Abstract

The calibration of a complete Stokes birefringent prismatic imaging polarimeter (BPIP) in the MWIR is demonstrated. The BPIP technique, originally developed by K. Oka, is implemented with a set of four Yttrium Vanadate (YVO4) crystal prisms. A mathematical model for the polarimeter is presented in which diattenuation due to Fresnel effects and dichroism in the crystal are included. An improved polarimetric calibration technique is introduced to remove the diattenuation effects, along with the relative radiometric calibration required for the BPIP operating with a thermal background and large detector offsets. Data demonstrating emission polarization are presented from various blackbodies, which are compared to data from our Fourier transform infrared spectropolarimeter.

©2008 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Infrared hyperspectral imaging polarimeter using birefringent prisms

Julia Craven-Jones, Michael W. Kudenov, Maryn G. Stapelbroek, and Eustace L. Dereniak
Appl. Opt. 50(8) 1170-1185 (2011)

Modeling and optimization for a prismatic snapshot imaging polarimeter

Haitao Luo, Kazuhiko Oka, Nathan Hagen, Tomasz Tkaczyk, and Eustace L. Dereniak
Appl. Opt. 45(33) 8400-8409 (2006)

Fourier transform channeled spectropolarimetry in the MWIR

Michael W. Kudenov, Nathan A. Hagen, Eustace L. Dereniak, and Grant R. Gerhart
Opt. Express 15(20) 12792-12805 (2007)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. S. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45 No.  22, 5453–5469 (2006).
    [Crossref]
  2. K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express 11, 1510–1519 (2003).
    [Crossref] [PubMed]
  3. 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]
  4. M. Kudenov, N. Hagen, E. Dereniak, and G. Gerhart, “Fourier transform channeled spectropolarimetry in the MWIR,” Opt. Express 15, 12792–12805 (2007).
    [Crossref] [PubMed]
  5. H. Luo, T. Tkaczyk, and E. Dereniak, “High birefringence of the yttrium Vanadate crystal in the middle wavelength infrared,” Opt. Lett. 31, 616–618 (2006).
    [Crossref] [PubMed]
  6. L. DeShazer, “Improved mid-infrared polarizers using yttrium vanadate,” Proc. SPIE.Polarization Analysis, Measurement, & Remote Sensing IV  4481, 10–16 (2002).
    [Crossref]
  7. R. Blumer, M. Miranda, J. Howe, and M. Stevens, “LWIR Polarimeter Calibration,” Proc. SPIE.Polarization analysis, Measurement, & Remote Sensing IV,  4481, 37–45 (2002).
    [Crossref]
  8. L. Pezzaniti and D. Chenault, “A Division of Aperture MWIR Imaging Polarimeter,” Proc. of SPIE.Polarization Science and Remote Sensing II,  58880V-1 (2005).
    [Crossref]
  9. Y-T. Gau, et. al., “256×256 InSb Focal Plane Arrays,” Proc. of SPIE.Optoelectronic Materials and Devices II,  4078, 467–479 (2000).
    [Crossref]
  10. K. C. Lapworth, T. J. Quinn, and L. A. Allnutt, “A black-body source of radiation covering a wavelength range from the ultraviolet to the infrared,” J. Phys. E. 3, 116–120 (1970).
    [Crossref]
  11. D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8, 914–925 (2006).
    [Crossref]
  12. D. Bowers, J. Boger, D. Wellems, and S. Ortega, et. al., “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47046403 (2008).
    [Crossref]

2008 (1)

D. Bowers, J. Boger, D. Wellems, and S. Ortega, et. al., “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47046403 (2008).
[Crossref]

2007 (1)

2006 (4)

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8, 914–925 (2006).
[Crossref]

H. Luo, T. Tkaczyk, and E. Dereniak, “High birefringence of the yttrium Vanadate crystal in the middle wavelength infrared,” Opt. Lett. 31, 616–618 (2006).
[Crossref] [PubMed]

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]

S. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45 No.  22, 5453–5469 (2006).
[Crossref]

2005 (1)

L. Pezzaniti and D. Chenault, “A Division of Aperture MWIR Imaging Polarimeter,” Proc. of SPIE.Polarization Science and Remote Sensing II,  58880V-1 (2005).
[Crossref]

2003 (1)

2002 (2)

L. DeShazer, “Improved mid-infrared polarizers using yttrium vanadate,” Proc. SPIE.Polarization Analysis, Measurement, & Remote Sensing IV  4481, 10–16 (2002).
[Crossref]

R. Blumer, M. Miranda, J. Howe, and M. Stevens, “LWIR Polarimeter Calibration,” Proc. SPIE.Polarization analysis, Measurement, & Remote Sensing IV,  4481, 37–45 (2002).
[Crossref]

2000 (1)

Y-T. Gau, et. al., “256×256 InSb Focal Plane Arrays,” Proc. of SPIE.Optoelectronic Materials and Devices II,  4078, 467–479 (2000).
[Crossref]

1970 (1)

K. C. Lapworth, T. J. Quinn, and L. A. Allnutt, “A black-body source of radiation covering a wavelength range from the ultraviolet to the infrared,” J. Phys. E. 3, 116–120 (1970).
[Crossref]

Allnutt, L. A.

K. C. Lapworth, T. J. Quinn, and L. A. Allnutt, “A black-body source of radiation covering a wavelength range from the ultraviolet to the infrared,” J. Phys. E. 3, 116–120 (1970).
[Crossref]

Blumer, R.

R. Blumer, M. Miranda, J. Howe, and M. Stevens, “LWIR Polarimeter Calibration,” Proc. SPIE.Polarization analysis, Measurement, & Remote Sensing IV,  4481, 37–45 (2002).
[Crossref]

Boger, J.

D. Bowers, J. Boger, D. Wellems, and S. Ortega, et. al., “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47046403 (2008).
[Crossref]

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8, 914–925 (2006).
[Crossref]

Bowers, D.

D. Bowers, J. Boger, D. Wellems, and S. Ortega, et. al., “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47046403 (2008).
[Crossref]

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8, 914–925 (2006).
[Crossref]

Chenault, D.

S. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45 No.  22, 5453–5469 (2006).
[Crossref]

L. Pezzaniti and D. Chenault, “A Division of Aperture MWIR Imaging Polarimeter,” Proc. of SPIE.Polarization Science and Remote Sensing II,  58880V-1 (2005).
[Crossref]

Dereniak, E.

DeShazer, L.

L. DeShazer, “Improved mid-infrared polarizers using yttrium vanadate,” Proc. SPIE.Polarization Analysis, Measurement, & Remote Sensing IV  4481, 10–16 (2002).
[Crossref]

Fetrow, M.

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8, 914–925 (2006).
[Crossref]

Gau, Y-T.

Y-T. Gau, et. al., “256×256 InSb Focal Plane Arrays,” Proc. of SPIE.Optoelectronic Materials and Devices II,  4078, 467–479 (2000).
[Crossref]

Gerhart, G.

Goldstein, D.

S. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45 No.  22, 5453–5469 (2006).
[Crossref]

Hagen, N.

Hayakawa, M.

Howe, J.

R. Blumer, M. Miranda, J. Howe, and M. Stevens, “LWIR Polarimeter Calibration,” Proc. SPIE.Polarization analysis, Measurement, & Remote Sensing IV,  4481, 37–45 (2002).
[Crossref]

Kaneko, T.

Kudenov, M.

Lapworth, K. C.

K. C. Lapworth, T. J. Quinn, and L. A. Allnutt, “A black-body source of radiation covering a wavelength range from the ultraviolet to the infrared,” J. Phys. E. 3, 116–120 (1970).
[Crossref]

Luo, H.

Miranda, M.

R. Blumer, M. Miranda, J. Howe, and M. Stevens, “LWIR Polarimeter Calibration,” Proc. SPIE.Polarization analysis, Measurement, & Remote Sensing IV,  4481, 37–45 (2002).
[Crossref]

Oka, K.

Okabe, H.

Ortega, S.

D. Bowers, J. Boger, D. Wellems, and S. Ortega, et. al., “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47046403 (2008).
[Crossref]

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8, 914–925 (2006).
[Crossref]

Pezzaniti, L.

L. Pezzaniti and D. Chenault, “A Division of Aperture MWIR Imaging Polarimeter,” Proc. of SPIE.Polarization Science and Remote Sensing II,  58880V-1 (2005).
[Crossref]

Quinn, T. J.

K. C. Lapworth, T. J. Quinn, and L. A. Allnutt, “A black-body source of radiation covering a wavelength range from the ultraviolet to the infrared,” J. Phys. E. 3, 116–120 (1970).
[Crossref]

Shaw, J.

S. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45 No.  22, 5453–5469 (2006).
[Crossref]

Stevens, M.

R. Blumer, M. Miranda, J. Howe, and M. Stevens, “LWIR Polarimeter Calibration,” Proc. SPIE.Polarization analysis, Measurement, & Remote Sensing IV,  4481, 37–45 (2002).
[Crossref]

Taniguchi, A.

Tkaczyk, T.

Tyo, S.

S. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45 No.  22, 5453–5469 (2006).
[Crossref]

Wellems, D.

D. Bowers, J. Boger, D. Wellems, and S. Ortega, et. al., “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47046403 (2008).
[Crossref]

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8, 914–925 (2006).
[Crossref]

Appl. Opt. (1)

S. Tyo, D. Goldstein, D. Chenault, and J. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45 No.  22, 5453–5469 (2006).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

D. Wellems, S. Ortega, D. Bowers, J. Boger, and M. Fetrow, “Long wave infrared polarimetric model: theory, measurements and parameters,” J. Opt. A: Pure Appl. Opt. 8, 914–925 (2006).
[Crossref]

J. Phys. E. (1)

K. C. Lapworth, T. J. Quinn, and L. A. Allnutt, “A black-body source of radiation covering a wavelength range from the ultraviolet to the infrared,” J. Phys. E. 3, 116–120 (1970).
[Crossref]

Opt. Eng. (1)

D. Bowers, J. Boger, D. Wellems, and S. Ortega, et. al., “Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters,” Opt. Eng. 47046403 (2008).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Proc. of SPIE. (2)

L. Pezzaniti and D. Chenault, “A Division of Aperture MWIR Imaging Polarimeter,” Proc. of SPIE.Polarization Science and Remote Sensing II,  58880V-1 (2005).
[Crossref]

Y-T. Gau, et. al., “256×256 InSb Focal Plane Arrays,” Proc. of SPIE.Optoelectronic Materials and Devices II,  4078, 467–479 (2000).
[Crossref]

Proc. SPIE. (2)

L. DeShazer, “Improved mid-infrared polarizers using yttrium vanadate,” Proc. SPIE.Polarization Analysis, Measurement, & Remote Sensing IV  4481, 10–16 (2002).
[Crossref]

R. Blumer, M. Miranda, J. Howe, and M. Stevens, “LWIR Polarimeter Calibration,” Proc. SPIE.Polarization analysis, Measurement, & Remote Sensing IV,  4481, 37–45 (2002).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1.
Fig. 1. Prism polarimeter wedge set. Glue is located on each common interface of the prisms (between P1–P2, P2–P3, and P3–P4).

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2. Optical setup for the BPIP. A relay lens is used to transfer the fringes, that have been superimposed onto the image of a scene by the prisms, from the intermediate image plane to the FPA. A bandpass filter maintains temporal coherence for the interference effects.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. Fourier spectra of the 2D interference pattern generated by the prisms. Seven channels carrying various portions of the Stokes vector information are present.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4. P1 and P2 prism geometry for approximation verification of εγ/(ε 2+γ 2)~ε/γ and Eq. 38. The prisms are simulated assuming YVO4. Assumed indices of refraction are ne =2.11, no =1.9 (B=0.21) [5] with a glue index of ng =1.58.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5. YVO4 absorption coefficients spanning the MWIR spectral region for the ordinary and extraordinary axes [4].

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6. Comparison of εγ(ε 2+γ 2)to the approximation ε/γ with percent error as calculated from simulation. Left: Error due to absorption only. Right: Error with both absorption and Fresnel losses.

Download Full Size | PPT Slide | PDF

Fig. 7.
Fig. 7. Percent error between the exact calibration that uses γS0 to the calibration using S0,approx and S0,sample . Left: Error due to absorption only. Right: Error with absorption and Fresnel losses.

Download Full Size | PPT Slide | PDF

Fig. 8.
Fig. 8. Calibration setup for the prismatic polarimeter illustrating sources of emission.

Download Full Size | PPT Slide | PDF

Fig. 9.
Fig. 9. Image of the experimental setup for the MWIR BPIP. An objective lens is followed by the prisms, analyzer, a relay lens, the bandpass filter (BPF), and the InSb FPA.

Download Full Size | PPT Slide | PDF

Fig. 10.
Fig. 10. Experimental setup for the BPIP accuracy assessment.

Download Full Size | PPT Slide | PDF

Fig. 11.
Fig. 11. Raw data and DOLP for the tilted plate with the improved calibration. The top row contains the raw images, while the bottom indicates the DOLP.

Download Full Size | PPT Slide | PDF

Fig. 12.
Fig. 12. (Left) Estimated spatial distribution of ε/γ obtained from an unpolarized blackbody. (Right) Expected value of ε/γ using a prism similar to Fig. 4 where ne =2.11, no =1.9, ng =1.58, α o =0.2164 cm-1, α e =0.4042 cm-1, and λ=4.5 µm.

Download Full Size | PPT Slide | PDF

Fig. 13.
Fig. 13. Cross section of ε/γ, per Fig. 12, extracted from the region under the dashed line.

Download Full Size | PPT Slide | PDF

Fig. 14.
Fig. 14. DOLP of the central pixel from the imaging polarimeter vs. data from our FTSP.

Download Full Size | PPT Slide | PDF

Fig. 15.
Fig. 15. Raw data of polarized emission from a spherical light bulb. Temperature is 175 °C.

Download Full Size | PPT Slide | PDF

Fig. 16.
Fig. 16. Reconstructed Stokes vectors of the spherical blackbody.

Download Full Size | PPT Slide | PDF

Fig. 17.
Fig. 17. DOLP, orientation, and ellipticity angles calculated from the Stokes parameters of Fig. 16.

Download Full Size | PPT Slide | PDF

Equations (68)

Equations on this page are rendered with MathJax. Learn more.

I ( x , y ) = 1 2 S 0 ( x , y ) + 1 2 S 1 ( x , y ) cos ⁡ ( 2 π U x ) + 1 4 ∣ S 23 ( x , y ) ∣ cos ⁡ ( 2 π U ( x − y ) + arg [ S 23 ( x , y ) ] )
− 1 4 ∣ S 23 ( x , y ) ∣ cos ⁡ ( 2 π U ( x + y ) − arg [ S 23 ( x , y ) ] )
S 23 ( x , y ) = S 2 ( x , y ) + j S 3 ( x , y )
U = 2 B λ tan ⁡ ( β )
ℱ ( C 0 ) = 1 2 S 0 ( x , y )
ℱ ( C 1 ) = 1 4 S 1 ( x , y ) exp ( j 2 π U x )
ℱ ( C 2 ) = 1 8 ( − S 2 ( x , y ) + j S 3 ( x , y ) ) exp ( j 2 π Ux ) exp ( j 2 π Uy )
S 0 , sample ( x , y ) = ∣ ℱ ( C 0 , sample ) ∣
S 1 , sample ( x , y ) = ℜ [ ℱ ( C 1 , sample ) ℱ ( C 1 , reference , 0 ° ) ∣ ℱ ( C 0 , reference , 0 ° ) ∣ S 0 , sample ]
S 2 , sample ( x , y ) = ℜ [ ℱ ( C 2 , sample ) ℱ ( C 2 , reference , 45 ° ) ∣ ℱ ( C 0 , reference , 45 ° ) ∣ S 0 , sample ]
S 3 , sample ( x , y ) = ℑ [ ℱ ( C 2 , sample ) ℱ ( C 2 , reference , 45 ° ) ∣ ℱ ( C 0 , reference , 45 ° ) ∣ S 0 , sample ]
M D ( T x , T y , θ ) = 1 2 R ( − θ ) [ ( T x + T y ) ( T x − T y ) 0 0 ( T x − T y ) ( T x + T y ) 0 0 0 0 2 T x T y 0 0 0 0 2 T x T y ] R ( θ )
R ( θ ) = [ 1 0 0 0 0 cos ⁡ ( 2 θ ) sin ⁡ ( 2 θ ) 0 0 − sin ⁡ ( 2 θ ) cos ⁡ ( 2 θ ) 0 0 0 0 1 ]
M pn ( δ n ( x , y ) , θ n ) = R ( − θ n ) [ 1 0 0 0 0 1 0 0 0 0 cos ⁡ ( δ n ( x , y ) ) − sin ⁡ ( δ n ( x , y ) ) 0 0 sin ⁡ ( δ n ( x , y ) ) cos ⁡ ( δ n ( x , y ) ) ] R ( θ n )
δ 1 ( x , y ) = 2 π B λ [ d T + tan ⁡ ( β ) ( d y 2 + y ) ] δ 2 ( x , y ) = 2 π B λ [ d T + tan ⁡ ( β ) ( d y 2 − y ) ]
δ 3 ( x , y ) = 2 π B λ [ d T + tan ⁡ ( β ) ( d x 2 + x ) ] δ 4 ( x , y ) = 2 π B λ [ d T + tan ⁡ ( β ) ( d x 2 − x ) ]
T x , Fca = 1 − [ ( n o − 1 ) ( n o + 1 ) ] 2 T y , Fca = 1 − [ n e − 1 n e + 1 ] 2
T x , Fcg = 1 − [ ( n o − n g ) ( n o − n g ) ] 2 T y , Fcg = 1 − [ ( n e − n g ) ( n e − n g ) ] 2
T xn , a = exp ( − d zn α o ) T yn , α = exp ( − d zn α e )
d z 1 = d T + ( d y 2 + y ) tan ⁡ ( β ) d z 2 = d T + ( d y 2 − y ) tan ⁡ ( β )
d z 3 = d T + ( d x 2 + x ) tan ⁡ ( β ) d z 4 = d T + ( d x 2 − x ) tan ⁡ ( β )
M sys = A · M D ( T x , Fca , T y , Fca , 135 ° ) · M p 4 · M D ( T x 4 , α , T y 4 , α , 135 ° ) · M D ( T x , Fcg , T y , Fcg , 135 ° ) · …
M D ( T x , Fcg , T y , Fcg , 45 ° ) · M p 3 · M D ( T x 3 , α , 45 ° ) · M D ( T x , Fcg , T y , Fcg , 45 ° ) · …
M D ( T x , Fcg , T y , Fcg , 90 ° ) · M p 2 · M D ( T x 2 , α , T y 2 , α , 90 ° ) · M D ( T x , Fcg , T y , Fcg , 90 ° ) · …
M D ( T x , Fcg , T y , Fcg , 0 ° ) · M p 1 · M D ( T x 1 , α , 0 ° ) · M D ( T x , Fca , T y , Fca , 0 ° )
T x 1 = T x 1 , α T x , Fca T x , Fcg T y 1 = T y 1 , α T y , Fca T y , Fcg T x 2 = T x 2 , α T x , Fcg 2 T y 2 = T y 2 , α T y , Fcg 2 T x 3 = T x 3 , α T x , Fcg 2 T y 3 = T y 3 , α T y , Fcg 2 T x 4 = T x 4 , α T x , Fca T x , Fcg T y 4 = T y 4 , α T y , Fca T y , Fcg
M sys = A · M D ( T x 4 , T y 4 , 135 ° ) · M p 4 · M D ( T x 3 , T y 3 , 45 ° ) · M p 3 · …
M D ( T x 2 , T y 2 , 90 ° ) · M p 2 · M D ( T x 1 , T y 1 , 0 ° ) · M p 1
I = [ 1 8 ( T x 4 T y 3 + T x 3 T y 4 ) ( T x 1 T y 2 + T x 2 T y 1 ) + 1 4 ( T x 1 T y 2 − T x 2 T y 1 ) T x 3 T y 3 T x 4 T y 4 cos ⁡ ( δ 3 − δ 4 ) ] S 0 +
[ 1 8 ( T x 4 T y 3 + T x 3 T y 4 ) ( T x 1 T y 2 − T x 2 T y 1 ) + 1 4 ( T x 1 T y 2 + T x 2 T y 1 ) T x 3 T y 3 T x 4 T y 4 cos ⁡ ( δ 3 − δ 4 ) ] S 1 +
[ 1 4 T x 1 T y 1 T x 2 T y 2 ( ( T x 3 T y 4 − T x 4 T y 3 ) cos ⁡ ( δ 1 + δ 2 ) + 2 T x 3 T y 3 T x 4 T y 4 sin ⁡ ⁡ ⁡ ⁡ ( δ 1 − δ 2 ) sin ⁡ ( δ 3 − δ 4 ) ) ] S 2 +
[ 1 4 T x 1 T y 1 T x 2 T y 2 ( ( T x 3 T y 4 − T x 4 T y 3 ) sin ⁡ ( δ 1 − δ 2 ) − 2 T x 3 T y 3 T x 4 T y 4 cos ⁡ ( δ 1 − δ 2 ) sin ⁡ ( δ 3 − δ 4 ) ) ] S 3
S 0 , sample ( x , y ) = 1 8 ( T x 4 T y 3 + T x 3 T y 4 ) [ ( T x 1 T y2 + T x 2 T y 1 ) S 0 + ( T x 1 T y 2 − T x 2 T y 1 ) S 1 ]
S 1 , sample ( x , y ) = [ ( T x 1 T y 2 − T x 2 T y 1 ) S 0 + ( T x 1 T y 2 + T x 2 T y 1 ) S 1 ( T x 1 T y 2 + T x 2 T y 1 ) S 0 + ( T x 1 T y 2 − T x 2 T y 1 ) S 1 ]
S 2 , sample ( x , y ) = ( T x 1 T y 2 + T x 2 T y 1 ) S 2 ( T x 1 T y 2 + T x 2 T y 1 ) S 0 + ( T x 1 T y 2 − T x 2 T y 1 ) S 1
S 3 , sample ( x , y ) = ( T x 1 T y 2 + T x 2 T y 1 ) S 3 ( T x 1 T y 2 + T x 2 T y 1 ) S 0 + ( T x 1 T y 2 − T x 2 T y 1 ) S 1
S 0 , sample ( x , y ) = ( S 0 γ + S 1 ε )
S 1 , sample ( x , y ) = S 0 ε + S 1 γ S 0 γ + S 1 ε
S 0 ( x , y ) = S 0 , sample ε S 1 , sample − γ ( ε 2 − γ 2 )
S 1 ( x , y ) = S 0 , sample ε − γ S 1 , sample ( ε 2 − γ 2 )
S 1 , corrected ( x , y ) = S 1 S 0 = ε − γ S 1 , sample ε S 1 , sample − γ
S 0 , reference , 45 ° ( x , y ) = ∣ ℱ ( C 0 , reference , 45 ° ) ∣ = S ' 0 γ
S 2 , corrected ( x , y ) = ℛ [ ℱ ( C 2 , sample ) ℱ ( C 2 , reference , 45 ° ) S 0 , reference , 45 ° ( x , y ) S 0 ( x , y ) ] 1 γ
S 3 , corrected ( x , y ) = ℑ [ ℱ ( C 2 , sample ) ℱ ( C 2 , reference , 45 ° ) S 0 , reference , 45 ° ( x , y ) S 0 ( x , y ) ] 1 γ
S 1 , corrected ( x , y ) = ε γ − S 1 , sample ε γ S 1 , sample − 1
S measured ( x , y ) = M D ( T x 1 T y 2 , T x 2 T y 1 , 0 ° ) = [ 1 0 0 0 ] = 1 2 [ ( T x 1 T y 2 + T x 2 T y 1 ) ( T x 1 T y 2 − T x 2 T y 1 ) 0 0 ] = 1 2 [ γ ε 0 0 ]
S 1 , sample ( x , y ) = ε γ ε 2 + γ 2
S 1 , sample ( x , y ) ≈ ε γ
S 0 , approx ( x , y ) = S 0 , sample ( 1 − ε S 1 , sample γ )
S 0 , approx ( x , y ) = S 0 ( γ 2 − ε 2 ) γ
S 0 , approx ( x , y ) ≈ S 0 γ
S 2 , corrected ( x , y ) = ℛ [ ℱ ( C 2 , sample ) ℱ ( C 2 , reference , 45 ° ) S 0 , reference , 45 ° ( x , y ) S 0 , approx ( x , y ) ]
S 3 , corrected ( x , y ) = ℑ [ ℱ ( C 2 , sample ) ℱ ( C 2 , reference , 45 ° ) S 0 , reference , 45 ° ( x , y ) S 0 , approx ( x , y ) ]
S 0 , total = S 0 , bb , T + S 0 , bb , R + S 0 , r + S 0 , e + S 0 , optics + S 0 , objective
S 0 , optics = S 0 , prism + S 0 , R & F + S 0 , FPA
I ref ∝ [ S 0 , optics + ( S 0 , bb , T + S 0 , bb , R + S 0 , r + S 0 , e + S 0 , objective ) γ + S 1 , bb , T ε ]
+ [ ( S 0 , bb , T + S 0 , bb , R + S 0 , r + S 0 , e + S 0 , objective ) ε + S 1 , bb , T γ ] cos ⁡ ( 2 π Ux )
+ S 2 , bb , T [ cos⁡ ( 2 π U ( x + y ) ) − cos⁡ ( 2 π U ( x − y ) ) ]
I ref , T 2 − I ref , T 1 = [ ( S 0 , bb , T 2 − S 0 , bb , T 1 ) γ + ( S 1 , bb , T 2 − S 1 , bb , T 1 ) ε ] +
[ ( S 0 , bb , T 2 − S 0 , bb , T 1 ) ε + ( S 1 , bb , T 2 − S 1 , bb , T 1 ) γ ] cos ⁡ ( 2 π Ux ) +
( S 2 , bb , T 2 − S 2 , bb , T 1 ) [ cos⁡ ( 2 π U ( x + y ) ) − cos ⁡ ( 2 π U ( x − y ) ) ]
S 1 , bb , T 1 S 0 , bb , T 1 = S 1 , bb , T 2 S 0 , bb , T 2 = S 1 , bb , T 2 − S 1 , bb , T 1 S 0 , bb , T 2 − S 0 , bb , T 1
I sample ∝ [ S 0 , optics + ( S 0 , sample + S 0 , objective ) γ + S 1 , sample ε ] +
[ ε ( S 0 , sample + S 0 , objective ) + γ S 1 , sample ] cos ⁡ ( 2 π Ux )
+ ∣ S 23 , sample ∣ [ cos ⁡ ( 2 π U ( x + y ) + arg [ S 23 , sample ] ) − cos ⁡ ( 2 π U ( x − y ) − arg [ S 23 , sample ] ) ]
I offset ∝ S 0 , optics + γ S 0 , objective + ε S 0 , objective cos ⁡ ( 2 π Ux )
I sample − I offset = ( S 0 , sample γ + S 1 , sample ε ) + ( S 0 , sample ε + S 1 , sample γ ) cos ( 2 π Ux )
+ ∣ S 23 , sample ∣ [ cos ⁡ ( 2 π U ( x + y ) + arg [ S 23 , sample ] ) − cos ⁡ ( 2 π U ( x − y ) − arg [ S 23 , sample ] ) ]

Metrics