Abstract

Calculating the evolution of polarization for all polarization states of light in optical systems, in global coordinates, is an important, yet challenging task. This calculation exists for completely polarized light, but has not yet been developed for partially polarized light. A 3 × 3 coherency matrix for partially polarized light, in global coordinates, is presented to calculate the transformation of its polarization as it passes through an optical system. This matrix is a three-dimensional generalization of the coherency matrix. A new coherency matrix calculus method in three dimensions is suggested and validated for two cases. A double Gauss optical lens is introduced to compare this method’s performance with two-dimensional calculus.

© 2017 Optical Society of America

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References

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  1. R. A. Chipman, “Polarization analysis of optical systems,” Proc. SPIE 891, 10–31 (1988).
  2. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28(2), 090–099 (1989).
  3. J. Atwood, W. Skidmore, and G. C. Anupama, “Polarimetric analysis of the Thirty Meter Telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).
  4. Y. Yang, C. Yan, C. Hu, and C. Wu, “Modified heterodyne efficiency for coherent laser communication in the presence of polarization aberrations,” Opt. Express 25(7), 7567–7591 (2017).
    [PubMed]
  5. 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(22), 5453–5469 (2006).
    [PubMed]
  6. B. Yang, X. Ju, C. Yan, and J. Zhang, “Alignment errors calibration for a channeled spectropolarimeter,” Opt. Express 24(25), 28923–28935 (2016).
    [PubMed]
  7. G. Yun, K. Crabtree, and R. A. Chipman, “Skew aberration: a form of polarization aberration,” Opt. Lett. 36(20), 4062–4064 (2011).
    [PubMed]
  8. X. Xu, W. Huang, and M. Xu, “Orthogonal polynomials describing polarization aberration for rotationally symmetric optical systems,” Opt. Express 23(21), 27911–27919 (2015).
    [PubMed]
  9. Y. Yang and C. Yan, “Polarization property analysis of a periscopic scanner with three-dimensional polarization ray-tracing calculus,” Appl. Opt. 55(6), 1343–1350 (2016).
    [PubMed]
  10. H. Di, D. Hua, L. Yan, X. L. Hou, and X. Wei, “Polarization analysis and corrections of different telescopes in polarization lidar,” Appl. Opt. 54(3), 389–397 (2015).
  11. O. Morel, R. Seulin, and D. Fofi, “Handy method to calibrate division-of-amplitude polarimeters for the first three Stokes parameters,” Opt. Express 24(12), 13634–13646 (2016).
    [PubMed]
  12. H. Y. Zhang, J. Q. Zhang, B. Yang, and C. X. Yan, “Calibration for polarization remote detection system of multi-linear focal plane divided,” Acta Opt. Sin. 36(11), 1128003 (2016).
  13. H. Y. Zhang, Y. Li, C. X. Yan, and J. Q. Zhang, “Calibration of polarized effect for time-divided polarization spectral measurement system,” Opt. Precis. Eng. 25(2), 325–333 (2017).
  14. W. S. Tiffany Lam and R. Chipman, “Balancing polarization aberrations in crossed fold mirrors,” Appl. Opt. 54(11), 3236–3245 (2015).
    [PubMed]
  15. J. Wolfe and R. Chipman, “Reducing symmetric polarization aberrations in a lens by annealing,” Opt. Express 12(15), 3443–3451 (2004).
    [PubMed]
  16. J. P. McGuire and R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33(22), 5080–5100 (1994).
    [PubMed]
  17. R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34(6), 1636–1645 (1995).
  18. R. C. Jones, “A new calculus for the treatment of optical systems I,” J. Opt. Soc. Am. 31, 488–493 (1941).
  19. R. Kalibjian, “Stokes polarization vector and Mueller matrix for a corner-cube reflector,” Opt. Commun. 240, 39–68 (2004).
  20. G. Yun, K. Crabtree, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus I: definition and diattenuation,” Appl. Opt. 50(18), 2855–2865 (2011).
    [PubMed]
  21. G. Yun, S. C. McClain, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus II: retardance,” Appl. Opt. 50(18), 2866–2874 (2011).
    [PubMed]
  22. J. J. Gil and I. S. José, “3D polarimetric purity,” Opt. Commun. 283(22), 4430–4434 (2010).
  23. J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).
  24. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  25. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  26. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  27. R. A. Chipman, “Mueller Matrices,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 2009).
  28. W. S. T. Lam, S. McClain, G. A. Smith, and R. A. Chipman, “Ray tracing in biaxial materials,” Proc. SPIE 7652, 76521R (2010).
  29. W. S. T. Lam, “Anisotropic Ray Trace,” PhD Dissertation, University of Arizona (2015).

2017 (2)

H. Y. Zhang, Y. Li, C. X. Yan, and J. Q. Zhang, “Calibration of polarized effect for time-divided polarization spectral measurement system,” Opt. Precis. Eng. 25(2), 325–333 (2017).

Y. Yang, C. Yan, C. Hu, and C. Wu, “Modified heterodyne efficiency for coherent laser communication in the presence of polarization aberrations,” Opt. Express 25(7), 7567–7591 (2017).
[PubMed]

2016 (4)

2015 (3)

2014 (1)

J. Atwood, W. Skidmore, and G. C. Anupama, “Polarimetric analysis of the Thirty Meter Telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).

2011 (3)

2010 (2)

W. S. T. Lam, S. McClain, G. A. Smith, and R. A. Chipman, “Ray tracing in biaxial materials,” Proc. SPIE 7652, 76521R (2010).

J. J. Gil and I. S. José, “3D polarimetric purity,” Opt. Commun. 283(22), 4430–4434 (2010).

2006 (1)

2004 (3)

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

R. Kalibjian, “Stokes polarization vector and Mueller matrix for a corner-cube reflector,” Opt. Commun. 240, 39–68 (2004).

J. Wolfe and R. Chipman, “Reducing symmetric polarization aberrations in a lens by annealing,” Opt. Express 12(15), 3443–3451 (2004).
[PubMed]

1995 (1)

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34(6), 1636–1645 (1995).

1994 (1)

1989 (1)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28(2), 090–099 (1989).

1988 (1)

R. A. Chipman, “Polarization analysis of optical systems,” Proc. SPIE 891, 10–31 (1988).

1941 (1)

Anupama, G. C.

J. Atwood, W. Skidmore, and G. C. Anupama, “Polarimetric analysis of the Thirty Meter Telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).

Atwood, J.

J. Atwood, W. Skidmore, and G. C. Anupama, “Polarimetric analysis of the Thirty Meter Telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).

Chenault, D. B.

Chipman, R.

Chipman, R. A.

Correas, J. M.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Crabtree, K.

Di, H.

Ferreira, C.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Fofi, D.

Gil, J. J.

J. J. Gil and I. S. José, “3D polarimetric purity,” Opt. Commun. 283(22), 4430–4434 (2010).

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Goldstein, D. L.

Hou, X. L.

Hu, C.

Hua, D.

Huang, W.

Jones, R. C.

José, I. S.

J. J. Gil and I. S. José, “3D polarimetric purity,” Opt. Commun. 283(22), 4430–4434 (2010).

Ju, X.

Kalibjian, R.

R. Kalibjian, “Stokes polarization vector and Mueller matrix for a corner-cube reflector,” Opt. Commun. 240, 39–68 (2004).

Lam, W. S. T.

W. S. T. Lam, S. McClain, G. A. Smith, and R. A. Chipman, “Ray tracing in biaxial materials,” Proc. SPIE 7652, 76521R (2010).

Li, Y.

H. Y. Zhang, Y. Li, C. X. Yan, and J. Q. Zhang, “Calibration of polarized effect for time-divided polarization spectral measurement system,” Opt. Precis. Eng. 25(2), 325–333 (2017).

McClain, S.

W. S. T. Lam, S. McClain, G. A. Smith, and R. A. Chipman, “Ray tracing in biaxial materials,” Proc. SPIE 7652, 76521R (2010).

McClain, S. C.

McGuire, J. P.

Melero, P. A.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Morel, O.

Seulin, R.

Shaw, J. A.

Skidmore, W.

J. Atwood, W. Skidmore, and G. C. Anupama, “Polarimetric analysis of the Thirty Meter Telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).

Smith, G. A.

W. S. T. Lam, S. McClain, G. A. Smith, and R. A. Chipman, “Ray tracing in biaxial materials,” Proc. SPIE 7652, 76521R (2010).

Tiffany Lam, W. S.

Tyo, J. S.

Wei, X.

Wolfe, J.

Wu, C.

Xu, M.

Xu, X.

Yan, C.

Yan, C. X.

H. Y. Zhang, Y. Li, C. X. Yan, and J. Q. Zhang, “Calibration of polarized effect for time-divided polarization spectral measurement system,” Opt. Precis. Eng. 25(2), 325–333 (2017).

H. Y. Zhang, J. Q. Zhang, B. Yang, and C. X. Yan, “Calibration for polarization remote detection system of multi-linear focal plane divided,” Acta Opt. Sin. 36(11), 1128003 (2016).

Yan, L.

Yang, B.

B. Yang, X. Ju, C. Yan, and J. Zhang, “Alignment errors calibration for a channeled spectropolarimeter,” Opt. Express 24(25), 28923–28935 (2016).
[PubMed]

H. Y. Zhang, J. Q. Zhang, B. Yang, and C. X. Yan, “Calibration for polarization remote detection system of multi-linear focal plane divided,” Acta Opt. Sin. 36(11), 1128003 (2016).

Yang, Y.

Yun, G.

Zhang, H. Y.

H. Y. Zhang, Y. Li, C. X. Yan, and J. Q. Zhang, “Calibration of polarized effect for time-divided polarization spectral measurement system,” Opt. Precis. Eng. 25(2), 325–333 (2017).

H. Y. Zhang, J. Q. Zhang, B. Yang, and C. X. Yan, “Calibration for polarization remote detection system of multi-linear focal plane divided,” Acta Opt. Sin. 36(11), 1128003 (2016).

Zhang, J.

Zhang, J. Q.

H. Y. Zhang, Y. Li, C. X. Yan, and J. Q. Zhang, “Calibration of polarized effect for time-divided polarization spectral measurement system,” Opt. Precis. Eng. 25(2), 325–333 (2017).

H. Y. Zhang, J. Q. Zhang, B. Yang, and C. X. Yan, “Calibration for polarization remote detection system of multi-linear focal plane divided,” Acta Opt. Sin. 36(11), 1128003 (2016).

Acta Opt. Sin. (1)

H. Y. Zhang, J. Q. Zhang, B. Yang, and C. X. Yan, “Calibration for polarization remote detection system of multi-linear focal plane divided,” Acta Opt. Sin. 36(11), 1128003 (2016).

Appl. Opt. (7)

J. Opt. Soc. Am. (1)

Monografías del Seminario Matemático García de Galdeano (1)

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Opt. Commun. (2)

J. J. Gil and I. S. José, “3D polarimetric purity,” Opt. Commun. 283(22), 4430–4434 (2010).

R. Kalibjian, “Stokes polarization vector and Mueller matrix for a corner-cube reflector,” Opt. Commun. 240, 39–68 (2004).

Opt. Eng. (2)

R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34(6), 1636–1645 (1995).

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28(2), 090–099 (1989).

Opt. Express (5)

Opt. Lett. (1)

Opt. Precis. Eng. (1)

H. Y. Zhang, Y. Li, C. X. Yan, and J. Q. Zhang, “Calibration of polarized effect for time-divided polarization spectral measurement system,” Opt. Precis. Eng. 25(2), 325–333 (2017).

Proc. SPIE (3)

J. Atwood, W. Skidmore, and G. C. Anupama, “Polarimetric analysis of the Thirty Meter Telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).

R. A. Chipman, “Polarization analysis of optical systems,” Proc. SPIE 891, 10–31 (1988).

W. S. T. Lam, S. McClain, G. A. Smith, and R. A. Chipman, “Ray tracing in biaxial materials,” Proc. SPIE 7652, 76521R (2010).

Other (5)

W. S. T. Lam, “Anisotropic Ray Trace,” PhD Dissertation, University of Arizona (2015).

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

R. A. Chipman, “Mueller Matrices,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 2009).

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Figures (7)

Fig. 1
Fig. 1 The eigen-vibration coordinates of the propagating light.
Fig. 2
Fig. 2 Global coordinates O-XYZ and the eigen-vibration coordinates O-X’Y’Z’.
Fig. 3
Fig. 3 Polarization paraxial approximation for an optical system.
Fig. 4
Fig. 4 The double Gauss optical lens. The entrance pupil diameter is 33 mm, the focal length is 100 mm, and the field of view is 28 deg.
Fig. 5
Fig. 5 The DoP pupil maps for the three-dimensional (a) and two-dimensional (b) calculi, and their deviations (c) for the field coordinates (0, 0). The color code corresponds to the values.
Fig. 6
Fig. 6 The DoP pupil maps for the three-dimensional (a) and two-dimensional (b) calculi, and their deviations (c) for the field coordinates (0, 1). The color code corresponds to the values.
Fig. 7
Fig. 7 Average DoP (a) and Orient (b) calculated from the two calculi, vs. the Y field of view. The blue curves with triangles are calculated from the two-dimensional calculation. The red curves with circles are calculated from the three-dimensional calculation.

Tables (2)

Tables Icon

Table 1 The previous polarization calculus.

Tables Icon

Table 2 A numerical example of the two three-dimensional calculi (θ = 45°).

Equations (34)

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E=A e i( kzwt )
J P = E× E = [ E x E y ]×[ E x , E y ] =[ J xx J xy J yx J yy ].
Tr( J P )= E x E x + E y E y = J xx + J yy .
P= 1 4| J P | ( J xx + J yy ) 2 ,0P1.
θ= 1 2 arctan( J xy + J yx J xx J yy ),ε= 1 2 arcsin( j( J xy J yx ) P( J xx + J yy ) ).
J P = I 2 [ 1+Pcos2θcos2εP(sin2θcos2εjsin2ε) P(sin2θcos2ε+jsin2ε)1Pcos2θcos2ε ].
J P3L = E L × E L = [ E x' E y' 0 ]×[ E x' , E y' ,0 ] =[ J P 0 00 ].
P= 1 4( J xx J yy J xy J yx ) ( J xx + J yy ) 2 .
ω y =(90arccos( K x K x 2 + K y 2 + K z 2 )), ω x =arctan( K y K z ).
R= R x × R y ,
R y =[ cos ω y 0sin ω y 010 sin ω y 0cos ω y ], R x =[ 100 0cos ω x sin ω x sin ω x 0cos ω x ].
E G =R× E L .
J P3 = E G × E G = R× E L × (R× E L * ) T =R× J P3L × R T .
J CP3 = E 3 × E 3 = E 3 × ( E 3 * ) T = R× E 3L × (R× E 3L * ) T = R× E 3L × ( E 3L * ) T × R T =R× J CP3L × R T .
J UP = I up 2 [ 10 00 ]+ I up 2 [ 00 01 ].
J UP3L = I up 2 [ 100 000 000 ]+ I up 2 [ 000 010 000 ].
J UP3 =R× J UP3L × R T .
J P3L = I 2 [ 1+Pcos2θcos2εP(sin2θcos2εjsin2ε)0 P(sin2θcos2ε+jsin2ε)1Pcos2θcos2ε0 000 ], J CP3L = PI 2 [ 1+cos2θcos2εsin2θcos2εjsin2ε0 sin2θcos2ε+jsin2ε1cos2θcos2ε0 000 ], J UP3L = (1P)I 2 [ 100 010 000 ], J P3L = J CP3L + J UP3L .
J P3 = J CP3 + J UP3 ,
J CP3 =R× J CP3L × R T , J UP3 =R× J UP3L × R T .
J P3 =R× J CP3L × R T +R× J UP3L × R T =R×( J CP3L + J UP3L )× R T =R× J P3L × R T .
J P3(out,q) = P 3(q) × J P3(in,q) × P 3(q) .
P 3(q) =[ p (out,q) , s (out,q) , k (out,q) ]× T 3(q) × [ p (in,q) , s (in,q) , k (in,q) ] T ,
s (in,q) = k (in,q) × k (out,q) | k (in,q) × k (out,q) | , p (in,q) = k (in,q) × s (in,q) , s (out,q) = s (in,q) , p (out,q) = k (out,q) × s (out,q) .
T 3(q) =[ T (q) 0 01 ], T (q) =[ t p(q) 0 0 t s(q) ]or[ r p(q) 0 0 r s(q) ].
P 3(total) = q=1 Q P 3(q) .
J P3(out) = P 3(total) × J P3(in) × P 3(total) ,
k (in) = [ 0,0,1 ] T , k (out) = [ 0,0,1 ] T .
J P3(in) =[ J P(in) 0 00 ], P 3 =[ T0 01 ].
J P3(out) = P 3 × J P3(in) × P 3 =[ T× J P(in) × T 0 00 ].
J P(out) =T× J P(in) × T .
E 3L(in) = [ cosθ,sinθ,0 ] T , J CP3L(in) = 1 2 [ 1+cos2θsin2θ0 sin2θ1cos2θ0 000 ].
E 3L(out) = R (out) -1 × P 3 × R (in) × E 3L(in) , J CP3L(out) = R (out) -1 ×[ P 3 ×( R (in) × J CP3L(in) × R (in) T )× P 3 ]× ( R (out) -1 ) T .
θ 1( E 3 ) =arctan[ E 3L(out) (2,1) E 3L(out) (1,1) ], θ 1( J 3 ) = 1 2 arctan[ J CP3L(out) (1,2)+ J CP3L(out) (2,1) J CP3L(out) (1,1) J CP3L(out) (2,2) ],

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