Abstract

A white-light snapshot channeled linear imaging (CLI) polarimeter is demonstrated by utilizing polarization gratings (PGs). The CLI polarimeter is capable of measuring the two-dimensional distribution of the linear Stokes polarization parameters by incorporating two identical PGs, in series, along the optical axis. In this configuration, the general optical shearing functionality of a uniaxial crystal-based Savart plate is realized. However, unlike a Savart plate, the diffractive nature of the PGs creates a linear dependence of the shear versus wavelength, thus providing broadband functionality. Consequently, by incorporating the PG-based Savart plate into a Savart plate channeled imaging polarimeter, white-light interference fringes can be generated. This enables polarimetric image data to be acquired at shorter exposure times in daylight conditions, making it more appealing over the quasi-monochromatic channeled imaging polarimeters previously described in the literature. Furthermore, the PG-based device offers significantly more compactness, field of view, optical simplicity, and vibration insensitivity than previously described white-light CLI polarimeters based on Sagnac interferometers. Included in this paper are theoretical descriptions of the linear (S0, S1, and S2) and complete (S0, S1, S2, and S3) channeled Stokes imaging polarimeters. Additionally, descriptions of our calibration procedures and our experimental proof of concept CLI system are provided. These are followed by laboratory and outdoor polarimetric measurements of S0, S1, and S2.

© 2011 Optical Society of America

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References

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2010 (2)

C. Oh, J. Kim, J. F. Muth, S. Serati, and M. J. Escuti, “High-throughput, continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

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

2009 (1)

2008 (2)

2007 (2)

R. K. Komanduri, W. M. Jones, C. Oh, and M. J. Escuti, “Polarization-independent modulation for projection displays using small-period LC polarization gratings,” J. Soc. Inf. Disp. 15, 589–594 (2007).
[CrossRef]

C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76, 043815 (2007).
[CrossRef]

2006 (3)

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

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

M. J. Escuti, C. Oh, C. Sanchez, C. W. M. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207(2006).
[CrossRef]

2005 (1)

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. C. Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98, 123102 (2005).
[CrossRef]

2004 (1)

2003 (1)

2000 (1)

1998 (1)

1992 (1)

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

1984 (1)

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31, 579–588 (1984).
[CrossRef]

1975 (1)

1956 (1)

S. Pancharatnam, “Generalized theory of interference, and its applications. Part I. Coherent pencils,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Ai, C.

Baleine, E.

Bastiaansen, C. W. M.

M. J. Escuti, C. Oh, C. Sanchez, C. W. M. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207(2006).
[CrossRef]

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

Broer, D. J.

M. J. Escuti, C. Oh, C. Sanchez, C. W. M. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207(2006).
[CrossRef]

Chenault, D. B.

Cochran, E. R.

Craven, J.

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

Crawford, G. P.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. C. Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98, 123102 (2005).
[CrossRef]

Dereniak, E. L.

Dogariu, A.

Eakin, J. N.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. C. Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98, 123102 (2005).
[CrossRef]

Engheta, N.

Escuti, M. J.

C. Oh, J. Kim, J. F. Muth, S. Serati, and M. J. Escuti, “High-throughput, continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33, 2287–2289(2008).
[CrossRef] [PubMed]

C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76, 043815 (2007).
[CrossRef]

R. K. Komanduri, W. M. Jones, C. Oh, and M. J. Escuti, “Polarization-independent modulation for projection displays using small-period LC polarization gratings,” J. Soc. Inf. Disp. 15, 589–594 (2007).
[CrossRef]

M. J. Escuti, C. Oh, C. Sanchez, C. W. M. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207(2006).
[CrossRef]

Gerhart, G. R.

Goldstein, D.

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

Goldstein, D. L.

Jones, A. C.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. C. Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98, 123102 (2005).
[CrossRef]

Jones, W. M.

R. K. Komanduri, W. M. Jones, C. Oh, and M. J. Escuti, “Polarization-independent modulation for projection displays using small-period LC polarization gratings,” J. Soc. Inf. Disp. 15, 589–594 (2007).
[CrossRef]

Jungwirth, M. E. L.

Kaneko, T.

Kim, J.

C. Oh, J. Kim, J. F. Muth, S. Serati, and M. J. Escuti, “High-throughput, continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Komanduri, R. K.

R. K. Komanduri, W. M. Jones, C. Oh, and M. J. Escuti, “Polarization-independent modulation for projection displays using small-period LC polarization gratings,” J. Soc. Inf. Disp. 15, 589–594 (2007).
[CrossRef]

Kudenov, M. W.

Miller, D.

K. Oka, R. Suda, M. Ohnuki, D. Miller, and E. L. Dereniak, “Snapshot imaging polarimeter for polychromatic light using Savart plates and diffractive lenses,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThF4.

Mujat, M.

Muth, J. F.

C. Oh, J. Kim, J. F. Muth, S. Serati, and M. J. Escuti, “High-throughput, continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Nikolova, L.

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31, 579–588 (1984).
[CrossRef]

Oh, C.

C. Oh, J. Kim, J. F. Muth, S. Serati, and M. J. Escuti, “High-throughput, continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

C. Oh and M. J. Escuti, “Achromatic diffraction from polarization gratings with high efficiency,” Opt. Lett. 33, 2287–2289(2008).
[CrossRef] [PubMed]

C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76, 043815 (2007).
[CrossRef]

R. K. Komanduri, W. M. Jones, C. Oh, and M. J. Escuti, “Polarization-independent modulation for projection displays using small-period LC polarization gratings,” J. Soc. Inf. Disp. 15, 589–594 (2007).
[CrossRef]

M. J. Escuti, C. Oh, C. Sanchez, C. W. M. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207(2006).
[CrossRef]

Ohnuki, M.

K. Oka, R. Suda, M. Ohnuki, D. Miller, and E. L. Dereniak, “Snapshot imaging polarimeter for polychromatic light using Savart plates and diffractive lenses,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThF4.

Oka, K.

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

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

K. Oka, R. Suda, M. Ohnuki, D. Miller, and E. L. Dereniak, “Snapshot imaging polarimeter for polychromatic light using Savart plates and diffractive lenses,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThF4.

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference, and its applications. Part I. Coherent pencils,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Pelcovits, R. A.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. C. Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98, 123102 (2005).
[CrossRef]

Pezzaniti, L.

Pugh, E. N.

Radcliffe, M. D.

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. C. Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98, 123102 (2005).
[CrossRef]

Saito, N.

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

Sanchez, C.

M. J. Escuti, C. Oh, C. Sanchez, C. W. M. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207(2006).
[CrossRef]

Serati, S.

C. Oh, J. Kim, J. F. Muth, S. Serati, and M. J. Escuti, “High-throughput, continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

Shaw, J. A.

Suda, R.

K. Oka, R. Suda, M. Ohnuki, D. Miller, and E. L. Dereniak, “Snapshot imaging polarimeter for polychromatic light using Savart plates and diffractive lenses,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThF4.

Tervo, J.

Todorov, T.

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31, 579–588 (1984).
[CrossRef]

Turunen, J.

Tyo, J. S.

VanDelden, J.

J. VanDelden, “Interferometric polarization interrogating filter assembly and method,” U.S. patent 6,674,532 B2 (6 January 2004).

Wyant, J. C.

Appl. Opt. (3)

IEEE Photon. Technol. Lett. (1)

C. Oh, J. Kim, J. F. Muth, S. Serati, and M. J. Escuti, “High-throughput, continuous beam steering using rotating polarization gratings,” IEEE Photon. Technol. Lett. 22, 200–202 (2010).
[CrossRef]

J. Appl. Phys. (1)

G. P. Crawford, J. N. Eakin, M. D. Radcliffe, A. C. Jones, and R. A. Pelcovits, “Liquid-crystal diffraction gratings using polarization holography alignment techniques,” J. Appl. Phys. 98, 123102 (2005).
[CrossRef]

J. Mod. Opt. (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407 (1987).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Soc. Inf. Disp. (1)

R. K. Komanduri, W. M. Jones, C. Oh, and M. J. Escuti, “Polarization-independent modulation for projection displays using small-period LC polarization gratings,” J. Soc. Inf. Disp. 15, 589–594 (2007).
[CrossRef]

Opt. Acta (1)

L. Nikolova and T. Todorov, “Diffraction efficiency and selectivity of polarization holographic recording,” Opt. Acta 31, 579–588 (1984).
[CrossRef]

Opt. Eng. (1)

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

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (1)

C. Oh and M. J. Escuti, “Numerical analysis of polarization gratings using the finite-difference time-domain method,” Phys. Rev. A 76, 043815 (2007).
[CrossRef]

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, “Generalized theory of interference, and its applications. Part I. Coherent pencils,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Proc. SPIE (2)

M. J. Escuti, C. Oh, C. Sanchez, C. W. M. Bastiaansen, and D. J. Broer, “Simplified spectropolarimetry using reactive mesogen polarization gratings,” Proc. SPIE 6302, 630207(2006).
[CrossRef]

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

Other (3)

J. VanDelden, “Interferometric polarization interrogating filter assembly and method,” U.S. patent 6,674,532 B2 (6 January 2004).

K. Oka, R. Suda, M. Ohnuki, D. Miller, and E. L. Dereniak, “Snapshot imaging polarimeter for polychromatic light using Savart plates and diffractive lenses,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThF4.

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

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Figures (12)

Fig. 1
Fig. 1

Schematic of the CLI polarimeter. Right circular (RC) and left circular (LC) polarizations diffract into the 1 st and + 1 st diffraction orders, respectively.

Fig. 2
Fig. 2

Experimental setup for establishing the measurement accuracy of the CLI polarimeter in white light.

Fig. 3
Fig. 3

Measured zero-order and total first-order ( T ± 1 = T + 1 + T 1 ) transmission spectra of the PGs.

Fig. 4
Fig. 4

White-light interference fringes generated in the central 100 × 100 pixels on the focal plane array for generating polarizer orientations of (a)  θ = 0 ° , (b)  θ = 50 ° , and (c)  θ = 90 ° .

Fig. 5
Fig. 5

Measured and theoretical results of the polarimetric reconstructions.

Fig. 6
Fig. 6

Experimental setup for viewing outdoor targets with the CLI polarimeter. An afocal telescope is included to allow the scene to be defocused while maintaining focus on the interference fringes.

Fig. 7
Fig. 7

Photo of the CLI polarimeter on the bench top. This configuration of the polarimeter is optimized for viewing scenes outdoors.

Fig. 8
Fig. 8

Raw image of a moving vehicle. Interference fringes are located in areas of the scene that are linearly po larized. These fringes are particularly evident in the hood of the vehicle.

Fig. 9
Fig. 9

Processed polarization data of the vehicle, calculated from the raw data in Fig. 8. (a)  S 0 , (b) DOLP, (c)  S 1 / S 0 , and (d)  S 2 / S 0 .

Fig. 10
Fig. 10

Color fusion image of the vehicle, generated from the spectrally broadband polarization data. Magenta and green represent linearly polarized light with θ L 10 ° and 45 ° , respectively. Full saturation corresponds to a DOLP of 0.4.

Fig. 11
Fig. 11

Schematic of the full imaging Stokes polarimeter. Note that PG 1 and PG 2 diffract in the y z plane, while PG 3 and PG 4 diffract in the x z plane. The coordinate system has been rotated to portray this clearly in two dimensions.

Fig. 12
Fig. 12

Three-dimensional schematic of the polarization gratings PG 1 through PG 4 , in addition to the two QWPs.

Equations (43)

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

S ( x , y ) = [ S 0 ( x , y ) S 1 ( x , y ) S 2 ( x , y ) S 3 ( x , y ) ] = [ I 0 ( x , y ) + I 90 ( x , y ) I 0 ( x , y ) I 90 ( x , y ) I 45 ( x , y ) I 135 ( x , y ) I R ( x , y ) I L ( x , y ) ] ,
η ± 1 = ( 1 2 S 3 2 S 0 ) K ,
η 0 = ( 1 K ) ,
E inc = [ E ¯ X E ¯ Y ] = [ E X ( ξ , η ) e j φ x ( ξ , η ) E Y ( ξ , η ) e j φ y ( ξ , η ) ] ,
J + 1 , RC = 1 2 [ 1 i i 1 ] ,
J 1 , LC = 1 2 [ 1 i i 1 ] .
E A = J 1 , LC E inc = 1 2 [ E ¯ X ( ξ , η α ) j E ¯ Y ( ξ , η α ) j E ¯ X ( ξ , η α ) + E ¯ Y ( ξ , η α ) ] ,
E B = J + 1 , RC E inc = 1 2 [ E ¯ X ( ξ , η + α ) + j E ¯ Y ( ξ , η + α ) j E ¯ X ( ξ , η + α ) + E ¯ Y ( ξ , η + α ) ] ,
α m λ Λ t ,
E LP + = E A + E B = 1 2 [ E ¯ X ( ξ , η + α ) + j E ¯ Y ( ξ , η + α ) + E ¯ X ( ξ , η α ) j E ¯ Y ( ξ , η α ) j E ¯ X ( ξ , η + α ) + E ¯ Y ( ξ , η + α ) + j E ¯ X ( ξ , η α ) + E ¯ Y ( ξ , η α ) ] .
E LP = [ 1 0 0 0 ] E LP + = 1 2 [ E ¯ X ( ξ , η + α ) + j E ¯ Y ( ξ , η + α ) + E ¯ X ( ξ , η α ) j E ¯ Y ( ξ , η α ) 0 ] .
E f = F [ E LP ] ξ = x λ f , η = y λ f = 1 2 [ E ¯ X e j 2 π λ f α y + j E ¯ Y e j 2 π λ f α y + E ¯ X e j 2 π λ f α y j E ¯ Y e j 2 π λ f α y ] ,
I = | E f | 2 = 1 2 ( | E ¯ X | 2 + | E ¯ Y | 2 ) + 1 4 ( E ¯ X E ¯ X * E ¯ Y E ¯ Y * ) e j 2 π λ f 2 α y + 1 4 ( E ¯ X E ¯ X * E ¯ Y E ¯ Y * ) e j 2 π λ f 2 α y + j 1 4 ( E ¯ X E ¯ Y * + E ¯ Y E ¯ X * ) e j 2 π λ f 2 α y j 1 4 ( E ¯ X E ¯ Y * + E ¯ Y E ¯ X * ) e j 2 π λ f 2 α y .
I ( x , y ) = 1 2 [ S 0 ( x , y ) + S 1 ( x , y ) cos ( 2 π λ f 2 α y ) + S 2 ( x , y ) sin ( 2 π λ f 2 α y ) ] .
I ( x , y ) = 1 2 [ S 0 ( x , y ) + S 1 ( x , y ) cos ( 2 π 2 m t f Λ y ) + S 2 ( x , y ) sin ( 2 π 2 m t f Λ y ) ] .
U = 2 m t f Λ .
I ( ξ , η ) = F [ I ( x , y ) ] = 1 2 S 0 ( ξ , η ) + 1 4 S 1 ( ξ , η ) * [ δ ( ξ , η + U ) + δ ( ξ , η U ) ] + i 1 4 S 2 ( ξ , η ) * [ δ ( ξ , η + U ) δ ( ξ , η U ) ] ,
C 0 = 1 2 S 0 ( x , y ) ,
C 1 = 1 4 ( S 1 ( x , y ) i S 2 ( x , y ) ) e i 2 π U y .
S 0 ( x , y ) = | C 0 , sample | ,
S 1 ( x , y ) S 0 ( x , y ) = R [ C 1 , sample C 1 , reference C 0 , reference C 0 , sample ( S 1 , ref ( x , y ) i S 2 , ref ( x , y ) S 0 , ref ( x , y ) ) ] ,
S 2 ( x , y ) S 0 ( x , y ) = I [ C 1 , sample C 1 , reference C 0 , reference C 0 , sample ( S 1 , ref ( x , y ) i S 2 , ref ( x , y ) S 0 , ref ( x , y ) ) ] .
S 0 ( x , y ) = | C 0 , sample | ,
S 1 ( x , y ) S 0 ( x , y ) = R [ C 1 , sample C 1 , reference C 0 , reference C 0 , sample ] ,
S 2 ( x , y ) S 0 ( x , y ) = I [ C 1 , sample C 1 , reference C 0 , reference C 0 , sample ] .
S n ( x , y ) = λ 1 λ 2 DE 2 ( λ ) S n ( x , y , λ ) d λ ,
I ( x , y ) = 1 2 [ Δ offset ( x , y ) + S 0 ( x , y ) + S 1 ( x , y ) cos ( 2 π 2 m t f Λ y ) + S 2 ( x , y ) sin ( 2 π 2 m t f Λ y ) ] .
S 0 ( x , y ) = S 0 ( x , y ) + Δ offset ( x , y ) ,
S n ( x , y ) S 0 ( x , y ) = S n ( x , y ) S 0 ( x , y ) + Δ offset ( x , y ) ,
DOLP ( x , y ) = S 1 2 ( x , y ) + S 2 2 ( x , y ) S 0 ( x , y ) .
θ L ( x , y ) = 1 2 tan 1 ( S 2 ( x , y ) S 1 ( x , y ) ) .
E inc = [ E ¯ X E ¯ Y ] = [ E X ( ξ , η ) e j ϕ x ( ξ , η ) E Y ( ξ , η ) e j ϕ y ( ξ , η ) ] .
E A = [ 1 0 0 j ] E A = 1 2 [ E ¯ X ( ξ , η α ) j E ¯ Y ( ξ , η α ) E ¯ X ( ξ , η α ) j E ¯ Y ( ξ , η α ) ] ,
E B = [ 1 0 0 j ] E B = 1 2 [ E ¯ X ( ξ , η + α ) + j E ¯ Y ( ξ , η + α ) E ¯ X ( ξ , η + α ) j E ¯ Y ( ξ , η + α ) ] .
E C ( ξ + α , η α ) = J 1 , LC E A ( ξ , η α ) = 1 4 [ ( E ¯ X E ¯ Y ) j ( E ¯ X + E ¯ Y ) ( E ¯ X + E ¯ Y ) + j ( E ¯ X E ¯ Y ) ] ,
E D ( ξ α , η α ) = J + 1 , RC E A ( ξ , η α ) = 1 4 [ ( E ¯ X + E ¯ Y ) + j ( E ¯ X E ¯ Y ) ( E ¯ X E ¯ Y ) j ( E ¯ X + E ¯ Y ) ] ,
E E ( ξ + α , η + α ) = J 1 , LC E B ( ξ , η + α ) = 1 4 [ ( E ¯ X E ¯ Y ) + j ( E ¯ X + E ¯ Y ) ( E ¯ X + E ¯ Y ) + j ( E ¯ X E ¯ Y ) ] ,
E F ( ξ α , η + α ) = J + 1 , RC E B ( ξ , η + α ) = 1 4 [ ( E ¯ X + E ¯ Y ) j ( E ¯ X E ¯ Y ) ( E ¯ X E ¯ Y ) j ( E ¯ X + E ¯ Y ) ] ,
E X L = E Y L = 1 4 ( E ¯ X ( ξ + α , η α ) j E ¯ Y ( ξ + α , η α ) ) + ( E ¯ X ( ξ α , η α ) j E ¯ Y ( ξ α , η α ) ) + ( j E ¯ X ( ξ + α , η + α ) E ¯ Y ( ξ + α , η + α ) ) + ( j E ¯ X ( ξ α , η + α ) + E ¯ Y ( ξ α , η + α ) ) .
E L = F [ E X L ] ξ = x λ f , η = y λ f = 1 4 ( E ¯ X j E ¯ Y ) e j 2 π λ f α ( x y ) + ( j E ¯ X + E ¯ Y ) e j 2 π λ f α ( x y ) + ( j E ¯ X E ¯ Y ) e j 2 π λ f α ( x + y ) + ( E ¯ X j E ¯ Y ) e j 2 π λ f α ( x + y ) ,
I ( x , y ) = 1 2 S 0 ( x , y ) + 1 2 S 3 ( x , y ) cos ( 2 π 2 m t f Λ x ) + 1 4 S 2 ( x , y ) [ cos ( 2 π 2 m t f Λ ( x y ) ) cos ( 2 π 2 m t f Λ ( x + y ) ) ] + 1 4 S 1 ( x , y ) [ sin ( 2 π 2 m t f Λ ( x y ) ) + sin ( 2 π 2 m t f Λ ( x + y ) ) ] .
U 1 = 2 m t f Λ ,
U 2 = 2 2 m t f Λ .

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