Static Fourier transform imaging spectropolarimeter based on quarter-wave plate array

Accepted Manuscript Title: Static Fourier transform imaging spectropolarimeter based on quarter-wave plate array Author: Naicheng Quan Chunmin Zhang Tingkui Mu PII: DOI: Reference:

S0030-4026(16)30824-5 http://dx.doi.org/doi:10.1016/j.ijleo.2016.07.058 IJLEO 57978

To appear in: Received date: Accepted date:

23-5-2016 22-7-2016

Please cite this article as: Naicheng Quan, Chunmin Zhang, Tingkui Mu, Static Fourier transform imaging spectropolarimeter based on quarter-wave plate array, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.07.058 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

Static Fourier transform imaging spectropolarimeter based on quarter-wave plate array Naicheng Quan, Chunmin Zhang*, Tingkui Mu Institute of Space optics, School of Science, Xi’an Jiao tong University, No.28 Xian ning West Road, Xi’an 710049, Shaanxi, PR China

ABSTRACT A compact, static Fourier transform imaging spectropolarimeter based on quarter-wave plate array is conceptually described. It improves a Savart interferometer by replacing front polarizer with a Wollaston prism and an aperture divided by a quarter-wave plate array, and can simultaneously acquire six interferograms corresponding to six linearly polarized lights on a single CCD. The spectral dependence of all Stokes parameters can be recovered by Fourier transform. Since there is no rotating or moving part, the system is relatively robust. The interference model is proved. The performance of the system is demonstrated through a numerical simulation. The orientation of quarter-wave plate array is optimized. Alignment errors of the elements respect to the ideal position are discussed and a procedure for compensating the alignment errors is also developed. Keywords: Interference imaging spectropolarimeter; Birefringent beam splitter; Stokes paramters

*Corresponding author. Tel.: +86 2982668016; fax: +86 82668016 E-mail address: [email protected]

1. INTRODUCTION Imaging spectropolarimeter (ISP) is a novel instrument that integrates the function of camera, spectrometer and polarimeter. It can acquire the image, spectrum and polarization data of a target simultaneously [1-3]. ISP provides richer information for target detection and identification. Stokes imaging spectropolarimetry is usually employed for ground, airborne and spaceborne remote sensing. It collects the spectral variation of Stokes parameters coming from the reflection of sunlight by a scene or from its own radiation. The spatial and spectral dependence of Stokes parameters is defined by [3]

 S0 ( x, y,  )   I 0 ( x, y,  )  I 90 ( x, y,  )   S ( x , y ,  )   I ( x , y ,  )  I ( x, y ,  )  90  0  S ( x, y ,  )   1  S2 ( x, y,  )   I 45 ( x, y,  )  I135 ( x, y,  )       S 3 ( x , y ,  )   I R ( x , y ,  )  I L ( x, y ,  )  Where  is spectral variable, (x, y) is the spatial coordinates of image, S0 is the total intensity of the light, S1 is the difference between linear polarizations of 0° and 90°, S2 is the difference between linear polarizations of ± 45°, and S3 is the difference between right and left circular polarization. Since rotating polarization elements are typically used in the conventional ISP, the moving mechanical components increases the complexity and decreases the reliability of the measurement system [4-5]. Recently, novel ISP based on channeled polarimetric technique arouse wide interest due to their snapshot imaging capability [6-9]. The most typical one is combination of Fourier transform imaging spectrometer and channeled polarimetric technique [8]. Such sensor adds two thick retarders and an

analyzer before a Fourier transform imaging spectrometer, and can determine four Stokes parameters from only a single interferogram. Since the decoded or recorded interferogram usually contains more than three interference channels, the spectral resolution of each spectral Stokes parameters is lower than that of the spectrometer. The spectral resolution of the instrument needs to be much higher than the required spectral resolution of the data product to carry the additional polarization information in the spectrum. Besides, when a narrow-band spectrum is measured by this way, there may be aliasing among the fringes of different channels. A complicated method should be utilized to revise the retrieve spectrum [10]. Recent progress in the ISP based on birefringent elements has remarkably enhanced the performance of static, compactness, and robustness, especially overcoming the drawbacks of channeled polarimetric technique [11-16]. However, such instruments can only acquire the linear polarized components. In this paper, we propose a static ISP based on quarter-wave plate array (QWPAISP) that can record the spectral variation of all four Stokes parameters with the interferometer’s high spectral resolution. We describe the configuration and interference model of the QWPAISP in section 2. In section 3, The features of the system are characterized, the orientation of the quarter-wave plate array (QWPA) is optimized. Alignment errors of QWPA, WP and LA respect to the ideal position are discussed and a procedure for compensating the alignment errors is also developed. Our conclusions are contained in section 4. 2. THEORY

2.1 Optical layout The optical layout of a QWPAISP is depicted in Fig.1. The QWPA are consists of three quarter-wave plates (QWP1, QWP2 and QWP3) with different azimuth angles. The Wollaston prism (WP) contains two orthogonally oriented birefringent crystal prisms with optic axes orientations of 0° and 90° with respect to the x axis, respectively. A Savart polariscope (SP), consists of two identical uniaxial crystal plates with orthogonally oriented principal sections, is positioned behind the WP. The optic axes of the two plates are oriented at 45° relative to the z axis and their projections on the x-y plane is oriented at ± 45° respectively relative to the x axis. A linear analyzer (LA) follows the group with its transmission axis at 90°. A single charge coupled device (CCD) is placed on the back focal plane of three imaging lenses (L2, L3 and L4) with same focal length. Light from the object is imaged on intermediate image plane M by lens L0 and then collimated by lens L1. The incident electric field vectors modulated by different QWPs are separated by the WP along y axis with the vibration direction oriented at 0° and 90°, respectively. The 0° and 90° oriented electric fields are laterally sheared respectively by the SP into two pairs of equal-amplitude but orthogonally polarized components. The LA extracts the identical linearly polarized components. Each pair of the equal-amplitude polarized components are reunited on the six parts of the CCD camera, and six interference images in the spatial domain can be recorded simultaneously. The optical path difference (OPD) is introduced by the SP. Complete interferogram for the same object pixel can be collected by employing tempo-spatially

mixed modulated mode (also called windowing mode).

Fig1. Optical layout of the system (a) three dimensional graph (b) side view graph

2.2 Optical path difference As described above, the optical path difference (OPD) is supplied by the SP and is given by [11]  sp  t

2 2 no 2 ( )  ne 2 ( ) t no ( )  no ( )  ne ( )  (cos   sin  )sin i  (cos 2   sin 2  )sin 2 i+terms in sin 4i, etc (1) 3/2 no 2 ( )  ne 2 ( ) 2 ne ( )  no 2 ( )  ne 2 ( ) 

here the coefficient of sini term is lateral displacement produced by the SP; ne and no are refractive indices of the uniaxial crystal material of SP for extraordinary and ordinary rays, respectively; i is the incident angle; ω is the angle between the incident plane and the principal section of the plate, usually ω=0°; t is the thickness of the single crystal plate. The interference fringes formed on the detector is equivalent to that of equal angles of incidence. For small angles of incidence, the second and higher power terms of

sini can be vanished, the OPD can be expressed as

 sp  t

no 2 ( )  ne 2 ( ) (cos   sin  )sin i no 2 ( )  ne 2 ( )

(2)

The sini term in Eq.(2) produces straight interference fringes along x axis with constant spatial frequency. Since SP is usually made of birefringent uniaxial crystal materials, there is chromatic variation in the lateral displacement and OPD, which leads to image blurring in spectral images and nonlinear effects in the process of recovering spectrum. Fortunately, the achromatization of the traditional SP by combining two SPs of similar and opposite chromatic dispersions are proposed [17] and it can be used to overcome the above drawback over a spectral region of 400 nm ~1000 nm. Such an achromatic SP can reduce chromatic variation in the lateral displacement and OPD by an order of magnitude while retaining the key conveniences of SPs. 2.3 Interference intensities As described above, the WP acts as orthogonal polarizers and its corresponding Mueller matrices are MWP1 (ε =0°) (ε denotes the polarization direction with the x axis) and MWP2 (ε =90°). The Mueller matrix of each channel in the optical system can be represented as

M1  M LA (90 )M SP MWP1 (0 )M QWP (1 )

(3a)

M 2  M LA (90 )M SP MWP1 (90 )M QWP (1 )

(3b)

M 3  M LA (90 )M SP MWP1 (0 )M QWP (2 )

(3c)

M 4  M LA (90 )M SP MWP1 (90 )M QWP (2 )

(3d)

M 5  M LA (90 )M SP MWP1 (0 )M QWP (3 )

(3e)

M 6  M LA (90 )M SP MWP1 (90 )M QWP (3 )

(3f)

where, M LA , M SP , M WP , M QWP are the ideal Mueller matrices of LA, SP, WP and QWP. If the incident Stokes vector is  S0 ( x, y,  ) S1 ( x, y,  ) S2 ( x, y,  ) S3 ( x, y,  ) , the T

six intensities measured by the CCD would be 1  cos  S0  cos2 21S1  cos 21 sin 21S2  sin 21S3   4 1  cos  I2   S0  cos2 21S1  cos 21 sin 21S2 +sin 21S3  4 1  cos  I3   S0  cos2 22 S1  cos 22 sin 22 S2  sin 22 S3  4 1  cos  I4   S0  cos2 22 S1  cos 22 sin 22 S2  sin 22S3  4 1  cos  I5  S0  cos2 23 S1  cos 23 sin 23 S2  sin 23 S3   4 1  cos  I6  S0  cos2 23 S1  cos 23 sin 23 S2  sin 23S3   4 I1 

(4a) (4b) (4c) (4d) (4e) (4f)

After remove the background intensity of each interferogram, the pure interference fringes corresponding to each channel can be obtained cos   S0  cos2 21S1  cos 21 sin 21S2  sin 21S3  4 cos  I 2p  S0  cos 2 21S1  cos 21 sin 21S2  sin 21S3   4 c o s I3p  S0 c o 2s 2 2 S 1  c os 22 si nS2 2 2 s iS2n 23  4 cos  I 4p   S0  S1 cos2 22  cos 22 sin 22 S2  sin 22 S3  4 cos  I5p   S0  S1 cos2 23  cos 23 sin 23S2  sin 23S3  4 cos  I 6p   S0  S1 cos2 23  cos 23 sin 23S2  sin 23S3  4 I 1p 

(5a) (5b) (5c) (5d) (5e) (5f)

From Eqs.(5a)—(5f), we can get I 2p  I 1p  I 4p  I 3p =I 6p  I 5p =

cos  S0 2

cos    cos2 21S1  cos 21 sin 21S2 +sin 21S3  2 cos  I3p +I 4p    cos2 22 S1  cos 22 sin 22 S2  sin 22 S3  2 I1p +I 2p 

(6a) (6b) (6c)

I 5p +I 6p 

cos    cos2 23S1  cos 23 sin 23S2  sin 23S3  2

(6e)

By taking Fourier transform of Eqs.(6a)—(6e), it has





(7a)

 cos2 21S1  cos 21 sin 21S2 +sin 21S3 =2 FT I1p +I 2p   F1

(7b)

 cos2 22 S1  cos 22 sin 22 S2 +sin 22 S3 =2 FT I3p +I 4p   F2

(7c)

S0 =2FT I 2p  I 1p  F0

 cos2 23 S1  cos 23 sin 23 S2 +sin 23 S3 =2 FT I5p +I 6p   F3

(7e)

The Eqs. (7a)— (7e) can be expressed by matrix A Sin  F

 1  0 A 0  0

Where

F   F0

F1

F2

0 a1 a2 a3

F3 

T

0 0   b1 c1  b2 c2   b3 c3 

with

(8)  ai   cos 2 2i  bi   cos 2i sin 2i  ci  sin 2i 

i  1, 2,3

Sin   S0 ( x, y,  ) S1 ( x, y,  ) S2 ( x, y,  ) S3 ( x, y,  ) 

T

The spatial-spectral variation of the all four Stokes parameters can be calculated by Sin  A1F

(9)

The four Stokes parameters are separated without spatial aliasing, so both broad-band spectrum and narrow-band spectrum can be detected. Besides, the spectral resolution is the same as that of the spectrometer. 2.4 Optimization of the orientation angles of the quarter-wave plate array Similar to the rotatable retarder fixed polarizer (RRFP) Stokes polarimeter, matrix A is the measurement matrix of the optical system [18]. Errors can be introduced in the measurements by misalignments of the quarter wave-plates or random fluctuations in the detected intensity. We assume the WP is in the ideal position, consistent with Ambirajan and Look’s error equation for a Stokes polarimeter, Eq.(8) becomes

 F  F    A  A Sin  S 

(10)

F is the error term which is introduced by the Fourier transform of the random fluctuations in the detected intensity [18]. A is the error term introduced by misalignments of the quarter wave-plates. S is the resultant error due to the perturbations F and A in the estimated incident Stokes parameters Sin. The minimization of S / Sin

(||

∙ || denotes the norm of a vector, whether is a 1,

a 2, a ∞ and a Frobenius norm) will minimize the propagation of error and amplification of variance in the estimate. Thus, the following procedure to minimize

S / Sin

can be taken:

We denote the 4X4 identity matrix by 1.Matrix multiplication yields

A1F  A1F  1  A1A  S  S 

(11)

Noting that no element of A has an absolute value greater than 1, then S can be obtained by using the fact that S  A1F .

S  1  A1A

1

A

F  A1ASin 

1

(12)

By taking the norm and using the triangle and consistency inequalities, we obtain S  1  A1A

1

A

1

F + A1 A Sin



(13)

By using

1  A

A

1

1



1 1

1 A

A

(14)

We get S S



  A 1

1 A

The condition number is defined as

 F A  +   A  A  S A

(15)

  A = A1 A for any matrix norm, such

that

1  A = A1

 2  A = A1    A = A1

 Frob  A = A1 are

condition

numbers

1

A1

(16a)

2

A2

(16b)



A

(16c)

Frob

corresponding to

A

(16d)

Frob

1-norm,

2-norm,  -norm

and

Frobenius-norm, respectively. Clearly, if the condition number of A is minimized, then the right-hand side of Eq.(15) is minimized, hence the left-hand side will be constrained. Table 1—6 show the optimization results of minimizing the four condition numbers by different angle sets. Considering the convenience of angle adjustment and the inverting of matrix A, the angle sets are consist of a pair of fixed angles by (0° 30° 45° 60° ) and anther variable one. It can be seen that the values of κ1 κ2 κ∞ κFrob in Table 4 and 5 are obviously larger than others. The minimum values of κ1 κ2 κ∞ κFrob are 2.8133，2.0953, 3.5774 and 4.5890. The angle sets (0°

60° 25.5° ) and (0° 30°

64.5°) have the same optimization results with (2.8133 2.0970 3.9188 4.5913), so do the angle sets (0°

30°

60° ) and (30°

60°

90° ) with (3 2.1814 3.5774 4.6547).

They are all very close to the minimum values of κ1, κ2, κ∞ and κFrob. The advantage of angle sets (0°

30°

60° ) and (30°

60°

90°) is that they can lead to a matrix

A that is simple to invert and yields more simple calculation process.

Table 1 Minimization results of condition numbers of the measurement matrix with fixed (θ1 =0°, θ2=45°) and a variable θ3 minimization

(θ1 , θ2)

θ3

κ1

κ2

κ∞

κ1

(0°,45°)

161°

κ2

(0°,45°)

κ∞ κFrob

κFrob

3.6957

3.8636

7.9377

6.4799

22.5°

4.1213

3.7321

7.5355

6.3246

(0°,45°)

63.5

4.7873

3.9051

7.3796

6.5293

(0°,45°)

67.5

4.1213

3.7321

7.5355

6.3246

Table 2 Minimization results of condition numbers of the measurement matrix with fixed (θ1 =0°, θ2=60°) and a variable θ3 minimization

(θ1 , θ2)

θ3

κ1

κ2

κ∞

κ1

(0°,60°)

25.5°

κ2

(0°,60°)

κ∞ κFrob

κFrob

2.8133

2.0970

3.9188

4.5913

153.5°

2.8661

2.0895

4.6913

4.7187

(0°,60°)

30°

3

2.1814

3.5774

4.6547

(0°,60°)

26°

2.8221

2.0953

3.8741

4.5890

Table 3 Minimization results of condition numbers of the measurement matrix with fixed (θ1 =0°, θ2=30°) and a variable θ3 minimization

(θ1 , θ2)

θ3

κ1

κ2

κ∞

κ1

(0°,30°)

64.5°

κ2

(0°,30°)

κ∞ κFrob

κFrob

2.8133

2.0970

3.9188

4.5913

116.5°

2.8661

2.0895

4.6913

4.7187

(0°,30°)

60°

3

2.1814

3.5774

4.6547

(0°,30°)

64°

2.8221

2.0953

3.8741

4.5890

Table 4 Minimization results of condition numbers of the measurement matrix with fixed (θ1 =45°, θ2=30°) and a variable θ3 minimization

(θ1 , θ2)

θ3

κ1

κ2

κ∞

κ1

(45°,30°)

90°

κ2

(45°,30°)

κ∞ κFrob

κFrob

5.5981

4.3911

7.5698

7.1181

80.5°

7.3883

3.9714

6.1312

6.4225

(45°,30°)

74.5°

8.5648

4.1593

5.8896

6.5608

(45°,30°)

79.5°

7.5804

3.9781

6.0633

6.4155

Table 5 Minimization results of condition numbers of the measurement matrix with fixed (θ1 =45°, θ2=60°) and a variable θ3 minimization

(θ1 , θ2)

θ3

κ1

κ2

κ∞

κ1

(45°,60°)

90°

κ2

(45°,60°)

κ∞ κFrob

κFrob

5.5981

4.3911

7.5698

7.1181

9.5°

7.3883

3.9714

6.1312

6.4225

(45°,60°)

15.5°

8.5648

4.1593

5.8896

6.5608

(45°,60°)

10.5°

7.5804

3.9781

6.0633

6.4155

Table 6 Minimization results of condition numbers of the measurement matrix with fixed (θ1 =30°, θ2=60°) and a variable θ3 minimization

(θ1 , θ2)

θ3

κ1

(30°,60°)

90°

κ2

(30°,60°)

κ∞ κFrob

κ1

κ2

κ∞

κFrob

3

2.1814

3.5774

4.6547

176°

3.5533

2.1449

3.5774

4.6418

(30°,60°)

1.5°

3.3027

2.2174

3.5774

4.6849

(30°,60°)

92.5°

3.3424

2.1489

3.5774

4.6359

3. SIMULATIOIN AND ANALYISIS 3.1 Reconstruction of polarimetric spectrum To demonstrate the versatility of the QWPAISP system, a mathematical model for simulation and reconstruction is developed. The simulation considers only the reconstruction of polarimetric spectral data from extended source. The spectral region is 0.48 m ~0.96 m. The angle set of QWPA is (30° 60° 90°) . The CCD is a 16 bit monochrome camera with a resolution of 512 × 512, and the pixel size is 16 m × 16 m. Then each interferogram will occupy 85 × 512 pixels. According to the Nyquist sampling theorem, to avoid spectrum aliasing the sampling interval of the

=min/2 = 0.24 m. The maximum OPD is effectively limited by the Nyquist criterion that requires at least two data points per fringe period. Hence, if the interferogram is symmetrically recorded about zero OPD, the maximum OPD is max =256× =61.44m. Correspondingly, the highest spectral resolution with rectangular function is =/2max, which is about 1.6 nm at the

wavelength of 0.48 m. The number of spectral bands is about 256. To realize compactness of the polarization elements, the WP and SP can be made of calcite with high transmittance over a wide waveband. If the focal length of the L1, L2 and L3 is f = 80 mm, to fully utilize the spatial resolution of the CCD along the y direction, the apex angle of WP should be 2.7° and the corresponding splitting angle is about 1°. To achieve maximum OPD along the x direction, the lateral displacement produced by the SP should be d=1 mm, then the thickness of the single plate is t = 6.5 mm. Therefore, the attainable incident angle of the polarization integration is ± 3° in the x direction and ± 0.5° in the y direction. Figure 2 shows the simulated interferograms for each polarized electric field that emitted from the object. Applying Fourier transformation with hamming apodization, the input and reconstructed spectra of Stokes parameters are depicted in Fig. 3. As can be seen, the reconstructed data consist with the input data.

Fig. 2 The simulated interferograms

1

S0

0.9 0.8

Intensity/a.u

0.7

S1

0.6 0.5

S

0.4

2

0.3

S3

0.2 0.1 0 0.48

0.6

0.72

0.84

0.96

Wavelength/m

Fig. 3 Superimposed original spectra (solid lines) and reconstructed spectra (dotted lines)

3.2 Error analysis The Stokes parameters contained in each interferogram may distorted by systematic errors inherent in the optical system. Herein, we consider the alignment errors (ε1, ε2, ε3) of quarter-wave plates array with respect to the optimized angle sets, ς and δ of WP and LA with respect to the ideal position. Inserting the error terms into the Mueller calculus, the six interference intensities can be calculated by

1 I a  (1  sin 2 sin 2  cos 2 cos 2 cos  ) Sa 8

(17a)

1 Ib  (1+sin 2 sin 2 + cos 2 cos 2 cos  ) Sb 8

(17b)

1 I c  (1  sin 2 sin 2  cos 2 cos 2 cos  ) Sc 8

(17c)

1 I d  (1+sin 2 sin 2 + cos 2 cos 2 cos  ) Sd 8

(17d)

1 I e  (1  sin 2 sin 2  cos 2 cos 2 cos  ) Se 8

(17e)

1 I f  (1+sin 2 sin 2 + cos 2 cos 2 cos  ) S f 8

(17f)

where Sa  2S0 ( )+ cos  41  41  2   cos 2  S1 ( )+ sin  41  41  2   sin 2  S2 ( )  2sin  41  21  2  S3 ( ) Sb  2S0 ( )  cos  41  41  2   cos 2  S1 ( )  sin  41  41  2   sin 2  S2 ( )  2sin  41  21  2  S3 ( )

Sc  2S0 ( )+ cos  42  41  2   cos 2  S1 ( )+ sin  42  41  2   sin 2  S2 ( )  2sin  42  21  2  S3 ( )

(18a) (18b) (18c)

Sd  2S0 ( )  cos  42  41  2   cos 2  S1 ( )  sin  42  41  2   sin 2  S2 ( )  2sin  42  21  2  S3 ( )

(18d)

Se  2S0 ( )  cos  43  41  2   cos 2  S1 ( )  sin  43  41  2   sin 2  S2 ( )  2sin  43  21  2  S3 ( )

(18e)

S f  2S0 ( )  cos  43  41  2   cos 2  S1 ( )  sin  43  41  2   sin 2  S2 ( )  2sin  43  21  2  S3 ( )

(18f)

The fringe visibilities, pure interference fringes and the recovery equations with alignment errors can be expressed as

Vm  Vn 

cos 2 cos 2 1  sin 2 sin 2

cos 2 cos 2 1+sin 2 sin 2

(m  a, c, e)

(n  b, d , f )

1 I mint er   cos 2 cos 2 cos  Sm 8 1 I nint er  cos 2 cos 2 cos  Sm 8

(m  a, c, e)

(n  b, d , f )

(19a)

(19b)

(20a)

(20b)

 S0   A1 0     S1    0 B1  S2   0 B2     S3   0 B3 where

0   D1  D2   D3 

0 C1 C2 C3

1

 F0 e     F1e   F2 e     F3e 

(21)

F0e  2FT Ib  I a  F1e  2FT I a +Ib  F2e  2FT I c  I d  F3e  2FT I e +I f 

A1  cos 2 cos 2 Bi =  cos 2 cos 2

cos  4i  4 i  2   cos 2 2

Ci =  cos 2 cos 2

sin  4i  4 i  2   sin 2 2

Di  cos 2 cos 2 sin  2i  2 i  2 

As can be seen, the fringe visibilities are only affected by the alignment errors of WP and LA. WP herein not only acts as a beam splitter, but also as two orthogonal linear polarizers. The alignment errors of WP and QWP complicate the Stokes parameter content of the each channel. Therefore, we should develop a procedure for compensating the alignment errors.

1 0.9

1

(a)  =0 deg

(b)  =5 deg

Vm

Vm

Vn

0.8

Vn

0.8

0.7

0.6

0.5

V

V

0.6

0.4

0.4 0.3

0.2

0.2 0.1 0

-45

-22.5

0  /deg

22.5

1 0.9

0

45

1

Vm

(c) =0 deg

Vn

0.8

-45

-22.5

0  /deg

22.5

(d)  =5 deg

45

Vm Vn

0.8

0.7

0.6

0.5

V

V

0.6

0.4

0.4 0.3

0.2

0.2 0.1 0

-45

-22.5

0  /deg

22.5

45

0

-45

-22.5

0  /deg

22.5

45

Fig.4 The visibilities of the interferograms vary with the alignment errors of the LA when the WP is placed at (a) the ideal direction and (b) the nonideal direction and vary with the alignment errors of WP when the LA is placed at (c) the ideal direction and (d) the nonideal direction, respectively

Fig.4 shows the relationships between the fringe visibilities and alignment errors of WP and LA. The modulations of WP on the visibilities of the interference fringes are similar to each other when LA is at an ideal position. The visibility of the six interferograms will drop to zero when WP deviates from the ideal position with an angle of 45°. If the LA deviates from the ideal position, the difference between Vm and Vn will be increased. The effect of the LA on the visibility is similar to that of the

WP. According to the above discussions, the reconstructed polarimetric spectrum would deviate from the input ones when different misalignments are introduced. Fig.5 shows the absolute deviation between the ideal input and the reconstructed polarimetric spectrum when the misalignment is 5°. As can be seen, the deviations in S0 are not affected by the misalignment in QWPA.

S3 is most insensitive to the

misalignment in the system than other Stokes parameters when the misalignment in WP equals to that of QWPA. In other cases, S2 is most sensitive to the misalignments of WP and QWPA. S1 and S3 are distorted most in Fig.5(c), and S2 distorted most in Fig.5 (b). The maximum deviations of S0, S1, S2 and S3 are 0.0302, 0.1740, 0.4737 and 0.1608, respectively. To reduce the distortion of the reconstructed Stokes parameters, a precise physical adjustment can be employed. In experiment, an alignment of 1° can be achieved easily with a common physical adjustment. We assume that the minimum input misalignment of 1° is the result of the mechanical correction. As is shown in Fig.6, a misalignment as small as 1° can still deteriorate the reconstructed Stokes spectra. The maximum deviations of S0, S1, S2 and S3 are 0.0012, 0.0263, 0.0925 and 0.0264, respectively. An even more precise physical adjustment may be needed. However, if an effective software correction is developed to account for the alignment errors in post-processing, manufacturing tolerances can be loosened. In the following, a compensation procedure will be proposed.

0.5

(a)

Absolute deviation/a.u

0.14

S0

0.4

S1

0.12

S2

0.1

(b)

0.45

Absolute deviation /a.u

0.16

S3

0.08 0.06 0.04

S0 S1

0.35

S2

0.3

S3

0.25 0.2 0.15 0.1

0.02

0.05

0 0.48

0.4

0.6

0.72 Wavelength/m

0.84

0 0.48

0.96

0.35

0.2

0.84

0.96

(d)

0.1

S0 S1

Absolute deviation/a.u

Absolute deviation/a.u

0.25

0.72 Wavelength/m

0.12

(c)

0.3

0.6

S2 S3

0.15

0.08

S0 S1

0.06

S2 S3

0.04

0.1 0.02

0.05 0 0.48

0.6

0.72 Wavelength/m

0.84

0.96

0 0.48

0.6

0.72 Wavelength/m

0.84

0.96

Fig. 5 The absolute deviation between the ideal input and the reconstructed Stokes spectra when the alignment error is 5°. (a) ς= δ=5°, ε1=ε2=ε3= 5° (b) ς= -5°, δ=5°, ε1=ε2=ε3= 5° (c) ς= δ=5°, ε1=ε2=ε3= -5° (d) ς= -5°, δ=5°, ε1=ε2=ε3= -5°

0.03

0.1

(a)

0.08 x 10

0.02

-3

Absolute deviation/a.u

Absolute deviation/a.u

0.025

(b)

S0

1

S1 0.5

S2

0.015 0 0.48

0.6

0.72

0.84

S3

0.96

0.01

S0 0.06

x 10

S1

1.2

S2

1.1

S3

0.04

-3

1 0.9 0.8 0.48

0.6

0.72

0.84

0.96

0.02

0.005 0 0.48

0.6

0.72 Wavelength/m

0.84

0.09

0 0.48

0.96

0.6

0.72 Wavelength/m

0.84

0.96

0.03

(c)

(d)

0.08 0.025

0.06

x 10

-3

1.2

S0

1.1

S1

0.05

1

S2

0.04

0.9

S3

0.03

0.8 0.48

0.6

0.72

0.84

Absolute deviation/a.u

Absolute deviation/a.u

0.07

0.96

0.02

0.015

S0 S1 S2 S3 x 10

0.01

1

0.02

0.5

0.005

0.01 0 0.48

0 0.48

0.6

0.72 Wavelength/m

0.84

0.96

0 0.48

-3

0.6

0.72 Wavelength/m

0.6

0.72 0.84 0.96

0.84

0.96

Fig. 6 The absolute deviation between the ideal input and the reconstructed Stokes spectra when the alignment error is 1°. (a) ς=δ=1°, ε1=ε2=ε3=1° (b) ς= -1°, δ=1°, ε1=ε2=ε3= 1° (c) ς= δ=1°, ε1=ε2=ε3= -1° (d) ς= -1°, δ=1°, ε1=ε2=ε3= -1°

3.3 Compensation of the alignment errors The Stokes parameters can be calculated from Eq.(9). As mentioned in section 2.1, random fluctuations in the detected intensity are another error source which is mainly caused by the additive noise on the detector and this additive noise can be removed by many methods [15-16, 19]. Hence, a more accurate vector F can be obtained by the Fourier transform of the intensity vector I which is read from the detector, and the

modulated spectrum only with alignment errors can be obtained. The matrix A can be determined by the ―reference beam calibration technique‖ [20]. Any known polarization state can be used as a reference calibration state. In order to determine all elements of matrix A, at least three reference calibration states with S1≠0, S2≠0 or S3≠0 are needed. This particular constraint in reference calibration states will complicate the calibration procedure. However, for the QWPAISP system, the element Di corresponding to the coefficient of S3 in Eq.(21) can be expressed by the other three elements, as shown in Eq.(23). Hence, by taking two linearly polarization states as the reference data, all the elements in matrix A can be determined. This is another advantage we expect. The pair of linearly polarization reference calibration states is denoted as  S01 , S11 , S21 , S31  =S01 1, cos 21 ,sin 21 , 0 T

T

 S02 , S12 , S22 , S32  =S02 1, cos 2 2 ,sin 2 2 , 0 T

T

where α1 and α2 are polarization angles of the two reference calibration states, respectively. Taking the pair of known polarization states as the incident spectrum, the following two equations can be obtained

 A1 0  B1 1  0 S0  0 B2   0 B3

0 C1 C2 C3

0  1   F01     D1  cos 21   F11  = D2  sin 21   F21      D3   0   F31 

(22a)

 A1 0  B1 2  0 S0  0 B2   0 B3

0 C1 C2 C3

0  1   F0 2     D1  cos 2 2   F1 2  = D2  sin 2 2   F2 2      D3   0   F3 2 

(22b)

The element in matrix A with errors can be determined as

 F01 F0 2 A =  2 1  1 S S0 0   S 1 sin 21 Fi 2  S0 2 sin 2 2 Fi1  Bi = 0  S0 2 S01 sin  21  2 2    S0 2 cos 2 2 Fi1  S01 cos 21 Fi 2 Ci = S0 2 S01 sin  21  2 2    Di = A12   Bi 2  Ci 2  

i  1, 2,3

(23)

Hence, the recovery polarimetric spectrum can be calculated from Eq.(10) by the acquired intensities on the detector. Fig.7 shows the compensated result by α1 =30°, α2=45°when misalignment is 1°. The absolute deviations of the four Stokes parameters in Fig.7, compared to that in Fig.6, are all reduced to zero. Because the measurement matrix A can be determined precisely by two linearly polarization reference calibration states, any errors introduced by the misalignment can be compensated effectively.

x 10 1

-14

S0

Absolute deviation/a.u

S1 S2 0.5

S3

0

-0.5

-1 0.48

0.6

0.72 Wavelength/m

0.84

0.96

Fig. 7 The absolute deviation between the input and the compensated Stokes parameters when misalignment is 1°.

4. SUMMARY In conclusion, we proposed a static Fourier transform imaging spectropolarimeter based on quarter-wave plate array, which can detect the image and polarimetric spectral data of a target simultaneously.

The orientation angle set of the

quarter-wave plate array is (30° 60° 90°), which is optimized by minimization of the condition numbers of the measurement matrix. The alignment errors in WP and LA only affected the fringe visibilities. The reconstructed wavelength depended Stokes parameters would be distorted by the alignment errors in QWPA and WP.

Duto the system characteristics of the QWPAISP, the elements in the fourth row corresponding to S3 of the measurement matrix can be calculated by other elements. Therefore, all the elements of measurement matrix can be determined by two linearly polarization states as the reference data and the precise reconstructed polarimetric spectral data can be obtained. The future study on the QWPAISP includes experimental verification, reconstruction techniques and prototyping of systems for the visible and near-infrared spectral bands.

Acknowledgments The authors thank the anonymous reviewers for their helpful comments and constructive suggestions. This work was supported by the State Key Program of National Natural Science Foundation of China (Grant No. 41530422); the National Natural Science Foundation of China (Grant No. 61540018；61275184, 61405153)； the National Major Project (Grant No. 32-Y30B08-9001-13/15); the National High Technology Research and Development Program of China 863 (Grant No. 2012AA121101), and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130201120047).

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