Abstract

A compact birefringent interferometer (CBI) for Fourier transform hyperspectral imaging is presented. The CBI employs only two birefringent crystal plates: a shearing plate (SP) and a compensation plate (CP). The SP generates the optical path difference (OPD) associated with the field of view for broadband interference. The CP compensates the constant term and square term OPDs of the SP to adjust the position of the zero-order fringe pattern and suppress inconsistent total OPDs and other nonlinear OPDs. This paper details the theoretically deduced OPDs and then presents simulation analyses and verification experiments conducted to investigate the OPD distribution characteristics. To verify the CBI performance, experimental spectral measurements and hyperspectral imaging were performed. The experimental results demonstrate that the CBI can suppress inconsistent total OPDs and other nonlinear OPDs with only two birefringent crystal plates, and therefore offers much promise for miniature and high-precision Fourier transform hyperspectral imaging.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2016 (5)

2015 (2)

C. Coudrain, S. Bernhardt, M. Caes, R. Domel, Y. Ferrec, R. Gouyon, D. Henry, M. Jacquart, A. Kattnig, P. Perrault, L. Poutier, M. Tauvy, and J. Primot, “SIELETERS, an airborne infrared dual-band spectro-imaging system for measurement of scene spectral signatures,” Opt. Express 23(12), 16164–16176 (2015).

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

2014 (4)

J. Li, X. Meng, D. Xu, H. Song, L. Wang, and R. Zhu, “Near-infrared Fourier transform imaging spectrometer for remote sensing,” Proc. SPIE 9298, 929812 (2014).

P. G. Lucey, J. Akagi, A. L. Bingham, J. L. Hinrichs, and E. T. Knobbe, “A compact Fourier transform imaging spectrometer employing a variable gap Fabry-Perot interferometer,” Proc. SPIE 9101, 910110 (2014).

T. Mu, C. Zhang, Q. Li, L. Zhang, Y. Wei, and Q. Chen, “Achromatic Savart polariscope: choice of materials,” Opt. Express 22(5), 5043–5051 (2014).

W. Wang, J. Liang, Z. Liang, J. Lü, Y. Qin, C. Tian, and W. Wang, “Design of spatio-temporally modulated static infrared imaging Fourier transform spectrometer,” Opt. Lett. 39(16), 4911–4914 (2014).

2013 (3)

2012 (2)

E. Puckrin, C. S. Turcotte, M. A. Gagnon, J. Bastedo, V. Farley, and M. Chamberland, “Airborne infrared hyperspectral imager for intelligence, surveillance and reconnaissance applications,” Proc. SPIE 8360, 836004 (2012).

S. T. Crites, P. G. Lucey, R. Wright, H. Garbeil, K. A. Horton, and M. Wood, “A low cost thermal infrared hyperspectral imager for small satellites,” Proc. SPIE 8385, 838509 (2012).

2011 (4)

D. Cabib, M. Lavi, A. Gil, and U. Milman, “Long wave infrared (8 to 14 microns) hyperspectral imager based on an uncooled thermal camera and the traditional CI block interferometer (SI-LWIR-UC),” Proc. SPIE 8012, 80123H (2011).

J. Craven, M. W. Kudenov, M. G. Stapelbroek, and E. L. Dereniak, “Infrared hyperspectral imaging polarimeter using birefringent prisms,” Appl. Opt. 50(8), 1170–1185 (2011).

J. Li, J. Zhu, and X. Hou, “Comment on the paper ‘Design and analysis of wide-field-of-view polarization imaging spectrometer’,” Opt. Eng. 50(1), 019702 (2011).

J. Li, J. Zhu, and X. Hou, “Field-compensated birefringent Fourier transform spectrometer,” Opt. Commun. 284, 1127–1131 (2011).

2010 (2)

C. Zhang, T. Mu, W. Ren, L. Zhang, and N. Liu, “Design and analysis of wide-field-of-view polarization imaging spectrometer,” Opt. Eng. 49(4), 043002 (2010).

A. Barducci, D. Guzzi, C. Lastri, P. Marcoionni, V. Nardino, and I. Pippi, “Theoretical aspects of Fourier Transform Spectrometry and common path triangular interferometers,” Opt. Express 18(11), 11622–11649 (2010).

2009 (1)

2007 (2)

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in wide-field-of-view polarization interference imaging spectrometer,” Opt. Commun. 273, 67–73 (2007).

L. Moreau, F. Grandmont, S. Lantagne, R. Desbiens, and M. Soucy, “An interferometer for compact imaging spectrometer,” Proc. SPIE 6661, 666109 (2007).

2006 (1)

R. D. Alcock and J. M. Coupland, “A compact, high numerical aperture imaging Fourier transform spectrometer and its application,” Meas. Sci. Technol. 17, 2861–2868 (2006).

2004 (1)

2003 (2)

J. Yang, P. Ruan, B. Xiangli, and B. Zhao, “Large aperture static imaging spectroscopy (LASIS),” Proc. SPIE 4897, 318–324 (2003).

B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite massions,” Acta Astronaut. 52(9-12), 803–811 (2003).

2002 (2)

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).

G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Birefringent imaging spectrometer,” Appl. Opt. 41(4), 734–738 (2002).

1998 (1)

W. J. Slough, J. B. Rafert, C. A. Rohde, and C. L. Hart, “THRIFTI: Tomographic hyperspectral remote imaging Fourier transform interferometer,” Proc. SPIE 3393, 207–216 (1998).

1996 (3)

R. F. Horton, “Optical design for a high etendue imaging Fourier transform spectrometer,” Proc. SPIE 2819, 300–315 (1996).

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

W. H. Smith and P. D. Hammer, “Digital array scanned interferometer: sensors and results,” Appl. Opt. 35(16), 2902–2909 (1996).

1994 (1)

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

1993 (1)

C. L. Bennett, M. R. Carter, D. J. Fields, and J. A. M. Hernandez, “Imaging Fourier transform spectrometer,” Proc. SPIE 1937, 191–200 (1993).
[Crossref]

1966 (1)

Akagi, J.

P. G. Lucey, J. Akagi, A. L. Bingham, J. L. Hinrichs, and E. T. Knobbe, “A compact Fourier transform imaging spectrometer employing a variable gap Fabry-Perot interferometer,” Proc. SPIE 9101, 910110 (2014).

Alcock, R. D.

R. D. Alcock and J. M. Coupland, “A compact, high numerical aperture imaging Fourier transform spectrometer and its application,” Meas. Sci. Technol. 17, 2861–2868 (2006).

Baba, N.

Bai, C.

Bar-Am, I.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Barducci, A.

Bastedo, J.

E. Puckrin, C. S. Turcotte, M. A. Gagnon, J. Bastedo, V. Farley, and M. Chamberland, “Airborne infrared hyperspectral imager for intelligence, surveillance and reconnaissance applications,” Proc. SPIE 8360, 836004 (2012).

Bennett, C. L.

C. L. Bennett, M. R. Carter, D. J. Fields, and J. A. M. Hernandez, “Imaging Fourier transform spectrometer,” Proc. SPIE 1937, 191–200 (1993).
[Crossref]

Bernhardt, S.

Bingham, A. L.

P. G. Lucey, J. Akagi, A. L. Bingham, J. L. Hinrichs, and E. T. Knobbe, “A compact Fourier transform imaging spectrometer employing a variable gap Fabry-Perot interferometer,” Proc. SPIE 9101, 910110 (2014).

Blatt, J.

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

Cabib, D.

D. Cabib, M. Lavi, A. Gil, and U. Milman, “Long wave infrared (8 to 14 microns) hyperspectral imager based on an uncooled thermal camera and the traditional CI block interferometer (SI-LWIR-UC),” Proc. SPIE 8012, 80123H (2011).

Caes, M.

Carter, M. R.

C. L. Bennett, M. R. Carter, D. J. Fields, and J. A. M. Hernandez, “Imaging Fourier transform spectrometer,” Proc. SPIE 1937, 191–200 (1993).
[Crossref]

Chamberland, M.

E. Puckrin, C. S. Turcotte, M. A. Gagnon, J. Bastedo, V. Farley, and M. Chamberland, “Airborne infrared hyperspectral imager for intelligence, surveillance and reconnaissance applications,” Proc. SPIE 8360, 836004 (2012).

Chan, J.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

Chen, Q.

Coudrain, C.

Coupland, J. M.

R. D. Alcock and J. M. Coupland, “A compact, high numerical aperture imaging Fourier transform spectrometer and its application,” Meas. Sci. Technol. 17, 2861–2868 (2006).

Craven, J.

Crites, S. T.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

S. T. Crites, P. G. Lucey, R. Wright, H. Garbeil, K. A. Horton, and M. Wood, “A low cost thermal infrared hyperspectral imager for small satellites,” Proc. SPIE 8385, 838509 (2012).

Cui, X.

D’almeida, O.

Dereniak, E. L.

Desbiens, R.

L. Moreau, F. Grandmont, S. Lantagne, R. Desbiens, and M. Soucy, “An interferometer for compact imaging spectrometer,” Proc. SPIE 6661, 666109 (2007).

Domel, R.

Durham, S.

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

Farley, V.

E. Puckrin, C. S. Turcotte, M. A. Gagnon, J. Bastedo, V. Farley, and M. Chamberland, “Airborne infrared hyperspectral imager for intelligence, surveillance and reconnaissance applications,” Proc. SPIE 8360, 836004 (2012).

Ferguson-Smith, M. A.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Ferrec, Y.

Fields, D. J.

C. L. Bennett, M. R. Carter, D. J. Fields, and J. A. M. Hernandez, “Imaging Fourier transform spectrometer,” Proc. SPIE 1937, 191–200 (1993).
[Crossref]

Fletcher-Holmes, D. W.

Fossi, A. P.

Gabrieli, A.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

Gagnon, M. A.

E. Puckrin, C. S. Turcotte, M. A. Gagnon, J. Bastedo, V. Farley, and M. Chamberland, “Airborne infrared hyperspectral imager for intelligence, surveillance and reconnaissance applications,” Proc. SPIE 8360, 836004 (2012).

Gao, B.

Garbeil, H.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

S. T. Crites, P. G. Lucey, R. Wright, H. Garbeil, K. A. Horton, and M. Wood, “A low cost thermal infrared hyperspectral imager for small satellites,” Proc. SPIE 8385, 838509 (2012).

Garini, Y.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Gil, A.

D. Cabib, M. Lavi, A. Gil, and U. Milman, “Long wave infrared (8 to 14 microns) hyperspectral imager based on an uncooled thermal camera and the traditional CI block interferometer (SI-LWIR-UC),” Proc. SPIE 8012, 80123H (2011).

Gouyon, R.

Grandmont, F.

L. Moreau, F. Grandmont, S. Lantagne, R. Desbiens, and M. Soucy, “An interferometer for compact imaging spectrometer,” Proc. SPIE 6661, 666109 (2007).

Guerineau, N.

Guzzi, D.

Hammer, P. D.

Harnisch, B.

B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite massions,” Acta Astronaut. 52(9-12), 803–811 (2003).

Hart, C. L.

W. J. Slough, J. B. Rafert, C. A. Rohde, and C. L. Hart, “THRIFTI: Tomographic hyperspectral remote imaging Fourier transform interferometer,” Proc. SPIE 3393, 207–216 (1998).

Harvey, A. R.

Henry, D.

Hernandez, J. A. M.

C. L. Bennett, M. R. Carter, D. J. Fields, and J. A. M. Hernandez, “Imaging Fourier transform spectrometer,” Proc. SPIE 1937, 191–200 (1993).
[Crossref]

Hilliard, R. L.

Hinrichs, J. L.

P. G. Lucey, J. Akagi, A. L. Bingham, J. L. Hinrichs, and E. T. Knobbe, “A compact Fourier transform imaging spectrometer employing a variable gap Fabry-Perot interferometer,” Proc. SPIE 9101, 910110 (2014).

Holbert, E.

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

Holota, K.

B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite massions,” Acta Astronaut. 52(9-12), 803–811 (2003).

Horton, K. A.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

S. T. Crites, P. G. Lucey, R. Wright, H. Garbeil, K. A. Horton, and M. Wood, “A low cost thermal infrared hyperspectral imager for small satellites,” Proc. SPIE 8385, 838509 (2012).

Horton, R. F.

R. F. Horton, “Optical design for a high etendue imaging Fourier transform spectrometer,” Proc. SPIE 2819, 300–315 (1996).

Hou, X.

J. Li, J. Zhu, C. Qi, C. Zheng, B. Gao, Y. Zhang, and X. Hou, “Compact static imaging spectrometer combining spectral zooming capability with a birefringent interferometer,” Opt. Express 21(8), 10182–10187 (2013).

J. Li, J. Zhu, and X. Hou, “Comment on the paper ‘Design and analysis of wide-field-of-view polarization imaging spectrometer’,” Opt. Eng. 50(1), 019702 (2011).

J. Li, J. Zhu, and X. Hou, “Field-compensated birefringent Fourier transform spectrometer,” Opt. Commun. 284, 1127–1131 (2011).

Imai-Hong, A. K. R.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

Ishigaki, T.

Jacquart, M.

Kattnig, A.

Knobbe, E. T.

P. G. Lucey, J. Akagi, A. L. Bingham, J. L. Hinrichs, and E. T. Knobbe, “A compact Fourier transform imaging spectrometer employing a variable gap Fabry-Perot interferometer,” Proc. SPIE 9101, 910110 (2014).

Kudenov, M. W.

Lantagne, S.

L. Moreau, F. Grandmont, S. Lantagne, R. Desbiens, and M. Soucy, “An interferometer for compact imaging spectrometer,” Proc. SPIE 6661, 666109 (2007).

Lastri, C.

Lavi, M.

D. Cabib, M. Lavi, A. Gil, and U. Milman, “Long wave infrared (8 to 14 microns) hyperspectral imager based on an uncooled thermal camera and the traditional CI block interferometer (SI-LWIR-UC),” Proc. SPIE 8012, 80123H (2011).

Ledbetter, D. H.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Li, J.

J. Li, C. Bai, Y. Shen, and D. Xu, “Optical path squeezing interferometry: boosting the resolution for Fourier transform imaging spectrometers,” Opt. Lett. 41(22), 5329–5332 (2016).

C. Bai, J. Li, Y. Shen, and J. Zhou, “Birefringent Fourier transform imaging spectrometer with a rotating retroreflector,” Opt. Lett. 41(15), 3647–3650 (2016).

C. Liu, J. Li, R. Zhu, and X. Cui, “Large field-of-view Fourier transform imaging spectrometer using dual-channel stitching,” Opt. Express 24(25), 28473–28490 (2016).

J. Li, X. Meng, D. Xu, H. Song, L. Wang, and R. Zhu, “Near-infrared Fourier transform imaging spectrometer for remote sensing,” Proc. SPIE 9298, 929812 (2014).

J. Li, J. Zhu, C. Qi, C. Zheng, B. Gao, Y. Zhang, and X. Hou, “Compact static imaging spectrometer combining spectral zooming capability with a birefringent interferometer,” Opt. Express 21(8), 10182–10187 (2013).

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

J. Li, W. Zhou, X. Meng, D. Liu, and R. Zhu, “Fourier transform imaging spectrometry using Sagnac interferometer,” Proc. SPIE 8910, 89101Y (2013).

J. Li, J. Zhu, and X. Hou, “Comment on the paper ‘Design and analysis of wide-field-of-view polarization imaging spectrometer’,” Opt. Eng. 50(1), 019702 (2011).

J. Li, J. Zhu, and X. Hou, “Field-compensated birefringent Fourier transform spectrometer,” Opt. Commun. 284, 1127–1131 (2011).

Li, Q.

Liang, J.

Liang, Z.

Liu, C.

Liu, D.

J. Li, W. Zhou, X. Meng, D. Liu, and R. Zhu, “Fourier transform imaging spectrometry using Sagnac interferometer,” Proc. SPIE 8910, 89101Y (2013).

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

Liu, N.

C. Zhang, T. Mu, W. Ren, L. Zhang, and N. Liu, “Design and analysis of wide-field-of-view polarization imaging spectrometer,” Opt. Eng. 49(4), 043002 (2010).

Lü, J.

Lucey, P. G.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

P. G. Lucey, J. Akagi, A. L. Bingham, J. L. Hinrichs, and E. T. Knobbe, “A compact Fourier transform imaging spectrometer employing a variable gap Fabry-Perot interferometer,” Proc. SPIE 9101, 910110 (2014).

S. T. Crites, P. G. Lucey, R. Wright, H. Garbeil, K. A. Horton, and M. Wood, “A low cost thermal infrared hyperspectral imager for small satellites,” Proc. SPIE 8385, 838509 (2012).

Manoir, S.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Marcoionni, P.

Meng, X.

J. Li, X. Meng, D. Xu, H. Song, L. Wang, and R. Zhu, “Near-infrared Fourier transform imaging spectrometer for remote sensing,” Proc. SPIE 9298, 929812 (2014).

J. Li, W. Zhou, X. Meng, D. Liu, and R. Zhu, “Fourier transform imaging spectrometry using Sagnac interferometer,” Proc. SPIE 8910, 89101Y (2013).

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

Milman, U.

D. Cabib, M. Lavi, A. Gil, and U. Milman, “Long wave infrared (8 to 14 microns) hyperspectral imager based on an uncooled thermal camera and the traditional CI block interferometer (SI-LWIR-UC),” Proc. SPIE 8012, 80123H (2011).

Moreau, L.

L. Moreau, F. Grandmont, S. Lantagne, R. Desbiens, and M. Soucy, “An interferometer for compact imaging spectrometer,” Proc. SPIE 6661, 666109 (2007).

Mu, T.

N. Quan, C. Zhang, and T. Mu, “Principle and analysis of a birefringent beam splitter,” Optik (Stuttg.) 127, 6735–6741 (2016).

T. Mu, C. Zhang, Q. Li, L. Zhang, Y. Wei, and Q. Chen, “Achromatic Savart polariscope: choice of materials,” Opt. Express 22(5), 5043–5051 (2014).

C. Zhang, T. Mu, W. Ren, L. Zhang, and N. Liu, “Design and analysis of wide-field-of-view polarization imaging spectrometer,” Opt. Eng. 49(4), 043002 (2010).

Nardino, V.

Newby, H.

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

Ning, Y.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Oka, K.

Perrault, P.

Pilger, E. J.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

Pippi, I.

Pisani, M.

Posselt, W.

B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite massions,” Acta Astronaut. 52(9-12), 803–811 (2003).

Poutier, L.

Primot, J.

Puckrin, E.

E. Puckrin, C. S. Turcotte, M. A. Gagnon, J. Bastedo, V. Farley, and M. Chamberland, “Airborne infrared hyperspectral imager for intelligence, surveillance and reconnaissance applications,” Proc. SPIE 8360, 836004 (2012).

Qi, C.

Qin, Y.

Quan, N.

N. Quan, C. Zhang, and T. Mu, “Principle and analysis of a birefringent beam splitter,” Optik (Stuttg.) 127, 6735–6741 (2016).

Rafert, J. B.

W. J. Slough, J. B. Rafert, C. A. Rohde, and C. L. Hart, “THRIFTI: Tomographic hyperspectral remote imaging Fourier transform interferometer,” Proc. SPIE 3393, 207–216 (1998).

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

Ren, W.

C. Zhang, T. Mu, W. Ren, L. Zhang, and N. Liu, “Design and analysis of wide-field-of-view polarization imaging spectrometer,” Opt. Eng. 49(4), 043002 (2010).

Ried, T.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Rohde, C. A.

W. J. Slough, J. B. Rafert, C. A. Rohde, and C. L. Hart, “THRIFTI: Tomographic hyperspectral remote imaging Fourier transform interferometer,” Proc. SPIE 3393, 207–216 (1998).

Rost, M.

B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite massions,” Acta Astronaut. 52(9-12), 803–811 (2003).

Roux, N.

Ruan, P.

J. Yang, P. Ruan, B. Xiangli, and B. Zhao, “Large aperture static imaging spectroscopy (LASIS),” Proc. SPIE 4897, 318–324 (2003).

Sauer, H.

Schoell, B.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Schröck, E.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Sellar, R. G.

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

Shen, Y.

Shepherd, G. G.

Slough, W. J.

W. J. Slough, J. B. Rafert, C. A. Rohde, and C. L. Hart, “THRIFTI: Tomographic hyperspectral remote imaging Fourier transform interferometer,” Proc. SPIE 3393, 207–216 (1998).

Smith, W. H.

Soenksen, D.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Song, H.

J. Li, X. Meng, D. Xu, H. Song, L. Wang, and R. Zhu, “Near-infrared Fourier transform imaging spectrometer for remote sensing,” Proc. SPIE 9298, 929812 (2014).

Soucy, M.

L. Moreau, F. Grandmont, S. Lantagne, R. Desbiens, and M. Soucy, “An interferometer for compact imaging spectrometer,” Proc. SPIE 6661, 666109 (2007).

Stapelbroek, M. G.

Tauvy, M.

Tian, C.

Tittel, H. O.

B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite massions,” Acta Astronaut. 52(9-12), 803–811 (2003).

Turcotte, C. S.

E. Puckrin, C. S. Turcotte, M. A. Gagnon, J. Bastedo, V. Farley, and M. Chamberland, “Airborne infrared hyperspectral imager for intelligence, surveillance and reconnaissance applications,” Proc. SPIE 8360, 836004 (2012).

Tyler, D. W.

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

Veldman, T.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Wang, L.

J. Li, X. Meng, D. Xu, H. Song, L. Wang, and R. Zhu, “Near-infrared Fourier transform imaging spectrometer for remote sensing,” Proc. SPIE 9298, 929812 (2014).

Wang, W.

Wei, Y.

Wienberg, J.

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Wood, M.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

S. T. Crites, P. G. Lucey, R. Wright, H. Garbeil, K. A. Horton, and M. Wood, “A low cost thermal infrared hyperspectral imager for small satellites,” Proc. SPIE 8385, 838509 (2012).

Wright, R.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

S. T. Crites, P. G. Lucey, R. Wright, H. Garbeil, K. A. Horton, and M. Wood, “A low cost thermal infrared hyperspectral imager for small satellites,” Proc. SPIE 8385, 838509 (2012).

Wu, L.

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in wide-field-of-view polarization interference imaging spectrometer,” Opt. Commun. 273, 67–73 (2007).

Xiangli, B.

J. Yang, P. Ruan, B. Xiangli, and B. Zhao, “Large aperture static imaging spectroscopy (LASIS),” Proc. SPIE 4897, 318–324 (2003).

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).

Xu, D.

J. Li, C. Bai, Y. Shen, and D. Xu, “Optical path squeezing interferometry: boosting the resolution for Fourier transform imaging spectrometers,” Opt. Lett. 41(22), 5329–5332 (2016).

J. Li, X. Meng, D. Xu, H. Song, L. Wang, and R. Zhu, “Near-infrared Fourier transform imaging spectrometer for remote sensing,” Proc. SPIE 9298, 929812 (2014).

Yang, J.

J. Yang, P. Ruan, B. Xiangli, and B. Zhao, “Large aperture static imaging spectroscopy (LASIS),” Proc. SPIE 4897, 318–324 (2003).

Yoneshige, L.

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

Yuan, X.

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).

Zhan, G.

Zhang, C.

N. Quan, C. Zhang, and T. Mu, “Principle and analysis of a birefringent beam splitter,” Optik (Stuttg.) 127, 6735–6741 (2016).

T. Mu, C. Zhang, Q. Li, L. Zhang, Y. Wei, and Q. Chen, “Achromatic Savart polariscope: choice of materials,” Opt. Express 22(5), 5043–5051 (2014).

C. Zhang, T. Mu, W. Ren, L. Zhang, and N. Liu, “Design and analysis of wide-field-of-view polarization imaging spectrometer,” Opt. Eng. 49(4), 043002 (2010).

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in wide-field-of-view polarization interference imaging spectrometer,” Opt. Commun. 273, 67–73 (2007).

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).

Zhang, L.

T. Mu, C. Zhang, Q. Li, L. Zhang, Y. Wei, and Q. Chen, “Achromatic Savart polariscope: choice of materials,” Opt. Express 22(5), 5043–5051 (2014).

C. Zhang, T. Mu, W. Ren, L. Zhang, and N. Liu, “Design and analysis of wide-field-of-view polarization imaging spectrometer,” Opt. Eng. 49(4), 043002 (2010).

Zhang, Y.

Zhao, B.

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in wide-field-of-view polarization interference imaging spectrometer,” Opt. Commun. 273, 67–73 (2007).

J. Yang, P. Ruan, B. Xiangli, and B. Zhao, “Large aperture static imaging spectroscopy (LASIS),” Proc. SPIE 4897, 318–324 (2003).

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).

Zheng, C.

Zhou, J.

Zhou, W.

J. Li, W. Zhou, X. Meng, D. Liu, and R. Zhu, “Fourier transform imaging spectrometry using Sagnac interferometer,” Proc. SPIE 8910, 89101Y (2013).

Zhu, J.

J. Li, J. Zhu, C. Qi, C. Zheng, B. Gao, Y. Zhang, and X. Hou, “Compact static imaging spectrometer combining spectral zooming capability with a birefringent interferometer,” Opt. Express 21(8), 10182–10187 (2013).

J. Li, J. Zhu, and X. Hou, “Comment on the paper ‘Design and analysis of wide-field-of-view polarization imaging spectrometer’,” Opt. Eng. 50(1), 019702 (2011).

J. Li, J. Zhu, and X. Hou, “Field-compensated birefringent Fourier transform spectrometer,” Opt. Commun. 284, 1127–1131 (2011).

Zhu, R.

C. Liu, J. Li, R. Zhu, and X. Cui, “Large field-of-view Fourier transform imaging spectrometer using dual-channel stitching,” Opt. Express 24(25), 28473–28490 (2016).

J. Li, X. Meng, D. Xu, H. Song, L. Wang, and R. Zhu, “Near-infrared Fourier transform imaging spectrometer for remote sensing,” Proc. SPIE 9298, 929812 (2014).

J. Li, W. Zhou, X. Meng, D. Liu, and R. Zhu, “Fourier transform imaging spectrometry using Sagnac interferometer,” Proc. SPIE 8910, 89101Y (2013).

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

Zucco, M.

Acta Astronaut. (1)

B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite massions,” Acta Astronaut. 52(9-12), 803–811 (2003).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Meas. Sci. Technol. (1)

R. D. Alcock and J. M. Coupland, “A compact, high numerical aperture imaging Fourier transform spectrometer and its application,” Meas. Sci. Technol. 17, 2861–2868 (2006).

Opt. Commun. (3)

C. Zhang, B. Xiangli, B. Zhao, and X. Yuan, “A static polarization imaging spectrometer based on a Savart polariscope,” Opt. Commun. 203, 21–26 (2002).

J. Li, J. Zhu, and X. Hou, “Field-compensated birefringent Fourier transform spectrometer,” Opt. Commun. 284, 1127–1131 (2011).

L. Wu, C. Zhang, and B. Zhao, “Analysis of the lateral displacement and optical path difference in wide-field-of-view polarization interference imaging spectrometer,” Opt. Commun. 273, 67–73 (2007).

Opt. Eng. (2)

C. Zhang, T. Mu, W. Ren, L. Zhang, and N. Liu, “Design and analysis of wide-field-of-view polarization imaging spectrometer,” Opt. Eng. 49(4), 043002 (2010).

J. Li, J. Zhu, and X. Hou, “Comment on the paper ‘Design and analysis of wide-field-of-view polarization imaging spectrometer’,” Opt. Eng. 50(1), 019702 (2011).

Opt. Express (7)

Opt. Lett. (5)

Optik (Stuttg.) (1)

N. Quan, C. Zhang, and T. Mu, “Principle and analysis of a birefringent beam splitter,” Optik (Stuttg.) 127, 6735–6741 (2016).

Proc. SPIE (13)

E. Puckrin, C. S. Turcotte, M. A. Gagnon, J. Bastedo, V. Farley, and M. Chamberland, “Airborne infrared hyperspectral imager for intelligence, surveillance and reconnaissance applications,” Proc. SPIE 8360, 836004 (2012).

J. Yang, P. Ruan, B. Xiangli, and B. Zhao, “Large aperture static imaging spectroscopy (LASIS),” Proc. SPIE 4897, 318–324 (2003).

S. T. Crites, R. Wright, P. G. Lucey, J. Chan, A. Gabrieli, H. Garbeil, K. A. Horton, A. K. R. Imai-Hong, E. J. Pilger, M. Wood, and L. Yoneshige, “The thermal infrared compact imaging spectrometer (TIRCIS): A follow-on to the space ultra compact hyperspectral imager (SUCHI),” Proc. SPIE 9469, 94690R (2015).

C. L. Bennett, M. R. Carter, D. J. Fields, and J. A. M. Hernandez, “Imaging Fourier transform spectrometer,” Proc. SPIE 1937, 191–200 (1993).
[Crossref]

J. Li, X. Meng, D. Xu, H. Song, L. Wang, and R. Zhu, “Near-infrared Fourier transform imaging spectrometer for remote sensing,” Proc. SPIE 9298, 929812 (2014).

J. B. Rafert, R. G. Sellar, E. Holbert, J. Blatt, D. W. Tyler, S. Durham, and H. Newby, “Hyperspectral imaging Fourier transform spectrometers for astronomical and remote sensing observations,” Proc. SPIE 2198, 338–349 (1994).

L. Moreau, F. Grandmont, S. Lantagne, R. Desbiens, and M. Soucy, “An interferometer for compact imaging spectrometer,” Proc. SPIE 6661, 666109 (2007).

J. Li, W. Zhou, X. Meng, D. Liu, and R. Zhu, “Fourier transform imaging spectrometry using Sagnac interferometer,” Proc. SPIE 8910, 89101Y (2013).

R. F. Horton, “Optical design for a high etendue imaging Fourier transform spectrometer,” Proc. SPIE 2819, 300–315 (1996).

P. G. Lucey, J. Akagi, A. L. Bingham, J. L. Hinrichs, and E. T. Knobbe, “A compact Fourier transform imaging spectrometer employing a variable gap Fabry-Perot interferometer,” Proc. SPIE 9101, 910110 (2014).

D. Cabib, M. Lavi, A. Gil, and U. Milman, “Long wave infrared (8 to 14 microns) hyperspectral imager based on an uncooled thermal camera and the traditional CI block interferometer (SI-LWIR-UC),” Proc. SPIE 8012, 80123H (2011).

S. T. Crites, P. G. Lucey, R. Wright, H. Garbeil, K. A. Horton, and M. Wood, “A low cost thermal infrared hyperspectral imager for small satellites,” Proc. SPIE 8385, 838509 (2012).

W. J. Slough, J. B. Rafert, C. A. Rohde, and C. L. Hart, “THRIFTI: Tomographic hyperspectral remote imaging Fourier transform interferometer,” Proc. SPIE 3393, 207–216 (1998).

Science (1)

E. Schröck, S. Manoir, T. Veldman, B. Schoell, J. Wienberg, M. A. Ferguson-Smith, Y. Ning, D. H. Ledbetter, I. Bar-Am, D. Soenksen, Y. Garini, and T. Ried, “Multicolor spectral karyotyping of human chromosomes,” Science 273(5274), 494–497 (1996).

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

M. Françon and S. Mallick, Polarization Interferometers: Applications in Microscopy and Macroscopy (Wiley-Interscience, 1971).

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Figures (18)

Fig. 1
Fig. 1

Schematic of the compact birefringent interferometer (CBI) for hyperspectral imaging.

Fig. 2
Fig. 2

Wave normals and rays in the uniaxial crystal SP.

Fig. 3
Fig. 3

Wave normals and rays in the uniaxial crystal CP.

Fig. 4
Fig. 4

Distribution of pixel coordinates on the image plane.

Fig. 5
Fig. 5

Distribution of the nonlinear OPD for the CBI and three other interferometers.

Fig. 6
Fig. 6

Optimal fitting curves of the linear OPD for the CBI and three other interferometers.

Fig. 7
Fig. 7

Constant term OPD at different wavelengths for the CBI.

Fig. 8
Fig. 8

Asymmetric fringe patterns of the reference solar spectra.

Fig. 9
Fig. 9

Fringe patterns generated by the CBI for a 650-nm laser.

Fig. 10
Fig. 10

Experimental nonlinear OPD results for the CBI at three wavelengths.

Fig. 11
Fig. 11

Correction of the wavelength position. (a) Curve of the total linear OPDs. (b) Wavelength position of original result and corrected result.

Fig. 12
Fig. 12

Fringe patterns generated by the CBI for a white light source.

Fig. 13
Fig. 13

Spectral measurement results using the CBI and the Flame-T.

Fig. 14
Fig. 14

Outdoor imaging scene and corresponding interferograms.

Fig. 15
Fig. 15

Spectral images of Mount Zijin.

Fig. 16
Fig. 16

CSP structure.

Fig. 17
Fig. 17

ASP structure.

Fig. 18
Fig. 18

WSP structure.

Tables (5)

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Table 1 Thickness (mm) of these four interferometers

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Table 2 Total linear OPD (μm) at different rows for the four interferometers.

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Table 3 Spectral resolution (nm) at different rows for the four interferometers.

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Table 4 Wavelength position (nm) at different rows for the four interferometers.

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Table 5 Experimentally determined total linear OPDs and maximum nonlinear OPDs of the CBI.

Equations (29)

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I(x,y,σ)= S(σ)cos( 2πσΔ(x,y,σ) ) dσ,
sini= n o sin r o = n e sin θ e , and
n e = n o n e n o 2 sin 2 θ+ n e 2 cos 2 θ ,
cos θ Ι = K ^ eo_Ι w ^ Ι =cos β Ι cos θ eo_Ι +sin β Ι cos ω Ι sin θ eo_Ι ,
Δ Ι = n eo d I / cos r eo_I ( n o d I / cos r oe_I + Δ Ι ),
n eo = n e ( cos r eo_Ι cos θ eo_Ι +sin r eo_Ι sin θ eo_Ι cos( ω Ι ω Ι ) ), and
Δ Ι = d Ι ( tan r eo_Ι cos( ω Ι ω Ι )tan r oe_Ι )sini,
Δ Ι = d Ι (cot θ eo_Ι cot r oe_Ι )sini .
cot r oe_Ι = n o sini sini 2 n o , and
cot θ eo_Ι = n o 2 n e 2 n o 2 + n e 2 cos ω Ι + 2 n o n e sini n o 2 + n e 2 ( 1 n o 2 sin 2 ω Ι + n e 2 (1+ cos 2 ω Ι ) 2 n e 2 ( n o 2 + n e 2 ) sin 2 i ).
Δ Ι = k 0_Ι + k 1_Ι sini+ k 2_Ι sin 2 i ,
k 0_Ι =( 2 n e n o 2 + n e 2 1 ) n o d Ι ,
k 1_Ι = n o 2 n e 2 n o 2 + n e 2 cos ω Ι d Ι , and
k 2_Ι =( 1 2 n o n o ( n o 2 sin 2 ω Ι + n e 2 cos 2 ω Ι + n e 2 ) 2 n e ( n o 2 + n e 2 ) 3/2 ) d Ι .
Δ ΙΙ = d ΙΙ (cot r eo_ΙΙ cot θ oe_ΙΙ )sini ,
cot r eo_ΙΙ = n o sini sini 2 n o , and
cot θ oe_ΙΙ = n e sini ( 1 1 2 ( cos 2 ω ΙΙ n o 2 + sin 2 ω ΙΙ n e 2 ) sin 2 i ).
Δ ΙΙ = k 0_ΙΙ + k 2_ΙΙ sin 2 i ,
k 0_ΙΙ =( n o n e ) d ΙΙ , and
k 2_ΙΙ =( 1 2 n o + sin 2 ω ΙΙ 2 n e + n e cos 2 ω ΙΙ 2 n o 2 ) d ΙΙ .
Δ(i,ω,σ, d Ι , d ΙΙ )= Δ I + Δ II =( 2 n o n e n o 2 + n e 2 n o ) d Ι +( n o n e ) d ΙΙ + n o 2 n e 2 n o 2 + n e 2 cosω d Ι sini +( ( 1 2 n o n o ( n o 2 sin 2 ω+ n e 2 cos 2 ω+ n e 2 ) 2 n e ( n o 2 + n e 2 ) 3/2 ) d Ι ( 1 2 n o + cos 2 ω 2 n e + n e sin 2 ω 2 n o 2 ) d ΙΙ ) sin 2 i.
δσ= 1 Δ(l/2,0,σ, d Ι , d ΙΙ )Δ(l/2,0,σ, d Ι , d ΙΙ ) ,
d Ι d ΙΙ = n o ( σ c ) n e ( σ c ) n o ( σ c ) 2 n o ( σ c ) n e ( σ c )/ n o 2 ( σ c )+ n e 2 ( σ c ) .
n o 2 (σ)=2.69705+0.0192064/( σ 2 0.01820)0.0151624 σ 2 , n e 2 (σ)=2.18438+0.0087309/( σ 2 0.01018)0.0024411 σ 2 .
Δ CSP = n o 2 n e 2 n o 2 + n e 2 (cosω+sinω) d _CSP sini+ n o ( n o 2 n e 2 ) 2 n e ( n o 2 + n e 2 ) 3/2 ( cos 2 ω sin 2 ω) d _CSP sin 2 i,
Δ ASP =( n o_P 2 n e_P 2 n o_P 2 + n e_P 2 d P_ASP + n o_N 2 n e_N 2 n o_N 2 + n e_N 2 d N_ASP )(cosω+sinω)sini +( n o_P ( n o_P 2 n e_P 2 ) 2 n e_P ( n o_P 2 + n e_P 2 ) 3/2 d P_ASP + n o_N ( n o_N 2 n e_N 2 ) 2 n e_N ( n o_N 2 + n e_N 2 ) 3/2 d N_ASP )( cos 2 ω sin 2 ω) sin 2 i,
d P_ASP d N_ASP = n o_N ( n o_N 2 n e_N 2 ) 2 n e_N ( n o_N 2 + n e_N 2 ) 3/2 n o_P ( n o_P 2 n e_P 2 ) 2 n e_P ( n o_P 2 + n e_P 2 ) 3/2 .
Δ WSP = n o_P 2 n e_P 2 n o_P 2 + n e_P 2 (cosω+sinω) d P_WSP sini +( n o_P ( n o_P 2 n e_P 2 ) 2 n e_P ( n o_P 2 + n e_P 2 ) 3/2 d P_WSP + n o_N 2 n e_N 2 2 n o_N 2 n e_N 3 d N_WSP )( cos 2 ω sin 2 ω) sin 2 i,
d P_WSP d N_WSP = n o_N 2 n e_N 2 2 n o_N 2 n e_N 3 n o_P ( n o_P 2 n e_P 2 ) 2 n e_P ( n o_P 2 + n e_P 2 ) 3/2 .

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