1. Introduction

Fourier transform imaging spectrometry (FTIS) is an important technology for subdivision spectral imaging. In recent years, various types of FTIS instruments have been developed, including those based on Michelson interferometers [1–7], Sagnac interferometers [8-17], Mach–Zehnder interferometers [18,19], Fabry–Perot interferometers [20-22], and birefringent interferometers [23-28]. These instruments have been explored for their potential to improve target recognition capacity in various applications, such as, remote sensing, environmental monitoring, and biomedical diagnosis.

Birefringent FTISs have attracted particular attention because of their compact structure and low complexity. The key components of birefringent interferometers are based on either Wollaston or Savart prisms. Harvey proposed a birefringent interferometer for hyperspectral imaging that consists of a pair of Wollaston prisms with identical apex angles [29]. Li proposed another FTIS with a new lateral shearing birefringent interferometer that includes a Wollaston prism and a rotating retroreflector [30]. This equipment splits an incident light beam into two shearing parallel components to obtain interference fringe patterns of an imaging target, which helps to reduce problems associated with optical alignment and manufacturing precision. However, these two types of birefringent interferometers based on Wollaston prisms generate large nonlinear optical path differences (OPDs). When the interference data is reconstructed by directly adopting a conventional Fourier transform, the nonlinear OPD leads to broadening and distortion of the restored spectrum. Therefore, these interferometers are unsuitable for high-precision spectral measurements.

Like Wollaston interferometers, the Savart interferometers exhibit the similar characteristic of beam shearing. Zhang proposed a birefringent FTIS based on a conventional Savart polariscope (CSP), which consists of two Savart plates with identical parameters [31]. The CSP not only generates OPD associated with the field of view (FOV) for broadband interference, but also adjusts the zero-order fringe pattern to the center of the FOV by compensating the constant term OPD. Nevertheless, large differences between the total OPDs in different vertical FOVs are introduced in the CSP when the square term OPDs of each Savart plate are combined. The total OPD is the difference between the maximum OPD and the minimum OPD. Because there is a reciprocal relationship between the total OPD and the spectral resolution, the difference in the total OPD generated by the CSP can lead to inconsistent spectral resolution and wavelength positions at different positions on the image plane. As a result, it is also unfavorable to slice the representation of the 3D spectral data cube.

Based on the previous CSP research, several improved shearing interferometers have been studied. For instance, Zhang employed a modified Savart polariscope (MSP), which consists of two identical Savart plates and an achromatic half-wave plate [32]. The principal sections of the two Savart plates are parallel, and their optic axes are perpendicular to each other. An achromatic half-wave plate is interposed between the two Savart plates to cause a π/2 rotation of the linear polarization states for the incident rays. Theoretically, the difference between the total OPDs can be suppressed by introducing the achromatic half-wave plate into the CSP to eliminate its square term OPD. However, it is very difficult to design and manufacture an achromatic half-wave plate with a sufficiently large aperture, wide FOV, and wide spectral range. To avoid using an achromatic half-wave plate, Mu proposed an achromatic Savart polariscope (ASP) with a four-plate structure for compensating the square term OPD. The ASP is formed by combining two Savart polariscopes that are made of negative and positive birefringent materials, respectively [33,34]. Li also proposed another field-compensated interferometer based on two Savart polariscopes for Fourier transform hyperspectral imaging [35]. In addition, Zhang also proposed a Savart polariscope that achieved a wide FOV (WSP) with the four-plate structure. The WSP comprises a CSP and a pair of uniaxial crystal plates that are cut parallel to the optic axis and are fabricated from birefringent materials with opposite signs relative to that of the CSP, with their principal sections perpendicular to each other [36]. This method can suppress the nonlinear OPDs and the difference between the total OPDs. However, the OPD characteristics in birefringent interferometers are closely related to the azimuth angle of the fast axis for each birefringent crystal, and to achieve the optimal OPD conditions, the alignment accuracy of their fast axes must be strictly guaranteed. Therefore, increasing the number of birefringent crystals enhances the alignment difficulty and can introduce error in the OPD evaluation. Both the ASP and the WSP use four birefringent crystals, which complicates the alignment of the interferometer. Thus, designing a birefringent interferometer with fewer birefringent crystals, less thickness, and less complexity would significantly facilitate resolving the difference between the total OPDs.

In this paper, we present a FTIS with a compact birefringent interferometer (CBI), which consists of a shearing plate (SP) and a compensation plate (CP). To the best of our knowledge, it is an effective way to suppress the difference of the total OPDs for the birefringent FTIS only employs two birefringent crystal plates. The remainder of this report is organized as follows. First, the principle of the CBI is briefly introduced, and then the theoretical deductions of the OPDs for the CBI are detailed. The distribution characteristics of the OPDs are investigated both by simulation analyses and by verification experiments. Lastly, to verify the measurement performance of the presented method, the results of two experimental demonstrations are presented.

2. Theory of the compact birefringent interferometer (CBI)

A schematic of the CBI for hyperspectral imaging is shown in Fig. 1. The incident light is collimated by the fore-optical system, consisting of the objective lens (L1), field stop (S), and collimation lens (L2). The collimated beam then feeds into the birefringent lateral shearing splitter, which consists of a linear polarizer (P1), shearing plate (SP), compensation plate (CP), and analyzer (P2). The transmission axis of P1 is at 45° with respect to the X-axis; as a result, the linearly polarized light emerging from P1 is at 45° to the X-axis. The SP, with fast axis orientations at 45° relative to the X-axis and perpendicular to the Y-axis, is installed behind P1. The linearly polarized light is divided into two orthogonally-oriented polarized light rays with equal amplitude: the eo ray and the oe ray. Because the CP’s fast axis is parallel to the Y-axis and orthogonal to the SP’s fast axis, the eo ray changes from an extraordinary ray (e ray) in SP to an ordinary ray (o ray) in CP, and the oe ray is orthogonal with respect to the eo ray. Passing through P2, the eo and oe rays with the same propagation direction are resolved into linearly polarized light with the same polarized orientation. They are then focused by the imaging lens (L3) onto the same position at the detector (D). Because the OPD of the birefringent interferometer is a function of the position on the image plane, the fringe patterns are produced in the spatial domain and are superimposed on the image. To assemble all of the fringe patterns for each object point, the system requires a relative continuous motion to scan the target across the fringe patterns. Therefore, in window scanning mode, a series of time-sequential interferograms are recorded, where the fringe patterns are fixed and the target is moving. The spatial-spectral data cube of the target can then be obtained by employing Fourier-transform-based spectral recovery of the fringe patterns extracted from the time-sequential interferograms.

 

Fig. 1 Schematic of the compact birefringent interferometer (CBI) for hyperspectral imaging.

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The interference pattern can be calculated by integrating the total flux for all frequencies, and it is expressed as

3. Analysis of the CBI

3.1 Optical path difference (OPD) of the CBI

The CBI is used to generate the OPD for broadband interference based on birefringence characteristics of uniaxial crystals. To analyze the OPD accurately, the exact expressions of the OPD produced by the SP and the CP in the CBI are deduced by the ray-tracing method [37].

In general, when a light beam enters a uniaxial crystal, it is divided into an o ray and an e ray. The refractive index of the o ray is a constant, but the refractive index of the e ray is related to the direction of its wave normal. According to Snell’s law, the refractive angles of the o ray and e ray wave normals, $r_o$ and ${\theta }_e$ respectively, satisfy the following relationships:

The coordinate system established in SP is shown in Fig. 2. The Z-axis is parallel to the normal of the crystal surface and the XZ-plane is the principal section of the crystal. The direction cosines of the fast axis are ${\widehat{w}}_{{\rm I}}=(\sin {\beta }_{{\rm I}},0,\cos {\beta }_{{\rm I}})$, where ${\beta }_{{\rm I}}$ is the angle between the SP’s fast axis and the normal of the crystal surface. The oe ray $S_{oe\_I}$ and its wave normal $K_{oe\_I}$ have the same direction, but the directions of the eo ray $S_{eo\_I}$ and eo wave normal $K_{eo\_I}$ are different: the direction cosines of the eo wave normal are ${\widehat{K}}_{eo\_{\rm I}}=(\cos {\omega }_{{\rm I}}\sin {\theta }_{eo\_{\rm I}},\sin {\omega }_{{\rm I}}\sin {\theta }_{eo\_{\rm I}},\cos {\theta }_{eo\_{\rm I}})$, where ${\omega }_{{\rm I}}$ is the angle between the incident plane and the principal section, and ${\theta }_{eo\_{\rm I}}$ is the angle that the eo wave normal makes with the optical axis. Thus, the refractive angle ${\theta }_{eo\_{\rm I}}$ of the eo wave normal in SP is determined by

 

Fig. 2 Wave normals and rays in the uniaxial crystal SP.

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Analyzing the exact optical path of the eo and oe rays with the ray-tracing method, the OPD between the two rays can be expressed as

According to Eqs. (5)‒(7), the OPD generated by the SP is given by

On the basis of Eqs. (2)‒(4) and ${\beta }_{{\rm I}}=$45°, the cotangents of the refractive angles for the oe and eo wave normals are respectively expressed as

Combining analysis of Eqs. (8)‒(10), the OPD generated by the SP can be expressed as the sum of three parts:

In the above equations, the constant term $k_{0\_{\rm I}}$ in the expression for the OPD is unrelated to the incident angle $i$, which determines the position of the zero-order fringe. With regard to the incident angle, the primary term $k_{1\_{\rm I}}\sin i$ and square term $k_{2\_{\rm I}}{\sin }^{\text{2}}i$ determine the distribution of the OPD on the image plane. The primary term $k_{1\_{\rm I}}\sin i$, which has a large magnitude, significantly contributes to modulating the OPD. Owing to the presence of the constant term $k_{0\_{\rm I}}$ in the OPD, the zero-order fringe pattern deviates from the center of the FOV. Moreover, the square term $k_{2\_{\rm I}}{\sin }^{\text{2}}i$ in the OPD introduces a nonlinear OPD that cannot be ignored. Therefore, to adjust the position of the zero-order fringe pattern and suppress the nonlinear OPD, another birefringent crystal plate, CP, is employed to compensate the constant term OPD and the square term OPD of the SP.

To analyze the characteristics of the OPD generated by the CP, the same coordinate system used for SP is established in CP as shown in Fig. 3. The Z-axis is parallel to the normal of the crystal surface, and the YZ-plane is the principal section of the crystal. The CP’s fast axis is parallel to the Y-axis, and its direction cosines are ${\widehat{w}}_{{\rm I}{\rm I}}=(0,1,0)$. The direction cosines of the oe wave normal are ${\widehat{K}}_{oe\_{\rm I}I}=(\sin {\omega }_{{\rm I}{\rm I}}\sin {\theta }_{oe\_{\rm I}{\rm I}},\cos {\omega }_{{\rm I}{\rm I}}\sin {\theta }_{oe\_{\rm I}{\rm I}},\cos {\theta }_{oe\_{\rm I}{\rm I}})$, where ${\omega }_{{\rm I}{\rm I}}$ is the angle between the incident plane and the principal section, and ${\theta }_{oe\_{\rm I}{\rm I}}$ is the refractive angle of the oe wave normal in CP. Using a similar analysis method as was applied for the SP, the OPD generated by the CP can be given by

 

Fig. 3 Wave normals and rays in the uniaxial crystal CP.

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Combining analysis of Eq. (15)‒(17), the OPD generated by the CP can be expressed as the sum of only two parts

In the above equations, the OPD ${\Delta }_{{\rm I}{\rm I}}$ includes only the constant term $k_{0\_{\rm I}{\rm I}}$ and the square term $k_{2\_{\rm I}}{\sin }^{\text{2}}i$, which are mainly used to compensate the OPD generated by the SP. Numerical calculations and analysis of the OPD demonstrate that the signs of $k_{0\_{\rm I}}$ and $k_{0\_{\rm I}{\rm I}}$ are opposite. In particular, the signs of $k_{2\_{\rm I}}{\sin }^{\text{2}}i$ and $k_{2\_{\rm I}{\rm I}}{\sin }^{\text{2}}i$ are also opposite. This result means that the OPD generated by the CP cannot only adjust the zero-order fringe pattern to the center of the FOV by compensating the constant term OPD of the SP, but can also suppress the nonlinear OPD and the inconsistency in the total OPD in different vertical FOVs by compensating the square term OPD of the SP.

The above analysis shows that the ${\Delta }_I$ and ${\Delta }_{II}$ are functions of parameters, such as, the angles $i$, ${\omega }_{{\rm I}}$, ${\omega }_{{\rm I}{\rm I}}$, wavenumber $\sigma$, and thicknesses $d_{{\rm I}}$ and $d_{{\rm I}{\rm I}}$. Because the directions of the fast axes of the SP and CP are orthogonal to each other, the relationship ${\omega }_{{\rm I}{\rm I}}={\omega }_{{\rm I}}+\pi /2$ can be obtained. Defining the variable symbol $\omega ={\omega }_{{\rm I}}$, the OPD generated by the CBI can be expressed as

According to the geometric relation shown in Fig. 4, the angles $i$ and $\omega$ can be written as $i={\tan }^{-\text{1}}\left({(x^2+y^2)}^{1/2}/f\right)$ and $\omega ={\tan }^{-\text{1}}(y/x)$, respectively, where $f$ is the focal length of the lens L3. Thus, the OPD $\Delta (x,y,\sigma ,d_{{\rm I}},d_{{\rm I}{\rm I}})$ for the position $(x,y)$ on the image plane can be obtained by Eq. (21).

 

Fig. 4 Distribution of pixel coordinates on the image plane.

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Denoting $\delta \sigma$ and ${\Delta }_{\max }$ as the wavenumber resolution and total OPD, respectively, and employing the physical equation $\delta \sigma =1/{\Delta }_{\max }$ from classical Fourier transform spectrometry [38], the wavenumber resolution $\delta \sigma$ can be calculated as

To adjust the zero-order fringe pattern of the center wavelength ${\lambda }_c$ to the center of the FOV, the constant term OPD of the ${\lambda }_c$ should satisfy $k_{0\_I}+k_{0\_II}=0$. Therefore, the thicknesses of the SP and the CP are designed to fit the following condition:

Combining analysis of Eq. (22) and Eq. (23), the thicknesses $d_{{\rm I}}$ and $d_{{\rm I}{\rm I}}$ can be obtained when the refractive indices $n_o$ and $n_e$, focal length $f$, dimension $l$, and wavenumber resolution $\delta \sigma$ have already been given.

3.2 Analysis of the OPD

To intuitively analyze the OPD distribution characteristics of the presented interferometer, the OPDs for several interferometers were calculated and compared, including the CSP, the ASP, the WSP, and the presented CBI. The effective spectral range was from 400 to 1000 nm, and the wavenumber resolution was 91.2 cm^‒1 (5.37 nm) at the center wavelength 700 nm. The pixel number and pixel size of the detector array were 1024 × 1024 and 6.5 μm, respectively. The focal length of the lens L3 was 75 mm. The material used for the SP and the CP was calcite, and its refractive indices corresponding to the range of 400−1000 nm can be calculated by the Sell Meier equation [33]:

According to the OPD formulas of the four interferometers, the thickness of each crystal plate can be designed as shown in Table 1. Compared with the ASP and WSP, the total thickness of the CBI was the smallest, which is conducive to miniaturizing the interferometer. The OPD distributions on the image plane for the several interferometers were obtained by analyzing the relationships between the OPD and the incident angle, based on the parameters of the birefringent crystal plates. The differences of the OPDs in the horizontal FOV, the total OPDs in the vertical FOV, and the dispersion of the total OPDs for these four interferometers are analyzed and compared below.

Table 1. Thickness (mm) of these four interferometers

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3.2.1 OPD distributions

The different OPD distributions for the CBI, CSP, ASP, and WSP refer primarily to the nonlinear OPDs for certain wavelengths and vertical FOVs. The nonlinear OPDs can be given by eliminating the optimal linear OPDs from the original OPDs. The distributions of the nonlinear OPDs at 700 nm for the CBI and other interferometers are shown in Fig. 5. Figure 6 shows the optimal fitting curves of their linear OPDs.

 

Fig. 5 Distribution of the nonlinear OPD for the CBI and three other interferometers.

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Fig. 6 Optimal fitting curves of the linear OPD for the CBI and three other interferometers.

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The nonlinear OPD can lead to linewidth broadening of the restored spectrum. Therefore, the effect of the nonlinear OPDs for these four interferometers should be analyzed and discussed. The maximum nonlinear OPDs of these four interferometers are relatively small, as the value is 65.4 nm for the CBI. The nonlinear OPD of the presented CBI can clearly be treated as a quasi-zero OPD, because it is much smaller than $\lambda /2$ [39]. Therefore, the effect of nonlinear OPD on the CBI’s spectral reconstruction can be neglected.

3.2.2 Total OPDs

The total OPDs for the CBI, CSP, ASP, and WSP refer primarily to the linear OPDs for certain wavelengths and vertical FOVs. According to the optimal fitting curves of the linear OPDs in Fig. 6, the difference between the total OPDs for the CSP in different vertical FOVs is quite clear, whereas the differences for the ASP, the WSP, and the CBI must be analyzed in detail by numerical calculation. Table 2 lists the total OPDs of the four interferometers at 400 nm, 700 nm, and 1000 nm, which are on different rows of the image plane. From the analysis of the CSP presented in Table 2, the total OPDs of the upper FOV are clearly greater than the total OPDs of the lower FOV, with a maximum difference at 400 nm of 5.232 μm, and differences of 4.868 μm and 4.743 μm at 700 nm and 1000 nm, respectively. For the other interferometers, the differences of the total OPDs for the ASP, the WSP, and the CBI are clearly improved over that of the CSP. The maximum differences of the total OPDs for the ASP and the WSP are 0.273 μm and 0.128 μm, respectively. The maximum difference of the total OPD for the CBI is 0.116 μm, which is the smallest of these four interferometers.

Table 2. Total linear OPD (μm) at different rows for the four interferometers.

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Differences between the total OPDs can lead to differences in the spectral resolution. The spectral resolutions of these four interferometers are listed in Table 3. For the CSP, the spectral resolution of the upper FOV is clearly higher than the spectral resolution of the lower FOV, with a maximum difference of at 400 nm of 0.060 nm, and differences of 0.199 nm and 0.417 nm at 700 nm and 1000 nm, respectively. For the other interferometers, the maximum differences of the spectral resolution for the ASP and the WSP are 0.012 nm and 0.010 nm, respectively. The maximum difference of the spectral resolution for the CBI is 0.009 nm, which is the smallest of these four interferometers.

Table 3. Spectral resolution (nm) at different rows for the four interferometers.

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Compared to its influence on spectral resolution, the difference between the total OPDs has an even greater impact on the wavelength positions of the restored spectra. Table 4 lists the wavelength positions of the restored spectra for the four interferometers at different positions on the image plane. The maximum difference of the wavelength position for the CSP at 400 nm is 16.16 nm, whereas the differences are 28.59 nm and 41.14 nm at 700 nm and 1000 nm, respectively. The maximum differences of the wavelength positions for the ASP and the WSP are 1.16 nm and 0.87 nm, respectively. The maximum difference of the wavelength position for the CBI is 0.85 nm, which is also the smallest of these four interferometers. From the results presented in Tables 2 through 4, the difference between the total OPDs for the CBI has the least influence on both the spectral resolution and the wavelength position of the restored spectra.

Table 4. Wavelength position (nm) at different rows for the four interferometers.

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In addition, as shown in Table 2, there are also differences between the total linear OPDs at different wavelengths for the same vertical FOV, which is mainly caused by the dispersive effect of birefringent crystals. The OPD $\Delta$ is related to the wavenumber $\sigma$ in Eq. (21). For a certain vertical position y, the total linear OPDs at different wavelengths are unequal, because the refractive indices of the birefringent crystals vary with the wavelength. Although the dispersion difference of the total linear OPD for the CBI is the smallest among these four interferometers, as shown in Table 2, the difference between the total linear OPDs at 400 nm and 700 nm is still 8.832 μm, whereas the difference is 2.934 μm between 700 nm and 1000 nm. Thus, adopting a conventional Fourier transform directly in the spectral reconstruction process causes a deviation of the wavelength position. To solve this problem, the wavelength position of the restored spectra should be corrected by the wavenumber sampling equation in optical path squeezing interferometry [40].

3.2.3 Zero-order fringe distributions

The constant term OPD of the CBI designed according to Eqs. (22) and (23) can be eliminated only at the center wavelength as shown in Fig. 7. The positions of the zero-order fringe patterns for different wavelengths are not coincident, because their constant term OPDs are different. As a result, the fringe patterns of white light are asymmetric. Figure 8 shows the asymmetric fringe patterns of the reference solar spectra from 400 nm to 1000 nm. The density of the fringe patterns on the left is clearly higher than that on the right. Nonetheless, the phase correction can effectively eliminate the influence of the asymmetric fringe patterns on the spectral measurement accuracy.

 

Fig. 7 Constant term OPD at different wavelengths for the CBI.

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Fig. 8 Asymmetric fringe patterns of the reference solar spectra.

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4. Experiment demonstration and results

4.1 OPD verification and wavelength correction

To test the distribution characteristics of the OPDs for the CBI, a verification experiment was performed using the simulated parameters. The focal lengths of lenses L1, L2, and L3 were 75 mm each. The birefringent crystal plates were made of calcite, and the thicknesses of the SP and the CP were 11.55 mm and 6.24 mm, respectively. The OPDs of the CBI were analyzed with three lasers of 405 nm, 650 nm, and 980 nm. As a light source, the expanded laser beam from a diffuser entered the CBI for interference. The high linearity and contrast of the fringe patterns produced by the 650-nm laser are shown in Fig. 9.

 

Fig. 9 Fringe patterns generated by the CBI for a 650-nm laser.

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The intensities were extracted from the interferogram at the 1st, 512th, and 1024th rows. The OPD distributions can be obtained by processing the intensity information with the phase extraction method. Figure 10 shows the nonlinear OPDs of the three wavelengths, and Table 5 lists their total linear OPDs and maximum nonlinear OPDs. Owing to the manufacturing error of the birefringent crystal and the aberration of the imaging lens, there were some deviations between the experimental values and the theoretical values of the total linear OPDs and the nonlinear OPDs. However, the maximum deviations of the total OPDs and the nonlinear OPDs were less than 0.361 μm and 0.090 μm, respectively.

 

Fig. 10 Experimental nonlinear OPD results for the CBI at three wavelengths.

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Table 5. Experimentally determined total linear OPDs and maximum nonlinear OPDs of the CBI.

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To restore the spectra with high precision, the wavelength positions of the spectra should be corrected to address the dispersive effects of the birefringent crystals. For a certain wavelength $\lambda$, the differences among the total linear OPDs of different vertical FOVs are very small. Therefore, the total linear OPD of the wavelength $\lambda$ can be represented by the mean value $L(\lambda )$ of all total OPDs. The curve of the total linear OPDs for the wavelength range from 400 nm to 1000 nm is shown in Fig. 11(a). According to the wavenumber sampling equation from optical path squeezing interferometry, the corrected wavelength position must satisfy $k=L(\lambda )/\lambda$, where $k$ is the spectral sampling number. As shown in Fig. 11(b), the difference between the original wavelength position and the corrected wavelength position is quite clear, indicating that the wavelength position must be corrected.

 

Fig. 11 Correction of the wavelength position. (a) Curve of the total linear OPDs. (b) Wavelength position of original result and corrected result.

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4.2 Spectral measurements

To verify the accuracy of the presented method, an experiment was conducted to compare the spectral measurements of the presented method (spectral range: 400–1000 nm, spectral resolution: 91.2 cm^‒1 @ 700 nm) and a commercial spectrometer, the Ocean Optics Flame-T (spectral range: 340–1040 nm, spectral resolution: 0.19 nm). Figure 12(a) shows the interferogram of a white light source collected by the CBI, and Fig. 12(b) details the fringes shown in the red rectangle in Fig. 12(a). The intensity information of one row from the interferogram is shown in Fig. 12(c), where the asymmetric distribution coincides with the theoretical result presented in Section 3.2.3. The spectral measurement results obtained using the CBI and the Flame-T are shown in Fig. 13. The two curves are quite consistent, indicating that the presented method can feasibly recover spectral data.

 

Fig. 12 Fringe patterns generated by the CBI for a white light source.

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Fig. 13 Spectral measurement results using the CBI and the Flame-T.

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4.3 Hyperspectral imaging experiments

To further verify the imaging performance of the presented method, another experiment was performed in which a high-precision motorized stage was used to rotate the CBI continually. A sequence of interferograms of Mount Zijin (Nanjing, China) was recorded when the stage was rotating. Figure 14(a) shows the image of the mount, on which we can find the Purple Mountain Observatory and the Zhongshan Mountain National Park. Two interferograms at different sampling positions are shown in Figs. 14(b) and 14(c), where the fringe patterns were fixed and the target was moving in the horizontal direction. Figure 14(d) details the fringe patterns from the white rectangle shown in Fig. 14(c). To recover the spectral images, an image registration with sub-pixel accuracy was employed to extract the interference signal at each target point. The spectra were then calculated using the fast Fourier transform (FFT) of the interference data, and the wavelength positions were calculated by the wavelength correction shown in Fig. 11(b). The recovered spectral images of 451 nm, 629 nm, 694 nm, and 754 nm are shown in Figs. 15(a)−15(d), and the color-fusion and infrared-enhanced images are shown in Figs. 15(e) and 15(f), respectively. This experiment demonstrates the ability of the presented method to acquire 3D spatial and spectral data cubes.

 

Fig. 14 Outdoor imaging scene and corresponding interferograms.

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Fig. 15 Spectral images of Mount Zijin.

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5. Discussion

Compared with the MSP, ASP and WSP, the presented CBI employs only two birefringent crystal plates without a broadband achromatic half-wave plate. In addition to its advantages of compact structure and low complexity, the CBI can also effectively improve the OPD distribution characteristics.

First, the presented CBI can suppress different OPD distributions. The OPDs generated by the CP cannot only adjust the zero-order fringe pattern to the center of the FOV by compensating the constant term OPD of the SP, but can also suppress the nonlinear OPDs by compensating the square term OPD of the SP. Both the ASP and the WSP employ four birefringent crystals. Although these two schemes demonstrate good performance with regard to suppressing nonlinear OPDs, the increase in the number of birefringent crystals incurs alignment difficulty and nonlinear OPD error. Moreover, despite the birefringent crystal manufacturing error and the aberration of the imaging lens, the experimental nonlinear OPDs of the CBI were still less than 91 nm, and because this value is much smaller than $\lambda /2$, the nonlinear OPDs of the CBI can be treated as a quasi-zero OPD. Thus, the CBI exhibits favorable OPD linearity.

Second, the presented CBI can suppress the differences between the total linear OPDs for certain wavelengths and vertical FOVs. In the CSP, the square term OPDs of the two birefringent crystals cannot compensate each other. Large differences between the total linear OPDs are generated by the square term OPD, which are several times the long wavelength or even tens of times the short wavelength. As a result, the spectral resolution of the upper FOV for the CSP is significantly higher than the spectral resolution of the lower FOV. In contrast, the difference between the total linear OPDs for the CBI is very small. Moreover, the maximum differences of the spectral resolution and the wavelength position for the CBI are 0.009 nm and 0.85 nm, respectively, which are both the smallest of the four interferometers analyzed here. Thus, the presented CBI can consistently maintain the spectral resolution and the wavelength position of the restored spectra.

Third, the presented CBI can suppress the differences between the total linear OPDs at different wavelengths for the same vertical FOV, which are mainly caused by the dispersive effects of birefringent crystals on the refractive indices. For the CSP, ASP, and WSP, the total short-wavelength linear OPDs were clearly larger than the total long-wavelength linear OPDs. As a result, large differences in the wavenumber resolution at different wavelengths were generated. The maximum differences in the spectral resolutions for CSP, ASP, and WSP were 13.248 cm⁻¹, 19.892 cm⁻¹, and 14.526 cm⁻¹, respectively. In contrast, the maximum difference of the spectral resolution for the presented CBI was only 9.377 cm⁻¹, and therefore the CBI demonstrates an improved ability to suppress the differences in wavenumber resolution caused by the dispersive effect.

6. Conclusions

In summary, we presented a compact birefringent interferometer for Fourier transform hyperspectral imaging that employs only two birefringent crystal plates, the SP and CP. The SP is used mainly to generate an OPD associated with the FOV for broadband interference. By compensating for the constant term OPD and the square term OPD of the SP, the CP can adjust the position of the zero-order fringe pattern and suppress the inconsistent total OPDs. Both simulations and experiments were performed to demonstrate the feasibility of the presented method. Compared with the CSP, MSP, ASP, and WSP, the presented CBI offers improved suppression of OPD differences, as well as other significant advantages, including a compact structure and low complexity. Therefore, the presented CBI is considered highly suitable for measuring spectral information in miniaturized and high-precision hyperspectral imaging applications.

Appendix

1. Conventional Savart polariscope (CSP)

The structure of the CSP that comprises two Savart plates with identical parameters is shown in Fig. 16 [31]. The material used for these Savart plates is calcite, which is a negative uniaxial crystal. The OPD ${\Delta }_{CSP}$ generated by the CSP can be expressed as

(25)$${\Delta }_{CSP}=\frac{n_o^2-n_e^2}{n_o^2+n_e^2}(\cos \omega +\sin \omega )d_{\_CSP}\sin i+\frac{n_o(n_o^2-n_e^2)}{\sqrt{2}{n}_e{(n_o^2+n_e^2)}^{3/2}}({\cos }^2\omega -{\sin }^2\omega )d_{\_CSP}{\sin }^{2}i,$$

where $i$ is the incident angle; $\omega$ is the angle between the incident plane of the first Savart plate and its principal section; $n_o$ and $n_e$ are the refractive indices of Savart plates for the o ray and e ray, respectively; and $d_{\_CSP}$ is the thickness of each Savart plate.

 

Fig. 16 CSP structure.

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2. Achromatic Savart polariscope (ASP)

The structure of the ASP is shown in Fig. 17. The ASP is formed by combining two CSPs [33-35]. The first CSP is made of a positive birefringent material, yttrium vanadate (YVO4), and the second CSP is made of a negative birefringent material, calcite. The OPD ${\Delta }_{ASP}$ generated by the ASP can be expressed as

(26)$$\begin{array}{l}{\Delta }_{ASP}=\left(\frac{n_{o\_P}^2-n_{e\_P}^2}{n_{o\_P}^2+n_{e\_P}^2}{d}_{P\_ASP}+\frac{n_{o\_N}^2-n_{e\_N}^2}{n_{o\_N}^2+n_{e\_N}^2}{d}_{N\_ASP}\right)(\cos \omega +\sin \omega )\sin i\\ +\left(\frac{n_{o\_P}(n_{o\_P}^2-n_{e\_P}^2)}{\sqrt{2}{n}_{e\_P}{(n_{o\_P}^2+n_{e\_P}^2)}^{3/2}}{d}_{P\_ASP}+\frac{n_{o\_N}(n_{o\_N}^2-n_{e\_N}^2)}{\sqrt{2}{n}_{e\_N}{(n_{o\_N}^2+n_{e\_N}^2)}^{3/2}}{d}_{N\_ASP}\right)({\cos }^2\omega -{\sin }^2\omega ){\sin }^{2}i,\end{array}$$

where $i$ is the incident angle; $\omega$ is the angle between the incident plane of the first Savart plate and its principal section; $n_{o\_P}$ and $n_{e\_P}$ are the refractive indices of the first CSP for the o ray and e ray, respectively; $n_{o\_N}$ and $n_{e\_N}$ are the refractive indices of the second CSP for the o ray and e ray, respectively; and $d_{P\_ASP}$ and $d_{N\_ASP}$ are the thicknesses of each Savart plate for the first CSP and the second CSP, respectively. The thicknesses $d_{P\_ASP}$ and $d_{N\_ASP}$ satisfy the following relationship:

 

Fig. 17 ASP structure.

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(27)$$\frac{d_{P\_ASP}}{d_{N\_ASP}}=-\frac{\frac{n_{o\_N}(n_{o\_N}^2-n_{e\_N}^2)}{\sqrt{2}{n}_{e\_N}{(n_{o\_N}^2+n_{e\_N}^2)}^{3/2}}}{\frac{n_{o\_P}(n_{o\_P}^2-n_{e\_P}^2)}{\sqrt{2}{n}_{e\_P}{(n_{o\_P}^2+n_{e\_P}^2)}^{3/2}}}.$$

3. Savart polariscope with wide field of view (WSP)

Figure 18 shows the structure of the WSP, which comprises a CSP and a pair of uniaxial crystal plates cut parallel to the optic axis with their principal sections perpendicular to each other [36]. The pair of uniaxial crystal plates are made of a negative birefringent material, calcite, and the CSP is made of a positive birefringent material, yttrium vanadate (YVO4). The OPD ${\Delta }_{WSP}$ generated by the WSP can be expressed as

(28)$$\begin{array}{l}{\Delta }_{WSP}=\frac{n_{o\_P}^2-n_{e\_P}^2}{n_{o\_P}^2+n_{e\_P}^2}(\cos \omega +\sin \omega )d_{P\_WSP}\sin i\\ +\left(\frac{n_{o\_P}(n_{o\_P}^2-n_{e\_P}^2)}{\sqrt{2}{n}_{e\_P}{(n_{o\_P}^2+n_{e\_P}^2)}^{3/2}}{d}_{P\_WSP}+\frac{n_{o\_N}^2-n_{e\_N}^2}{2n_{o\_N}^2{n}_{e\_N}^3}{d}_{N\_WSP}\right)({\cos }^2\omega -{\sin }^2\omega ){\sin }^{2}i,\end{array}$$

where $i$ is the incident angle; $\omega$ is the angle between the incident plane of the first Savart plate and its principal section; $n_{o\_P}$ and $n_{e\_P}$ are the refractive indices of the CSP for the o ray and e ray, respectively; $n_{o\_N}$ and $n_{e\_N}$ are the refractive indices of the pair of uniaxial crystal plates for the o ray and e ray, respectively; and $d_{P\_WSP}$ and $d_{N\_WSP}$ are the thicknesses of each crystal plate for the CSP and the pair of uniaxial crystal plates, respectively. The thicknesses $d_{P\_WSP}$ and $d_{N\_WSP}$ satisfy the following relationship:

 

Fig. 18 WSP structure.

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(29)$$\frac{d_{P\_WSP}}{d_{N\_WSP}}=-\frac{\frac{n_{o\_N}^2-n_{e\_N}^2}{2n_{o\_N}^2{n}_{e\_N}^3}}{\frac{n_{o\_P}(n_{o\_P}^2-n_{e\_P}^2)}{\sqrt{2}{n}_{e\_P}{(n_{o\_P}^2+n_{e\_P}^2)}^{3/2}}}.$$

Funding

National Natural Science Foundation of China (NSFC) (61475072); National Key Scientific Instrument and Equipment Development Projects of China (2013YQ150829); Fundamental Research Funds for the Central Universities (30916014112-010); Graduate Student Innovation Project of Jiangsu Province of China (KYLX16_0426).

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