Abstract

A theoretical analysis of eigenpolarizations and eigenvalues pertaining to the Jones matrices of dichroic, birefringent, and degenerate polarization elements is presented. The analysis is carried out employing a general model of a polarization element. Expressions for the corresponding polarization elements are derived and analyzed. It is shown that, despite the presence of birefringence, a polarization element can, in a general case, demonstrate a totally dichroic behavior. Moreover, it is proved that birefringence necessarily accompanies dichroic elements with orthogonal eigenpolarizations. A transition between degenerate, dichroic, and birefringent eigenvalues is studied, and examples of synthesis of polarization elements are given.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).
  2. R. A. Chipman, "Polarimetry," in Handbook of Optics, M.Bass, ed. (McGraw-Hill, 1995), Vol. II, Chap. 22.
  3. P. Huard, Polarization of Light (Wiley, 1997).
  4. C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).
  5. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge U. Press, 1999).
    [PubMed]
  6. S.-Y. Lu and R. A. Chipman, "Homogeneous and inhomogeneous Jones matrices," J. Opt. Soc. Am. A 11, 766-773 (1994).
    [CrossRef]
  7. S. N. Savenkov, O. I. Sydoruk, and R. S. Muttiah, "Conditions for polarization elements to be dichroic and birefringent," J. Opt. Soc. Am. A 22, 1447-1452 (2005).
    [CrossRef]
  8. H. de Lang, "Polarization properties of optical resonators passive and active," Ph.D. dissertation (University of Utrecht, 1966).
  9. H. Hurwitz and R. C. Jones, "A new calculus for the treatment of optical systems. 2. Proof of three general equivalence theorems," J. Opt. Soc. Am. 31, 493-499 (1941).
  10. C. Whitney, "Pauli-algebraic operators in polarization optics," J. Opt. Soc. Am. 61, 1207-1213 (1971).
    [CrossRef]
  11. J. J. Gil and E. Bernabeu, "Obtainment of the polarizing and retardation parameters of nondepolarizing optical system from polar decomposition of its Mueller matrix," Optik (Stuttgart) 76, 67-71 (1987).
  12. S.-Y. Lu and R. Chipman, "Interpretation of the Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996).
    [CrossRef]
  13. H. Hammer, "Characteristic parameters in integrated photoelasticity: an application of Poincare's equivalence theorem," J. Mod. Opt. 51, 597-618 (2004).
  14. Sudha and A. V. Gopala Rao, "Polarization elements: a group-theoretical study," J. Opt. Soc. Am. A 18, 3130-3134 (2001).
    [CrossRef]
  15. T. Tudor, "Generalized observables in polarization optics," J. Phys. A 36, 9577-9590 (2003).
    [CrossRef]
  16. M. V. Berry and M. R. Dennis, "Black polarization sandwiches are square roots of zero," J. Opt. A 6,S24-S25 (2004).
  17. T. Tudor, "Non-Hermitian polarizers: a biorthogonal analysis," J. Opt. Soc. Am. A 23, 1513-1522 (2006).
    [CrossRef]
  18. J. R. L. Moxon and A. R. Renshow, "The simultaneous measurement of optical activity and circular dichroism in birefringent linearly dichroic crystal section: 1. Introduction and description of the method," J. Phys. Condens. Matter 2, 6807-6836 (1990).
    [CrossRef]
  19. S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, "Generalized matrix equivalence theorem for polarization theory," Phys. Rev. E 74, 056607 (2006).
    [CrossRef]
  20. T. Tudor and A. Gheondea, "Pauli algebraic forms of normal and nonnormal operators," J. Opt. Soc. Am. A 24, 204-210 (2007).
    [CrossRef]
  21. P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985).
  22. R. Barakat, "Jones matrix equivalence theorems for polarization theory," Eur. J. Phys. 19, 209-216 (1998).
    [CrossRef]
  23. K. Lu and B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-246 (1990).
    [CrossRef]
  24. J. A. Davis, I. Moreno, and P. Tsai, "Polarization eigenstates for twisted-nematic liquid-crystal displays," Appl. Opt. 37, 937-945 (1998).
    [CrossRef]
  25. X. Zhu, Q. Hong, Y. Huang, and S.-T. Wu, "Eigenmodes of a reflective twisted-nematic liquid-crystal cell," J. Appl. Phys. 94, 2868-2873 (2003).
    [CrossRef]
  26. M. Yamauchi, "Jones-matrix models for twisted-nematic liquid-crystal devices," Appl. Opt. 44, 4484-4493 (2005).
    [CrossRef] [PubMed]
  27. D. Tentori, C. Ayala-Díaz, F. Treveñino-Martínez, F. J. Mendieta-Jiménez, and H. Soto-Ortriz, "Birefringence evaluation of helically wound optical fibers," J. Mod. Opt. 48, 1767-1780 (2001).
  28. H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, "Jones matrix for second-order polarization mode dispersion," Opt. Lett. 25, 19-21 (2000).
    [CrossRef]
  29. F. Heismann, "Extended Jones matrix for first-order polarization mode dispersion," Opt. Lett. 30, 1111-1113 (2005).
    [CrossRef] [PubMed]
  30. D. J. Donohue, B. J. Stoyanov, R. L. McCally, and R. A. Farrell, "Numerical modeling of the cornea's lamellar structure and birefringence properties," J. Opt. Soc. Am. A 12, 1425-1438 (1995).
    [CrossRef]
  31. V. F. Izotova, I. L. Maksimova, I. S. Nefedov, and S. V. Romanov, "Investigation of Mueller matrices of anisotropic nonhomogeneous layers in application to an optical model of the cornea," Appl. Opt. 36, 164-169 (1997).
    [CrossRef] [PubMed]
  32. R. A. Farrell, D. Rouseff, and R. L. McCally, "Propagation of polarized light through two- and three-layer anisotropic stacks," J. Opt. Soc. Am. A 22, 1981-1992 (2005).
    [CrossRef]

2007 (1)

2006 (2)

T. Tudor, "Non-Hermitian polarizers: a biorthogonal analysis," J. Opt. Soc. Am. A 23, 1513-1522 (2006).
[CrossRef]

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, "Generalized matrix equivalence theorem for polarization theory," Phys. Rev. E 74, 056607 (2006).
[CrossRef]

2005 (4)

2004 (2)

H. Hammer, "Characteristic parameters in integrated photoelasticity: an application of Poincare's equivalence theorem," J. Mod. Opt. 51, 597-618 (2004).

M. V. Berry and M. R. Dennis, "Black polarization sandwiches are square roots of zero," J. Opt. A 6,S24-S25 (2004).

2003 (2)

T. Tudor, "Generalized observables in polarization optics," J. Phys. A 36, 9577-9590 (2003).
[CrossRef]

X. Zhu, Q. Hong, Y. Huang, and S.-T. Wu, "Eigenmodes of a reflective twisted-nematic liquid-crystal cell," J. Appl. Phys. 94, 2868-2873 (2003).
[CrossRef]

2001 (2)

D. Tentori, C. Ayala-Díaz, F. Treveñino-Martínez, F. J. Mendieta-Jiménez, and H. Soto-Ortriz, "Birefringence evaluation of helically wound optical fibers," J. Mod. Opt. 48, 1767-1780 (2001).

Sudha and A. V. Gopala Rao, "Polarization elements: a group-theoretical study," J. Opt. Soc. Am. A 18, 3130-3134 (2001).
[CrossRef]

2000 (1)

1998 (2)

R. Barakat, "Jones matrix equivalence theorems for polarization theory," Eur. J. Phys. 19, 209-216 (1998).
[CrossRef]

J. A. Davis, I. Moreno, and P. Tsai, "Polarization eigenstates for twisted-nematic liquid-crystal displays," Appl. Opt. 37, 937-945 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

1994 (1)

1990 (2)

J. R. L. Moxon and A. R. Renshow, "The simultaneous measurement of optical activity and circular dichroism in birefringent linearly dichroic crystal section: 1. Introduction and description of the method," J. Phys. Condens. Matter 2, 6807-6836 (1990).
[CrossRef]

K. Lu and B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-246 (1990).
[CrossRef]

1987 (1)

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

1971 (1)

1941 (1)

Appl. Opt. (3)

Eur. J. Phys. (1)

R. Barakat, "Jones matrix equivalence theorems for polarization theory," Eur. J. Phys. 19, 209-216 (1998).
[CrossRef]

J. Appl. Phys. (1)

X. Zhu, Q. Hong, Y. Huang, and S.-T. Wu, "Eigenmodes of a reflective twisted-nematic liquid-crystal cell," J. Appl. Phys. 94, 2868-2873 (2003).
[CrossRef]

J. Mod. Opt. (2)

D. Tentori, C. Ayala-Díaz, F. Treveñino-Martínez, F. J. Mendieta-Jiménez, and H. Soto-Ortriz, "Birefringence evaluation of helically wound optical fibers," J. Mod. Opt. 48, 1767-1780 (2001).

H. Hammer, "Characteristic parameters in integrated photoelasticity: an application of Poincare's equivalence theorem," J. Mod. Opt. 51, 597-618 (2004).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

J. Phys. A (1)

T. Tudor, "Generalized observables in polarization optics," J. Phys. A 36, 9577-9590 (2003).
[CrossRef]

J. Phys. Condens. Matter (1)

J. R. L. Moxon and A. R. Renshow, "The simultaneous measurement of optical activity and circular dichroism in birefringent linearly dichroic crystal section: 1. Introduction and description of the method," J. Phys. Condens. Matter 2, 6807-6836 (1990).
[CrossRef]

Opt. Eng. (1)

K. Lu and B. E. A. Saleh, "Theory and design of the liquid crystal TV as an optical spatial phase modulator," Opt. Eng. 29, 240-246 (1990).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

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

Phys. Rev. E (1)

S. N. Savenkov, V. V. Marienko, E. A. Oberemok, and O. Sydoruk, "Generalized matrix equivalence theorem for polarization theory," Phys. Rev. E 74, 056607 (2006).
[CrossRef]

Other (8)

P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985).

M. V. Berry and M. R. Dennis, "Black polarization sandwiches are square roots of zero," J. Opt. A 6,S24-S25 (2004).

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

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).

R. A. Chipman, "Polarimetry," in Handbook of Optics, M.Bass, ed. (McGraw-Hill, 1995), Vol. II, Chap. 22.

P. Huard, Polarization of Light (Wiley, 1997).

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light (Cambridge U. Press, 1999).
[PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

General polarization element can be presented as a combination of the four primitive ones (from right to left): a linear polarizer, a circular polarizer, a retarder, and a rotator.

Fig. 2
Fig. 2

(a), (b) Ellipticity of eigenpolarizations and (c), (d) the corresponding normalized eigenvalues of the Jones matrix of the general dichroic polarization element with orthogonal polarizations.

Fig. 3
Fig. 3

Evolution in the complex plane of V 1 / V 2 with the change of rotation angle, ϕ, for the example in Subsection 5A. The eigenvalues transit from dichroic to degenerate and birefringent depending on the rotation angle.

Fig. 4
Fig. 4

Evolution in the complex plane of χ 1 χ 2 * with the change of retardance, Δ, for the example in Subsection 5B. The value χ 1 χ 2 * = 1 corresponding to orthogonal eigenpolarizations is reached when Eq. (17) is satisfied.

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

T = ( T 11 T 12 T 21 T 22 ) .
χ 1 , 2 = 1 2 T 22 T 11 ± ( T 22 T 11 ) 2 + 4 T 21 T 12 T 12 ,
V 1 , 2 = 1 2 ( T 22 + T 11 ± ( T 22 T 11 ) 2 + 4 T 21 T 12 ) .
Im   V 1 V 2 = 0 , Re   V 1 V 2 > 0 .
| V 1 V 2 | = 1 , χ 1 χ 2 * = 1 ,
V 1 = V 2 , χ 1 = χ 2 ,
T = i = 1 N T N + 1 i .
T G e n = T C P T L P T C A T L A .
T L P = ( cos 2 α + exp ( i Δ ) sin 2 α [ 1 exp ( i Δ ) ] cos   α   sin   α [ 1 exp ( i Δ ) ] cos   α   sin   α sin 2 α + exp ( i Δ ) cos 2 α ) ,
T L A = ( cos 2 γ + P sin 2 γ ( 1 P ) cos   γ   sin   γ ( 1 P ) cos   γ   sin   γ sin 2 γ + P cos 2 γ ) ,
T C P = ( cos   ϕ sin   ϕ sin   ϕ cos   ϕ ) ,
T C A = ( 1 i R i R 1 ) ,
0 P 1 , 1 R 1 ,
π / 2 α π / 2 , π / 2 γ π / 2 ,
0 Δ 2 π , 0 ϕ 2 π .
( 1 + P ) R   cos ( Δ 2 ) sin   ϕ + ( 1 P ) sin ( Δ 2 ) × cos [ ϕ 2 ( α γ ) ] = 0 ,
{ ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) R   sin ( Δ 2 ) × sin [ ϕ 2 ( α γ ) ] } 2 4 P ( 1 R 2 ) > 0.
V 1 , 2 = 1 2  exp ( i   Δ 2 ) { ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) R  sin ( Δ 2 ) sin [ ϕ 2 ( α γ ) ] ± { ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) R   sin ( Δ 2 ) sin [ ϕ 2 ( α γ ) ] } 2 4 P ( 1 R 2 ) } ,
V 1 = X   exp ( i   Δ 2 ) , V 2 = Y   exp ( i   Δ 2 ) .
ϕ = 0 , π , α = π / 4 + γ ,
tan   Δ 2 = 1 P 1 + P R ,
ϕ = 0 , π , α = π / 4 + γ ,
tan   Δ 2 = 1 P 1 + P R .
T = ± 1 2  exp ( i   Δ 2 ) H ,
H = 1 Q 1 ( Q 1 + Q 2   cos   2 γ Q 2   sin   2 γ 4i P R Q 2   sin   2 γ + 4i P R Q 1 Q 2   cos   2 γ ) ,
Q 1 = ( 1 + P ) 2 + ( 1 P ) 2 R 2 ,
Q 2 = ( 1 P 2 ) ( 1 + R 2 ) .
V 1 , 2 = ± 1 2  exp ( i   Δ 2 ) [ Q 1 ± Q 1 4 P ( 1 R 2 ) ] ,
χ 1 , 2 = Q 2   cos   2 γ Q 1 ( Q 1 4 P ( 1 R 2 ) Q 2   sin   2 γ 4 i P R ,
( 1 + P ) R   cos ( Δ 2 ) sin   ϕ + ( 1 P ) sin ( Δ 2 ) × cos [ ϕ 2 ( α γ ) ] = 0 ,
{ ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) R   sin ( Δ 2 ) × sin [ ϕ 2 ( α γ ) ] } 2 4 P ( 1 R 2 ) < 0.
V 1 , 2 = 1 2  exp ( i   Δ 2 ) { ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) R   sin ( Δ 2 ) sin [ ϕ 2 ( α γ ) ] ± i 4 P ( 1 R 2 ) { ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) R   sin ( Δ 2 ) sin [ ϕ 2 ( α γ ) ] } 2 } .
V 1 = Z   exp ( i   Δ 2 ) , V 2 = Z *   exp ( i   Δ 2 ) ,
P = 1 , R = 0 .
( 1 + P ) R   cos ( Δ 2 ) sin   ϕ + ( 1 P ) sin ( Δ 2 ) × cos [ ϕ 2 ( α γ ) ] = 0 ,
{ ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) R   sin ( Δ 2 ) × sin [ ϕ 2 ( α γ ) ] } 2 4 P ( 1 R 2 ) = 0.
V 1 , 2 = 1 2  exp ( i   Δ 2 ) { ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) × R   sin ( Δ 2 ) sin [ ϕ 2 ( α γ ) ] } .
V 1 , 2 = 1 + P 2 [ cos   ϕ ± cos 2 ϕ 4 P ( 1 + P ) 2 ] .
cos   ϕ = ± 2 P 1 + P ,
T = ( i 1 i 1 i 2 + i ) .
R = 0.781 , P = 0.527 , γ = 67.5 ° ,
Δ = 136.6 9 ° , α = 22.5 ° , ϕ = 0 .
T = ( 1 0 2 ( 1 i ) 1 ) .
R = 0.618 , P = 0.236 , γ = 13.3 ° ,
Δ = 63.4 ° , α = 13.3 ° , ϕ = 31.7 ° .
T = ( i   2 P 1 + P P 1 i   2 P P 1 + P ) .
χ 1 = χ 2 = i   1 P ,
V 1 = V 2 = i P 1 P 1 + P .
T 22 + T 11 = ( C 1 + i C 2 ) exp ( i   Δ 2 ) ,
C 1 = ( 1 + P ) cos ( Δ 2 ) cos   ϕ + ( 1 P ) R   sin ( Δ 2 ) × sin   [ ϕ 2 ( α γ ) ] ,
C 2 = ( 1 + P ) R   cos ( Δ 2 ) sin   ϕ + ( 1 P ) sin ( Δ 2 ) × cos [ ϕ 2 ( α γ ) ] ,
T 22 T 11 = ( C 3 + i C 4 ) exp ( i   Δ 2 ) ,
C 3 = ( 1 + P ) R   sin ( Δ 2 ) sin ( 2 α ϕ ) ( 1 P ) × cos ( Δ 2 ) cos ( 2 γ ϕ ) ,
C 4 = ( 1 + P ) sin ( Δ 2 ) cos ( 2 α ϕ ) + ( 1 P ) R × cos ( Δ 2 ) sin [ 2 γ ϕ ] ,
T 21 + T 12 = ( C 5 + i C 6 ) exp ( i   Δ 2 ) ,
C 5 = ( 1 + P ) R   sin ( Δ 2 ) cos ( 2 α ϕ ) + ( 1 P ) × cos ( Δ 2 ) sin ( 2 γ ϕ ) ,
C 6 = ( 1 + P ) sin ( Δ 2 ) sin ( 2 α ϕ ) + ( 1 P ) R × cos ( Δ 2 ) cos [ 2 γ ϕ ] ,
T 21 T 12 = ( C 7 + i C 8 ) exp ( i   Δ 2 ) ,
C 7 = ( 1 + P ) cos ( Δ 2 ) sin   ϕ + ( 1 P ) R   sin ( Δ 2 ) × cos [ ϕ 2 ( α γ ) ] ,
C 8 = ( 1 + P ) R   cos ( Δ 2 ) cos   ϕ ( 1 P ) sin ( Δ 2 ) × sin [ ϕ 2 ( α γ ) ] .
| T G e n | = T 22 T 11 T 12 T 21 = P ( 1 R 2 ) exp ( i Δ ) .
V 1 , 2 = 1 2 [ C 1 + i C 2 ± ( C 1 + i C 2 ) 2 4 P ( 1 R 2 ) ] × exp ( i   Δ 2 ) .
C 2 = 0 , C 1 2 4 P ( 1 R 2 ) < 0.
C 2 = 0 , C 1 2 4 P ( 1 R 2 ) > 0.
C 2 = 0 , C 1 2 4 P ( 1 R 2 ) = 0.
C 2 = 0 , C 3 = 0 , C 5 = 0 , C 8 = 0.
C 2 = 0 , C 4 = 0 , C 6 = 0 , C 7 = 0.

Metrics