Abstract

This paper presents a method to calibrate alignment errors for a channeled spectropolarimeter. A calibration model, including an alignment errors determination model and an alignment errors compensation model, is derived firstly. To determine the exact alignment errors of the high-order retarders and polarizer included in the spectropolarimeter, an auxiliary high-order retarder and a reference beam are used. The auxiliary high-order retarder does not affect the normal use of the spectropolarimeter and the polarization state of the reference beam needs not to be controlled accurately. Based on the determination results, the alignment errors are compensated by using a correction algorithm without any precise mechanical adjustments. Simulation results show that the alignment errors can be determined accurately and the errors of the reconstructed Stokes parameters due to the alignment errors are reduced effectively by the presented method. Finally, experimental results are summarized and analyzed to demonstrate the effectiveness of the calibration method.

© 2016 Optical Society of America

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References

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    [Crossref]
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2015 (1)

A. A. Thyparambil, Y. Wei, and R. A. Latour, “Experimental characterization of adsorbed protein orientation, conformation, and bioactivity,” Biointerphases 10(1), 019002 (2015).
[Crossref] [PubMed]

2013 (1)

T. Mu, C. Zhang, C. Jia, W. Ren, L. Zhang, and Q. Li, “Alignment and retardance errors, and compensation of a channeled spectropolarimeter,” Opt. Commun. 294, 88–95 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (2)

2009 (1)

2008 (1)

R. W. Aumiller, C. Vandervlugt, E. L. Dereniak, R. Sampson, and R. W. McMillan, “Snapshot imaging spectropolarimetry in the visible and infrared,” Proc. SPIE 6972, 69720D (2008).
[Crossref]

2007 (1)

2006 (2)

2004 (1)

2003 (1)

2002 (2)

F. Cremer, W. de Jong, and K. Schutte, “Infrared polarization measurements and modeling applied to surface-laid antipersonnel landmines,” Opt. Eng. 41(5), 1021–1032 (2002).
[Crossref]

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

1999 (2)

K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24(21), 1475–1477 (1999).
[Crossref] [PubMed]

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric-spectral intensity modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474–1477 (1999).
[Crossref]

Aumiller, R. W.

R. W. Aumiller, C. Vandervlugt, E. L. Dereniak, R. Sampson, and R. W. McMillan, “Snapshot imaging spectropolarimetry in the visible and infrared,” Proc. SPIE 6972, 69720D (2008).
[Crossref]

Chenault, D. B.

Craven-Jones, J.

Cremer, F.

F. Cremer, W. de Jong, and K. Schutte, “Infrared polarization measurements and modeling applied to surface-laid antipersonnel landmines,” Opt. Eng. 41(5), 1021–1032 (2002).
[Crossref]

de Jong, W.

F. Cremer, W. de Jong, and K. Schutte, “Infrared polarization measurements and modeling applied to surface-laid antipersonnel landmines,” Opt. Eng. 41(5), 1021–1032 (2002).
[Crossref]

Dereniak, E. L.

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

R. W. Aumiller, C. Vandervlugt, E. L. Dereniak, R. Sampson, and R. W. McMillan, “Snapshot imaging spectropolarimetry in the visible and infrared,” Proc. SPIE 6972, 69720D (2008).
[Crossref]

N. Hagen, K. Oka, and E. L. Dereniak, “Snapshot Mueller matrix spectropolarimeter,” Opt. Lett. 32(15), 2100–2102 (2007).
[Crossref] [PubMed]

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

Descour, M. R.

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

Dubreuil, M.

Dupont, L.

Garcia, J. P.

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

Goldstein, D. L.

Hagen, N.

Hamilton, T.

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

Hayakawa, M.

Iannarilli, F.

Iannarilli, F. J.

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric-spectral intensity modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474–1477 (1999).
[Crossref]

Jia, C.

T. Mu, C. Zhang, C. Jia, W. Ren, L. Zhang, and Q. Li, “Alignment and retardance errors, and compensation of a channeled spectropolarimeter,” Opt. Commun. 294, 88–95 (2013).
[Crossref]

T. Mu, C. Zhang, C. Jia, and W. Ren, “Static hyperspectral imaging polarimeter for full linear Stokes parameters,” Opt. Express 20(16), 18194–18201 (2012).
[Crossref] [PubMed]

Jones, S.

Jones, S. H.

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric-spectral intensity modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474–1477 (1999).
[Crossref]

Kaneko, T.

Karalidi, T.

Kato, T.

Kebabian, P.

S. Jones, F. Iannarilli, and P. Kebabian, “Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter,” Opt. Express 12(26), 6559–6573 (2004).
[Crossref] [PubMed]

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric-spectral intensity modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474–1477 (1999).
[Crossref]

Keller, C. U.

Kudenov, M. W.

Latour, R. A.

A. A. Thyparambil, Y. Wei, and R. A. Latour, “Experimental characterization of adsorbed protein orientation, conformation, and bioactivity,” Biointerphases 10(1), 019002 (2015).
[Crossref] [PubMed]

Le Jeune, B.

Li, J.

Li, Q.

T. Mu, C. Zhang, C. Jia, W. Ren, L. Zhang, and Q. Li, “Alignment and retardance errors, and compensation of a channeled spectropolarimeter,” Opt. Commun. 294, 88–95 (2013).
[Crossref]

Locke, M.

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

McMillan, R. W.

R. W. Aumiller, C. Vandervlugt, E. L. Dereniak, R. Sampson, and R. W. McMillan, “Snapshot imaging spectropolarimetry in the visible and infrared,” Proc. SPIE 6972, 69720D (2008).
[Crossref]

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

Mu, T.

T. Mu, C. Zhang, C. Jia, W. Ren, L. Zhang, and Q. Li, “Alignment and retardance errors, and compensation of a channeled spectropolarimeter,” Opt. Commun. 294, 88–95 (2013).
[Crossref]

T. Mu, C. Zhang, C. Jia, and W. Ren, “Static hyperspectral imaging polarimeter for full linear Stokes parameters,” Opt. Express 20(16), 18194–18201 (2012).
[Crossref] [PubMed]

Oka, K.

Okabe, H.

Ren, W.

T. Mu, C. Zhang, C. Jia, W. Ren, L. Zhang, and Q. Li, “Alignment and retardance errors, and compensation of a channeled spectropolarimeter,” Opt. Commun. 294, 88–95 (2013).
[Crossref]

T. Mu, C. Zhang, C. Jia, and W. Ren, “Static hyperspectral imaging polarimeter for full linear Stokes parameters,” Opt. Express 20(16), 18194–18201 (2012).
[Crossref] [PubMed]

Rivet, S.

Sabatke, D. S.

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

Sampson, R.

R. W. Aumiller, C. Vandervlugt, E. L. Dereniak, R. Sampson, and R. W. McMillan, “Snapshot imaging spectropolarimetry in the visible and infrared,” Proc. SPIE 6972, 69720D (2008).
[Crossref]

Schutte, K.

F. Cremer, W. de Jong, and K. Schutte, “Infrared polarization measurements and modeling applied to surface-laid antipersonnel landmines,” Opt. Eng. 41(5), 1021–1032 (2002).
[Crossref]

Scott, H. E.

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric-spectral intensity modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474–1477 (1999).
[Crossref]

Shaw, J. A.

Snik, F.

Stapelbroek, M. G.

Taniguchi, A.

Thyparambil, A. A.

A. A. Thyparambil, Y. Wei, and R. A. Latour, “Experimental characterization of adsorbed protein orientation, conformation, and bioactivity,” Biointerphases 10(1), 019002 (2015).
[Crossref] [PubMed]

Tyo, J. S.

Vandervlugt, C.

R. W. Aumiller, C. Vandervlugt, E. L. Dereniak, R. Sampson, and R. W. McMillan, “Snapshot imaging spectropolarimetry in the visible and infrared,” Proc. SPIE 6972, 69720D (2008).
[Crossref]

Wei, Y.

A. A. Thyparambil, Y. Wei, and R. A. Latour, “Experimental characterization of adsorbed protein orientation, conformation, and bioactivity,” Biointerphases 10(1), 019002 (2015).
[Crossref] [PubMed]

Wu, H.

Zhang, C.

T. Mu, C. Zhang, C. Jia, W. Ren, L. Zhang, and Q. Li, “Alignment and retardance errors, and compensation of a channeled spectropolarimeter,” Opt. Commun. 294, 88–95 (2013).
[Crossref]

T. Mu, C. Zhang, C. Jia, and W. Ren, “Static hyperspectral imaging polarimeter for full linear Stokes parameters,” Opt. Express 20(16), 18194–18201 (2012).
[Crossref] [PubMed]

Zhang, L.

T. Mu, C. Zhang, C. Jia, W. Ren, L. Zhang, and Q. Li, “Alignment and retardance errors, and compensation of a channeled spectropolarimeter,” Opt. Commun. 294, 88–95 (2013).
[Crossref]

Zhu, J.

Appl. Opt. (3)

Biointerphases (1)

A. A. Thyparambil, Y. Wei, and R. A. Latour, “Experimental characterization of adsorbed protein orientation, conformation, and bioactivity,” Biointerphases 10(1), 019002 (2015).
[Crossref] [PubMed]

Opt. Commun. (1)

T. Mu, C. Zhang, C. Jia, W. Ren, L. Zhang, and Q. Li, “Alignment and retardance errors, and compensation of a channeled spectropolarimeter,” Opt. Commun. 294, 88–95 (2013).
[Crossref]

Opt. Eng. (1)

F. Cremer, W. de Jong, and K. Schutte, “Infrared polarization measurements and modeling applied to surface-laid antipersonnel landmines,” Opt. Eng. 41(5), 1021–1032 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Proc. SPIE (3)

F. J. Iannarilli, S. H. Jones, H. E. Scott, and P. Kebabian, “Polarimetric-spectral intensity modulation (P-SIM): Enabling simultaneous hyperspectral and polarimetric imaging,” Proc. SPIE 3698, 474–1477 (1999).
[Crossref]

R. W. Aumiller, C. Vandervlugt, E. L. Dereniak, R. Sampson, and R. W. McMillan, “Snapshot imaging spectropolarimetry in the visible and infrared,” Proc. SPIE 6972, 69720D (2008).
[Crossref]

M. Locke, D. S. Sabatke, E. L. Dereniak, M. R. Descour, J. P. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot Imaging Spectropolarimeter,” Proc. SPIE 4481, 64–72 (2002).
[Crossref]

Other (2)

B. Henderson, “Spectroscopic measurements,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995).

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Figures (9)

Fig. 1
Fig. 1 Schematic of a channeled spectropolarimeter.
Fig. 2
Fig. 2 Schematic of the configuration for determining the alignment errors of a channeled spectropolarimeter. R3 is marked with a dashed box means that it will be removed after determining the alignment errors.
Fig. 3
Fig. 3 Magnitude of the autocorrelation function of the obtained spectrum when determining the alignment errors. The part in the dashed box is enlarged.
Fig. 4
Fig. 4 Determination results of the alignment errors of R1, R2, and A.
Fig. 5
Fig. 5 Magnitude of the autocorrelation function of the obtained spectrum when measuring the target light.
Fig. 6
Fig. 6 Reconstructed normalized Stokes parameters. Theoretical values are S 1 / S 0 =1/2, S 2 / S 0 = 3 /2, and S 3 / S 0 =0.
Fig. 7
Fig. 7 Photograph of the experimental configuration. The polarizers, P and A, and the high-order retarders, R1, R2, and R3, are installed in precision adjusting racks.
Fig. 8
Fig. 8 Measured result in the progress of determining the alignment errors. (a) measured spectral intensity; (b) magnitude of the desired part of the measured spectrum’s autocorrelation function.
Fig. 9
Fig. 9 Stokes parameters reconstructed from experimental measurements under variations of the alignment errors using (a) the traditional method and (b) the presented method. Theoretical reference values are S 1 / S 0 =1/2, S 2 / S 0 = 3 /2, and S 3 / S 0 =0.

Tables (1)

Tables Icon

Table 1 Averages and Errors of the Determination Results of θ 1 , θ 2 , and ε

Equations (38)

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B(σ)=(1/2) S 0 (σ)+(1/4)| S 23 (σ) |cos{ φ 2 (σ) φ 1 (σ)+arg[ S 23 (σ)]} (1/4)| S 23 (σ) |cos{ φ 2 (σ)+ φ 1 (σ)arg[ S 23 (σ)]} +(1/2)| S 1 (σ) |cos{ φ 2 (σ)+arg[ S 1 (σ)]},
S out (σ)= M A (ε) M R2 [ 45 o + θ 2 , φ 2 (σ)] M R1 [ θ 1 , φ 1 (σ)] M R3 [ 90 o , φ 3 (σ)] S (σ),
B (σ)=(1/2) S 0 (σ)+(1/2) Γ 1 | S 1 (σ) |cos{ φ 2 (σ)+arg[ S 1 (σ)]} +(1/4) Γ 2 | S 1 (σ) |cos{ φ 1 (σ)+ φ 2 (σ)+arg[ S 1 (σ)]} (1/4) Γ 2 | S 1 (σ) |cos{ φ 2 (σ) φ 1 (σ)+arg[ S 1 (σ)]} +(1/4) Γ 3 | S 23 (σ) |cos{ φ 2 (σ) φ 3 (σ)+arg[ S 23 (σ)]} +(1/4) Γ 3 | S 23 (σ) |cos{ φ 2 (σ)+ φ 3 (σ)arg[ S 23 (σ)]} +(1/4)( Γ 4 + Γ 5 Γ 6 Γ 7 )| S 23 (σ) |cos{ φ 1 (σ) φ 3 (σ)arg[ S 23 (σ)]} +(1/8)( Γ 8 Γ 9 + Γ 10 + Γ 6 Γ 3 + Γ 7 )| S 23 (σ) |cos{ φ 2 (σ)+ φ 3 (σ) φ 1 (σ)+arg[ S 23 (σ)]} (1/8)( Γ 8 + Γ 9 + Γ 10 Γ 6 + Γ 3 Γ 7 )| S 23 (v) |cos{ φ 1 (σ)+ φ 2 (σ) φ 3 (σ)arg[ S 23 (σ)]},
{ Γ 1 = b 2 d 2 f Γ 2 =adf Γ 3 =ab d 2 f Γ 4 =b d 2 e Γ 5 = b 2 d 2 e Γ 6 =bcdf Γ 7 = b 2 cdf Γ 8 =df Γ 9 =a d 2 f Γ 10 =bdf ,
C(h)= C 0 (h)+ C 1 [ h( L 1 L 3 ) ]+ C 1 * [ h( L 1 L 3 ) ] + C 2 [ h( L 2 L 1 ) ]+ C 2 * [ h( L 2 L 1 ) ] + C 3 [ h( L 2 L 3 ) ]+ C 3 * [ h( L 2 L 3 ) ] + C 4 [ h( L 2 + L 3 L 1 ) ]+ C 4 * [ h( L 2 + L 3 L 1 ) ] + C 5 (h L 2 )+ C 5 * (h L 2 ) + C 6 [ h( L 1 + L 2 L 3 ) ]+ C 6 * [ h( L 1 + L 2 L 3 ) ] + C 7 [ h( L 2 + L 3 ) ]+ C 7 * [ h( L 2 + L 3 ) ] + C 8 [ h( L 1 + L 2 ) ]+ C 8 * [ h( L 1 + L 2 ) ],
C 0 = 1 [(1/2) S 0 (σ)],
C 1 = 1 {(1/8)( Γ 4 + Γ 5 Γ 6 Γ 7 ) S 23 (σ)exp{i[ φ 1 (σ) φ 3 (σ)]}},
C 2 = 1 {(1/8) Γ 2 S 1 (σ)exp{i[ φ 2 (σ) φ 1 (σ)]}},
C 3 = 1 {(1/8) Γ 3 S 23 (σ)exp{i[ φ 2 (σ) φ 3 (σ)]}},
C 4 = 1 {(1/16)( Γ 8 Γ 9 + Γ 10 + Γ 6 Γ 3 + Γ 7 ) S 23 (σ)exp{i[ φ 2 (σ)+ φ 3 (σ) φ 1 (σ)]}},
C 5 = 1 {(1/4) Γ 1 S 1 (σ)exp{i[ φ 2 (σ)]}},
C 6 = 1 {(1/16)( Γ 8 + Γ 9 + Γ 10 Γ 6 + Γ 3 Γ 7 ) S 23 (σ)exp{i[ φ 1 (σ)+ φ 2 (σ) φ 3 (σ)]}},
C 7 = 1 {(1/8) Γ 3 S 23 (σ)exp{i[ φ 2 (σ)+ φ 3 (σ)]}},
C 8 = 1 {(1/8) Γ 2 S 1 (σ)exp{i[ φ 1 (σ)+ φ 2 (σ)]}},
abs[ ( C 8 ) ( C 5 ) ]= a 2 b 2 d ,
abs[ ( C 4 ) ( C 3 ) ]+abs[ ( C 6 ) ( C 7 ) ]= b+1 abd ,
abs[ ( C 6 ) ( C 7 ) ]abs[ ( C 4 ) ( C 3 ) ]= (adbc)(b+1) abd ,
abs[ ( C 1 ) ( C 7 ) ]= bd(b+1)(cfde) ab d 2 f ,
b= Q Q+2N ,
c= Wb Q + ( Wb Q ) 2 ( W Q ) 2 b 2 +1 ,
e= (QbcH) 2 Q 2 b 2 (1 c 2 )+ (QbcH) 2 ,
{ Q=abs[ ( C 4 ) ( C 3 ) ]+abs[ ( C 6 ) ( C 7 ) ] N=abs[ ( C 8 ) ( C 5 ) ] W=abs[ ( C 6 ) ( C 7 ) ]abs[ ( C 4 ) ( C 3 ) ] H=abs[ ( C 1 ) ( C 7 ) ] .
S out (σ)= M A (ε) M R2 [ 45 o + θ 2 , φ 2 (σ)] M R1 [ θ 1 , φ 1 (σ)]S(σ).
B(σ)=(1/2) S 0 (σ)+(1/2)b d 2 f(b S 1 (σ)+a S 2 (σ))cos[ φ 2 (σ)] (1/8)( Γ 8 + Γ 6 Γ 9 )[a S 1 (σ)b S 2 (σ)i S 3 (σ)]exp{i[ φ 2 (σ) φ 1 (σ)]} (1/8)( Γ 8 + Γ 6 Γ 9 )[a S 1 (σ)b S 2 (σ)+i S 3 (σ)]exp{i[ φ 2 (σ) φ 1 (σ)]} +(1/4)( Γ 6 Γ 4 )[b S 2 (σ)+i S 3 (σ)]exp[i φ 1 (σ)] +(1/4)( Γ 6 Γ 4 )[b S 2 (σ)i S 3 (σ)]exp[i φ 1 (σ)] +(1/8)( Γ 8 Γ 6 + Γ 9 )[a S 1 (σ)b S 2 (σ)+i S 3 (σ)]exp{i[ φ 2 (σ)+ φ 1 (σ)]} +(1/8)( Γ 8 Γ 6 + Γ 9 )[a S 1 (σ)b S 2 (σ)i S 3 (σ)]exp{i[ φ 2 (σ)+ φ 1 (σ)]}.
A(h)= A 0 (h)+ A 1 (h L 1 )+ A 1 (h L 1 ) + A 2 [h( L 2 L 1 )]+ A 2 [h( L 2 L 1 )] + A 3 (h L 2 )+ A 3 (h L 2 ) + A 4 [h( L 2 + L 1 )]+ A 4 [h( L 2 + L 1 )],
A 0 = 1 [(1/2) S 0 (σ)],
A 1 = 1 {(1/4)( Γ 6 Γ 4 )[b S 2 (σ)+i S 3 (σ)]exp[i φ 1 (σ)]},
A 2 = 1 {(1/8)( Γ 8 + Γ 6 Γ 9 )[a S 1 (σ)b S 2 (σ)i S 3 (σ)]exp{i[ φ 2 (σ) φ 1 (σ)]}},
A 3 = 1 {(1/4) Γ 10 d[b S 1 (σ)+a S 2 (σ)]exp[i φ 2 (σ)]},
A 4 = 1 {(1/8)( Γ 8 Γ 6 + Γ 9 )[a S 1 (σ)b S 2 (σ)+i S 3 (σ)]exp{i[ φ 2 (σ)+ φ 1 (σ)]}},
( A 1 )=(1/2) S 0 (σ),
( A 3 )=(1/4)d Γ 10 [b S 1 (σ)+a S 2 (σ)]exp[i φ 2 (σ)],
( A 4 )=(1/8)( Γ 8 Γ 6 + Γ 9 )[a S 1 (σ)b S 2 (σ)+i S 3 (σ)]exp{i[ φ 2 (σ)+ φ 1 (σ)],
E 1 (σ)= 4( A 3 ) d Γ 10 exp[i φ 2 (σ)] ,
E 2 (σ)= 8Re{( A 4 )/exp{i[ φ 2 (σ)+ φ 1 (σ)]}} Γ 8 Γ 6 + Γ 9 ,
E 3 (σ)= 8Im{( A 4 )/exp{i[ φ 2 (σ)+ φ 1 (σ)]}} Γ 8 Γ 6 + Γ 9 ,
exp[i φ 2 (σ)]= 2 2 ( Γ 1 + Γ 3 ) ( A 3, 22.5 o ) ( A 0, 22.5 o ) ,
exp{i[ φ 1 (σ)+ φ 2 (σ)]}= 4 2 (ab)( Γ 8 Γ 6 + Γ 9 ) ( A 4, 22.5 o ) ( A 0, 22.5 o ) .

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