1. Introduction

Imaging spectropolarimeters can measure spatial, spectral, and polarized four-dimensional data cubes and provide much information regarding various target features. These instruments have been explored for their potential to improve target recognition capacity in various applications, such as biomedical science, remote sensing, and military surveillance [1–15].

In recent years, various types of FTISPs have been developed. Three commonly used approaches for spectropolarimetric measurement are based on time-division, aperture-division, or channeled modulation. Among time-division methods, Tyo has proposed a variable retardance FTISP that features a four-mirror Sagnac splitter as the angular shearing interferometer [16]. The polarimetric components ahead of the interferometer consist of two voltage-controlled liquid crystal variable retarders, followed by a linear polarizer. However, the narrow slit placed after the objective lens limits this system’s throughput and spatial resolution. Li proposed another time-division FTISP in which a polarization modulator, composed of a rotating retarder and a fixed polarizer, is incorporated into a slitless Sagnac interferometer [17]. This system achieves high spatial resolution at the expense of the required measurement time. Among aperture-division methods, Li has presented a FTISP based on light field imaging that uses a polarization array and a micro-lens array to feed four separate polarized spectra into the Sagnac interferometer simultaneously [18]. This system reduces measurement errors because it has no movable elements; however, its spatial resolution is reduced because of the aperture division. Among the channeled modulation methods, the channeled spectropolarimetric technique was first proposed by Oka [19]. A pair of birefringent retarders and an analyzer are incorporated as a channel phase modulator in front of the dispersive spectrometer. No mechanically movable components are used for the polarization modulation, and the full set of Stokes parameters can be determined at one time from only the single spectrum. Based on this polarization modulation method, Li proposed a static FTISP with a slit and a Wollaston interferometer [20]. Julia proposed another compact infrared hyperspectral imaging polarimeter, in which a pair of Wollaston prisms serve as a birefringent interferometer [21]. Unlike diffraction-based channeled spectropolarimeters, using a Fourier transform spectrometer in channeled spectropolarimetry enables direct measurements of the spectral carrier frequencies containing the Stokes parameters information. Zhang also proposed a similar high throughput FTISP that combines two birefringent retarders with a Savart polariscope [22].

In this paper, we propose a FTISP that combines FLCs and a Wollaston interferometer. To the best of our knowledge, this is the first time FLCs have been used for rapid polarization modulation in the Fourier transform imaging spectrometer to obtain the spectropolarimetric information. The remainder of this report is organized as follows. First, the principle of the proposed FTISP is briefly introduced, and then the optimization method and results for the FLC-PSA, the Wollaston interferometer design, and the method of recovering spectral and polarized information are discussed in detail. Lastly, the results of two experimental demonstrations are presented.

2. Principle of the proposed FTISP

The spectral Stokes parameters $S(\sigma )$ are commonly used to characterize the full set of spectropolarimetric properties. These parameters are defined as $S(\sigma )={[S_0(\sigma )S_1(\sigma )S_2(\sigma )S_3(\sigma )]}^{\text{T}}$, where $\sigma =1/\lambda$ is the wave number, $S_0(\sigma )$ is the total power of the light beam, $S_{\text{1}}(\sigma )$ represents the preference for linear 0° over 90°, $S_{\text{2}}(\sigma )$ represents the preference linear 45° over 135°, and $S_{\text{3}}(\sigma )$ represents the preference right circular over left circular polarization states [23]. The aim of the proposed FTISP, depicted in Fig. 1, is to measure $S(\sigma )$ for each point of the target. The fore-optics includes a telescopic arrangement of an objective lens (L1), a field stop (FS), and a lens (L2), which work together to collimate a light beam and send the collimated beam into the FLC-PSA comprised of FLC1, a half-wave plate (HWP), FLC2, and a quarter-wave plate (QWP). The interferometer, consisting of a polarization beam splitter (PBS), HWP, Wollaston prism (WP), and retroreflector (R), splits the modulated polarized light into two parallel parts [24]. The PBS transmits p-polarized light, whereas the s-polarized light is reflected, and the transmitted linearly p-polarized light is termed the reference axis at 0°. The HWP, with its extraordinary axis at 22.5°, turns the linearly p-polarized light to 45° with respect to the reference axis. The PBS performs the functions of both polarizer and analyzer, and the lens (L3) concentrates the beams on the detector array (D), where they interfere. To assemble the complete interferograms at each object point, the system requires a relative continuous motion to scan the target across the interference patterns.

 

Fig. 1 Schematic of the proposed FTISP using FLCs and Wollaston interferometer.

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This FTISP implementation is based on the principle of quadruple-sampling proposed in this work. The FLC-PSA is modulated into four states in a sampling cycle by changing the fast axis orientation of the FLCs. Therefore, the interference intensity of each object point is modulated by four different polarized states. The four adjacent signals are sampled and resorted into four interferogram groups, and the spectra of the four groups are recovered by the Fourier transform. Using the four group spectra, the full Stokes parameters $S(\sigma )$ are then extracted in various wavenumbers. To obtain the undistorted polarized spectra of each object point, the sampling frequency should be four times the Nyquist sampling frequency corresponding to each polarized state. Because the spatial scanning that generates the optical path difference (OPD) and the FLC-PSA modulation are conducted simultaneously, the signal peaks of the four interferograms have different small offsets in relation to the zero OPD position. These offsets do not affect the accuracy of the spectral recovery because a phase correction is employed.

The fringe patterns $I(\Delta )$ generated in the FTISP can be expressed as [17, 18]

The polarized spectra are recovered by Fourier transform:

With $B(\sigma )={[B_0(\sigma )B_1(\sigma )B_2(\sigma )B_3(\sigma )]}^{\text{T}}$ and $A(\sigma )={[A_0(\sigma )A_1(\sigma )A_2(\sigma )A_3(\sigma )]}^{\text{T}}$, Eq. (2) can be written as

Then the measured Stokes parameters can then be expressed by inversion as

Equation (4) is the system equation of the FTISP, and $A(\sigma )$ is the Stokes measurement matrix describing the FLC-PSA. The optimization of $A(\sigma )$ and the analysis of Δ are detailed as follows.

3. Optimal design for FLC-PSA

3.1 Stokes measurement matrix

Figure 2 shows a schematic diagram of the FLC-PSA. The initial fast axis azimuths of FLC1, HWP, FLC2, and QWP are defined as ${\theta }_{\text{FLC1}}$, ${\theta }_{\text{HWP}}$, ${\theta }_{\text{FLC2}}$, and ${\theta }_{\text{QWP}}$, respectively.

 

Fig. 2 Schematic diagram of the FLC-PSA.

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When a beam passes through the FLC-PSA, the Stokes vector of the transmitted beam is defined as

The LP’s transmission axis is set to 0° with respect to the reference axis, and the LP’s Mueller matrix is

Because FLC1, HWP, FLC2, and QWP are all phase retarders, their Mueller matrices can be expressed as

FLC can be modeled as a switchable retarder with two states because its fast axis has two stable orientations (at 0° and 45°) when the applied voltage changes. Therefore, with two FLCs, four different states can be analyzed by the FLC-PSA. The evolution of the four states is shown in Fig. 3, where $T_n$ represents the n-th measurement (n = 0, 1, 2, 3).

 

Fig. 3 Optical configurations used to achieve four different polarization states.

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The n-th measurement matrix $A_n$ is given by

Table 1. Variations of orientation azimuth for FLC1 and FLC2.

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In general, there are two types of noise that degrade the signal-to-noise ratio (SNR) of the measured Stokes vector: Poisson noise and Gaussian noise [26]. It has been shown that the quality of the measurements is strictly related to the condition number of $A$. By using a well-conditioned matrix $A$, the Stokes parameters $S(\sigma )$ can be retrieved accurately by means of Eq. (4). Therefore, to improve the performance of the FTISP, $A$ needs to be optimized. When the thickness d is determined, the optimization of $A$ reduces to searching the best orientation azimuths $({\theta }_{\text{FLC1}},{\theta }_{\text{HWP}},{\theta }_{\text{FLC2}},{\theta }_{\text{QWP}})$.

3.2 Figures of merit for the optimization

Using reasonable figures of merit is important to achieve optimized azimuths with immunity to both Gaussian and Poisson noises, thereby improving the SNR and minimizing noise propagation on the measured Stokes vector [27–29]. With the aim of minimizing the noise transmitted through the matrix inversion from the vector $B$ to the solution $S$ described in Eq. (4), different types of figures of merit have been demonstrated to be useful in optimizations. Among these, the condition number (CN) is widely used because this figure of merit allows error minimizations for all Stokes vectors. In this work, the condition number is given as $\text{CN}={\Vert A^{}\Vert }_2{\Vert A^{-1}\Vert }_2$. The CN is defined for the Euclidean norm because when $A$ is inverted, the recovery noise of the spectral intensity matrix $B$ is amplified by the CN, destabilizing the calculation, and by minimizing the CN of a set of possible matrices $A$, we obtain the best conditioned $A$, namely that closest to a unitary matrix. The best CN that can be achieved for a full Stokes spectropolarimeter is ${\text{CN}}_{\min }=\sqrt{3}$ [30]. The minimization of CN renders the columns of $A$ more linearly independent; minimizes the Euclidean length of the rows of $A^{-1}$; decreases the noise variance in $S_1$, $S_2$, and $S_3$; increasing the SNR. The error function e is defined as [31]

3.3 Optimization by genetic algorithm

As previously stated, the optimization requires a global search in a four-dimensional parameters space $({\theta }_{\text{FLC1}},{\theta }_{\text{HWP}},{\theta }_{\text{FLC2}},{\theta }_{\text{QWP}})$ to determine the best azimuth combination. In this work, a genetic algorithm (GA) is selected to manage this combinatorial optimization, using e as the objective function in a minimization procedure

Each azimuth combination determines a value of e as calculated by Eq. (12) with the wavelength increasing from ${\lambda }_{\min }$ to ${\lambda }_{\text{max}}$ by a constant wavelength step. The azimuth combination is optimal when the value of e approaches its minimum. The initial population ${\theta }_1,{\theta }_2,\mathrm{...},{\theta }_m$, with population size m is generated randomly. Each phase retarder is assigned an 8-bit gene for each θ by Gray code; hence an azimuth combination is represented by 32 bits of genetic variables. Upon initializing the variation, heritability, and selection procedures, the combination that minimizes e is more likely to be reserved, and the operations are repeated until the optimal combination is determined. After considering the computational resources required, the GA configuration parameters listed in Table 2 were determined to be reasonable for this implementation.

Table 2. GA configuration parameters.

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3.4 Result of the optimal design

Optimizing the design of the FLC-PSA over a broad bandwidth, requires detailed knowledge of the spectral behavior of the phase retarders. The wavelength dependence of the components is considered based on experimental data. In the FLC-PSA configuration, the ordering of the components proceeds from FLC1 ($\lambda /2$ at 470 nm), HWP ($\lambda /2$ at 633 nm), FLC2 ($\lambda /2$ at 470 nm) and QWP ($\lambda /4$ at 633 nm), where HWP and QWP are zero-order wave plates. All retarders were characterized individually in the range of 400–1000 nm by the commercially available Unisel 2 ellipsometer from HORIBA. The optimized azimuths of the four retarders are shown in Table 3, and the optimal ${\text{CN}}^{-1}\text{(}\lambda \text{)}$ plots for each wavelength are shown in Fig. 4.

Table 3. Optimized initial azimuths for the FLC-PSA.

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Fig. 4 Optimized curves of FLC-PSA.

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To further verify the performance of the resulting design, a common graphical method to depict the polarization state is the Poincaré sphere. The position can be obtained within the Poincaré sphere by plotting $S_1/S_0$, $S_2/S_0$, and$S_3/S_0$along orthogonal axes in three-dimensional Euclidean space. The last three elements on each row of $A$ are coordinates of eigenvectors of the polarization detectors in the PSA. These vectors, normalized to unit magnitude, are the three-dimensional coordinates on the surface of the Poincaré spheres [32]. The four rows of $A$ thus specify the four vertices of a tetrahedron inscribed in the sphere, and each of these four vertices corresponds to one of the four FLC-PSA states. The volume enclosed in the Poincaré sphere is maximized when the vertices are positioned to form a regular tetrahedron, and the maximized the volume indicates that $A$ is moving away from singular matrices and approaching unitary matrices. In this work, the maximized volume formed by the regular tetrahedron thus indicates the best conditioned solution that results in the minimum possible CN. This optimization generates a measurement matrix $A_{\text{CN}}$:

The ideal measurement matrix $A^{\ast }$ inscribes a regular tetrahedron [33]:

The coordinates of the tetrahedral vertices are obtained from $A_{\text{CN}}$ and $A^{\ast }$, and the tetrahedrons are plotted upon the Poincaré sphere in Fig. 5(a). The enclosed volume of the two tetrahedrons is calculated as $V_{\text{CN}}=0.4010$ and $V_{\text{ideal}}=0.5132$, respectively. To illustrate the performance over the broad bandwidth, the volume values of the two tetrahedrons at each wavelength are plotted in Fig. 5(b).

 

Fig. 5 Performance of the resulting design. (a) Diagram of optimized tetrahedrons and ideal tetrahedron. (b) Volume curves of tetrahedrons.

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4. Design of Wollaston interferometer

4.1 Theory of lateral shearing interference

A lateral shearing beam splitter is used to generate the OPD for broadband interference. To create an OPD that enables a reasonably high spectral resolution, the inner wedge angle, thickness of the Wollaston prism, mirror offset, and thickness of the retroreflector need to be optimized. To ensure an accurate OPD analysis, the propagation of beams in a single WP should first be analyzed by ray tracing [34]. The WP is assumed to consist of two orthogonally oriented birefringent crystal prisms with fast axis orientations of 0° and 90° with respect to the reference axis, and $c$denotes the inner wedge angle of the WP. The light beam with incident angle $\omega$is divided into two orthogonally oriented polarized light rays: the eo ray and the oe ray. In the WP crystal I, the eo ray is the extraordinary ray (e ray), the polarized orientation of which is 0° with respect to the optical axis; and the oe ray is the ordinary ray (o ray), which is orthogonal with respect to the eo ray. Suppose that the refractive index of air is $n_a$; $n_e(\lambda )$ and $n_o(\lambda )$ are the refractive indices corresponding to the eo and oe rays, respectively; and the angles of the eo ray and the oe ray with respect to the x axis are defined by refraction law as ${\gamma }_{eo\_I}$ and ${\gamma }_{oe\_I}$, respectively:

When the light beam enters the WP crystal II, the eo ray changes from the e ray to the o ray. The angle ${\gamma }_{eo\_II}$of the eo ray with respect to the x axis is related to the refractive angle ${{\gamma }^{\prime }}_{eo\_II}$in crystal II, the refractive index of which is denoted by n_o. The angle${\gamma }_{eo\_II}$ can be calculated out from$n_o\sin {{\gamma }^{\prime }}_{eo\_II}=n_e\sin ({\gamma }_{eo\_I}+c)$:

Meanwhile, the oe ray changes from the o ray to the e ray. Considering that the refractive index varies with the wave normal direction of the oe ray, the refractive angle of the oe wave normal can be written as

The direction of the oe ray is not consistent with the direction of its wave normal. As shown in Fig. 6, the angle of the oe ray with respect to the optical axis ${\gamma }_{oe\_oa}$ can be given by ${\gamma }_{oe\_oa}=\pi /2+c-{{\gamma }^{\prime }}_{oe\_II}$. The angle of the oe wave normal with respect to the optical axis ${\gamma }_{oe\_II}$ is then calculated by the relationship of $\tan {\gamma }_{oe\_II}=(n_o^2/n_e^2)\tan {\gamma }_{oe\_oa}$:

 

Fig. 6 Propagation of the beams in a single WP. (a) Propagation of beams in the WP. (b) Propagation of wave normal in the WP crystal II.

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According to Snell’s law of refraction, the refractive index of the oe wave normal in WP crystal II is ${n^{\prime }}_{oe}=n_o\sin (c+{\gamma }_{oe\_I})/\sin {{\gamma }^{\prime }}_{oe\_II}$. Denoting the angle of the oe ray with respect to the oe wave normal as ${\overline{\gamma }}_{oe\_II}$, which satisfies ${\overline{\gamma }}_{oe\_II}={\gamma }_{oe\_II}-{\gamma }_{oe\_oa}$, the refractive index in crystal II can be expressed as

The beams then exit from the WP into air. The angles of the eo ray and the oe ray with respect to the x-axis are defined by refraction law as${\xi }_{eo}$and${\xi }_{oe}$, respectively:

A schematic diagram of the retroreflector is shown in Fig. 7(a). The beam propagation characteristics in the retroreflector can be equally analyzed as in the equivalent parallel plate. With the vertex O of the retroreflector as the coordinate origin, the perpendicular lines OA, OB, and OC are set as the x-, y-, and z-axes, respectively. The equivalent coordinate system is shown in Fig. 7(b). The plane ABC is the incident plane of the equivalent parallel plate, and the plane A'B'C' is the exit plane. In the retroreflector, the refractive index is$n_r$, and the angles of the eo ray and the oe ray with respect to the x-axis are defined as ${\phi }_{eo}$ and ${\phi }_{oe}$, respectively:

 

Fig. 7 Analysis of retroreflector. (a) Schematic diagram of retroreflector. (b) Equivalent coordinate system of retroreflector.

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The optical paths of the beams in the WP and retroreflector are shown in Fig. 8. The coordinates of the light in the WP crystal I at the incident point A are $(0,y_{\text{A}})$. The thickness of the WP is $d_{\text{W}}$, the distance between the Wollaston prism and the retroreflector is $h_{\text{W}}$, the distance from the vertex P of the retroreflector to the x-axis is $h_r$, and the equivalent thickness of the retroreflector is 2d_r. From these definitions, the following relationships can be obtained:

 

Fig. 8 Beam propagation in the WP and the equivalent parallel plate.

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4.2 Design result and analysis

The material used for the WP is calcite, and its refractive indices are

The inner wedge angle of the WP is $c$ = 3°, and the thickness is $d_{\text{W}}$ = 5.0 mm. The thickness of the retroreflector is $d_r$ = 9.5 mm, the optical aperture is 12.7mm, the constant refractive index is $n_r$ = 1.5168 and the vertex of the retroreflector has a displacement of $h_r$ = 0.48 mm. The distance between the WP and the retroreflector is $h_{\text{W}}$ = 8.0 mm. Using the parameters defined above, the relationship between the OPD and the incident angle can be analyzed by Eq. (24). Figure 9(a) shows that the OPD curve varies with the incident angle in the wavelength range from 400 to 1000nm, and Fig. 9(b) shows the OPD curves at 400, 700, and 1000 nm. The results show a nonlinearity between the OPD and the incident angle. Moreover, a difference between the OPD at different wavelengths can be clearly observed, and the difference is attributed to the chromatic dispersion property of the WP.

 

Fig. 9 2D OPD distribution and curves. (a) OPD variations with the incident angle ω and wavelength λ from 400 to 1000 nm. (b) OPDs of 400, 700, and 1000 nm.

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To recover the nonlinear interference signal accurately, the difference between the OPDs at different bands should be characterized thoroughly. Using the OPD curve $\Delta (\lambda ,\omega )$ at 550 nm as a reference, the OPD squeezing ratio at the wavelength$\lambda$is defined as

Figure 10(a) shows the OPD squeezing ratio over the wavelength range of 400–1000 nm. Figure 10(b) illustrates the 2D distribution of the OPD difference. These figures clearly show that the OPD difference is less than 50 nm over the entire wavelength range. Because the OPD difference is very small, the OPDs of different bands can be considered to have the same degree of nonlinearity even though the OPD curves have unequal squeezing ratios. Therefore, the nonuniform Fourier transform (NUFFT) and the sampling wavenumber calculation by optical path squeezing interferometry can be used to obtain accurate spectra [34].

 

Fig. 10 Squeezing characteristic of OPD. (a) OPD squeezing ratio over the wavelength range. (b) Difference in OPDs for different wavelengths.

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5. Spectrum and polarization recovery

Several steps are performed to retrieve the spectra and Stokes parameters. First, the spectral images are recovered by two steps: the fringe patterns are extracted for a particular object point and then the NUFFT is conducted. In the window scanning mode, the scene appears to shift between subsequent images, while the fringe pattern remains stationary. The FTISP samples the entire set of interferograms for the four polarization states simultaneously in combination with window scanning. A fast sub-pixel image registration is necessary to extract the fringe pattern of a particular object point, following the steps shown in Fig. 11. The regions illustrated in red, yellow, green, and purple represent the four polarized intensities of the given object point. Interferograms named as image 1, image 2, …, image n, are acquired by the detector in the window scanning mode. These images are resorted into four groups, in which the images in each group exhibit the same polarization state. Because the positions of the polarized intensity regions for the given point are different in each group, these regions can be registered to the same coordinates. The four sets of fringe patterns $I(\Delta )$can be obtained after image registration by extracting the polarized intensities of the given point as it appears in different groups, and then the four sets of polarized spectra $B(\sigma )$can be calculated by Eq. (2). With the aim of resolving the nonlinear problems discussed in Sec. 4 and thus speed the computation, the NUFFT using fast Gaussian gridding (FGG) is adopted [35]. Before performing NUFFT, procedures should be implemented to pre-process interference data (i.e., removing the trend term, apodization and phase correction). In addition, because of the nonlinear OPD, the sampling wavenumbers also need to be pre-calibrated. The sampling wavenumbers of the spectra can be calculated by the wavenumber sampling equation in the optical path squeezing interferometry [36]. Because the Stokes measurement matrix$A(\sigma )$has been obtained as described in Sec. 3, the desired Stokes spectra $S(\sigma )$ can be calculated by Eq. (4) combined with $B(\sigma )$.

 

Fig. 11 Extraction of four polarized interferograms for one object point.

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6. Experimental demonstration and results

This section presents imaging results from both laboratory and outdoor tests of the proposed FTISP. In the experimental setup, three objective lenses, each with a focal length of 75 mm, were used: L1, L2, and L3. FLC1 ($\lambda /2$at 470 nm) and FLC2 ($\lambda /2$at 470 nm) were from Meadowlark Optics. The WP was made of calcite, and the inner wedge angle was 3°. The diameter of the retroreflector was 12.7 mm. The detector array used GS3-U3-23S6M-C from Point Gray Research to measure the scene intensity, with a resolution of 1920 × 1200 pixels and pixel sizes of 5.86 μm. The motorized rotation stage was implemented with a Zolix RAK100, which rotated the proposed FTISP continually with rotation angle intervals of 0.27°. The polarization states of the FLC-PSA were changed at a frequency of 76 Hz. When the FLC-PSA was sent a control signal, the detector was also controlled by an external trigger signal to synchronously capture an image. For each of the system’s polarization states, 1200 interference images were acquired; therefore, 4800 frames captured the four polarization states, and the complete acquisition time was approximately 64 seconds. By calibrating the wavelength, the spectral resolution of the system was found to be approximately 341.9 cm⁻¹ (7.0 nm at 453 nm, 14.4 nm at 649 nm). In the laboratory experiment, two pairs of 3D glasses illuminated by an incandescent lamp were used as the imaging targets. The interferometric images with the four polarization states are shown in Fig. 12(a), and the three recovered spectral images of the Stokes parameter$S_0$rendered in the spectral colors are shown in Fig. 12(b). The parameters $(S_1,S_2,S_3)$ are normalized to the value of$S_0$, so that the first parameter $S_0$ is equal to unity, and $(S_1,S_2,S_3)$ have values between −1 and 1. The Stokes parameter images normalized at 649 nm are shown in Fig. 12(c). The spectral signatures of the glass frames can be clearly distinguished. The orange 3D glasses obviously show stronger linear polarized information, and the circular polarized information implies that there are retarder films involved in the 3D glasses. The Stokes images indicate the good performance of the proposed FTISP.

 

Fig. 12 Experiment on two pairs of 3D glasses. (a) Interferometric images with four polarization states. (b) 3D glasses and three recovered spectral images. (c) Four Stokes parameter images at 649 nm.

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To further verify the imaging performance of the FTISP, an additional outdoor experiment was performed in which several cars were imaged by the FTISP in the afternoon on a clear day. The recovered spectral images of Stokes parameter$S_0$are shown in Fig. 13(a), and the color-fusion image is shown on the bottom right corner in Fig. 13(a). The Stokes parameter images normalized at 453 nm are shown in Fig. 13(b). The spectral properties can be clearly recognized from the intensity variations of the red car, and the white parts of the cars show stronger linear polarized information.

 

Fig. 13 Outdoor experiment on cars. (a) Spectral images of the scene at different wavelengths, and a fusion spectral image of 3 channels (RGB). (b) Four Stokes parameter images at 453 nm.

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7. Discussion

The experiments above show the effectiveness of the proposed FTISP, including several promising features. First, the FLC is an attractive alternative to the classical time-division based FTISP. An advantage of the physical characteristics of the material is that the orientations of the optical axes of the two FLCs can be promptly transformed without mechanically moving, requiring less than 100 μs, which is a significant improvement in comparison with liquid crystal variable retarders. By incorporating this material into a Fourier transform spectrometer, the image acquisition can easily comply with the kHz frame rate. This advantage can significantly increase the measuring speed.

Second, the interferometer that combines the WP with the retroreflector exhibits good lateral shearing performance for high interference modulation. The parallelism of the shearing beams is with respect to only the right-angle errors of the retroreflector, and because the error in manufacturing right angles can be guaranteed to be less than approximately 0.00056°, the offset error of the right angle is less than 0.00275°. The angle between the two coherent interfering beams is thus less than approximately 0.00072° as calculated from Eqs. (16)-(21). This error is far less than the incident angle of approximately 0.00447° with respect to a single pixel of the detector. The beam can still converge at the same pixel after passing through the interferometer, and therefore, the presented interferometer demonstrates high interference modulation that can reduce problems associated with optical alignment, which helps the FTISP maintain imaging performance reliability.

Finally, compared with the conventional channeled FTISP, there is no aliasing between the four separate polarized spectra for the proposed FTISP. Rapid time-division polarization modulation brings a higher spectral resolution without losing measurement efficiency. Unlike the channeled FTISPs that suffer from low SNRs in some channel fringes with small amplitude ranges, the proposed FTISP easily enhances the SNR through the single channel fringe.

Conventional time-division FTISPs have two acquisition modes: stepping mode and repeating mode. In the stepping mode, four interferometric images of the same object point with different polarization states are captured in each motion step. The system remains stationary when the four images are being acquired. In the repeating mode, the system needs to repeat the complete sampling operation for each polarization state, and therefore has to perform scanning four times. However, when the FTISP is installed in an aircraft, these data acquisition modes cannot function correctly because of the aircraft’s continuous motion. The proposed FTISP is not only capable of measuring the target in both of these acquisition modes, but also can operate continuously when installed in an aircraft. This feature is achieved by the rapid polarization modulation and the quadruple-sampling principle discussed in Sec. 2, overcoming the limitations of the classical time-division systems in airborne applications.

8. Conclusion

In summary, we present a time-division FTISP for acquiring spatial, spectral, and polarized information. The proposed imaging spectropolarimeter employs two FLCs and a Wollaston interferometer. The fast axes of the FLCs are controlled to switch quickly without mechanically moving, enabling the PSA to modulate the full set of Stokes parameters rapidly. The interferometer combines a WP with a retroreflector and is shown to demonstrate good lateral shearing performance, enabling high interference modulation and facilitating system alignment. The feasibility and measurement performance of the proposed FTISP were verified through both laboratory and outdoor experiments. Benefiting from the rapid polarization modulation and the quadruple-sampling principle, the proposed FTISP offer significant advantages, such as a much faster measuring speed, high throughput, improved SNR, and high spectral resolution without aliasing. Therefore, the proposed FTISP is considered very suitable for measuring spectral and polarized information for ground and airborne applications.