A compact, static hyperspectral imaging linear polarimeter (HILP) based on a Savart interferometer (SI) is conceptually described. It improves the existing SI by replacing front polarizer with two Wollaston prisms, and can simultaneously acquire four interferograms corresponding to four linearly polarized lights on a single CCD. The spectral dependence of linear Stokes parameters can be recovered with Fourier transformation. Since there is no rotating or moving parts, the system is relatively robust. The interference model of the HILP is proved. The performance of the system is demonstrated through a numerical simulation, and the methods for compensating the imperfection of the polarization elements are described.
©2012 Optical Society of America
Polarization information scattered or reflected from targets usually differs from that of background light and thus can offer targets’ surface features, shape, shading, and roughness with higher contrast , while spectral information can tell us about targets’ physical and chemical properties. Imaging spectropolarimeters are valuable technology for performing smart detection and recognition of objects and backgrounds. These techniques cover a wide application in the fields of earth observation, environment monitor, and biomedical diagnosis, etc [2–8]. Stokes imaging spectropolarimetry operating in the passive mode is usually employed for ground, airborne and spaceborne remote sensing. It collects the spectral variation of Stokes parameters coming from the reflection of sunlight by a scene or from its own radiation. The spatial and spectral dependence of Stokes parameters is defined by a series of intensity measurements and is expressed by 
Since conventional imaging spectropolarimeters usually combine polarization switching elements with imaging spectrometers, jitter noise and beam wander from the elements must be minimized [10–12]. Recently, novel imaging spectropolarimeters based on channeled spectrum arouse wide interest due to their snapshot imaging capability [13–16]. Such sensor adds two thick retarders and a analyzer before imaging spectrometers or interferometers, and can determine four Stokes parameters from only a single spectrum. Since the decoded or recorded interferogram usually contains more than three interference channels, the spectral resolution of each spectral Stokes parameters is lower than that of the spectrometer or interferometer. The spectral resolution of the instrument needs to be much higher than the required spectral resolution of the data product to carry the additional polarization information in the spectrum . Recent progress in the Fourier transform imaging spectrometers that based on birefringent interferometers has remarkably enhanced the performance of static, compactness, and robustness, especially polarization preserving capability [17–23]. However, such spectrometers can only acquire a liner polarized component due to the front fixed polarizer . As stated by Tyo et al. , very little circular polarization can be expected in most passive imaging scenarios, so obtaining spectral dependence of linear Stokes parameters would satisfy a number of applications [3, 6].
In this paper, we propose a static hyperspectral imaging linear polarimeter (HILP) that takes advantage of a birefringent interferometer. Notably, it can record the spectral variation of linear Stokes parameters with the interferometer’s high spectral resolution. We describe the configuration and interference model of the HILP in Sec. 2. The simulation for validating the system is implemented in Sec. 3. Sec. 4 proposed several methods to overcome the spectral limitation inherent in the polarization elements, and our conclusion is contained in Sec. 5.
2. Optical layout and interference model
The optical layout of a hyperspectral imaging linear polarimeter (HILP) based on a Savart interferometer is depicted in Fig. 1 . It consists of a pair of Wollaston prisms (WP1 and WP2) with identical apex angles and with varying thickness along y axis. The WP1 contains two orthogonally oriented birefringent crystal prisms with optic axes orientations of 0° and 90° with respect to the x axis, respectively. The optic axes of two orthogonally oriented prisms in the WP2 are oriented at ± 45° respectively relative to the x axis. An achromatic half-wave plate (HWP) with its fast axis oriented at 22.5° is placed behind the WP2. A Savart polariscope (SP), consists of two identical uniaxial crystal plates with orthogonally oriented principal sections, is positioned behind the HWP. The optic axes of the two plates are oriented at 45° relative to the z axis and their projections on the x-y plane is oriented at ± 45° respectively relative to the x axis. A linear analyzer (LA) follows the group with its transmission axis at 90°. A single charge coupled device (CCD) is placed on the back focal plane of two imaging lenses (L2 and L3) with same focal length.
Light from the object is imaged on intermediate image plane M by lens L0 and then collimated by lens L1. The two incident electric field vectors oriented at 0° and 90° and the two that oriented at ± 45° are angularly separated by the WP1 and WP2 along y axis, respectively. The 0° and 90° oriented electric fields are laterally sheared respectively by the SP into two pairs of equal-amplitude but orthogonally polarized components. The ± 45° oriented electric fields are first rotated 45° by the HWP and then laterally decomposed by the SP. All of the sheared directions are parallel to the x axis. The LA extracts the identical linearly polarized components. Each pair of the equal-amplitude polarized components are reunited on the four parts of the CCD camera, and four interference images in the spatial domain can be recorded simultaneously . The optical path difference (OPD) is introduced by the shear of SP, the fringe pattern is similar to that in the Young’s double slit setup and is straight line that parallel to the x axis. Complete interferogram for the same object pixel can be collected by employing tempo-spatially mixed modulated mode (also called windowing mode) [19, 25]. That is for each channel, the system acquires a full two dimensional image per frame, with successive frames associated with different OPD by scanning across a scene. Therefore, an interferogram datacube can be constructed for each pixel in the polarized image, and Fourier transformation of the interferogram is used to reconstruct polarized hyperspectral datacube . The selected magnitude, range, and spacing of the OPD determine spectral range and resolution.
The OPD produced by the SP is variable along x axis and is given byEq. (3) can reconstruct input spectral intensity , and with the same high spectral resolution. According to Eq. (1), the spatial-spectral variation of linear Stokes parameters can be calculated. Although the beam splitting of the WPs is equivalent to that in the conventional division of focal plane polarimeter, polarization response of the CCD cannot be considered, because the last analyzer is fixed.
3. Simulation and analysis
To demonstrate the versatility of the HILP system, a mathematical model for simulation and reconstruction is developed. The simulation considers only the reconstruction of polarimetric spectral data from extended source. The spectral region is 400 nm ~700 nm. The CCD is an 16 bit monochrome camera with a resolution of 512 × 512, and the pixel size is 16 μm × 16 μm. Then each interferogram will occupy 128 × 512 pixels. According to the Nyquist sampling theorem, to avoid spectrum aliasing the sampling interval of the interferogram is no more than δΔ = λmin/2 = 0.2 μm. The maximum OPD is effectively limited by the Nyquist criterion that requires at least two data points per fringe period. Hence, if the interferogram is symmetrically recorded about zero OPD, the maximum OPD is Δmax = 256 × δΔ = 51.2 μm. Correspondingly, the highest spectral resolution with rectangular function is δλ = λ2/2Δmax, which is about 1.6 nm at the wavelength of 400 nm. The number of spectral bands is about 256. To realize compactness of the polarization elements, the WPs and SP can be made of calcite with high transmittance over a wide waveband. If the focal length of the L2 and L3 is f = 80 mm, to fully utilize the spatial resolution of the CCD along the y direction, the apex angle of WPs should be 2° and the corresponding splitting angle is about 0.7°. To achieve maximum OPD along the x direction, the lateral displacement produced by the SP should be d = 1 mm, then the thickness of the single plate is t = 6.5 mm. Therefore, the attainable incident angle of the polarization integration is ± 3° in the x direction and ± 0.7° in the y direction. To increase the field of view, a CCD with larger spatial resolution is necessary.
Figure 2 shows the simulated interferograms for each polarized electric field that emitted from the object. Applying Fourier transformation with hamming apodization, the input and reconstructed spectra of Stokes parameters are depicted in Fig. 3 . As can be seen, the reconstructed data consist with the input data, mainly due to the simulation did not take noise into account. Noise would cause deviations, especially at the edges of the interferogram where the signal is small.
4. Limitation and solution
4.1. Wollaston prisms
The two WPs are used to decompose the unknown input beam into four polarized beams oriented at 0° and 90° and ± 45°, then four interferograms can be obtained along the y axis. Usually the WPs are made of uniaxial birefringent crystal and the splitting angle is variable due to the dispersion in birefringence , which would lead to interference image blurring. Fortunately, the achromatization of the traditional WP by combining two WPs of similar and opposite chromatic dispersions are proposed  and can be used to overcome the above drawback over a spectral region of 400 nm ~700 nm. Such achromatic WP can reduce chromatic variation in the splitting angle by an order of magnitude while retaining key conveniences of WPs, namely high extinction ratios and acute-angle polarization beam splitting.
4.2. Savart polariscope
The SP is used to laterally shear the four polarization beams along the x axis and produce the needed OPD. Since the lateral shear also is the function of the extraordinary and ordinary indices, the OPD would vary as a function of wavenumber. Therefore, Fourier transform variable is nonlinear with respect to wavenumber for the Fourier transformation spectroscopy. However, the dispersion of the birefringent crystal can be compensated by using the Sellmeier equation and a reference wavenumber [16,28]. Recently, we have proposed two effective methods, nonconstrained spectrum reconstruction (NSRM)  and constrained spectrum reconstruction method (CSRM) , to suppress the effects of dispersion. Therefore, calibration of the reference wavenumber can be avoided. Compared with the NSRM method, the CSRM method can effectively handle the effects of the noise and the measurement errors.
4.3. Achromatic half-wave plate
Achromatic HWP is used to change the ± 45° polarized beams emerged from the WP2 into the 0° and 90° polarized beams. Therefore, each beam can be laterally decomposed by the SP into another two polarized beams with equal amplitude, and then high-contrast interferograms can be produced. A commercial achromatic HWP made of Quartz-MgF2 can be used for implementing this goal. As shown in Fig. 4 , the retardance weakly depends on the wavelength over the spectral region of 400 nm ~700 nm . The remainder errors just reduce the visibility of the interferograms, but do not distort the reconstructed spectral profile. In addition, more precise achromatic HWP can be obtained over a wide spectral region by combining three different birefringent material .
4.4. Selectable layout
Usually, the OPD of the Savart interferometer is limited by the field of view. A layout of HILP based on a Wollaston interferometer may be a choice, as depicted in Fig. 5 . The front polarization beamsplitter is a modified Savart polariscope (SP1)  and a conventional Savart polariscope (SP2) . The CCD is placed on the image plane of the L2 instead of the focal plane. However, since a front slit (S) must be used to limit the field of view, instead of a field stop M, the signal-collection capabilities is relatively lower . The OPD of the Wollaston interferometer varies along the x axis, where β is the apex angle of WP. As can be seen, the OPD is the function of apex angle and shift x.
In conclusion, we proposed a static hyperspectral imaging linear polarimeter based on a Savart interferometer. By replacing front polarizer with two Wollaston prisms, the spectral dependence of linearly Stokes parameters emitted from object can be detected by the system. By using a single CCD with simultaneous acquisition, the HILP with high signal-collection capability offers a novel solution to the limitations of conventional spectrometers and polarimeters, especially avoids the perturbation noise and beam wandering produced by polarization switching elements. Since the data are measured with a single CCD, its precision is then expected to be mainly affected by the CCD and photon shot noise. The price to pay is the loss in terms of field of view. The future work on the development of a working HILP systems includes studies of optimal design, simulation, reconstruction techniques, and prototyping of systems for the visible and near-infrared spectral bands.
The work was supported by the 863 Program (2012AA121101), the National Major Project (E0310/1112/JC02), the National Natural Science Foundation (40875013), and the National Natural Science Foundation for Young Scholars of China.
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