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Design and analysis of a Fourier transform imaging spectropolarimetry based on polarization modulation array (PMAFTISP)
Journal Pre-proof Design and analysis of a Fourier transform imaging spectropolarimetry based on polarization modulation array (PMAFTISP) Yanqiang Wang, Chunmin Zhang, Tingkui Mu, Tingyu Yan, Zhengyi Chen, Zeyu Chen, Yifan He
PII: DOI: Reference:
S0030-4018(19)31117-4 https://doi.org/10.1016/j.optcom.2019.125101 OPTICS 125101
To appear in:
Optics Communications
Received date : 16 September 2019 Revised date : 15 November 2019 Accepted date : 8 December 2019 Please cite this article as: Y. Wang, C. Zhang, T. Mu et al., Design and analysis of a Fourier transform imaging spectropolarimetry based on polarization modulation array (PMAFTISP), Optics Communications (2019), doi: https://doi.org/10.1016/j.optcom.2019.125101. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
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MB-3713_revised_manuscript Click here to view linked References
Design and analysis of a Fourier transform imaging
(PMAFTISP)
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spectropolarimetry based on polarization modulation array
Yanqiang Wang1,2,3, Chunmin Zhang1,2,3*, Tingkui Mu1,2,3, Tingyu Yan1,2,3, Zhengyi Chen1,2,3,Zeyu Chen1,2,3, Yifan He1,2,3
Institute of Space Optics, Xi’an Jiaotong University, Xi’an 710049, China 2
3
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1
School of Science, Xi’an Jiaotong University, Xi’an 710049, China
Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter (Xi’an Jiaotong University), Ministry of Education, Xi’an, Shaanxi 710049, China
Abstract A tempo-spatially mixed modulation Fourier transform imaging
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spectropolarimeter based on polarization modulation array (PMAFTISP) without internal moving parts was presented, which can obtain the image, spectrum, and full-stokes parameters of the target at the same time. The principle and feasibility were validated with simulations. The system was simulated and the influence of
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retarder error on the recovery of the full-Stokes parameters was mainly analyzed. Compared with channeled interference imaging spectropolarimetry (CIISP), the system separated four fringe patterns in space, without channel crosstalk as well as simplifying the data processing process. Additionally, the advantages of the Fourier transform spectrometer are maintained, such as high throughput (Jacquinot) and multiplex (Fellgett) advantages.
Keywords: Polarization modulation array; Tempo-spatially mixed modulation; Retarder; Achromatic four-face pyramid prism；Multi-dimensional optical information acquisition
1. Introduction
Imaging spectropolarimeter (ISP) is an instrument that integrates the function of camera, spectrometer, and polarimeter, it can acquire the image, spectrum and
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polarization data of a target simultaneously [1-4]. Due to its rich information, it is given more and more attention in the field of remote sensing, atmospheric probing, *
Corresponding author. E-mail address: [email protected]
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biomedical science, material analysis, and military surveillance, especially in target detection and identity [2, 5-8]. However, in conventional ISP, rotating polarization elements, electrically controllable components are typically required, which cause
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apparatuses generally suffer from vibration, electrical noise, and alignment difficulty [3, 9].
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Oka and Kato first described the channeled polarimetric technique in 1999, where the full-Stokes parameters can be detected at once without movable polarization components [10]. However, for the channeled polarimetric technique, it’s difficult to avoid the phenomenon of channel crosstalk, which is particularly serious when monochromatic light is incident [11]. In addition, the channel interferometric
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imaging spectropolarimeter can also reduce the spectral resolution. In view of these shortcomings, people have made some explorations [11-13], but have still not been able to solve the problems thoroughly.
In this paper, we presented a Fourier transform imaging spectropolarimetry based
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on polarization modulation array (PMAFTISP). This method was a kind of imaging spectropolarimetry with aperture divided, which separate fringe pattern in different direction thus imaging on a different area of CCD camera. It avoided channel crosstalk in principle and simplifies the process of data processing. Meanwhile, it can achieve higher spectral resolution as the channel imaging spectropolarimetry. In addition, the optical elements that made up the system were relatively simple, and the whole system was compact.
This paper was organized as follows: firstly, the optical configuration and theoretical model and achromatic four-face pyramid prism were described in section 2; then, the Fourier transform imaging spectropolarimetry based on polarization array was simulated in Section 3; next, in Section 4, we deduced the formula of Stokes parameters error caused by the fast axis direction and retardance deviation of the
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retarder, simulated it by computer, and giving the concept of "polarization modulation array crosstalk", and also simulated retarder and linear polarizer mixed error. In Section 5, the spectral resolution, alignment error of the instrument and image registration were discussed. Finally, the conclusion was contained in Section 6.
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2. Theory
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2.1 Optical configuration
Fig. 1. (Color online) Optical layout of the PMAFTISP.
(a) The fast axis of retarders were 0°, 45° relative to x-axis respectively. (b) The transmission axis
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of the linear polarizers were 0°,90°,45°,90° relative to x-axis respectively. (c) The fast axis of the Superachromatic Half-Wave Plate (SAHWP) array were 0°,45°,22.5°,45° relative to x-axis respectively.
The optical layout of the system was shown in Fig. 1. Fore-optics consisted of
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lens L1, L2, and field stop M (located in the intermediate image plane). Polarization modulation array consisted of a retarders array, a linear polarizer array, and a Superachromatic Achromatic Half-Wave Plate (SAHWP) array. Retarders array was composed of two retarders whose fast axis was 0°, 45° relative to the x-axis respectively. Linear polarizer array was formed of four linear polarizers whose transmission axis was 0°, 90°, 45°, and 90° relative to the x-axis respectively. SAHWP array was composed of four SAHWP whose fast axis was 0°, 45°, 22.5°, and 45° relative to the x-axis respectively. These were followed by a birefringent Fourier-transform imaging spectrometer which composed of a Savart lateral shearing splitter SP, an analyzer P2, an achromatic four-face pyramid prism P3 , a reimaging lens L3, and a charge-coupled device(CCD) placed on the back focal plane of L3. The SP consisted of two identical uniaxial crystal plates with orthogonally oriented
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principal sections and was rotated clockwise along the z-axis by 45° in order to maximize the contrast of the fringes, the optic axes of the two plates were oriented at 45° relative to the z-axis and their projections on the x-y plane were oriented at ±
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45°relative to the x-axis respectively, the principal section of left plate was oriented at 45° relative to the y-z plane and the right plate was oriented at 45° relative to x-y plane[3]. The transmission axes of P2 was 0° relative to the x-axis.
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The incident light from target was imaged on field stop M by lens L1 and collimated by lens L2. The parallel light passed through polarization modulation array
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and become 0° oriented linear polarized light relative to the x-axis, which was identical to transmission axes of P2. The 0° oriented polarized light was parallel laterally sheared by the Savart polariscope into a pair of equal-amplitude but orthogonally polarized components [3]. After passing through P2 the identical linearly polarized components were extracted and recombined onto the CCD camera by P3 and
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L3 and forming four identical partitions. Note that the fringe pattern was vertical 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)[3]. The principle of obtaining polarization spectral information of the target by the
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system as shown below. 2.2 Theoretical model
As mentioned above, the CCD was divided into four partitions to record the interference fringes. The intensity of four polarization modulated fringe patterns
I n () (where n=1, 2, 3, 4) can be described as I n ()
2
1
(1 cos(2)) n) (M (SAHWP M (P1n ) M (Rn1 )S in ( )) d , n 1, 2,3, 4 2
(1)
Where σ was the wavenumber and was the optical path difference. Sin= (S0 (σ), S1 (σ), S2 (σ), S3 (σ))
T
was the Stokes parameters of the incident light, σ1~σ1 was the
n) measured spectrum band. M(SAHWP , M(P1n) , M(Rn1 ) was the Mueller matrix of SAHWP,
linear polarizers P1, and the retarders R1 respectively. Using the Mueller calculus, one can get the final intensity of the fringe patterns [3, 5]
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1 2 (1 cos(2)) S0 ( ) S1 ( ) d 4 1 1 2 I 2 () (1 cos(2)) S0 ( ) S1 ( ) d 4 1 I1 ()
(2-1) (2-2)
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1 2 (1 cos(2 )) S0 ( ) S 2 ( ) d 4 1
(2-3)
1 2 (1 cos(2)) S0 ( ) S1 ( ) cos( ( )) S3 ( ) sin( ( )) d 4 1
(2-4)
I 3 () I 4 ()
CCD
was
I1 (), I 2 (), I 3 (), I 4 ()
respectively.
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Where ( ) was retardance of the retarder. The intensity of the fringe pattern on After
subtracting
fringe
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background and carrying out inverse Fourier transform, then solving the equations, full-Stokes parameters on variety wavenumber can be acquired
1 S0 ( ) S1 ( ) 2 1 B2 ( ) 1 I 2 () S0 ( ) S1 ( ) 2 1 1 B3 ( ) I 3 () S0 ( ) S 2 ( ) 2 1 1 B4 ( ) I 4 () S0 ( ) S1 ( ) cos( ( )) S3 ( ) sin( ( )) 2
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B1 ( ) 1 I1 ()
(3-1) (3-2) (3-3) (3-4)
where F 1 denotes the operator of the spectrum recovery by Fourier transform, then
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full-Stokes parameters can be calculated by the equations below
S0 ( ) B1 ( ) B2 ( )
(4-1)
S1 ( ) B1 ( ) B2 ( )
(4-2)
S2 ( ) 2B3 ( ) B1 ( ) B2 ( )
S3 ( )
2 B4 ( ) S0 ( ) S1 ( ) cos( ( )) sin( ( ))
(4-3) (4-4)
The retardance ( ) should not be equal to k (where k=0, 1, 2…), avoiding
sin( ( )) 0 . This can be done with a superachromatic retarder or by making the retarder very thin.
2.3 Achromatic four-face pyramid prism
The achromatic four-face pyramid prism P3 was a key component in the optical
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layout of the PMAFTISP showed in Fig.1. The top and cross-sectional views of the achromatic four-face pyramid prism structure were shown in Fig.2. It deflected the incident beam, and more importantly, it corresponded to the four sub-partitions of the
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polarization modulation array and formed four images with the imaging lens together on the detector plane. In order to achieve that achromatic effect, as shown in Fig. 2(b), two different glass materials were used, the materials of the two parts are H-ZPK5
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(left) and H-ZLAF92 (right) respectively, and the wedge angle =8.59 , 1.20 . The ray tracing of the achromatic four-face pyramid prism was shown in Fig.3 (a).
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The achromatic four-face pyramid prism was almost diffraction limited as the RMS spot sizes approach to the size of Airy disk in Fig. 3(b), and the absolute value of the
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distortion was less than 1% at different FOVs and wavelengths in Fig. 3(c).
(a) Top view. (b) Cross-sectional view. Fig. 2. (Color online) Structure of the chromatic four-face pyramid prism.
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(a)
(c)
Fig. 3. (Color online) The ray tracing of the achromatic four-face pyramid prism modeled using Zemax optics software. (a) The layout. (b) The spot diagrams at different FOVs and wavelengths. (c) The field curvature and distortion diagrams at different FOVs and wavelengths.
3. Simulation
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When this paper was writing, there was no processed polarization modulation
array and achromatic four-face pyramid prism in the laboratory, so a simulation experiment was performed to demonstrate the reconstruction of each Stokes parameter and effect of retarder errors on the recovery of it. The spectral range was
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focused on 480 ~960 nm and the CCD size was set to 512×512 pixels, and the size of each pixel was 18 m . As mentioned above, CCD was divided into four equal-sized
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partitions, each with a spatial resolution of 256×256.The maximum OPD of the interferometer was 61.44μm. Based on the above analysis, the retardance ( ) of over all wavelength. Fortunately, the
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retarder should satisfy sin( ( )) 0
Superachromatic Quarter-Wave Plate (SAQWP) and the Superachromatic Half-Wave Plate (SAHWP) were available on the market. As shown in Fig.4, the maximum and minimum retardance of the SAQWP were approximate 1.63 rad and 1.54 rad respectively, and the value for SAHWP were 3.18 rad and 3.10 rad respectively. The
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actual size of the CCD for each sub-partition was 4.608mm×4.608mm. Based on the analysis and simulation of the achromatic four-face pyramid prism above, the focal length of the imaging lens L3 in Fig.1 was set to 38.70 mm, and the half field of view
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from the achromatic four-face pyramid prism was 3.4 °.
(a)
(b)
Fig. 4. (Color online) (a) The retardance of the Superachromatic Quarter-Wave Plate (SAQWP), the maximum and the minimum retardance are approximate 1.63 rad and 1.54 rad respectively. (b) The retardance of the Superachromatic Half-Wave Plate (SAHWP), the maximum and the minimum retardance are approximate 3.18 rad and 3.10 rad respectively.
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(a)
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Fig.5. (Color online) (a) Stocks spectrum (b) The output intensity of the polarization modulation module (c) The interference fringes of four-partitions on CCD (d) Restored Stokes spectrum.
The narrowband fully polarized incident light and the broadband fully polarized incident light were simulated, and the Stokes spectrum was shown in Fig. 5 and Fig. 6. Where Fig. 5(a) was a Stocks spectrum, Fig. 5(b) was the output intensity of the polarization modulation module, Fig. 5(c) was interference fringes which spectral information contained and Fig. 5(d) was a restored Stokes spectrum. The notion of Fig.6 was similar to Fig.5. As one can see, the restored Stokes spectrum coincided well with the incident light.
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Fig.6. (Color online) (a) Stocks spectrum (b) The output intensity of the polarization modulation module (c) The interference fringes of four-partitions on CCD (d) Restored Stokes spectrum.
4 Error analysis on the recovery of Stokes parameters 4.1 Retarder error
In the actual installation and adjustment, each optical element may deviate more or less from the design value. As the retarder array was one of the core elements of the polarization modulation module, analysis of it was helpful to understand the system, so the effects of retarder R1 error on the restored Stokes spectrum were mainly analyzed.
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Fig.7. (Color online) Schematic diagram of alignment error of retarder array R1. Red arrows
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indicate ideal alignment, and blue arrows indicate actual alignment.
As Fig.7 showed, R1 consisted of two retarders with the ideal fast axis direction 0° and 45° respectively, and actual fast axis direction deviated from the ideal fast axis
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direction 1 and 2 , respectively. The ideal retardance of two retarders was represented by ( ) , and the actual retardance was ( ) 1 , ( ) 2 respectively.
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By replacing the parameters of the retarder into formula (1), the final intensity of the fringe patterns
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S0 ( ) sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) 1 2 d I1' () (1 cos(2)) 1 cos ( ) 1 sin 21 cos 21S 2 ( ) 4 1 sin 21 sin ( ) 1 S3 ( )
(5-1)
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S0 ( ) sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) 1 2 d I 2 ' () (1 cos(2)) 1 cos ( ) 1 sin 21 cos 21S 2 ( ) 4 1 sin 21 sin ( ) 1 S3 ( )
I 3' ( )
I 4 ' ()
2
1 4 1
2
1 4 1
(5-2)
S0 ( ) 1 cos ( ) 2 cos 2 2 sin 2 2 S1 ( ) d (1 cos(2)) cos 2 2 2 cos ( ) 2 sin 2 2 2 S2 ( ) sin ( ) 2 sin 2 2 S3 ( ) (5-3)
S0 ( ) cos 2 2 2 cos ( ) 2 sin 2 2 2 S1 ( ) d (1 cos(2)) 1 cos ( ) 2 sin 2 2 cos 2 2 S2 ( ) sin ( ) 2 cos 2 2 S3 ( )
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(5-4)
After subtracting fringe background and carrying out inverse Fourier transform, then solving the equations, the Stokes parameters on variety wavenumber
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(6-1)
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S0 ( ) sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) 1 B1' ( ) 1 I1' () 1 cos ( ) 1 sin 21 cos 21S 2 ( ) 2 sin 21 sin ( ) 1 S3 ( )
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S0 ( ) sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) 1 B2 ' ( ) 1 I 2 ' () 1 cos ( ) 1 sin 21 cos 21S 2 ( ) 2 sin 21 sin ( ) 1 S3 ( )
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S0 ( ) 1 cos ( ) 2 cos 2 2 sin 2 2 S1 ( ) 1 B3' ( ) 1 I 3' () cos 2 2 2 cos ( ) 2 sin 2 2 2 S2 ( ) 2 sin ( ) 2 sin 2 2 S3 ( )
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S0 ( ) cos 2 2 2 cos ( ) 2 sin 2 2 2 S1 ( ) 1 B4 ' ( ) 1 I 4 ' ( ) 1 cos ( ) 2 sin2 2 cos 2 2 S 2 ( ) 2 sin ( ) 2 cos 2 2 S3 ( )
(6-2)
(6-3)
(6-4)
From equation (6-1) ~ (6-4), one can derive B1' ( ) B2 ' ( )=S0 ( )
sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) B1' ( ) B2 ' ( )= 1 cos ( ) 1 sin 21 cos 21S2 ( ) sin 21 sin ( ) 1 S3 ( )
cos ( ) 2 1 cos 2 2 sin 2 2 S1 ( ) 2 B3' ( ) B1' ( ) B2' ( ) cos 2 2 2 cos ( ) 2 sin 2 2 2 S2 ( ) sin ( ) 2 sin 2 2 S3 ( )
(7-1)
(7-2)
(7-3)
2 B4 ' ( ) S0 ( ) S1 ( ) cos( ( )) sin( ( ))
(cos( ( )) cos 2 2 2 cos ( ) 2 sin 2 2 2 ) S1 ( ) sin( ( ))
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=
1 cos ( ) sin2 2
2
cos 2 2 S2 ( ) sin ( ) 2 cos 2 2 S3 ( ) sin( ( ))
(7-4)
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Equation (7-1) indicated that the Stokes parameter S0 (σ) with immunity to retarder errors. From equation (7-2), we found that the Stokes parameter S1 (σ) was
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not only affected by retardance error of retarder 1 and its fast axis direction error
1 , but also the Stokes parameter S2 (σ) and S3 (σ), and the latter worked through the existence of former. We named this phenomenon "polarization modulation array
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crosstalk ", i.e., when the fast axis or the transmission axis of the polarization modulation array including the retarders and the line polarizers had an alignment error and deviated from the ideal design value, if the original formula was calculated, the restored Stokes parameter would be introduced into other Stokes components that did not exist in the previous formula. It was somewhat similar to the channel crosstalk in
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channeled spectropolarimetry [10, 11], but the causes and effects of the two were different. As far as we knew, the channel crosstalk in channeled spectropolarimetry can’t be avoided in principle by now, which not only caused the restored Stokes parameters to deviate from the true value but also reduced the spectral resolution.
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Fortunately, "polarization modulation array crosstalk" can be avoided in theory. In equation (7-2) when set 1 =0 , the error of S1 (σ) was zero, thus "polarization modulation array crosstalk " in S1 (σ) can be avoided. It meant that when a manufacture or adjust, one should pay more attention to the direction of the retarder's fast axis and try to achieve the designed value. Similarly, in equation (7-3) when set
2 =0 , the error of S2 (σ) was zero. From equation (7-4), the stokes parameter S3 (σ) was somewhat different which need to simultaneously set 2 =0 and 2 =0 to avoid the ＂polarization modulation array crosstalk＂. It was relatively strict. Using the narrowband Gaussian type fully polarized incident light shown in Fig.
5 (a), we simulated the maximum relative errors of retardance deviation and fast axis direction deviation of the retarder to Stokes parameters S1 (σ), S2 (σ), and S3 (σ)
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respectively. The results were shown in Fig. 8, Fig.9 and Fig.10. From Fig.8 and Fig.9, one can notice that the fast axis direction deviation of the retarder was more sensitive to the Stokes parameters S1 (σ), S2 (σ) recovery than its retardance deviation, and
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when the fast axis direction deviation of the retarder was very small, the maximum relative error of the recovery Stokes parameters was also very small. For S1 (σ) this
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value 1 was about 0.003rad, while this value 2 was approximate 0.001rad for S2 (σ). In Fig.10, the retardance deviation of the retarder and its fast axis direction deviation made an approximate equally contributed to the maximum relative error of
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the S3 (σ). Combined with these analyses, the priority order of retarder error concerned about was: fast axis direction deviation 2 , fast axis direction deviation
1 , retardance deviation 2 , retardance deviation 1 . Moreover, the allowable deviation can be further expanded by the subsequent spectrum and polarization
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calibration process.
Fig.8. (Color online) The maximum relative errors of the recovery Stokes parameters S1 (σ) by using the narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a), as a function of retardance deviation 1 and fast axis direction deviation 1 .
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Fig.9. (Color online) The maximum relative errors of the recovery Stokes parameters S2 (σ) by using the narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a), as a
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function of retardance deviation 2 and fast axis direction deviation 2 .
Fig.10. (Color online) The maximum relative errors of the recovery Stokes parameters S3 (σ) by
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using the narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a), as a function of retardance deviation 2 and fast axis direction deviation 2 .
4.2 Retarder and linear polarizer mixed error
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The alignment error occurred not only on the retarder array R1 but also on the linear polarization array P1 and SAHWP. SAHWP played a role in rotating the polarization direction of the four beams passing through it to the horizontal direction
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in the optical layout of the PMAFTISP showed in Fig. 1, and the alignment error of SAHWP mainly affected the visibility of interference fringes. So here, the mixed
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alignment errors of R1 and P1 were analyzed. Using the Monte Carlo method, the random error of the uniform distribution of the average value of 0 and the range between -0.005 rad and 0.005 rad was selected to simulate the two fast-axis alignment errors, two retardance errors of R1 and four transmission axis alignment errors of P1 .The random points were set to be 5000, and the incident light was still the
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narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a). The maximum relative error for the recovery Stokes parameter S0 (σ), S1 (σ), S2 (σ), S3 (σ) were shown in Fig.11(a), Fig.11(b), Fig.11(c) Fig.11(d) respectively. One can see that the effect of alignment errors on the recovery of a, b, c, d increased in turn, and c, d
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were severely affected, which was a disadvantage of our scheme.
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(a)
(b)
(c)
(d)
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Fig.11. (Color online) Using the Monte Carlo method, the random error of the uniform distribution of the average value of 0 and the range between -0.005 rad and 0.005 rad was selected to simulate
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the two fast-axis alignment errors, two retardance errors of R1, and the four transmission axis alignment errors of P1.The random points were set to be 5000, and the incident light was the narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a). The maximum
Fig.11(a), Fig.11(b), Fig.11(c) Fig.11(d) respectively.
5 Discussion
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relative error for the recovery Stokes parameter S0 (σ) , S1 (σ), S2 (σ), S3 (σ) were shown in
PMAFTISP is essentially a method based on divided aperture, which divided the aperture into four equal-sized partitions, at the expense of half the spatial resolution
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and spectral resolution. Let max represents the maximum optical path difference of the spectropolarimeter, for the CIISP in reference [3], the spectral resolution of the reconstructed spectrum of each channel is 7 / (2 max ) , and for five-channel spectropolarimeter showed in reference[13], the spectral resolution of S1, S2 and S3 is
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7 / (4 max ) ,and for PMAFTISP in this paper, the spectral resolution is 1/ ( max ) , so it maintains a relatively high spectral resolution.
According to the analysis of Section 4, the system has a relatively high alignment requirement for the polarization modulation module, but that doesn't mean it's not practical, and the subsequent calibration process can improve the stokes spectral reconstruction accuracy, and an increasing number of algorithms are proposed [14], which can also be used in PMAFTISP to improve the stokes spectral reconstruction accuracy.
Additional, in practical application, image registration is important and challenged for PMAFTISP. The misregistration of sub-images is mainly contributed by the polarization modulation array, achromatic four-face pyramid prism, and
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reimaging lens L3. Therefore, the optical system needs to be carefully optimized to ensure that the aberration of the system, especially the distortion, is very small. In data processing, a subpixel image registration algorithm-speeded up robust features (SURF) [15], is suggested to employ to register the images as done in reference [9].
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Reference [16] is another recommended paper. 6 Conclusion In conclusion, we presented a tempo-spatially mixed modulation Fourier
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transform imaging spectropolarimetry based on polarization modulation array (PMAFTISP) and verified that the design of polarization modulation array and the
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method of Stokes parameter recovery were correct and feasible in principle. As a key component, the achromatic pyramid prism was designed and simulated. We had derived the error formula of the recovery Stokes parameters when the retardance and fast axis direction of retarder has a deviation, and the concept of "polarization modulation array crosstalk " was put forward and simulated. Monte Carlo method is
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used to simulate the mixed alignment error of retarder and linear polarizer. The results were of great significance for the design of the imaging spectropolarimeter. The PMAFTISP maintains the advantages of high throughput and multiplex advantages, and avoids channel crosstalk in principle and can be widely used in remote sensing,
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target recognition, and other fields. It is the next step to design the optical system and to perform the laboratory calibration and to improve the spectral reconstruction accuracy.
Acknowledgments
We are very grateful to the anonymous reviewers for their constructive comments. This work is supported by the Key Program of National Natural Science Foundation of China (Grant No.41530422), National High Technology Research and Development Program of China (863 Program) (Grant No.2012AA121101), the General Program of National Natural Science Foundation of China (Grant No.61775176). References
[1] J.S. Tyo, T.S. Turner Jr., Variable-retardance, Fourier-transform imaging
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spectropolarimeters for visible spectrum remote sensing, Appl. Opt. 40 (9) (2001) 1450–1458. https://doi.org/10.1364/AO.40.001450 [2] K. Homma, H. Shingu, H. Yamamoto, H. Kurosaki, M. Shibayama, Application of an imaging spectropolarimeter to agro-environmental sciences, Proceedings of SPIE.
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5234 (2004) 638-647. https://doi.org/10.1117/12.510676 [3] Chunmin Zhang, Qiwei Li, Tingyu Yan, Tingkui Mu, and Yutong Wei, High throughput static channeled interference imaging spectropolarimeter based on a
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Savart polariscope, Opt. Express. 24 (2016) 23314-23332. https://doi.org/10.1364/OE.24. 023314
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[4] Naicheng Quan, Chunmin Zhang, Tingkui Mu, Static Fourier transform imaging spectropolarimeter based on quarter-wave plate array, Optik. 127 (20) (2016) 9763-9774. https://doi.org/10.1016/j.ijleo.2016.07.058
[5] Xin Meng, Jianxin Li, Defang Liu, and Rihong Zhu, Fourier transform imaging spectropolarimeter using simultaneous polarization modulation, Opt. Lett. 38 (2013)
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778-780. https://doi.org/10.1364/OL.38.000778
[6] Zhengyi Chen, Chunmin Zhang, Tingkui Mu, Tingyu Yan, Zeyu Chen and Yanqiang Wang. An Efficient Representation-Based Subspace Clustering Framework for Polarized Hyperspectral Images, Remote Sensing. 11 (1513) (2019) 1-15.
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https://doi.org/10.3390/rs11131513
[7] Qin Liu, Caixun Bai, Jie Liu, Jiale He, and Jianxin Li, Fourier transform imaging spectropolarimeter using ferroelectric liquid crystals and Wollaston interferometer, Opt. Express. 25 (17) (2017) 19904-19922. https://doi.org/10.1364/OE.25.019904 [8] T. H. Tsai and D. J. Brady, Coded aperture snapshot spectral polarization imaging, Appl. Opt. 52 (10) (2013) 2153–2161. https://doi.org/10.1364/AO.52.002153 [9] Tingkui Mu, Donghao Bao, Feng Han, Yuanyuan Sun, Zeyu Chen, Qian Tang, and Chunmin Zhang. Optimized design, calibration, and validation of an achromatic snapshot full-Stokes imaging polarimeter, Opt. Express. 27 (2019) 23009-23028. https://doi.org/10.6084/m9.figshare.c.4442960.v1 [10] K. Oka and T. Kato, Spectroscopic polarimetry with a channeled spectrum, Opt. Lett. 24 (21) (1999) 1475–1477. https://doi.org/10.1364/OL.24.001475
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[11] Qiwei Li, Fang Lu, Xiaobin Wang, and Changjiang Zhu, Low crosstalk polarization-difference channeled imaging spectropolarimeter using double-Wollaston prism,
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11734-11747.
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[12] Tingyu Yan, Chunmin Zhang, Jirui Zhang, Naicheng Quan, and Cuncun Tong, High resolution channeled imaging spectropolarimetry based on liquid crystal variable retarder, Opt. Express 26(8) (2018) 10382-10391.
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[13] Jirui Zhang, Chunmin Zhang, Tingyu Yan, Zeyu Chen, Spectrum reconstruction
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for five-channel autocorrelation function of spectropolarimeter, Optik. 157 (2018) 1259–1266. https://doi.org/10.1016/j.ijleo.2017.12.067
[14] Guodong Zhou, Yanqiu Li, Jianhui Li, Jiazhi Wang. Fast compressed channeled spectropolarimeter for full Stokes vector measurement. Proc.SPIE 11057, Modeling Aspects
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https://doi.org/10.1117/12.2526089
[15] Bay, Herbert, Tinne Tuytelaars, and Luc Van Gool. Surf: Speeded up robust features. European conference on computer vision, Springer, Berlin, Heidelberg. (2006) 404-417. https://link.springer.com/chapter/10.1007/11744023_32
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[16] Rubin N A, D’Aversa G, Chevalier P, et al. Matrix Fourier optics enables a compact full-Stokes polarization camera, Science. 365 (6448 eaax1839) (2019). https://science.sciencemag.org/content/365/6448/eaax1839.abstract
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MB-3713-Author Contribution Statement
Yanqiang Wang: Conceptualization, Methodology, Software, Writing - Original Draft Chunmin Zhang: Funding acquisition, Supervision, Resources, Project administration Tingkui Mu: Software, Formal analysis, Validation Tingyu Yan: Writing - Review &
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Editing, Resources, Validation, Investigation Zhengyi Chen: Software, Visualization
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Zeyu Chen: Data curation, Supervision Yifan He: Visualization
PII: DOI: Reference:
S0030-4018(19)31117-4 https://doi.org/10.1016/j.optcom.2019.125101 OPTICS 125101
To appear in:
Optics Communications
Received date : 16 September 2019 Revised date : 15 November 2019 Accepted date : 8 December 2019 Please cite this article as: Y. Wang, C. Zhang, T. Mu et al., Design and analysis of a Fourier transform imaging spectropolarimetry based on polarization modulation array (PMAFTISP), Optics Communications (2019), doi: https://doi.org/10.1016/j.optcom.2019.125101. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
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MB-3713_revised_manuscript Click here to view linked References
Design and analysis of a Fourier transform imaging
(PMAFTISP)
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spectropolarimetry based on polarization modulation array
Yanqiang Wang1,2,3, Chunmin Zhang1,2,3*, Tingkui Mu1,2,3, Tingyu Yan1,2,3, Zhengyi Chen1,2,3,Zeyu Chen1,2,3, Yifan He1,2,3
Institute of Space Optics, Xi’an Jiaotong University, Xi’an 710049, China 2
3
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1
School of Science, Xi’an Jiaotong University, Xi’an 710049, China
Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter (Xi’an Jiaotong University), Ministry of Education, Xi’an, Shaanxi 710049, China
Abstract A tempo-spatially mixed modulation Fourier transform imaging
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spectropolarimeter based on polarization modulation array (PMAFTISP) without internal moving parts was presented, which can obtain the image, spectrum, and full-stokes parameters of the target at the same time. The principle and feasibility were validated with simulations. The system was simulated and the influence of
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retarder error on the recovery of the full-Stokes parameters was mainly analyzed. Compared with channeled interference imaging spectropolarimetry (CIISP), the system separated four fringe patterns in space, without channel crosstalk as well as simplifying the data processing process. Additionally, the advantages of the Fourier transform spectrometer are maintained, such as high throughput (Jacquinot) and multiplex (Fellgett) advantages.
Keywords: Polarization modulation array; Tempo-spatially mixed modulation; Retarder; Achromatic four-face pyramid prism；Multi-dimensional optical information acquisition
1. Introduction
Imaging spectropolarimeter (ISP) is an instrument that integrates the function of camera, spectrometer, and polarimeter, it can acquire the image, spectrum and
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polarization data of a target simultaneously [1-4]. Due to its rich information, it is given more and more attention in the field of remote sensing, atmospheric probing, *
Corresponding author. E-mail address: [email protected]
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biomedical science, material analysis, and military surveillance, especially in target detection and identity [2, 5-8]. However, in conventional ISP, rotating polarization elements, electrically controllable components are typically required, which cause
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apparatuses generally suffer from vibration, electrical noise, and alignment difficulty [3, 9].
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Oka and Kato first described the channeled polarimetric technique in 1999, where the full-Stokes parameters can be detected at once without movable polarization components [10]. However, for the channeled polarimetric technique, it’s difficult to avoid the phenomenon of channel crosstalk, which is particularly serious when monochromatic light is incident [11]. In addition, the channel interferometric
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imaging spectropolarimeter can also reduce the spectral resolution. In view of these shortcomings, people have made some explorations [11-13], but have still not been able to solve the problems thoroughly.
In this paper, we presented a Fourier transform imaging spectropolarimetry based
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on polarization modulation array (PMAFTISP). This method was a kind of imaging spectropolarimetry with aperture divided, which separate fringe pattern in different direction thus imaging on a different area of CCD camera. It avoided channel crosstalk in principle and simplifies the process of data processing. Meanwhile, it can achieve higher spectral resolution as the channel imaging spectropolarimetry. In addition, the optical elements that made up the system were relatively simple, and the whole system was compact.
This paper was organized as follows: firstly, the optical configuration and theoretical model and achromatic four-face pyramid prism were described in section 2; then, the Fourier transform imaging spectropolarimetry based on polarization array was simulated in Section 3; next, in Section 4, we deduced the formula of Stokes parameters error caused by the fast axis direction and retardance deviation of the
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retarder, simulated it by computer, and giving the concept of "polarization modulation array crosstalk", and also simulated retarder and linear polarizer mixed error. In Section 5, the spectral resolution, alignment error of the instrument and image registration were discussed. Finally, the conclusion was contained in Section 6.
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2. Theory
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2.1 Optical configuration
Fig. 1. (Color online) Optical layout of the PMAFTISP.
(a) The fast axis of retarders were 0°, 45° relative to x-axis respectively. (b) The transmission axis
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of the linear polarizers were 0°,90°,45°,90° relative to x-axis respectively. (c) The fast axis of the Superachromatic Half-Wave Plate (SAHWP) array were 0°,45°,22.5°,45° relative to x-axis respectively.
The optical layout of the system was shown in Fig. 1. Fore-optics consisted of
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lens L1, L2, and field stop M (located in the intermediate image plane). Polarization modulation array consisted of a retarders array, a linear polarizer array, and a Superachromatic Achromatic Half-Wave Plate (SAHWP) array. Retarders array was composed of two retarders whose fast axis was 0°, 45° relative to the x-axis respectively. Linear polarizer array was formed of four linear polarizers whose transmission axis was 0°, 90°, 45°, and 90° relative to the x-axis respectively. SAHWP array was composed of four SAHWP whose fast axis was 0°, 45°, 22.5°, and 45° relative to the x-axis respectively. These were followed by a birefringent Fourier-transform imaging spectrometer which composed of a Savart lateral shearing splitter SP, an analyzer P2, an achromatic four-face pyramid prism P3 , a reimaging lens L3, and a charge-coupled device(CCD) placed on the back focal plane of L3. The SP consisted of two identical uniaxial crystal plates with orthogonally oriented
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principal sections and was rotated clockwise along the z-axis by 45° in order to maximize the contrast of the fringes, the optic axes of the two plates were oriented at 45° relative to the z-axis and their projections on the x-y plane were oriented at ±
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45°relative to the x-axis respectively, the principal section of left plate was oriented at 45° relative to the y-z plane and the right plate was oriented at 45° relative to x-y plane[3]. The transmission axes of P2 was 0° relative to the x-axis.
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The incident light from target was imaged on field stop M by lens L1 and collimated by lens L2. The parallel light passed through polarization modulation array
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and become 0° oriented linear polarized light relative to the x-axis, which was identical to transmission axes of P2. The 0° oriented polarized light was parallel laterally sheared by the Savart polariscope into a pair of equal-amplitude but orthogonally polarized components [3]. After passing through P2 the identical linearly polarized components were extracted and recombined onto the CCD camera by P3 and
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L3 and forming four identical partitions. Note that the fringe pattern was vertical 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)[3]. The principle of obtaining polarization spectral information of the target by the
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system as shown below. 2.2 Theoretical model
As mentioned above, the CCD was divided into four partitions to record the interference fringes. The intensity of four polarization modulated fringe patterns
I n () (where n=1, 2, 3, 4) can be described as I n ()
2
1
(1 cos(2)) n) (M (SAHWP M (P1n ) M (Rn1 )S in ( )) d , n 1, 2,3, 4 2
(1)
Where σ was the wavenumber and was the optical path difference. Sin= (S0 (σ), S1 (σ), S2 (σ), S3 (σ))
T
was the Stokes parameters of the incident light, σ1~σ1 was the
n) measured spectrum band. M(SAHWP , M(P1n) , M(Rn1 ) was the Mueller matrix of SAHWP,
linear polarizers P1, and the retarders R1 respectively. Using the Mueller calculus, one can get the final intensity of the fringe patterns [3, 5]
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1 2 (1 cos(2)) S0 ( ) S1 ( ) d 4 1 1 2 I 2 () (1 cos(2)) S0 ( ) S1 ( ) d 4 1 I1 ()
(2-1) (2-2)
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1 2 (1 cos(2 )) S0 ( ) S 2 ( ) d 4 1
(2-3)
1 2 (1 cos(2)) S0 ( ) S1 ( ) cos( ( )) S3 ( ) sin( ( )) d 4 1
(2-4)
I 3 () I 4 ()
CCD
was
I1 (), I 2 (), I 3 (), I 4 ()
respectively.
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Where ( ) was retardance of the retarder. The intensity of the fringe pattern on After
subtracting
fringe
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background and carrying out inverse Fourier transform, then solving the equations, full-Stokes parameters on variety wavenumber can be acquired
1 S0 ( ) S1 ( ) 2 1 B2 ( ) 1 I 2 () S0 ( ) S1 ( ) 2 1 1 B3 ( ) I 3 () S0 ( ) S 2 ( ) 2 1 1 B4 ( ) I 4 () S0 ( ) S1 ( ) cos( ( )) S3 ( ) sin( ( )) 2
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B1 ( ) 1 I1 ()
(3-1) (3-2) (3-3) (3-4)
where F 1 denotes the operator of the spectrum recovery by Fourier transform, then
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full-Stokes parameters can be calculated by the equations below
S0 ( ) B1 ( ) B2 ( )
(4-1)
S1 ( ) B1 ( ) B2 ( )
(4-2)
S2 ( ) 2B3 ( ) B1 ( ) B2 ( )
S3 ( )
2 B4 ( ) S0 ( ) S1 ( ) cos( ( )) sin( ( ))
(4-3) (4-4)
The retardance ( ) should not be equal to k (where k=0, 1, 2…), avoiding
sin( ( )) 0 . This can be done with a superachromatic retarder or by making the retarder very thin.
2.3 Achromatic four-face pyramid prism
The achromatic four-face pyramid prism P3 was a key component in the optical
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layout of the PMAFTISP showed in Fig.1. The top and cross-sectional views of the achromatic four-face pyramid prism structure were shown in Fig.2. It deflected the incident beam, and more importantly, it corresponded to the four sub-partitions of the
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polarization modulation array and formed four images with the imaging lens together on the detector plane. In order to achieve that achromatic effect, as shown in Fig. 2(b), two different glass materials were used, the materials of the two parts are H-ZPK5
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(left) and H-ZLAF92 (right) respectively, and the wedge angle =8.59 , 1.20 . The ray tracing of the achromatic four-face pyramid prism was shown in Fig.3 (a).
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The achromatic four-face pyramid prism was almost diffraction limited as the RMS spot sizes approach to the size of Airy disk in Fig. 3(b), and the absolute value of the
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distortion was less than 1% at different FOVs and wavelengths in Fig. 3(c).
(a) Top view. (b) Cross-sectional view. Fig. 2. (Color online) Structure of the chromatic four-face pyramid prism.
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(a)
(c)
Fig. 3. (Color online) The ray tracing of the achromatic four-face pyramid prism modeled using Zemax optics software. (a) The layout. (b) The spot diagrams at different FOVs and wavelengths. (c) The field curvature and distortion diagrams at different FOVs and wavelengths.
3. Simulation
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When this paper was writing, there was no processed polarization modulation
array and achromatic four-face pyramid prism in the laboratory, so a simulation experiment was performed to demonstrate the reconstruction of each Stokes parameter and effect of retarder errors on the recovery of it. The spectral range was
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focused on 480 ~960 nm and the CCD size was set to 512×512 pixels, and the size of each pixel was 18 m . As mentioned above, CCD was divided into four equal-sized
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partitions, each with a spatial resolution of 256×256.The maximum OPD of the interferometer was 61.44μm. Based on the above analysis, the retardance ( ) of over all wavelength. Fortunately, the
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retarder should satisfy sin( ( )) 0
Superachromatic Quarter-Wave Plate (SAQWP) and the Superachromatic Half-Wave Plate (SAHWP) were available on the market. As shown in Fig.4, the maximum and minimum retardance of the SAQWP were approximate 1.63 rad and 1.54 rad respectively, and the value for SAHWP were 3.18 rad and 3.10 rad respectively. The
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actual size of the CCD for each sub-partition was 4.608mm×4.608mm. Based on the analysis and simulation of the achromatic four-face pyramid prism above, the focal length of the imaging lens L3 in Fig.1 was set to 38.70 mm, and the half field of view
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from the achromatic four-face pyramid prism was 3.4 °.
(a)
(b)
Fig. 4. (Color online) (a) The retardance of the Superachromatic Quarter-Wave Plate (SAQWP), the maximum and the minimum retardance are approximate 1.63 rad and 1.54 rad respectively. (b) The retardance of the Superachromatic Half-Wave Plate (SAHWP), the maximum and the minimum retardance are approximate 3.18 rad and 3.10 rad respectively.
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(a)
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Fig.5. (Color online) (a) Stocks spectrum (b) The output intensity of the polarization modulation module (c) The interference fringes of four-partitions on CCD (d) Restored Stokes spectrum.
The narrowband fully polarized incident light and the broadband fully polarized incident light were simulated, and the Stokes spectrum was shown in Fig. 5 and Fig. 6. Where Fig. 5(a) was a Stocks spectrum, Fig. 5(b) was the output intensity of the polarization modulation module, Fig. 5(c) was interference fringes which spectral information contained and Fig. 5(d) was a restored Stokes spectrum. The notion of Fig.6 was similar to Fig.5. As one can see, the restored Stokes spectrum coincided well with the incident light.
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Fig.6. (Color online) (a) Stocks spectrum (b) The output intensity of the polarization modulation module (c) The interference fringes of four-partitions on CCD (d) Restored Stokes spectrum.
4 Error analysis on the recovery of Stokes parameters 4.1 Retarder error
In the actual installation and adjustment, each optical element may deviate more or less from the design value. As the retarder array was one of the core elements of the polarization modulation module, analysis of it was helpful to understand the system, so the effects of retarder R1 error on the restored Stokes spectrum were mainly analyzed.
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Fig.7. (Color online) Schematic diagram of alignment error of retarder array R1. Red arrows
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indicate ideal alignment, and blue arrows indicate actual alignment.
As Fig.7 showed, R1 consisted of two retarders with the ideal fast axis direction 0° and 45° respectively, and actual fast axis direction deviated from the ideal fast axis
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direction 1 and 2 , respectively. The ideal retardance of two retarders was represented by ( ) , and the actual retardance was ( ) 1 , ( ) 2 respectively.
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By replacing the parameters of the retarder into formula (1), the final intensity of the fringe patterns
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S0 ( ) sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) 1 2 d I1' () (1 cos(2)) 1 cos ( ) 1 sin 21 cos 21S 2 ( ) 4 1 sin 21 sin ( ) 1 S3 ( )
(5-1)
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S0 ( ) sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) 1 2 d I 2 ' () (1 cos(2)) 1 cos ( ) 1 sin 21 cos 21S 2 ( ) 4 1 sin 21 sin ( ) 1 S3 ( )
I 3' ( )
I 4 ' ()
2
1 4 1
2
1 4 1
(5-2)
S0 ( ) 1 cos ( ) 2 cos 2 2 sin 2 2 S1 ( ) d (1 cos(2)) cos 2 2 2 cos ( ) 2 sin 2 2 2 S2 ( ) sin ( ) 2 sin 2 2 S3 ( ) (5-3)
S0 ( ) cos 2 2 2 cos ( ) 2 sin 2 2 2 S1 ( ) d (1 cos(2)) 1 cos ( ) 2 sin 2 2 cos 2 2 S2 ( ) sin ( ) 2 cos 2 2 S3 ( )
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(5-4)
After subtracting fringe background and carrying out inverse Fourier transform, then solving the equations, the Stokes parameters on variety wavenumber
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(6-1)
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S0 ( ) sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) 1 B1' ( ) 1 I1' () 1 cos ( ) 1 sin 21 cos 21S 2 ( ) 2 sin 21 sin ( ) 1 S3 ( )
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S0 ( ) sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) 1 B2 ' ( ) 1 I 2 ' () 1 cos ( ) 1 sin 21 cos 21S 2 ( ) 2 sin 21 sin ( ) 1 S3 ( )
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S0 ( ) 1 cos ( ) 2 cos 2 2 sin 2 2 S1 ( ) 1 B3' ( ) 1 I 3' () cos 2 2 2 cos ( ) 2 sin 2 2 2 S2 ( ) 2 sin ( ) 2 sin 2 2 S3 ( )
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S0 ( ) cos 2 2 2 cos ( ) 2 sin 2 2 2 S1 ( ) 1 B4 ' ( ) 1 I 4 ' ( ) 1 cos ( ) 2 sin2 2 cos 2 2 S 2 ( ) 2 sin ( ) 2 cos 2 2 S3 ( )
(6-2)
(6-3)
(6-4)
From equation (6-1) ~ (6-4), one can derive B1' ( ) B2 ' ( )=S0 ( )
sin 2 21 cos ( ) 1 cos 2 21 S1 ( ) B1' ( ) B2 ' ( )= 1 cos ( ) 1 sin 21 cos 21S2 ( ) sin 21 sin ( ) 1 S3 ( )
cos ( ) 2 1 cos 2 2 sin 2 2 S1 ( ) 2 B3' ( ) B1' ( ) B2' ( ) cos 2 2 2 cos ( ) 2 sin 2 2 2 S2 ( ) sin ( ) 2 sin 2 2 S3 ( )
(7-1)
(7-2)
(7-3)
2 B4 ' ( ) S0 ( ) S1 ( ) cos( ( )) sin( ( ))
(cos( ( )) cos 2 2 2 cos ( ) 2 sin 2 2 2 ) S1 ( ) sin( ( ))
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=
1 cos ( ) sin2 2
2
cos 2 2 S2 ( ) sin ( ) 2 cos 2 2 S3 ( ) sin( ( ))
(7-4)
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Equation (7-1) indicated that the Stokes parameter S0 (σ) with immunity to retarder errors. From equation (7-2), we found that the Stokes parameter S1 (σ) was
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not only affected by retardance error of retarder 1 and its fast axis direction error
1 , but also the Stokes parameter S2 (σ) and S3 (σ), and the latter worked through the existence of former. We named this phenomenon "polarization modulation array
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crosstalk ", i.e., when the fast axis or the transmission axis of the polarization modulation array including the retarders and the line polarizers had an alignment error and deviated from the ideal design value, if the original formula was calculated, the restored Stokes parameter would be introduced into other Stokes components that did not exist in the previous formula. It was somewhat similar to the channel crosstalk in
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channeled spectropolarimetry [10, 11], but the causes and effects of the two were different. As far as we knew, the channel crosstalk in channeled spectropolarimetry can’t be avoided in principle by now, which not only caused the restored Stokes parameters to deviate from the true value but also reduced the spectral resolution.
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Fortunately, "polarization modulation array crosstalk" can be avoided in theory. In equation (7-2) when set 1 =0 , the error of S1 (σ) was zero, thus "polarization modulation array crosstalk " in S1 (σ) can be avoided. It meant that when a manufacture or adjust, one should pay more attention to the direction of the retarder's fast axis and try to achieve the designed value. Similarly, in equation (7-3) when set
2 =0 , the error of S2 (σ) was zero. From equation (7-4), the stokes parameter S3 (σ) was somewhat different which need to simultaneously set 2 =0 and 2 =0 to avoid the ＂polarization modulation array crosstalk＂. It was relatively strict. Using the narrowband Gaussian type fully polarized incident light shown in Fig.
5 (a), we simulated the maximum relative errors of retardance deviation and fast axis direction deviation of the retarder to Stokes parameters S1 (σ), S2 (σ), and S3 (σ)
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respectively. The results were shown in Fig. 8, Fig.9 and Fig.10. From Fig.8 and Fig.9, one can notice that the fast axis direction deviation of the retarder was more sensitive to the Stokes parameters S1 (σ), S2 (σ) recovery than its retardance deviation, and
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when the fast axis direction deviation of the retarder was very small, the maximum relative error of the recovery Stokes parameters was also very small. For S1 (σ) this
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value 1 was about 0.003rad, while this value 2 was approximate 0.001rad for S2 (σ). In Fig.10, the retardance deviation of the retarder and its fast axis direction deviation made an approximate equally contributed to the maximum relative error of
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the S3 (σ). Combined with these analyses, the priority order of retarder error concerned about was: fast axis direction deviation 2 , fast axis direction deviation
1 , retardance deviation 2 , retardance deviation 1 . Moreover, the allowable deviation can be further expanded by the subsequent spectrum and polarization
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calibration process.
Fig.8. (Color online) The maximum relative errors of the recovery Stokes parameters S1 (σ) by using the narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a), as a function of retardance deviation 1 and fast axis direction deviation 1 .
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Fig.9. (Color online) The maximum relative errors of the recovery Stokes parameters S2 (σ) by using the narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a), as a
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function of retardance deviation 2 and fast axis direction deviation 2 .
Fig.10. (Color online) The maximum relative errors of the recovery Stokes parameters S3 (σ) by
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using the narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a), as a function of retardance deviation 2 and fast axis direction deviation 2 .
4.2 Retarder and linear polarizer mixed error
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The alignment error occurred not only on the retarder array R1 but also on the linear polarization array P1 and SAHWP. SAHWP played a role in rotating the polarization direction of the four beams passing through it to the horizontal direction
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in the optical layout of the PMAFTISP showed in Fig. 1, and the alignment error of SAHWP mainly affected the visibility of interference fringes. So here, the mixed
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alignment errors of R1 and P1 were analyzed. Using the Monte Carlo method, the random error of the uniform distribution of the average value of 0 and the range between -0.005 rad and 0.005 rad was selected to simulate the two fast-axis alignment errors, two retardance errors of R1 and four transmission axis alignment errors of P1 .The random points were set to be 5000, and the incident light was still the
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narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a). The maximum relative error for the recovery Stokes parameter S0 (σ), S1 (σ), S2 (σ), S3 (σ) were shown in Fig.11(a), Fig.11(b), Fig.11(c) Fig.11(d) respectively. One can see that the effect of alignment errors on the recovery of a, b, c, d increased in turn, and c, d
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were severely affected, which was a disadvantage of our scheme.
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(a)
(b)
(c)
(d)
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Fig.11. (Color online) Using the Monte Carlo method, the random error of the uniform distribution of the average value of 0 and the range between -0.005 rad and 0.005 rad was selected to simulate
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the two fast-axis alignment errors, two retardance errors of R1, and the four transmission axis alignment errors of P1.The random points were set to be 5000, and the incident light was the narrowband Gaussian type fully polarized incident light shown in Fig. 5 (a). The maximum
Fig.11(a), Fig.11(b), Fig.11(c) Fig.11(d) respectively.
5 Discussion
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relative error for the recovery Stokes parameter S0 (σ) , S1 (σ), S2 (σ), S3 (σ) were shown in
PMAFTISP is essentially a method based on divided aperture, which divided the aperture into four equal-sized partitions, at the expense of half the spatial resolution
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and spectral resolution. Let max represents the maximum optical path difference of the spectropolarimeter, for the CIISP in reference [3], the spectral resolution of the reconstructed spectrum of each channel is 7 / (2 max ) , and for five-channel spectropolarimeter showed in reference[13], the spectral resolution of S1, S2 and S3 is
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7 / (4 max ) ,and for PMAFTISP in this paper, the spectral resolution is 1/ ( max ) , so it maintains a relatively high spectral resolution.
According to the analysis of Section 4, the system has a relatively high alignment requirement for the polarization modulation module, but that doesn't mean it's not practical, and the subsequent calibration process can improve the stokes spectral reconstruction accuracy, and an increasing number of algorithms are proposed [14], which can also be used in PMAFTISP to improve the stokes spectral reconstruction accuracy.
Additional, in practical application, image registration is important and challenged for PMAFTISP. The misregistration of sub-images is mainly contributed by the polarization modulation array, achromatic four-face pyramid prism, and
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reimaging lens L3. Therefore, the optical system needs to be carefully optimized to ensure that the aberration of the system, especially the distortion, is very small. In data processing, a subpixel image registration algorithm-speeded up robust features (SURF) [15], is suggested to employ to register the images as done in reference [9].
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Reference [16] is another recommended paper. 6 Conclusion In conclusion, we presented a tempo-spatially mixed modulation Fourier
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transform imaging spectropolarimetry based on polarization modulation array (PMAFTISP) and verified that the design of polarization modulation array and the
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method of Stokes parameter recovery were correct and feasible in principle. As a key component, the achromatic pyramid prism was designed and simulated. We had derived the error formula of the recovery Stokes parameters when the retardance and fast axis direction of retarder has a deviation, and the concept of "polarization modulation array crosstalk " was put forward and simulated. Monte Carlo method is
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used to simulate the mixed alignment error of retarder and linear polarizer. The results were of great significance for the design of the imaging spectropolarimeter. The PMAFTISP maintains the advantages of high throughput and multiplex advantages, and avoids channel crosstalk in principle and can be widely used in remote sensing,
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target recognition, and other fields. It is the next step to design the optical system and to perform the laboratory calibration and to improve the spectral reconstruction accuracy.
Acknowledgments
We are very grateful to the anonymous reviewers for their constructive comments. This work is supported by the Key Program of National Natural Science Foundation of China (Grant No.41530422), National High Technology Research and Development Program of China (863 Program) (Grant No.2012AA121101), the General Program of National Natural Science Foundation of China (Grant No.61775176). References
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MB-3713-Author Contribution Statement
Yanqiang Wang: Conceptualization, Methodology, Software, Writing - Original Draft Chunmin Zhang: Funding acquisition, Supervision, Resources, Project administration Tingkui Mu: Software, Formal analysis, Validation Tingyu Yan: Writing - Review &
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Editing, Resources, Validation, Investigation Zhengyi Chen: Software, Visualization
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Zeyu Chen: Data curation, Supervision Yifan He: Visualization