Abstract

A compact short-wavelength and middle-wavelength infrared hyperspectral imaging polarimeter (IHIP) is introduced. The sensor includes a pair of sapphire Wollaston prisms and several high-order retarders to form an imaging Fourier transform spectropolarimeter. The Wollaston prisms serve as a birefringent interferometer with reduced sensitivity to vibration versus an unequal path interferometer, such as a Michelson. Polarimetric data are acquired through the use of channeled spectropolarimetry to modulate the spectrum with the Stokes parameter information. The collected interferogram is Fourier filtered and reconstructed to recover the spatially and spectrally varying Stokes vector data across the image. The IHIP operates over a ±5° field of view and implements a dual-scan false signature reduction technique to suppress polarimetric aliasing artifacts. In this paper, the optical layout and operation of the IHIP sensor are presented in addition to the radiometric, spectral, and polarimetric calibration techniques used with the system. Spectral and spectropolarimetric results from the laboratory and outdoor tests with the instrument are also presented.

© 2011 Optical Society of America

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References

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  1. W. L. Wolfe, Introduction to Imaging Spectrometers (SPIE, 1997).
    [CrossRef]
  2. K. H. Nordsieck, “A simple polarimetric system for the Lick Observatory Image-Tube Scanner,” Publ. Astron. Soc. Pac. 86, 324–329 (1974).
    [CrossRef]
  3. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475–1477 (1999).
    [CrossRef]
  4. S. H. Jones, F. J. Iannarilli, and P. L. Kebabian, “Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter,” Opt. Express 12, 6559–6573 (2004).
    [CrossRef] [PubMed]
  5. T. Kusunoki and K. Oka, “Fourier spectroscopic measurement of polarization using birefringent retarders,” in The 61st Autumn Meeting of the Japan Society of Applied Physics (2000), p. 871.
  6. M. W. Kudenov, N. A. Hagen, E. L. Dereniak, and G. R. Gerhart, “Fourier transform channeled spectropolarimetry in the MWIR,” Opt. Express 15, 12792–12805 (2007).
    [CrossRef] [PubMed]
  7. J. Craven, M. W. Kudenov, and E. L. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419, 741909 (2009).
    [CrossRef]
  8. J. Craven and M. W. Kudenov, “False signature reduction in channeled spectropolarimetry,” Opt. Eng. 49, 053602 (2010).
    [CrossRef]
  9. 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, 5453–5469 (2006).
    [CrossRef] [PubMed]
  10. A. R. Harvey and D. W. Fletcher-Holmes, “Birefringent Fourier-transform imaging spectrometer,” Opt. Express 12, 5368–5374 (2004).
    [CrossRef] [PubMed]
  11. V. Saptari, Fourier Transform Spectroscopy Instrumentation Engineering (SPIE, 2004).
  12. L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 (1967).
    [CrossRef]
  13. F. Snik, T. Karalidi, and C. U. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48, 1337–1346 (2009).
    [CrossRef] [PubMed]
  14. R. Venkateswarlu, M. H. Er, Y. H. Gan, and Y. C. Fong. “Nonuniformity compensation for IR focal plane array sensors,” Proc. SPIE 3061, 915–926 (1997).
    [CrossRef]
  15. The Infrared Handbook, W.L.Wolfe and G.J.Zissis, eds. (Infrared Information Analysis (IRIA) Center, Environmental Research Institute of Michigan, 1993).
  16. M. Francon and S. Mallick, Polarization Interferomers: Applications in Microscopy and Macroscopy (Wiley Interscience, 1972).
  17. A. Taniguchi, K. Oka, H. Okabe, and M. Hayakawa, “Stabilization of a channeled spectropolarimeter by self-calibration,” Opt. Lett. 31, 3279–3281 (2006).
    [CrossRef] [PubMed]
  18. L. W. Schumann and T. S. Lomhein, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” Proc. SPIE 4480, 1–14 (2002).
    [CrossRef]
  19. E. Voigtman and J. D. Winefordner, “The multiplex disadvantage and excess low-frequency noise,” Appl. Spectrosc. 41, 1182–1184 (1987).
    [CrossRef]
  20. D. Goldstein, Polarized Light (Marcel Dekker, 2003).
    [CrossRef]
  21. M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48, 063201 (2009).
    [CrossRef]
  22. J. S. Tyo, E. N. Pugh, and N. Engheta, “Colorimetric representations for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A 15, 367–374(1998).
    [CrossRef]
  23. O. Sandus, “A review of emission polarization,” Appl. Opt. 4, 1634–1642 (1965).
    [CrossRef]
  24. C. Koike, H. Hasegawa, N. Asada, and T. Komatuzaki, “Optical constants of fine particles for the infrared region,” Mon. Not. R. Astr. Soc. 239, 127–137 (1989).
  25. O. Jacquot and P. Herve, “Determination of the temperature field in exhaust gases by infrared spectroscopy,” Proc. SPIE 3493, 71–78 (1998).
    [CrossRef]

2010 (1)

J. Craven and M. W. Kudenov, “False signature reduction in channeled spectropolarimetry,” Opt. Eng. 49, 053602 (2010).
[CrossRef]

2009 (3)

F. Snik, T. Karalidi, and C. U. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48, 1337–1346 (2009).
[CrossRef] [PubMed]

M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48, 063201 (2009).
[CrossRef]

J. Craven, M. W. Kudenov, and E. L. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419, 741909 (2009).
[CrossRef]

2007 (1)

2006 (2)

2004 (2)

2002 (1)

L. W. Schumann and T. S. Lomhein, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” Proc. SPIE 4480, 1–14 (2002).
[CrossRef]

1999 (1)

1998 (2)

J. S. Tyo, E. N. Pugh, and N. Engheta, “Colorimetric representations for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A 15, 367–374(1998).
[CrossRef]

O. Jacquot and P. Herve, “Determination of the temperature field in exhaust gases by infrared spectroscopy,” Proc. SPIE 3493, 71–78 (1998).
[CrossRef]

1997 (1)

R. Venkateswarlu, M. H. Er, Y. H. Gan, and Y. C. Fong. “Nonuniformity compensation for IR focal plane array sensors,” Proc. SPIE 3061, 915–926 (1997).
[CrossRef]

1989 (1)

C. Koike, H. Hasegawa, N. Asada, and T. Komatuzaki, “Optical constants of fine particles for the infrared region,” Mon. Not. R. Astr. Soc. 239, 127–137 (1989).

1987 (1)

1974 (1)

K. H. Nordsieck, “A simple polarimetric system for the Lick Observatory Image-Tube Scanner,” Publ. Astron. Soc. Pac. 86, 324–329 (1974).
[CrossRef]

1967 (1)

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 (1967).
[CrossRef]

1965 (1)

Asada, N.

C. Koike, H. Hasegawa, N. Asada, and T. Komatuzaki, “Optical constants of fine particles for the infrared region,” Mon. Not. R. Astr. Soc. 239, 127–137 (1989).

Chenault, D. B.

Craven, J.

J. Craven and M. W. Kudenov, “False signature reduction in channeled spectropolarimetry,” Opt. Eng. 49, 053602 (2010).
[CrossRef]

J. Craven, M. W. Kudenov, and E. L. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419, 741909 (2009).
[CrossRef]

Dereniak, E. L.

J. Craven, M. W. Kudenov, and E. L. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419, 741909 (2009).
[CrossRef]

M. W. Kudenov, N. A. Hagen, E. L. Dereniak, and G. R. Gerhart, “Fourier transform channeled spectropolarimetry in the MWIR,” Opt. Express 15, 12792–12805 (2007).
[CrossRef] [PubMed]

Engheta, N.

Er, M. H.

R. Venkateswarlu, M. H. Er, Y. H. Gan, and Y. C. Fong. “Nonuniformity compensation for IR focal plane array sensors,” Proc. SPIE 3061, 915–926 (1997).
[CrossRef]

Fletcher-Holmes, D. W.

Fong, Y. C.

R. Venkateswarlu, M. H. Er, Y. H. Gan, and Y. C. Fong. “Nonuniformity compensation for IR focal plane array sensors,” Proc. SPIE 3061, 915–926 (1997).
[CrossRef]

Francon, M.

M. Francon and S. Mallick, Polarization Interferomers: Applications in Microscopy and Macroscopy (Wiley Interscience, 1972).

Gan, Y. H.

R. Venkateswarlu, M. H. Er, Y. H. Gan, and Y. C. Fong. “Nonuniformity compensation for IR focal plane array sensors,” Proc. SPIE 3061, 915–926 (1997).
[CrossRef]

Gerhart, G. R.

M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48, 063201 (2009).
[CrossRef]

M. W. Kudenov, N. A. Hagen, E. L. Dereniak, and G. R. Gerhart, “Fourier transform channeled spectropolarimetry in the MWIR,” Opt. Express 15, 12792–12805 (2007).
[CrossRef] [PubMed]

Goldstein, D.

D. Goldstein, Polarized Light (Marcel Dekker, 2003).
[CrossRef]

Goldstein, D. L.

Hagen, N. A.

Harvey, A. R.

Hasegawa, H.

C. Koike, H. Hasegawa, N. Asada, and T. Komatuzaki, “Optical constants of fine particles for the infrared region,” Mon. Not. R. Astr. Soc. 239, 127–137 (1989).

Hayakawa, M.

Herve, P.

O. Jacquot and P. Herve, “Determination of the temperature field in exhaust gases by infrared spectroscopy,” Proc. SPIE 3493, 71–78 (1998).
[CrossRef]

Iannarilli, F. J.

Jacquot, O.

O. Jacquot and P. Herve, “Determination of the temperature field in exhaust gases by infrared spectroscopy,” Proc. SPIE 3493, 71–78 (1998).
[CrossRef]

Jones, S. H.

Karalidi, T.

Kato, T.

Kebabian, P. L.

Keller, C. U.

Koike, C.

C. Koike, H. Hasegawa, N. Asada, and T. Komatuzaki, “Optical constants of fine particles for the infrared region,” Mon. Not. R. Astr. Soc. 239, 127–137 (1989).

Komatuzaki, T.

C. Koike, H. Hasegawa, N. Asada, and T. Komatuzaki, “Optical constants of fine particles for the infrared region,” Mon. Not. R. Astr. Soc. 239, 127–137 (1989).

Kudenov, M. W.

J. Craven and M. W. Kudenov, “False signature reduction in channeled spectropolarimetry,” Opt. Eng. 49, 053602 (2010).
[CrossRef]

J. Craven, M. W. Kudenov, and E. L. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419, 741909 (2009).
[CrossRef]

M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48, 063201 (2009).
[CrossRef]

M. W. Kudenov, N. A. Hagen, E. L. Dereniak, and G. R. Gerhart, “Fourier transform channeled spectropolarimetry in the MWIR,” Opt. Express 15, 12792–12805 (2007).
[CrossRef] [PubMed]

Kusunoki, T.

T. Kusunoki and K. Oka, “Fourier spectroscopic measurement of polarization using birefringent retarders,” in The 61st Autumn Meeting of the Japan Society of Applied Physics (2000), p. 871.

Lomhein, T. S.

L. W. Schumann and T. S. Lomhein, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” Proc. SPIE 4480, 1–14 (2002).
[CrossRef]

Mallick, S.

M. Francon and S. Mallick, Polarization Interferomers: Applications in Microscopy and Macroscopy (Wiley Interscience, 1972).

Mertz, L.

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 (1967).
[CrossRef]

Nordsieck, K. H.

K. H. Nordsieck, “A simple polarimetric system for the Lick Observatory Image-Tube Scanner,” Publ. Astron. Soc. Pac. 86, 324–329 (1974).
[CrossRef]

Oka, K.

Okabe, H.

Pezzaniti, J. L.

M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48, 063201 (2009).
[CrossRef]

Pugh, E. N.

Sandus, O.

Saptari, V.

V. Saptari, Fourier Transform Spectroscopy Instrumentation Engineering (SPIE, 2004).

Schumann, L. W.

L. W. Schumann and T. S. Lomhein, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” Proc. SPIE 4480, 1–14 (2002).
[CrossRef]

Shaw, J. A.

Snik, F.

Taniguchi, A.

Tyo, J. S.

Venkateswarlu, R.

R. Venkateswarlu, M. H. Er, Y. H. Gan, and Y. C. Fong. “Nonuniformity compensation for IR focal plane array sensors,” Proc. SPIE 3061, 915–926 (1997).
[CrossRef]

Voigtman, E.

Winefordner, J. D.

Wolfe, W. L.

W. L. Wolfe, Introduction to Imaging Spectrometers (SPIE, 1997).
[CrossRef]

Appl. Opt. (3)

Appl. Spectrosc. (1)

Infrared Phys. (1)

L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 (1967).
[CrossRef]

J. Opt. Soc. Am. A (1)

Mon. Not. R. Astr. Soc. (1)

C. Koike, H. Hasegawa, N. Asada, and T. Komatuzaki, “Optical constants of fine particles for the infrared region,” Mon. Not. R. Astr. Soc. 239, 127–137 (1989).

Opt. Eng. (2)

M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48, 063201 (2009).
[CrossRef]

J. Craven and M. W. Kudenov, “False signature reduction in channeled spectropolarimetry,” Opt. Eng. 49, 053602 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (4)

J. Craven, M. W. Kudenov, and E. L. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419, 741909 (2009).
[CrossRef]

L. W. Schumann and T. S. Lomhein, “Infrared hyperspectral imaging Fourier transform and dispersive spectrometers: comparison of signal-to-noise based performance,” Proc. SPIE 4480, 1–14 (2002).
[CrossRef]

R. Venkateswarlu, M. H. Er, Y. H. Gan, and Y. C. Fong. “Nonuniformity compensation for IR focal plane array sensors,” Proc. SPIE 3061, 915–926 (1997).
[CrossRef]

O. Jacquot and P. Herve, “Determination of the temperature field in exhaust gases by infrared spectroscopy,” Proc. SPIE 3493, 71–78 (1998).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

K. H. Nordsieck, “A simple polarimetric system for the Lick Observatory Image-Tube Scanner,” Publ. Astron. Soc. Pac. 86, 324–329 (1974).
[CrossRef]

Other (6)

W. L. Wolfe, Introduction to Imaging Spectrometers (SPIE, 1997).
[CrossRef]

T. Kusunoki and K. Oka, “Fourier spectroscopic measurement of polarization using birefringent retarders,” in The 61st Autumn Meeting of the Japan Society of Applied Physics (2000), p. 871.

The Infrared Handbook, W.L.Wolfe and G.J.Zissis, eds. (Infrared Information Analysis (IRIA) Center, Environmental Research Institute of Michigan, 1993).

M. Francon and S. Mallick, Polarization Interferomers: Applications in Microscopy and Macroscopy (Wiley Interscience, 1972).

V. Saptari, Fourier Transform Spectroscopy Instrumentation Engineering (SPIE, 2004).

D. Goldstein, Polarized Light (Marcel Dekker, 2003).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Imaging Fourier transform spectrometer using a birefringent interferometer. Two Wollaston prisms, W 1 and W 2 , and a high-order retarder, R 3 , are used in series between two polarizers, P 1 and P 2 . An objective lens focuses the interference onto the focal plane array. To assemble an interferogram at each pixel, W 2 is translated while frames of data are collected.

Fig. 2
Fig. 2

Schematic of birefringent imaging spectropolarimeter implementation for the IHIP. Two high order retarders, R 1 and R 2 , are placed in series before the first polarizer ( P 1 ) of the interferometer. The retarders modulate the interferogram at each pixel with carrier frequencies containing the polarimetric information.

Fig. 3
Fig. 3

Interferogram for the on-axis pixel of the IHIP for a 22.5 ° linear incident polarization state with the polarimetric channels, C 0 2 , superimposed. Each channel is used to reconstruct one or two of the Stokes parameters.

Fig. 4
Fig. 4

IHIP sensor assembly. (a) Photograph of the IHIP sensor mounted on a tripod and interfaced with the closed-cycle camera. (b) ZEMAX wire mesh model of the CS retarders, birefringent interferometer, and imaging optics. The objective lens is modeled by a series of thin lenses.

Fig. 5
Fig. 5

Radiometric calibration of the on-axis pixel. The temperature of the blackbody was varied from 56 ° C to 90 ° C .

Fig. 6
Fig. 6

Frame of data captured with the IHIP. The center burst is captured in the image for the on-axis pixel, but a fringe field is present across the FPA.

Fig. 7
Fig. 7

Calculated SNR for the IHIP for four spectral bands, where the center wavenumber of each band is indicated in the legend. Least squares fits for each band are also included to indicate the correspondence to SNR K .

Fig. 8
Fig. 8

Spectropolarimetric reconstructions for an unpolarized source after the implementation of the ART. Residual modulations are reconstructed as false polarimetric signatures. The increased error at the edges of the band correspond to regions of diminished S 0 signal.

Fig. 9
Fig. 9

Transmission spectra experiment. (a) Experimental setup. A 106 ° C hotplate is used as a source behind a scene of various plastic targets. (b) The scene, as imaged by the Indigo camera. Each object is labeled.

Fig. 10
Fig. 10

Transmission spectra results for the three pixel locations marked in Fig. 9b, as reconstructed by the IHIP (dashed curves), and as recovered by a commercial FTS (solid curves).

Fig. 11
Fig. 11

Spectropolarimetric data acquired of an obsidian sphere. (a) Band integrated S 0 signature. Stokes parameter signatures obtained at σ = 2577 cm 1 ( λ = 3.88 μm ), with (b)  S 1 , (c)  S 2 , and (d)  S 3 .

Fig. 12
Fig. 12

Color-fusion images of the obsidian sphere. The S 0 , S 1 , and S 2 signatures from Fig. 11 are represented in (a) using saturation for DoLP (full saturation at DoLP = 0.3 ), hue for orientation, and gray-scale value for band integrated intensity. S 0 and S 3 from Fig. 11 are depicted in (b) using saturation for DoCP (full saturation at DoCP = 0.3 ), hue for handedness, and gray-scale value for band integrated intensity.

Fig. 13
Fig. 13

DoLP data for the obsidian sphere. (a) Experimental and theoretical radial cross section at σ = 2755 cm 1 ( λ = 3.63 μm ). Measured data exhibit 5.6% rms error versus theoretical calculations. (b) Reconstructed normalized Stokes parameters for the center pixel ( row , column ) = ( 143 , 143 ) , as labeled in Fig. 11. The errors at the edges of the spectral band values correspond regions of diminished S 0 signal for the polarimetric reference calibration source and/or the obsidian sphere spectrum.

Fig. 14
Fig. 14

Experimental S 0 spectral data obtained for the obsidian sphere scene at the ( x , y ) = ( 150 , 150 ) pixel. Single-scan spectrum exhibits relatively low spectral resolution ( Δ σ 182 cm 1 ). The dual-scan spectrum was acquired by implementing the ART and provides the full spectral resolution of the interferometer ( Δ σ 46 cm 1 ).

Fig. 15
Fig. 15

Outdoor data collection. (a)  S 0 , band integrated data. (b) Spectra recovered at three arbitrarily selected pixels from across the scene.

Fig. 16
Fig. 16

Emission results from a running pickup truck. (a)  S 0 data across the FOV for σ = 2273 cm 1 ( λ = 4.40 μm ). The emission from the exhaust pipe of the truck is circled. (b) Nine pixel averaged S 0 spectral data recovered at different pixel locations across the image shown in (a). The “exhaust” spectrum displays several emission peaks, which may indicate the presence of CO 2 .

Fig. 17
Fig. 17

Color-fusion DoLP data recovered in the SWIR, (a)  σ = 6618 cm 1 ( λ = 1.51 μm ) and (b)  σ = 4396 cm 1 ( λ = 2.32 μm ); recovered in the MWIR, (c)  σ = 2846 cm 1 ( λ = 3.51 μm ), and (d)  σ = 2401 cm 1 ( λ = 4.17 μm ). Maximum saturation occurs at a DoLP value of 0.4.

Fig. 18
Fig. 18

Color-fusion DoCP data recovered in (a) SWIR, σ = 4813 cm 1 ( λ = 2.08 μm ) and (b) MWIR σ = 2599 cm 1 ( λ = 3.85 μm ). Maximum saturation occurs for DoCP = 0.4 .

Tables (1)

Tables Icon

Table 1 Extinction Ratio and the Resulting Amplification Coefficient that Produces an Artificial Increase in the Measured Values of the S 1 , S 2 , and S 3 Stokes Parameters

Equations (35)

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S ( σ , x , y ) = [ S 0 ( σ , x , y ) S 1 ( σ , x , y ) S 2 ( σ , x , y ) S 3 ( σ , x , y ) ] = [ I 0 ( σ , x , y ) + I 90 ( σ , x , y ) I 0 ( σ , x , y ) I 90 ( σ , x , y ) I 45 ( σ , x , y ) I 135 ( σ , x , y ) I R ( σ , x , y ) I L ( σ , x , y ) ] ,
OPD ( σ , h ) = 2 B ( σ ) h tan θ + B ( σ ) d 3 ,
I ( n , m , h ) 1 2 S 0 ( n , m , σ ) cos ( 2 π σ OPD ( h ) ) d σ .
Δ σ = 1 / OPD max .
I 0 ° ( n , m , h ) ( 1 + cos ( φ h ) ) 2 [ S 0 + S 1 cos ( φ 2 ) + S 2 sin ( φ 1 ) sin ( φ 2 ) S 3 cos ( φ 1 ) sin ( φ 2 ) ] d σ .
ϕ 1 ( σ ) = 2 π B ( σ ) d 1 σ ,
ϕ 2 ( σ ) = 2 π B ( σ ) d 2 σ ,
ϕ h ( σ , h ) = 2 π σ OPD ( σ , h ) ,
I ( n , m , h ) { S 0 2 cos ( φ h ) + S 1 4 [ cos ( φ h + φ 2 ) + cos ( φ h φ 2 ) ] + S 2 8 [ cos ( φ h + φ 1 + φ 2 ) + cos ( φ h φ 1 φ 2 ) cos ( φ h + φ 1 φ 2 ) cos ( φ h φ 1 + φ 2 ) ] + S 3 8 [ sin ( φ h + φ 1 + φ 2 ) + sin ( φ h φ 1 φ 2 ) sin ( φ h + φ 1 φ 2 ) + sin ( φ h φ 1 + φ 2 ) ] } d σ .
C 0 = S 0 2 cos ( ϕ h ) d σ ,
C 1 = S 1 4 cos ( ϕ h ϕ 2 ) d σ ,
C 2 = [ S 2 8 cos ( ϕ h ϕ 1 + ϕ 2 ) + S 3 8 sin ( ϕ h ϕ 1 + ϕ 2 ) ] d σ .
I ( C 0 ) = 1 2 S 0 ( σ ) ,
I ( C 1 ) = 1 4 S 1 ( σ ) exp ( j ϕ 2 ) ,
I ( C 2 ) = 1 8 ( S 2 ( σ ) + j S 3 ( σ ) ) exp ( j ϕ 2 ) exp ( j ϕ 1 ) .
P ( n , m , Φ ) = R n , m Φ + O n , m ,
f ny ( σ ) = 1 2 Δ OPD ( σ ) = 1 4 B ( σ ) tan ( α ) Δ h .
γ = B ( σ UC ) B ( σ ref ) .
σ = γ σ UC ,
S 1 , sample ( σ ) = 1 2 Re [ I ( C 1 , sample ) I ( C 1 , reference ) S 0 , reference S 0 , sample ] ,
S 2 , sample ( σ ) = 1 2 Re [ I ( C 2 , sample ) I ( C 2 , reference ) S 0 , reference S 0 , sample ] ,
S 3 , sample ( σ ) = 1 2 Im [ I ( C 2 , sample ) I ( C 2 , reference ) S 0 , reference S 0 , sample ] ,
S 0 , reference ( σ ) = | I ( C 0 , reference ) | ,
S 0 , sample ( σ ) = | I ( C 0 , sample ) | .
DoP = S 1 2 + S 2 2 + S 3 2 S 0 .
S i = 1 , 2 , 3 meas = E + 1 E 1 S i = 1 , 2 , 3 act .
N ( σ ) = 1 S 0 a , K ( σ ) S 0 b , K ( σ ) ,
N rms = 1 n samp i = 1 n samp [ N ( σ i ) ] 2 .
SNR = 1 N rms .
SNR 2 = α 1 K + α 0 .
I 0 ° ( n , m , h ) ( 1 + cos ( φ h ) ) 2 [ S 0 S 1 cos ( φ 2 ) S 2 sin ( φ 1 ) sin ( φ 2 ) + S 3 cos ( φ 1 ) sin ( φ 2 ) ] d σ .
Γ n , m ( σ ) = S 0 plastic ( σ , n , m ) S 0 hotplate ( σ , n , m ) ,
DoLP ( n , m , σ ) = S 1 ( n , m , σ ) 2 + S 2 ( n , m , σ ) 2 S 0 ( n , m , σ ) ,
θ L ( n , m , σ ) = 1 2 tan 1 [ S 2 ( n , m , σ ) S 1 ( n , m , σ ) ] .
DoCP ( n , m , σ ) = | S 3 ( n , m , σ ) S 0 ( n , m , σ ) | .

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