum. The maximum OPD is related to the work mode of the system. There are two operating modes of the spectropolarimeter to obtain the polarized fringe pattern of each pixel. The first is rotating the spectropolarimeter, which is suitable in aerospace remote sensing because the push-broom is completed by the flight of the platform. There is no movable element in the first method, which makes the system robust and compact. The device acquires the image of the scene superimposed to a fixed pattern of across-track interference fringes. It introduces a relative motion between the sensor and the scene. The scene is moved over the fixed fringe pattern. And the data captured in this mode is shown in Fig. 7(a).The fringe patterm is modulated by time and space. It is better to move a pixel one time. Otherwise, image matching is necessary in the progress of extracting the fringe pattern of a point . The maximum OPD captured by this mode is shown as
The other mode is by rotating the Sagnac interferometer alone respect to the incoming beam. When the interferometer rotates, the OPD of every pixel in the camera changes but the scene of target is fixed. The data obtained in this mode is shown in Fig. 7(b). The fringe pattern of a point is only modulated by time. The sampling frequency is decided by the rotation speed of the lateral beam splitter. We can choose a suitable sampling frequency without thinking about images matching. The maximum OPD in this mode can be calculated as30].
Thinking of the system noise, such as photonic noise and instrumental noise, the effective maximum OPD is limited by the SNR . A high SNR brings us a good performance to resist instrumental noise. The signal intensity of the fringe pattern is related to the vignetting of the system, which will reduce the SNR of the fringe pattern of the points in a wide field of view. For eliminating the vignetting, the aperture of L2 is expressed as
As the same time, the aperture of L3 should be
The visibility of the fringe pattern is another important influence factor of the SNR of the fringe pattern, which presents the utilization efficiency of the input radiation. In our system, the visibility of the fringe pattern is first modulated by the splitting ratio of the Sagnac interferometer and can be expressed as
To obtain the fringe pattern with high visibility, it is also required that the wave front from a point is a plane wave when it passes through the Sagnac lateral shearing splitter. Otherwise, if the wave front from a pointof the polarized image is a spherical wave, it will be divided into two shearing spherical waves and generates two virtual pointswhen it passes through the lateral shearing splitter, as shown in Fig. 8.
The two virtual object points will create two Airy disks which are incomplete coincide in the imaging plane which is close to the focal plane of L3. The interference occurs only in the overlap region of the two Airy disks. When the coincided region becomes small, the visibility of the fringe pattern will reduce and the image quality will be low. For eliminating the error, the focal plane of the microlens array should be coincided with the front focal plane of L2.
5. Experiment and results
5.1 Experiment setup and calibration
An experimental setup was constructed according to the schematic of Fig. 2, as shown in Fig. 9.The SPM was composed of achromatic quarter waveplates in 430-650nm used as the retarders, achromatic linear polarizers in 400-700nm and a SUSS MiscroOptics micro-lens array with 200 × 200 pixels. The Sagnac interferometer consisted of two reflector mirrors, a beam splitter and a camera with the resolution of 644 × 488. We acquired the interferometric images by rotating the Sagnac lateral shearing splitter.
The radiation throughput of the four polarization channels is affected by the system errors, which will decrease the accuracy of Stokes parameters. Prominent among them is the error from the nonuniform throughput performances of the four channels. So a flat field calibration should be carried out before the experiment. A white plate was used to calibrate the channels errors. The Sagnac interferometer was replaced by a reflector mirror. We obtained the four calibrating matrices of the four channels. Acquiring and normalizing the four images of a point one by one, we calculated the calibration matricesof the four channels without the polarization elements, which was used to calibrate the polarized spectra
5.2 Experimental results
Two proven experiments have been carried out to verify this method in the laboratory. A colour picture of the London Olympic mascot illuminated by a white LED lamp, as shown in Fig. 10(a), was detected by the experimental system first. An achromatic polarizer in visible light, whose transmission axis is oriented at 30° relative to the x axis, was placed before the system to create a known polarization state. We obtained 600 interferometric images of the scene. Three of them are shown in Figs. 10(b)–10(d).
Several steps have been performed to retrieve the spectrum and Stokes parameters. Firstly, we reconstructed the spectral images by Fourier transform according to Eq. (4) with double-sided interferometric fringe. There is no phase error in trade off the spectral resolution in this way. Then the spectral cube was created conveniently, where we retrieved the spectral images in various wavenumber. Lastly, we reconstructed the four Stoke parameters images in various wavenumber by Eq. (6).
17 spectral images from 1.65 × 104 to 2.05 × 104cm−1 (490~610nm) were reconstructed. The recovery spectrum of Point A in Fig. 10(a) is shown in Fig. 11(a). Figure 11(b) shows the normalized Stokes parameters with the theoretically calculated values of Point A. It is evident that the experimental result is in agreement with the theoretical value and proves the principle of the method. Figure 11(c) shows four extracted images of the scene with different polarization states at 540nm, where we obtained the Stokes parameters images in Fig. 11(d).
Then some fruits were imaged by the system as shown in Fig. 12(a), where the mango and the orange are made of plastic. A linear polarizer whose transmission axis was oriented at 0° was placed before the apple to create a known polarization state. Three of the interferometric images is shown in Figs. 12(b)–12(d). We reconstructed the spectropolarimetric information of the fruits. Figure 13(a) shows the normalized Stokes parameters with the theoretically calculated values of Point B in Fig. 12(a), which proves the method again. Four spectral images in different spectral band with the same Stokes parameter S0 are shown in Fig. 13(b). And four Stokes parameters images at 540 nm of the fruits are shown in Fig. 13(c).
5.3 Error analysis
The error of the recovery spectropolarizaton information is mainly from followed aspects. The instrument error of the Fourier transform spectrometer reduces the accuracy of the recovery Stokes spectra. The setup is set up by ourselves with low cost components. The camera is IMPERX IGV-B0620 digital Camera which is an industrial camera. Compared to scientific cameras, the SNR of the industrial camera is low, especially in the condition of weak light intensity. Besides, the spectral intensity at both ends is lower than the middle spectrum intensity as shown in Fig. 10(a). So the SNR of the ends of the spectrum is low and the recovery spectral error is large. In addition, the response spectral band of the beamsplitter is 450nm~650nm. The difference of the s light splitting ratio and p light splitting ratio is big at both ends of the spectrum, which increases the error of the recovery spectrum.
Besides, the minor shock of the rotator increases the error when the Sagnac lateral shearing splitter rotates. The interferometric images captured by the system are not fixed due to the minor shock. There is a tiny movement between them, which brings aliasing between the fringe patterns of neighbour points and increases the error of the recovery spectrum.
Lastly, the cross-talk of the four channels can also increase the error of the experimental result. The cross-talk is mainly from the errors of the polarization elements, the error of the aberration of the imaging system and the error of the different OPD of the four polarization channels.
In conclusion, we propose a complete high throughput full Stokes Fourier transform imaging spectropolarimeter based on aperture division. Four polarized spectra are fed into the Fourier transform spectrometer by changing the angles of the polarized elements instead of division of interferometric fringe. The four polarized spectra are separated without spatial aliasing, so not only broad-band spectrum but narrow-band spectrum can be detected and the spectral resolution is increased. The retardance of the retarder is optimized using the condition number of the system matrix. An improved polarization modulator is designed to introduce the channeled spectra at once by the way of aperture division. And the advantages of the modulator are presented. Besides, the system structure and analysis are outlined. Two experiments have been completed to prove the good performance of the method with the error analysis.
It should be noted that the instrument error reduces the accuracy of the recovery Stokes spectra in the proven experiments because of the large instrument noise. And the space resolution of the reconstructed spectropolarimetric images is limited. So we will optimize the optical designs and set up an advanced system with more suitable optics elements later.
The research is partially supported by the National Natural Science Foundation of China (61205016, U123111 2), the Opening Project of Key Laboratory of Astronomical Optics & Technology (Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences), the Postgraduate train innovation project of Jiangsu province (CXZZ13_0193), and the Research Fund for the Doctoral Program of Higher Education of China (20123219120021, 20133219110008).
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