Broadband snapshot complete imaging polarimeter based on dual Sagnac-grating interferometers

Jie Li; Wenfeng Qu; Haiying Wu; Chun Qi

doi:10.1364/OE.26.025858

1. Introduction

Imaging polarimeters (IP, Also called polarization imager) are designed to measure the spatial and polarimetric information of an object. The obtained data set can be described by Stokes parameters:

S_{i} (x, y) = [\begin{matrix} S_{0} (x, y) \\ S_{1} (x, y) \\ S_{2} (x, y) \\ S_{3} (x, y) \end{matrix}] = [\begin{matrix} I_{0} (x, y) + I_{90} (x, y) \\ I_{0} (x, y) - I_{90} (x, y) \\ I_{+ 45} (x, y) - I_{- 45} (x, y) \\ I_{R} (x, y) - I_{L} (x, y) \end{matrix}],

where the x, y are the spatial coordinates in the scene. S₀ is the total intensity of the input light, S₁ denotes the part of linear polarized light 0° over 90°, S₂ for + 45° over −45°, and S₃ for right circular over left circular. These four parameters can describe the complete polarization state of light from the object.

IP device has rapidly developed in recent years, and become a valuable tool in many scientific fields, including astronomy, remote sensing and biomedical diagnosis [1,2]. To acquire the four complete spatial related Stokes parameters by a single focal plane array (FPA), multiple measurements are always needed. The conventional methods include multiple sequential measurements over time by use of a rotating polarization element or tunable filter (division of time, DoT), or multiple parallel measurements over space through lens array (division of aperture, DoA), beamsplitters (division of amplitude, DoAP) or micro polarizer array (division of FPA, DoF) [3–7].

In the above methods, polarization analyzing optics such as, rotation polarizer (waveplate), electrically controllable components (e.g. liquid crystal filter), and micro waveplate or polarizer arrays are typically used. The apparatuses based on these methods generally suffer from vibration, heat generation, electrical noise, alignment difficulty and registration error [8,9]. Additionally, the incorporation of parallel polarization and imaging elements typically increases the complexity both in optics and image registration algorithms [10].

The channeled polarimetric technique proposed by K. Oka and T. Kaneko [11] is an attractive approach for snapshot imaging polarimetry, in which 2D Stokes parameters are modulated onto spatial interference patterns and could be captured by a single FPA in a single frame. Based on this concept, a variety of imaging polarimeters have been developed for spatial Stokes parameters acquisition [12–15]. However, the early sensors based on this method only could work in monochromatic light condition [11,12]. Kudenov et al. extended the working spectral range to white light (460-700 nm) for partial Stokes parameters (three of four) imaging by using a single Sagnac-grating interferometer [13]. Recently, they improved the detectable Stokes parameters to linear polarization (S₀, S₁ and S₂) in 410-750 nm based on two polarization gratings [14,15]. The main drawback is its relatively high reconstruction errors caused by the non-ideal diffraction efficiency of the polarization gratings. Very recently, several digital holographic systems were proposed for polarization detection [16,17]. Lu et al. [16] developed an angular-multiplexing polarization holographic imaging system based on a common-path interference configuration to acquire the complex amplitude distributions of two orthogonal linear polarized components of an object in a single-shot. Gildardo-pablo et al. [17] presented a modification of Lu’s system, which could acquire complete polarization state of an optical field with reducing errors and simple configuration. However, both of the systems could only work with a monochromatic laser as the reference beam and target illumination source.

Developing the ideas above even further, we propose a broadband snapshot complete imaging polarimeter (BSCIP) combined channeled polarimetric technique with two cascade Sagnac-grating interferometers. Without any moving, electrically controllable parts or micro-array elements, the 2D complete Stokes parameters of a scene can be acquired simultaneously in a broadband. In Section 2, we present the optical structure, operation principle and system performance analysis of the developed BSCIP. Section 3 is experimental demonstration of the BSCIP prototype working in the visible band.

2. Theories

2.1 Instrument setup

The schematic setup of the BSCIP system is depicted in Fig. 1. It comprises of a fore-optics, a collimator, a IR blocking filter, two identical Sagnac-grating interferometers, SG₁ and SG₂ with their meridian plane perpendicular to each other, an achromatic half-wave plate AHP, an analyzer A, an imaging lens L and a digital CCD camera. The fast axis of AHP is in y-z plane and oriented at 22.5° to the z-axis. The transmission axis of A is in x-z plane and oriented at 45° to the x-axis. The SG₁ and SG₂ each consists of a polarization beamsplitter PBS, two mirrors M₁ and M₂, two identical transmission gratings G₁ and G₂. Light from a scene is collected and collimated by fore-optics and collimator, and then split into two orthogonally polarized components with a small wavelength dependent lateral displacement by SG₁. After passing through AHP, the polarization directions of the two component rays are rotated to ± 45° to the y-axis, respectively. Thus, SG₂ could resolve the two component rays into four linear polarized rays with equal wavelength dependent shearing. Then the four output rays are resolved into linearly polarized light in the same polarization direction by A and recombined onto the FPA of the camera by L. The camera then records the interference patterns.

Fig. 1 Schematic setup of the developed BSCIP.

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2.2 Operation principle

The obtained interference patterns of the BSCIP can be calculated by using the Mueller calculus [11,13,18]. We assume that the light from the scene has a broadband spectrum from $λ_{1}$ to $λ_{2}$ , and the input Stokes parameters at the input aperture is $S_{i n} (x', y')$ . After passing through the optics of the BSCIP, the output Stokes parameters can be given by:

S_{o u t} (x, y) = M_{A} M_{SG2} M_{AHP} M_{SG1} S_{i} (x', y'),

where the x, y are the spatial coordinates in the FPA.

M_{A}

,

M_{SG2}

,

M_{AHP}

and

M_{SG1}

are the Mueller matrices of the analyzer, the Sagnac-grating interferometer SG₂, the half-wave plate AHP and the Sagnac-grating interferometer SG₁, respectively. Thus, the interference patterns captured by the camera which is the first row of the Stokes vector

S_{o u t} (x, y)

, is described as:

\begin{matrix} I (x, y) = \frac{1}{2} S_{0} (x, y) + \frac{1}{2} S_{1} (x, y) \cos 2 π ϕ x \\ + \frac{1}{4} | S_{23} (x, y) | \cos [2 π ϕ (x - y) - \arg (S_{23} (x, y))] \\ - \frac{1}{4} | S_{23} (x, y) | \cos [2 π ϕ (x + y) + \arg (S_{23} (x, y))], \end{matrix}

with:

S_{23} (x, y) = S_{2} (x, y) + i S_{3} (x, y),

ϕ = \frac{Δ}{λ f} = \frac{2 L \sin (θ_{G})}{λ f} = \frac{2 m L}{f d} .

Where

ϕ

is the spatial frequency of the obtained vertical and horizontal interference patterns and its value is limited by the spatial frequency of the sampling pixels of the CCD camera since the Nyquist criterion should be obeyed.

Δ

is the lateral displacement induced by SG₁ and SG₂.

λ

is the wavelength of the incident beams. f is the focal length of the imaging lens. L is the face to face optical path of the grating G₁ and G₂ (also, G₃ and G₄).

θ_{G}

and m is the diffraction angle and diffraction order of the transmission gratings, respectively. d is the grating period. Noted that since the input light has a broadband spectrum from

λ_{1}

to

λ_{2}

, the captured Stokes parameters are integration over wavelength:

S_{j} (x, y) = \int_{λ_{1}}^{λ_{2}} D E^{4} (λ) S_{j} (x, y, λ) d λ,

where the subscript j = 0, 1, 2, 3 denotes the four Stokes parameters, respectively.

D E (λ)

is the diffraction efficiency of the four transmission gratings.

From Eq. (3), we can see that different spatial carrier frequencies are modulated on the input Stokes parameters. This allows us to segment them in the frequency domain by performing a Fourier transform operation. The 2D Fourier transformation of Eq. (3) can be written as:

\begin{matrix} I_{f} (f_{x}, f_{y}) = \frac{1}{2} A_{0} (f_{x}, f_{y}) + \frac{1}{4} A_{1} (f_{x} - ϕ, f_{y}) + \frac{1}{4} A_{1}^{*} (- f_{x}, - f_{y} - ϕ) \\ + \frac{1}{8} A_{23} (f_{x} - ϕ, f_{y} + ϕ) + \frac{1}{8} A_{23}^{*} (- f_{x} - ϕ, - f_{y} + ϕ), \end{matrix}

where

f_{x}

and

f_{y}

denote the coordinates in the frequency domain. It can be shown that in the frequency domain,

I_{f} (f_{x}, f_{y})

is separated into seven channels and centered at

(f_{x}, f_{y}) = (0, 0)

,

(\pm ϕ, 0)

,

(\pm ϕ, \mp ϕ)

, and

(\pm ϕ, \pm ϕ)

.

By filtering the desire channels and performing the inverse Fourier transform, the input Stokes parameters of the scene can be recovered simultaneously as:

S_{0} (x, y) = ℑ^{- 1} [A_{0} (f_{x}, f_{y})],

S_{1} (x, y) = ℑ^{- 1} [A_{1} (f_{x}, f_{y})],

S_{23} (x, y) = ℑ^{- 1} [A_{23} (f_{x}, f_{y})] .

2.3 Optical efficiency and interference visibility

As a theoretical estimation of the system performance, we analyze the optical efficiency and interference visibility of BSCIP. Light from a scene is split into four coherent component beams and interference on the FPA of BSCIP. Thus the wavelength related optical efficiency can be given by:

η (λ) = \frac{I_{1} (λ) + I_{2} (λ) + I_{3} (λ) + I_{4} (λ)}{I_{i n} (λ)},

where

I_{i n} (λ)

is the intensity of the input light and

I_{1} (λ)

,

I_{2} (λ)

,

I_{3} (λ)

and

I_{4} (λ)

are the intensity of the four coherent component beams as illustrated in inset II in Fig. 1, respectively. Using the ray tracing calculations, the intensity of the four component beams may be estimated as:

I_{1} (λ) = \frac{1}{2} I_{s} (λ) R_{s}^{4} (λ) D E^{4} (λ) R_{m}^{4} (λ) \sin^{2} θ,

I_{2} (λ) = \frac{1}{2} I_{s} (λ) R_{s}^{2} (λ) T_{p}^{2} (λ) D E^{4} (λ) R_{m}^{4} (λ) \cos^{2} θ,

I_{3} (λ) = \frac{1}{2} I_{p} (λ) T_{p}^{2} (λ) R_{s}^{2} (λ) D E^{4} (λ) R_{m}^{4} (λ) \sin^{2} θ,

I_{4} (λ) = \frac{1}{2} I_{p} (λ) T_{p}^{4} (λ) D E^{4} (λ) R_{m}^{4} (λ) \cos^{2} θ,

where

I_{s} (λ)

and

I_{p} (λ)

are the S and P polarization component of the input light.

R_{s} (λ)

and

T_{p} (λ)

are the reflectance and transmission of the PBS, respectively.

R_{m} (λ)

is the reflectance of the mirrors.

θ = 45^{o}

is the polarization orientation of the analyzer A. Then the optical efficiency of BSCIP can be expressed as:

η (λ) = \frac{D E^{4} (λ) R_{m}^{4} (λ) [R_{s}^{2} (λ) + T_{p}^{2} (λ)] [I_{s} (λ) R_{s}^{2} (λ) + I_{p} (λ) T_{p}^{2} (λ)]}{4 [I_{s} (λ) + I_{p} (λ)]} .

Note that since the above calculation makes no assumptions about the polarization state of the input light, this result is general to all states of polarization. From Eq. (16), we can see that the optical efficiency is not only associated with the parameters of the Gratings, mirrors and polarization beamsplitters, but also depends on the polarization state of the input light. Figure 2(a) shows the calculated optical efficiencies for 0°, 90° linear polarized, ± 45° polarized and circular polarized input light. The related parameters

D E (λ)

,

R_{m} (λ)

,

R_{s} (λ)

and

T_{p} (λ)

are depicted in Fig. 2(b). Since the intensity of S polarization component is equal to that of P polarization component in ± 45° and circular polarized light, the optical efficiencies are the same in the conditions. It should be indicated that the optical efficiency of the BSCIP can be highly improved by increasing the diffraction efficiency of the transmission gratings, the P component transmission and S component reflectance of the polarization beamsplitters.

Fig. 2 (a) Calculated optical efficiencies of the BSCIP for different input polarized light. (b) Related parameters used in the optical efficiencies calculations.

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The obtained interferogram described by Eq. (3) can be divided into four parts: one bias part is $S_{0} (x, y)$ and three interference parts (the cosine terms) denote $S_{1} (x, y)$ ~ $S_{3} (x, y)$ as illustrated in Fig. 3. The vertical interference fringes represent $S_{1} (x, y)$ , and can be explained as the superposition of the interference patterns of $I_{1} (λ)$ , $I_{2} (λ)$ and interference patterns of $I_{3} (λ)$ , $I_{4} (λ)$ . The two patterns have a phase difference of $π$ . Thus the interference visibility of the vertical fringes is given by:

\begin{matrix} V_{v} (λ) = \frac{2 (\sqrt{I_{1} (λ) I_{2} (λ)} - \sqrt{I_{3} (λ) I_{4} (λ)})}{I_{1} (λ) + I_{2} (λ) + I_{3} (λ) + I_{4} (λ)} \\ = \frac{2 R_{s} (λ) T_{p} (λ) [I_{s} (λ) R_{s}^{2} (λ) - I_{p} (λ) T_{p}^{2} (λ)]}{[R_{s}^{2} (λ) + T_{p}^{2} (λ)] [I_{s} (λ) R_{s}^{2} (λ) + I_{p} (λ) T_{p}^{2} (λ)]}, \end{matrix}

The ± 45° fringes represent

S_{2} (x, y)

and

S_{3} (x, y)

, and can be expressed as:

V_{+ 45^{o}} (λ) = \frac{2 \sqrt{I_{1} (λ) I_{4} (λ)}}{I_{1} (λ) + I_{4} (λ)} = \frac{2 R_{s}^{2} (λ) T_{p}^{2} (λ) \sqrt{I_{s} (λ) I_{p} (λ)}}{I_{s} (λ) R_{s}^{4} (λ) + I_{p} (λ) T_{p}^{4} (λ)},

V_{- 45^{o}} (λ) = \frac{2 \sqrt{I_{2} (λ) I_{3} (λ)}}{I_{2} (λ) + I_{3} (λ)} = \frac{2 \sqrt{I_{s} (λ) I_{p} (λ)}}{I_{s} (λ) + I_{p} (λ)} .

Again, the results make no assumptions about the polarization state of the input light, and are therefore general to all states of polarization.

Fig. 3 Structure of the obtained interferogram of the BSCIP. Typographical style.

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3. Experimental demonstration

The prototype of the BSCIP is shown in Fig. 4. The working spectral range of the system is from 400 nm to 700 nm. Two 25.4 mm wire grid polarizing beamsplitter cubes are used as the PBS in the system. The four mirrors are 25 mm × 36 mm with λ/10 @ 633 nm flatness. The four transmission gratings are 25 mm × 25 mm, 30 grooves/mm and blazed for a first order wavelength of $λ_{B} = 530 nm$ . The AHP has a clear aperture of 22 mm, enabling operation in 400-800 nm with a retardance accuracy of λ/40. A low cost 1288 × 964 CCD camera (Pointgrey BFLY-PGE-13S2M) with a lens of 70 mm focal length is used to take the interferogram.

Fig. 4 Photograph of the prototype of the BSCIP.

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3.1 Broadband Stokes parameters test

To verify the polarization measurement capability of the BSCIP, the experimental setup depicted in Fig. 5 was implemented. An integrating sphere uniform light source (Labsphere), a collimator (Shanghai Optics) and a visible wire grid polarizer P at 45°, followed by an achromatic quarter-wave plate AQP with fast axis intersecting at an angle $α$ to the transmission axis of P, are used to generate broadband polarization light being measured. In the experiment, the AQP was rotated from $α = 0 °$ to 180° in 5° increments.

Fig. 5 Experiment setup for laboratory testing of the BSCIP.

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Recovered Stokes parameters from the BSCIP system was compared to the theoretical values of the polarizer and wave plate shown in Fig. 6. Note that the recovered Stokes parameters are average values in the field of view of the BSCIP. Compared with the theoretical values, the experimental results were shown accuracy better than 3% over most of the AQP rotating angles. The RMS errors between the measured data and the theoretical values are 0.0169, 0.0365 and 0.0354 for normalized Stokes parameters S₁/S₀, S₂/S₀ and S₃/S₀, respectively. The residual errors can likely be attributed to the variances of retardation of the AQP in broadband wavelength [10]. The RMS error is calculated by:

R M S = \sqrt{\frac{1}{N} \sum_{k = 1}^{N} {(\frac{S_{i, M e a s} (k)}{S_{0, M e a s} (k)} - \frac{S_{i, T h e o} (k)}{S_{0, T h e o} (k)})}^{2},}

where integer i = 1, 2, 3 denotes the latter three Stokes parameters.

Fig. 6 Comparison of the measured Stokes parameters from the BSCIP and the theoretical values of AQP: (a) S₁/S₀, (b) S₂/S₀, (c) S₃/S₀.

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3.2 Polarimetric imaging test

The polarimetric imaging performance of the BSCIP system was evaluated by measuring a rotating polarization filter wheel and a 532 nm true-zero-order quarter-wave plate. The filter wheel is composed of six plastic linear polarization filters offset by about 30° and the fast axis of the quarter-wave plate is oriented −45° to the transmission axis of the bottom linear polarization filter. The samples are illuminated by an integrating sphere uniform light source shown in Fig. 7. Figure 8 shows a raw image of the filter wheel obtained by the BSCIP in a single frame. Clear interference fringes are observed at the positions of the polarization filters. The fringes change in phase and amplitude due to the varying amounts of Stokes parameters produced by the polarization filters. It should be indicated that since the optic axes of the fore-optics and imaging lens of the system are crossed to each other, the raw image is rotated by 90 degrees clockwise to match the target. Figure 9 is the reconstructed Stokes images. Note that the noise at the corners of the reconstructed Stokes images is mainly caused by the fringe contrast losing. According to Eqs. (17)-(19), the visibility of the interference fringe is mainly determined by the polarization components of the input light, and the reflectance $R_{s} (λ)$ and transmission $T_{p} (λ)$ of the PBS. However, most commercial PBSs are designed for normal incidence. $R_{s} (λ)$ and $T_{p} (λ)$ falls very fast when the incident angles are larger than the acceptance angle. To overcome this problem, a wire-grid PBS with an acceptance angle beyond 25° is planning to use in our next generation BSCIP.

Fig. 7 Polarization filter wheel for polarimetric imaging test of the BSCIP. The transmitted directions of the linear polarization filters are indicated by solid arrows. The fast axis of the quarter-wave plate is indicated by dash arrows.

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Fig. 8 Raw image of the rotating polarization filter wheel obtained by the BSCIP in a single frame.

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Fig. 9 Reconstructed Stokes images from Fig. 8. (a) S₀. (b) S₁/S₀. (c) S₂/S₀. (d) S₃/S₀. (e) RGB composite image (Red: |S₁|/S₀; Green: |S₂|/S₀; Blue: |S₃|/S₀;).

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4. Conclusion

We have presented a broadband snapshot complete imaging polarimeter BSCIP by using two crossed Sagnac-grating interferometers. To know the performance of the developed apparatus in detail, the optical efficiency and interference visibility has been calculated. Furthermore, laboratory test experiments of a polarizer and quarter-wave plate have proved the system could acquire full polarization information at a high accuracy. Test images and video (Visualization 1) of a rotating polarization filter wheel have demonstrated the polarimetric imaging capability of the instrument.

5. Funding

National Natural Science Foundation of China (Grant Nos. 61675161 and 61205187); Fundamental Research Funds for the Central Universities (Grant No. zdyf2017003).

References

1. D. A. Miller and E. L. Dereniak, “Selective polarization imager for contrast enhancements in remote scattering media,” Appl. Opt. 51(18), 4092–4102 (2012). [CrossRef] [PubMed]

2. W. L. Hsu, J. Davis, K. Balakrishnan, M. Ibn-Elhaj, S. Kroto, N. Brock, and S. Pau, “Polarization microscope using a near infrared full-Stokes imaging polarimeter,” Opt. Express 23(4), 4357–4368 (2015). [CrossRef] [PubMed]

3. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006). [CrossRef] [PubMed]

4. J. L. Pezzaniti, B. Hokr, J. F. Niles, R. Edmondson, M. Roche, and L. Blackwell, “A dual wave infrared imaging polarimeter,” SPIE 13106550X (2018).

5. J. D. Perreault, “Triple Wollaston-prism complete-Stokes imaging polarimeter,” Opt. Lett. 38(19), 3874–3877 (2013). [CrossRef] [PubMed]

6. X. Tu, O. J. Spires, X. Tian, N. Brock, R. Liang, and S. Pau, “Division of amplitude RGB full-Stokes camera using micro-polarizer arrays: erratum,” Opt. Express 26(4), 4192–4193 (2018). [CrossRef] [PubMed]

7. V. Gruev, R. Perkins, and T. York, “CCD polarization imaging sensor with aluminum nanowire optical filters,” Opt. Express 18(18), 19087–19094 (2010). [CrossRef] [PubMed]

8. A. Álvarez-Herrero, P. G. Parejo, and M. Silva-López, “Fine tuning method for optimization of liquid crystal based polarimeters,” Opt. Express 26(9), 12038–12048 (2018). [CrossRef] [PubMed]

9. S. Gao and V. Gruev, “Gradient-based interpolation method for division-of-focal-plane polarimeters,” Opt. Express 21(1), 1137–1151 (2013). [CrossRef] [PubMed]

10. J. L. Pezzaniti, D. Chenault, M. Roche, J. Reinhardt, J. P. Pezzaniti, and H. Schultz, “Four camera complete Stokes imaging polarimeter,” Proc. SPIE 6972, 69720J (2008). [CrossRef]

11. K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express 11(13), 1510–1519 (2003). [CrossRef] [PubMed]

12. H. Luo, K. Oka, E. DeHoog, M. Kudenov, J. Schiewgerling, and E. L. Dereniak, “Compact and miniature snapshot imaging polarimeter,” Appl. Opt. 47(24), 4413–4417 (2008). [CrossRef] [PubMed]

13. M. W. Kudenov, M. E. L. Jungwirth, E. L. Dereniak, and G. R. Gerhart, “White light Sagnac interferometer for snapshot linear polarimetric imaging,” Opt. Express 17(25), 22520–22534 (2009). [CrossRef] [PubMed]

14. M. W. Kudenov, M. J. Escuti, E. L. Dereniak, and K. Oka, “White-light channeled imaging polarimeter using broadband polarization gratings,” Appl. Opt. 50(15), 2283–2293 (2011). [CrossRef] [PubMed]

15. C. F. LaCasse, B. J. Redman, M. W. Kudenov, and J. M. Craven, “Maximum bandwidth snapshot channeled imaging polarimeter with polarization gratings,” SPIE 12 9853 (2016) in Proceedings of Polarization, Measurement, Analysis, and Remote Sensing XII.

16. L. Han, Z.-J. Cheng, Y. Yang, B.-Y. Wang, Q.-Y. Yue, and C.-S. Guo, “Double-channel angular-multiplexing polarization holography with common-path and off-axis configuration,” Opt. Express 25(18), 21877–21886 (2017). [CrossRef] [PubMed]

17. G.-P. Lemus-Alonso, C. Meneses-Fabian, and R. Kantun-Montiel, “One-shot carrier fringe polarimeter in a double-aperture common-path interferometer,” Opt. Express 26(13), 17624–17634 (2018). [CrossRef] [PubMed]

18. J. Li, B. Gao, C. Qi, J. Zhu, and X. Hou, “Tests of a compact static Fourier-transform imaging spectropolarimeter,” Opt. Express 22(11), 13014–13021 (2014). [CrossRef] [PubMed]

Broadband snapshot complete imaging polarimeter based on dual Sagnac-grating interferometers

Abstract

1. Introduction

2. Theories

2.1 Instrument setup

2.2 Operation principle

2.3 Optical efficiency and interference visibility

3. Experimental demonstration

3.1 Broadband Stokes parameters test

3.2 Polarimetric imaging test

4. Conclusion

5. Funding

References

Supplementary Material (1)

Cited By

Figures (9)

Equations (20)

Optics Express