Abstract

The polarization of light when it passes through optical media can change as a result of change in the amplitude (dichroism) or phase shift (birefringence) of the electric vector. The anisotropic properties of media can be determined from these two optical features. We derive the conditions required for polarization elements to be dichroic and birefringent. Our derivation starts from commonly accepted assumptions for dichroism and birefringence. Our main conclusions are that (i) the generalized Jones matrix for dichroic elements has in general nonorthogonal eigenpolarizations and (ii) in the general case, the birefringent and dichroic properties of polarization elements have no direct association with the corresponding phase and dichroic polar forms derived in the polar decomposition of the polarization elements’ Jones matrices.

© 2005 Optical Society of America

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References

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  1. R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).
  2. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
    [CrossRef]
  3. R. C. Jones, “A new calculus for the treatment of optical systems. IV,” J. Opt. Soc. Am. 32, 486–493 (1941).
    [CrossRef]
  4. S.-Y. Lu, R. Chipman, “Interpretation of the Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
    [CrossRef]
  5. S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  6. Sudha, A. V.Gopala Rao, “Polarization elements: a group-theoretical study,” J. Opt. Soc. Am. A 18, 3130–3134 (2001).
    [CrossRef]
  7. S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—I. Theoretical,” Proc. Indian Acad. Sci. Sect. A 42, 89–109 (1955).
  8. S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—II. Experimental,” Proc. Indian Acad. Sci. Sect. A Sect. A 42, 235–248 (1955).
  9. H. de Lang, “Polarization properties of optical resonators passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).
  10. L. C. Meira-Belo, U. A. Leitao, “Singular polarization eigenstates in anisotropic stratified structures,” Appl. Opt. 39, 2695–2704 (2000).
    [CrossRef]
  11. J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
    [CrossRef]
  12. P. Lankaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).
  13. C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. 61, 1207–1213 (1971).
    [CrossRef]
  14. J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of non-depolarizing optical system from polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).
  15. H. Hurwitz, R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of the three general equivalence theorems,” J. Opt. Soc. Am. 31, 493–499 (1941).
    [CrossRef]
  16. R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. II (McGraw Hill, New York, 1995).
  17. O. G. Vlokh, O. S. Kushnir, “Specific features of propagation of polarized light in purely dichroic crystals,” Opt. Spectrosc. 80, 71–73 (1996).

2002 (1)

J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
[CrossRef]

2001 (1)

2000 (1)

1996 (2)

S.-Y. Lu, R. Chipman, “Interpretation of the Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
[CrossRef]

O. G. Vlokh, O. S. Kushnir, “Specific features of propagation of polarized light in purely dichroic crystals,” Opt. Spectrosc. 80, 71–73 (1996).

1994 (1)

1987 (1)

J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of non-depolarizing optical system from polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).

1971 (1)

1955 (2)

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—I. Theoretical,” Proc. Indian Acad. Sci. Sect. A 42, 89–109 (1955).

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—II. Experimental,” Proc. Indian Acad. Sci. Sect. A Sect. A 42, 235–248 (1955).

1941 (2)

Azzam, R. M.A.

R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

Barbosa-Garcia, O.

J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
[CrossRef]

Bashara, N. M.

R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

Bernabeu, E.

J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of non-depolarizing optical system from polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).

Chipman, R.

Chipman, R. A.

S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. II (McGraw Hill, New York, 1995).

De la Rosa-Cruz, E.

J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
[CrossRef]

de Lang, H.

H. de Lang, “Polarization properties of optical resonators passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).

Diaz-Torres, L. A.

J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
[CrossRef]

Gil, J. J.

J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of non-depolarizing optical system from polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).

Hurwitz, H.

Jones, R. C.

Kushnir, O. S.

O. G. Vlokh, O. S. Kushnir, “Specific features of propagation of polarized light in purely dichroic crystals,” Opt. Spectrosc. 80, 71–73 (1996).

Lankaster, P.

P. Lankaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

Leitao, U. A.

Lu, S.-Y.

Meira-Belo, L. C.

Meneses-Nava, M. A.

J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
[CrossRef]

Mosino, J. F.

J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—I. Theoretical,” Proc. Indian Acad. Sci. Sect. A 42, 89–109 (1955).

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—II. Experimental,” Proc. Indian Acad. Sci. Sect. A Sect. A 42, 235–248 (1955).

Rao, A. V.Gopala

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
[CrossRef]

Sudha,

Tismenetsky, M.

P. Lankaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

Vega-Duran, J. T.

J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
[CrossRef]

Vlokh, O. G.

O. G. Vlokh, O. S. Kushnir, “Specific features of propagation of polarized light in purely dichroic crystals,” Opt. Spectrosc. 80, 71–73 (1996).

Whitney, C.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Spectrosc. (1)

O. G. Vlokh, O. S. Kushnir, “Specific features of propagation of polarized light in purely dichroic crystals,” Opt. Spectrosc. 80, 71–73 (1996).

Optik (Stuttgart) (1)

J. J. Gil, E. Bernabeu, “Obtainment of the polarizing and retardation parameters of non-depolarizing optical system from polar decomposition of its Mueller matrix,” Optik (Stuttgart) 76, 67–71 (1987).

Proc. Indian Acad. Sci. Sect. A (2)

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—I. Theoretical,” Proc. Indian Acad. Sci. Sect. A 42, 89–109 (1955).

S. Pancharatnam, “The propagation of light in absorbing biaxial crystals—II. Experimental,” Proc. Indian Acad. Sci. Sect. A Sect. A 42, 235–248 (1955).

Pure Appl. Opt. (1)

J. F. Mosino, O. Barbosa-Garcia, M. A. Meneses-Nava, L. A. Diaz-Torres, E. De la Rosa-Cruz, J. T. Vega-Duran, “Anisotropic media with orthogonal eigenpolarizations,” Pure Appl. Opt. 4, 419–423 (2002).
[CrossRef]

Other (5)

P. Lankaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, San Diego, Calif., 1985).

R. M.A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1979).

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962).
[CrossRef]

H. de Lang, “Polarization properties of optical resonators passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, Vol. II (McGraw Hill, New York, 1995).

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Equations (73)

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E e 1 + E e 2 = 0 ,
E = ( E x exp ( i δ x ) E y exp ( i δ y ) ) ,
TE i = E o ,
T = [ T 11 T 12 T 21 T 22 ]
χ = E y E x exp [ j ( δ y δ x ) ] .
χ o = T 21 + T 22 χ i T 11 + T 12 χ i .
TE e 1 = V 1 E e 1 , TE e 2 = V 2 E e 2 ,
E e 1 , 2 = ( 1 χ e 1 , 2 ) .
χ e 1 , 2 = 1 2 T 22 T 11 ± ( T 22 T 11 ) 2 + 4 T 12 T 21 T 12 ,
V 1 , 2 = 1 2 ( T 22 + T 11 ± ( T 22 T 11 ) 2 + 4 T 12 T 21 ) .
T 11 = 1 χ e 1 χ e 2 ( V 2 χ e 1 V 1 χ e 2 ) ,
T 12 = 1 χ e 1 χ e 2 ( V 1 V 2 ) ,
T 21 = χ e 1 χ e 2 χ e 1 χ e 2 ( V 1 V 2 ) ,
T 22 = 1 χ e 1 χ e 2 ( V 1 χ e 1 V 2 χ e 2 ) .
T eigen = [ V 1 0 0 V 2 ] .
T = FT eigen F 1 ,
F = [ 1 1 χ e 1 χ e 2 ] ,
T = 1 χ e 1 χ e 2 [ 1 1 χ e 1 χ e 2 ] [ V 1 0 0 V 2 ] [ χ e 2 1 χ e 1 1 ] .
I o I i = V 2 ,
I = 1 2 [ ( E u + E v ) ( E u + E v ) * ] .
I = E x E x * + E y E y * ,
V 1 = V 2 = V .
E o x E o x * + E o y E o y * = V 2 ( E i x E i x * + E i y E i y * ) .
E o x = T 11 E i x + T 12 E i y ,
E o y = T 21 E i x + T 22 E i y .
T 11 T 12 * + T 21 T 22 * = 0 ,
T 11 2 + T 21 2 = V 2 ,
T 12 2 + T 22 2 = V 2 .
( V 1 V 2 * V 2 ) χ e 2 ( 1 + χ e 1 χ e 2 * ) + ( V 1 V 2 * V 2 ) χ e 1 ( 1 + χ e 1 χ e 2 * ) = 0 ,
( V 1 V 2 * V 2 ) χ e 1 * χ e 2 ( 1 + χ e 1 χ e 2 * ) + ( V 1 V 2 * V 2 ) χ e 1 χ e 2 * ( 1 + χ e 1 χ e 2 * ) = 0 ,
( V 1 V 2 * V 2 ) ( 1 + χ e 1 χ e 2 * ) + ( V 1 V 2 * V 2 ) ( 1 + χ e 1 χ e 2 * ) = 0 .
χ e 1 χ e 2 * = 1 .
V 1 V 2 = 1 ,
χ e 1 χ e 2 * = 1 .
V 1 = V exp ( i ϕ 1 ) ,
V 2 = V exp ( i ϕ 2 ) ,
χ e 1 = χ e exp ( i ψ ) ,
χ e 2 = 1 χ e exp ( i ψ ) .
T = V χ e 2 + 1 [ exp ( i ϕ 1 ) + χ e 2 exp ( i ϕ 2 ) χ e [ exp ( i ϕ 1 ) exp ( i ϕ 2 ) ] exp ( i ψ ) χ e [ exp ( i ϕ 1 ) exp ( i ϕ 2 ) ] exp ( i ψ ) χ e 2 exp ( i ϕ 1 ) + exp ( i ϕ 2 ) ] .
det ( T ) = V 2 exp [ i ( ϕ 1 + ϕ 2 ) ] ,
det ( T ) = V 2 .
χ o χ o * = χ i χ i * .
T 12 = 0 ,
T 21 = 0 ,
arg ( T 11 ) = arg ( T 22 ) .
T = [ A e i ϕ 0 0 B e i ϕ ] .
Im ( V 1 V 2 ) = 0 ,
Re ( V 1 V 2 ) 0 .
T = e i ϕ χ e 1 χ e 2 [ B χ e 1 A χ e 2 A B χ e 1 χ e 2 ( A B ) A χ e 1 B χ e 2 ] .
( T 22 T 11 ) 2 + 4 T 12 T 21 = 0 .
T 21 = T 11 T 12 * T 22 * ,
T 11 2 = T 22 2 ,
T 12 2 = T 21 2 .
T 12 2 = ( T 11 T 22 ) 2 T 22 * 4 T 11 .
T 11 = T 11 exp ( i ϕ 11 ) ,
T 12 = T 12 exp ( i ϕ 12 ) ,
T 21 = T 21 exp ( i ϕ 21 ) ,
T 22 = T 22 exp ( i ϕ 22 ) ,
T 12 2 = T 11 2 4 [ exp ( i ϕ 11 ) exp ( i ϕ 22 ) ] 2 exp ( i ϕ 22 ) exp ( i ϕ 11 ) = T 11 2 4 ( 2 + exp [ i ( ϕ 11 ϕ 22 ) ] + exp [ i ( ϕ 11 ϕ 22 ) ] ) = T 11 2 2 ( cos ( ϕ 11 ϕ 22 ) 1 ) 0 .
T 12 = T 21 = 0 ,
T 11 = T 22 ,
[ 1 i R i R 1 ] ,
[ cos 2 γ + P sin 2 γ cos γ sin γ [ 1 P ] cos γ sin γ [ 1 P ] sin 2 γ + P cos 2 γ ] ,
[ exp ( i Δ ) 0 0 exp ( i Δ ) ] ,
T = [ exp ( i Δ ) 0 0 exp ( i Δ ) ] [ cos 2 γ + P sin 2 γ cos γ sin γ ( 1 P ) cos γ sin γ ( 1 P ) sin 2 γ + P cos 2 γ ] [ 1 i R i R 1 ] .
T = [ 0.712 0.513 i 0.003 0.34 i 0.286 + 0.005 i 0.437 0.452 i ] .
V 1 = 1.078 exp ( 40 0 i ) ,
V 2 = 0.422 exp ( 40 0 i ) .
χ e 1 = 0.527 + 0.338 i ,
χ e 2 = 0.7 1.148 i .
T = e i φ χ e 1 2 + 1 [ B χ e 1 2 + A ( A B ) χ e 1 * ( A B ) χ e 1 A χ e 1 2 + B ] .
T R = [ 0.817323 0.57406 i 0.0313745 0.038071 i 0.0321013 0.0374603 i 0.7062 0.706291 i ] .
T D = [ 0.86706 0.195634 0.248732 i 0.195634 + 0.248732 i 0.640891 ] .

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