Abstract

A complete full Stokes imaging spectropolarimeter is proposed. Four separate polarized spectra are fed into the Sagnac Fourier transform spectrometer without slit using different angle combinations of the polarized elements. The four polarized spectra are separated without spatial aliasing. And the system has a good performance to resist the instrument noise due to its high light throughput. The mathematical model for the approach is derived and an optimization of the retardance is discussed. For acquiring the four spectra simultaneously, an improved robust polarization modulator using aperture division is outlined. Then the system is discussed in detail including the imaging principle and spectral resolution. Lastly, two proven experiments are carried out and the experimental results in visible light are outlined.

© 2013 Optical Society of America

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2013 (1)

2012 (1)

2011 (3)

Y. Ferrec, J. Taboury, H. Sauer, P. Chavel, P. Fournet, C. Coudrain, J. Deschamps, and J. Primot, “Experimental results from an airborne static Fourier transform imaging spectrometer,” Appl. Opt.50(30), 5894–5904 (2011).
[CrossRef] [PubMed]

C. Zhang, H. Wu, and J. Li, “Fourier transform hyperspectral imaging polarimeter for remote sensing,” Opt. Eng.50(6), 066201 (2011).
[CrossRef]

S. Li, R. Zhu, and J. Li, “Research on new spectral reconstruction solutions for Fourier-transform spectrometer,” Opt. Appl.41, 121–133 (2011).

2010 (3)

2009 (1)

D. B. Cavanaugh, J. M. Lorenz, N. Unwin, M. Dombrowski, and P. Willson, “VNIR hypersensor camera system,” Proc. SPIE7457, 74570O (2009).
[CrossRef]

2008 (1)

2007 (2)

2006 (2)

2005 (2)

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

R. G. Sellar and G. D. Boreman, “Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer,” Appl. Opt.44(9), 1614–1624 (2005).
[CrossRef] [PubMed]

2004 (2)

2002 (1)

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based of a Savart polariscope,” Opt. Commun.203(1-2), 21–26 (2002).
[CrossRef]

2001 (3)

2000 (1)

1999 (1)

1996 (1)

D. G. Soenksen, Y. Garini, and I. Bar-Am, “Multicolor FIHS using a novel spectral bio-imaging system,” Proc. SPIE2678, 303–309 (1996).
[CrossRef]

1995 (2)

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: Part II,” Opt. Eng.34(6), 1656–1658 (1995).
[CrossRef]

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: Part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

1994 (1)

1991 (1)

A. Ballangrudm, T. Jaeger, and G. Wang, “High-resolution imaging interferometer,” Proc. SPIE1521, 89–96 (1991).

1983 (1)

J. O. Stenflo, D. Twerenbold, J. W. Harvey, and J. W. Brault, “Coherent scattering in the solar spectrum: survey of linear polarization in the range 4200-9950Å,” Astron. Astrophys. Suppl. Ser.54, 505–514 (1983).

Ambirajan, A.

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: Part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: Part II,” Opt. Eng.34(6), 1656–1658 (1995).
[CrossRef]

Backman, V.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Badizadegan, K.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Ballangrudm, A.

A. Ballangrudm, T. Jaeger, and G. Wang, “High-resolution imaging interferometer,” Proc. SPIE1521, 89–96 (1991).

Bar-Am, I.

D. G. Soenksen, Y. Garini, and I. Bar-Am, “Multicolor FIHS using a novel spectral bio-imaging system,” Proc. SPIE2678, 303–309 (1996).
[CrossRef]

Barducci, A.

Beaudry, N. A.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Bergstralh, J.

Boreman, G. D.

Brady, D. J.

Brault, J. W.

J. O. Stenflo, D. Twerenbold, J. W. Harvey, and J. W. Brault, “Coherent scattering in the solar spectrum: survey of linear polarization in the range 4200-9950Å,” Astron. Astrophys. Suppl. Ser.54, 505–514 (1983).

Cairns, B.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Cavanaugh, D. B.

D. B. Cavanaugh, J. M. Lorenz, N. Unwin, M. Dombrowski, and P. Willson, “VNIR hypersensor camera system,” Proc. SPIE7457, 74570O (2009).
[CrossRef]

Chavel, P.

Chenault, D. B.

Chipman, R. A.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Coudrain, C.

Craven, J.

J. Craven and M. W. Kudenov, “False signature reduction in channeled spectropolarimetry,” Opt. Eng.49(5), 053602 (2010).
[CrossRef]

Cunningham, T. J.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Dasari, R. R.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Dereniak, E. L.

Deschamps, J.

Diner, D. J.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Dombrowski, M.

D. B. Cavanaugh, J. M. Lorenz, N. Unwin, M. Dombrowski, and P. Willson, “VNIR hypersensor camera system,” Proc. SPIE7457, 74570O (2009).
[CrossRef]

Feld, M. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Ferrec, Y.

Fletcher-Holmes, D. W.

Foo, L. D.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Fournet, P.

Garini, Y.

D. G. Soenksen, Y. Garini, and I. Bar-Am, “Multicolor FIHS using a novel spectral bio-imaging system,” Proc. SPIE2678, 303–309 (1996).
[CrossRef]

Gehm, M. E.

Georgakoudi, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Gerhart, G. R.

Glenar, D. A.

Goldstein, D. L.

Gurjar, R. S.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Guzzi, D.

Hagen, N. A.

Harashima, H.

Harvey, A. R.

Harvey, J. W.

J. O. Stenflo, D. Twerenbold, J. W. Harvey, and J. W. Brault, “Coherent scattering in the solar spectrum: survey of linear polarization in the range 4200-9950Å,” Astron. Astrophys. Suppl. Ser.54, 505–514 (1983).

Hillman, J. J.

Horton, K. A.

Iannarilli, F. J.

Itzkan, I.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Jaeger, T.

A. Ballangrudm, T. Jaeger, and G. Wang, “High-resolution imaging interferometer,” Proc. SPIE1521, 89–96 (1991).

Jia, C.

John, R.

Jones, S. H.

Kato, T.

Kebabian, P. L.

Keller, C. U.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Kudenov, M. W.

Lastri, C.

Li, J.

X. Meng, J. Li, D. Liu, and R. Zhu, “Fourier transform imaging spectropolarimeter using simultaneous polarization modulation,” Opt. Lett.38(5), 778–780 (2013).
[CrossRef] [PubMed]

S. Li, R. Zhu, and J. Li, “Research on new spectral reconstruction solutions for Fourier-transform spectrometer,” Opt. Appl.41, 121–133 (2011).

C. Zhang, H. Wu, and J. Li, “Fourier transform hyperspectral imaging polarimeter for remote sensing,” Opt. Eng.50(6), 066201 (2011).
[CrossRef]

J. Li, J. Zhu, and H. Wu, “Compact static Fourier transform imaging spectropolarimeter based on channeled polarimetry,” Opt. Lett.35(22), 3784–3786 (2010).
[CrossRef] [PubMed]

Li, S.

S. Li, R. Zhu, and J. Li, “Research on new spectral reconstruction solutions for Fourier-transform spectrometer,” Opt. Appl.41, 121–133 (2011).

Liu, D.

Look, D. C.

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: Part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: Part II,” Opt. Eng.34(6), 1656–1658 (1995).
[CrossRef]

Lorenz, J. M.

D. B. Cavanaugh, J. M. Lorenz, N. Unwin, M. Dombrowski, and P. Willson, “VNIR hypersensor camera system,” Proc. SPIE7457, 74570O (2009).
[CrossRef]

Lucey, P. G.

Macenka, S. A.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Marcoionni, P.

Meng, X.

Mu, T.

Naemura, T.

Nardino, V.

Oka, K.

Perelman, L. T.

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Pippi, I.

Primot, J.

Pust, N. J.

Ren, W.

Saif, B.

Sauer, H.

Schulz, T. J.

Sellar, R. G.

Seshadri, S.

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

Shaw, J. A.

Soenksen, D. G.

D. G. Soenksen, Y. Garini, and I. Bar-Am, “Multicolor FIHS using a novel spectral bio-imaging system,” Proc. SPIE2678, 303–309 (1996).
[CrossRef]

Stenflo, J. O.

J. O. Stenflo, D. Twerenbold, J. W. Harvey, and J. W. Brault, “Coherent scattering in the solar spectrum: survey of linear polarization in the range 4200-9950Å,” Astron. Astrophys. Suppl. Ser.54, 505–514 (1983).

Taboury, J.

Turner, T. S.

Twerenbold, D.

J. O. Stenflo, D. Twerenbold, J. W. Harvey, and J. W. Brault, “Coherent scattering in the solar spectrum: survey of linear polarization in the range 4200-9950Å,” Astron. Astrophys. Suppl. Ser.54, 505–514 (1983).

Tyo, J. S.

Unwin, N.

D. B. Cavanaugh, J. M. Lorenz, N. Unwin, M. Dombrowski, and P. Willson, “VNIR hypersensor camera system,” Proc. SPIE7457, 74570O (2009).
[CrossRef]

Wang, G.

A. Ballangrudm, T. Jaeger, and G. Wang, “High-resolution imaging interferometer,” Proc. SPIE1521, 89–96 (1991).

Willett, R. M.

Williams, T.

Willson, P.

D. B. Cavanaugh, J. M. Lorenz, N. Unwin, M. Dombrowski, and P. Willson, “VNIR hypersensor camera system,” Proc. SPIE7457, 74570O (2009).
[CrossRef]

Wu, H.

C. Zhang, H. Wu, and J. Li, “Fourier transform hyperspectral imaging polarimeter for remote sensing,” Opt. Eng.50(6), 066201 (2011).
[CrossRef]

J. Li, J. Zhu, and H. Wu, “Compact static Fourier transform imaging spectropolarimeter based on channeled polarimetry,” Opt. Lett.35(22), 3784–3786 (2010).
[CrossRef] [PubMed]

Xiangli, B.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based of a Savart polariscope,” Opt. Commun.203(1-2), 21–26 (2002).
[CrossRef]

Yoshida, T.

Yuan, X.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based of a Savart polariscope,” Opt. Commun.203(1-2), 21–26 (2002).
[CrossRef]

Zhang, C.

T. Mu, C. Zhang, C. Jia, and W. Ren, “Static hyperspectral imaging polarimeter for full linear Stokes parameters,” Opt. Express20(16), 18194–18201 (2012).
[CrossRef] [PubMed]

C. Zhang, H. Wu, and J. Li, “Fourier transform hyperspectral imaging polarimeter for remote sensing,” Opt. Eng.50(6), 066201 (2011).
[CrossRef]

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based of a Savart polariscope,” Opt. Commun.203(1-2), 21–26 (2002).
[CrossRef]

Zhao, B.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based of a Savart polariscope,” Opt. Commun.203(1-2), 21–26 (2002).
[CrossRef]

Zhu, J.

Zhu, R.

X. Meng, J. Li, D. Liu, and R. Zhu, “Fourier transform imaging spectropolarimeter using simultaneous polarization modulation,” Opt. Lett.38(5), 778–780 (2013).
[CrossRef] [PubMed]

S. Li, R. Zhu, and J. Li, “Research on new spectral reconstruction solutions for Fourier-transform spectrometer,” Opt. Appl.41, 121–133 (2011).

Appl. Opt. (7)

Astron. Astrophys. Suppl. Ser. (1)

J. O. Stenflo, D. Twerenbold, J. W. Harvey, and J. W. Brault, “Coherent scattering in the solar spectrum: survey of linear polarization in the range 4200-9950Å,” Astron. Astrophys. Suppl. Ser.54, 505–514 (1983).

Nat. Med. (1)

R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light-scattering spectroscopy,” Nat. Med.7(11), 1245–1248 (2001).
[CrossRef] [PubMed]

Opt. Appl. (1)

S. Li, R. Zhu, and J. Li, “Research on new spectral reconstruction solutions for Fourier-transform spectrometer,” Opt. Appl.41, 121–133 (2011).

Opt. Commun. (1)

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based of a Savart polariscope,” Opt. Commun.203(1-2), 21–26 (2002).
[CrossRef]

Opt. Eng. (4)

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: Part I,” Opt. Eng.34(6), 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look., “Optimum angles for a polarimeter: Part II,” Opt. Eng.34(6), 1656–1658 (1995).
[CrossRef]

J. Craven and M. W. Kudenov, “False signature reduction in channeled spectropolarimetry,” Opt. Eng.49(5), 053602 (2010).
[CrossRef]

C. Zhang, H. Wu, and J. Li, “Fourier transform hyperspectral imaging polarimeter for remote sensing,” Opt. Eng.50(6), 066201 (2011).
[CrossRef]

Opt. Express (7)

Opt. Lett. (4)

Proc. SPIE (4)

D. J. Diner, R. A. Chipman, N. A. Beaudry, B. Cairns, L. D. Foo, S. A. Macenka, T. J. Cunningham, S. Seshadri, and C. U. Keller, “An integrated multiangle, multispectral, and polarimetric imaging concept for aerosol remote sensing from space,” Proc. SPIE5659, 88–96 (2005).
[CrossRef]

D. B. Cavanaugh, J. M. Lorenz, N. Unwin, M. Dombrowski, and P. Willson, “VNIR hypersensor camera system,” Proc. SPIE7457, 74570O (2009).
[CrossRef]

A. Ballangrudm, T. Jaeger, and G. Wang, “High-resolution imaging interferometer,” Proc. SPIE1521, 89–96 (1991).

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[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Schematic of the measurement configuration of spectropolarization information.

Fig. 2
Fig. 2

(a) Condition number of the polarization system with the retardance from 0° to 180°. (b) Optimization of the retardance.

Fig. 3
Fig. 3

(a) Structure of the SPM where PA is the polarized array and MLA represents the micro-lens array. (b) Structure of the polarization array composed of two rectangular retarders and two rectangular polarizers. (c) Polarization states of the polarization array where the orange arrow represents the fast axis of the retarder and the green arrow represents the axis of the polarizers.

Fig. 4
Fig. 4

Schematic of the complete spectropolarimeter. The back focal plane of the micro-lens array and the front focal plane of the collimate lens L2 are constrained to be coincident.

Fig. 5
Fig. 5

Architecture of the system composed of the simultaneous polarization modulator (SPM) and a Sagnac Fourier transform spectrometer without slit.

Fig. 6
Fig. 6

(a) Schematic image captured by the SPM. (b) four decomposed polarized images of the original image. (c) Four polarized images of a point in a wide field of view when the polarization array is not the aperture stop. (d) Aliasing between the images of neighbor points

Fig. 7
Fig. 7

Data captured by the two operation mode, The number 1,2,…,6 represent the pixels of a line perpendicular to the fringes on the detector, the An, Bn,…, Fn represent the radiation intensity of the points of a line on the target, n represents the number of the captured interferometric image.

Fig. 8
Fig. 8

Schematic of the Fourier transform spectrometer when the light from L2 is a spherical wave. The Sagnac lateral shearing splitter is unfolded into a flat in the schematic

Fig. 9
Fig. 9

Experimental setup consists of the SPM and Sagnac Fourier transform spectrometer. The electronic control box is used to control the rotating of the Sagnac interferometer. The computer is used to capture the interferometric images.

Fig. 10
Fig. 10

Experimental target and three of the captured images. (a) The colour picture as the target. (b) ~(d) three of the captured interferometric images.

Fig. 11
Fig. 11

Experimental results of the colour picture. (a) The spectrum of point A with Stokes parameter S0. (b) The normalized Stokes parameters. (c) Four extracted images of the scene with different polarization states at 540nm. (d) Four stokes parameters images at 540nm.

Fig. 12
Fig. 12

Experimental fruits and three of the captured images. (a) The experimental fruits where a linear polarizer is placed before the apple. (b) ~(d) Three of the interferometric images.

Fig. 13
Fig. 13

Experimental results of the fruits. (a) The normalized Stokes parameters spectra of Point B. (b) Four spectral images with different spectral bands with the same Stokes parameter S0. (c) Four stokes parameters images at 540nm.

Equations (21)

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S out = M LP (β) M R (α,δ(σ)) S in ,
S in = ( S 0 , S 1 , S 2 , S 3 ) T ,
I(Δ)= 0 σ max R(1+cos2πσΔ) ( S out0 (σ))dσ = 0 σ max R(1+cos2πσΔ) ( 1 2 ( a 0 (σ) S 0 (σ)+ a 1 (σ) S 1 (σ)+ a 2 (σ) S 2 (σ)+ a 3 (σ) S 3 (σ)))dσ,
a 0 (σ)=1, a 1 (σ)=( cos 2 2α+ sin 2 2αcos(δ(σ)))cos2β+(sin2αcos2α(1cos(δ(σ))))sin2β, a 2 (σ)=(sin2αcos2α(1cos(δ(σ))))cos2β+( sin 2 2α+ cos 2 2αcos(δ(σ)))sin2β, a 3 (σ)=( sin2αsin(δ(σ)) )cos2β+(cos2αsin(δ(σ)))sin2β.
a 0 n (σ) S 0 + a 1 n (σ) S 1 + a 2 n (σ) S 2 + a 3 n (σ) S 3 =2 B n (σ)=2 -1 ( I n (Δ)),
A S in =B,
A=[ a 0 0 (σ) a 1 0 (σ) a 2 0 (σ) a 3 0 (σ) a 0 1 (σ) a 1 1 (σ) a 2 1 (σ) a 3 1 (σ) a 0 2 (σ) a 1 2 (σ) a 2 2 (σ) a 3 2 (σ) a 0 3 (σ) a 1 3 (σ) a 2 3 (σ) a 3 3 (σ) ],B=[ B 0 (σ) B 1 (σ) B 2 (σ) B 3 (σ) ].
S in = A 1 B.
Δ S i n S i n L ν ( A ) ( Δ A A + Δ B B ) ,
L ν = A ν A 1 ν ( v = 1 , 2 o r ) ,
I(θ)= σ 1 σ 2 B(σ)(1+cos2πσdsin(θ)) dσ,
SNR= S eff ( N p + N i ) ,
R space = D micro l 1 ,
D 1 f 1 D micro f micro ,
ω max =2arctan( H f 2 2 f 3 f 1 ),
Δ max d l c a m f 3 ,
Δ max =dsin θ max ,
D 2 H f 2 f 3 + D 1 f 2 f 1 .
D 3 D 1 f 2 f 1 H f 3 (2 f 2 + f 3 ).
K 2 = I max - I min I max + I min = r s 1- r s 2 + r p 1- r p 2 ,
B ^ n (σ)= C n B n (σ),

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