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

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  1. S. Tyo, D. Goldstein, D. Chenault, and J. Shaw, "Review of passive imaging polarimetry for remote sensing applications," Appl. Opt. 45, 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 4481, 10-16 (2002).
    [CrossRef]
  7. R. Blumer, M. Miranda, J. Howe, and M. Stevens, "LWIR Polarimeter Calibration," Proc. SPIE 4481, 37-45 (2002).
    [CrossRef]
  8. L. Pezzaniti and D. Chenault, "A Division of Aperture MWIR Imaging Polarimeter," Proc. SPIE 5888, 239-250 (2005).
    [CrossRef]
  9. Y-T. Gau,  et al., "256x256 InSb Focal Plane Arrays," Proc. SPIE. 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, S. Ortega,  et al., "Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters," Opt. Eng. 47, 046403 (2008).
    [CrossRef]

2008

D. Bowers, J. Boger, D. Wellems, S. Ortega,  et al., "Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters," Opt. Eng. 47, 046403 (2008).
[CrossRef]

2007

2006

2005

L. Pezzaniti and D. Chenault, "A Division of Aperture MWIR Imaging Polarimeter," Proc. SPIE 5888, 239-250 (2005).
[CrossRef]

2003

2002

L. DeShazer, "Improved mid-infrared polarizers using yttrium vanadate," Proc. SPIE 4481, 10-16 (2002).
[CrossRef]

R. Blumer, M. Miranda, J. Howe, and M. Stevens, "LWIR Polarimeter Calibration," Proc. SPIE 4481, 37-45 (2002).
[CrossRef]

1970

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 4481, 37-45 (2002).
[CrossRef]

Boger, J.

D. Bowers, J. Boger, D. Wellems, S. Ortega,  et al., "Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters," Opt. Eng. 47, 046403 (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, S. Ortega,  et al., "Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters," Opt. Eng. 47, 046403 (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.

Dereniak, E.

DeShazer, L.

L. DeShazer, "Improved mid-infrared polarizers using yttrium vanadate," Proc. SPIE 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]

Gerhart, G.

Goldstein, D.

Hagen, N.

Hayakawa, M.

Howe, J.

R. Blumer, M. Miranda, J. Howe, and M. Stevens, "LWIR Polarimeter Calibration," Proc. SPIE 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 4481, 37-45 (2002).
[CrossRef]

Oka, K.

Okabe, H.

Ortega, S.

D. Bowers, J. Boger, D. Wellems, S. Ortega,  et al., "Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters," Opt. Eng. 47, 046403 (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. SPIE 5888, 239-250 (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.

Stevens, M.

R. Blumer, M. Miranda, J. Howe, and M. Stevens, "LWIR Polarimeter Calibration," Proc. SPIE 4481, 37-45 (2002).
[CrossRef]

Taniguchi, A.

Tkaczyk, T.

Tyo, S.

Wellems, D.

D. Bowers, J. Boger, D. Wellems, S. Ortega,  et al., "Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters," Opt. Eng. 47, 046403 (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.

J. Opt. A: Pure Appl. Opt.

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.

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.

D. Bowers, J. Boger, D. Wellems, S. Ortega,  et al., "Unpolarized calibration and nonuniformity correction for long-wave infrared microgrid imaging polarimeters," Opt. Eng. 47, 046403 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

L. DeShazer, "Improved mid-infrared polarizers using yttrium vanadate," Proc. SPIE 4481, 10-16 (2002).
[CrossRef]

R. Blumer, M. Miranda, J. Howe, and M. Stevens, "LWIR Polarimeter Calibration," Proc. SPIE 4481, 37-45 (2002).
[CrossRef]

L. Pezzaniti and D. Chenault, "A Division of Aperture MWIR Imaging Polarimeter," Proc. SPIE 5888, 239-250 (2005).
[CrossRef]

Other

Y-T. Gau,  et al., "256x256 InSb Focal Plane Arrays," Proc. SPIE. 4078, 467-479 (2000).
[CrossRef]

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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).

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.

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.

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.

Fig. 5.
Fig. 5.

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

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.

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.

Fig. 8.
Fig. 8.

Calibration setup for the prismatic polarimeter illustrating sources of emission.

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.

Fig. 10.
Fig. 10.

Experimental setup for the BPIP accuracy assessment.

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.

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.

Fig. 13.
Fig. 13.

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

Fig. 14.
Fig. 14.

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

Fig. 15.
Fig. 15.

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

Fig. 16.
Fig. 16.

Reconstructed Stokes vectors of the spherical blackbody.

Fig. 17.
Fig. 17.

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

Equations (68)

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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 )
( C 0 ) = 1 2 S 0 ( x , y )
( C 1 ) = 1 4 S 1 ( x , y ) exp ( j 2 π U x )
( 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 ) = ( C 0 , sample )
S 1 , sample ( x , y ) = [ ( C 1 , sample ) ( C 1 , reference , 0 ° ) ( C 0 , reference , 0 ° ) S 0 , sample ]
S 2 , sample ( x , y ) = [ ( C 2 , sample ) ( C 2 , reference , 45 ° ) ( C 0 , reference , 45 ° ) S 0 , sample ]
S 3 , sample ( x , y ) = [ ( C 2 , sample ) ( C 2 , reference , 45 ° ) ( 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 ) = ( C 0 , reference , 45 ° ) = S ' 0 γ
S 2 , corrected ( x , y ) = [ ( C 2 , sample ) ( C 2 , reference , 45 ° ) S 0 , reference , 45 ° ( x , y ) S 0 ( x , y ) ] 1 γ
S 3 , corrected ( x , y ) = [ ( C 2 , sample ) ( 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 ) = [ ( C 2 , sample ) ( C 2 , reference , 45 ° ) S 0 , reference , 45 ° ( x , y ) S 0 , approx ( x , y ) ]
S 3 , corrected ( x , y ) = [ ( C 2 , sample ) ( 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 ] ) ]

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