Abstract

Procedures for performing polarization ray tracing through birefringent media are presented in a form compatible with the standard methods of geometrical ray tracing. The birefringent materials treated include the following: anisotropic optically active materials such as quartz, non-optically active uniaxial materials such as calcite, and isotropic optically active materials such as mercury sulfide and organic liquids. Refraction and reflection algorithms are presented that compute both ray directions and wave directions. Methods for computing polarization modes, refractive indices, optical path lengths, and Fresnel transmission and reflection coefficients are also specified. A numerical example of these algorithms is given for analyzing the field of view of a quartz rotator.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1966), pp. 71, 87–108.
  2. B. H. Billings, “Monochromatic depolarizer,”J. Opt. Soc. Am. 38, 819–829 (1948).
    [CrossRef] [PubMed]
  3. J. D. McGuire, R. A. Chipman, “Analysis of spatial pseudo-depolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
    [CrossRef]
  4. A. M. Title, W. J. Rosenberg, “Improvements in birefringent filters. 5: Field of view effects,” Appl. Opt. 18, 3443–3456 (1979).
    [CrossRef] [PubMed]
  5. J. E. Greivenkamp, “Color dependent optical prefilter for suppression of aliasing artifacts,” Appl. Opt. 29, 676–684 (1990).
    [CrossRef] [PubMed]
  6. J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, ed. (McGraw-Hill, New York, 1978).
  7. O. N. Stavroudis, “Ray-tracing formulas for uniaxial crystals,”J. Opt. Soc. Am. 52, 187–191 (1962).
    [CrossRef]
  8. W. Swindell, “Extraordinary-ray and -wave tracing in uniaxial crystals,” Appl. Opt. 14, 2298–2301 (1975).
    [CrossRef] [PubMed]
  9. M. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
    [CrossRef] [PubMed]
  10. J. D. Trolinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466 (1991).
    [CrossRef]
  11. Q.-T. Liang, “Simple ray tracing formulas for uniaxial optical crystals,” Appl. Opt. 29, 1008–1010 (1990).
    [CrossRef] [PubMed]
  12. E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
    [CrossRef]
  13. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  14. R. A. Chipman, “Polarization ray tracing,” in Recent Trends in Optical Design; Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 61–68 (1987).
    [CrossRef]
  15. T. J. Bruegge, “Analysis of polarization in optical systems,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 165–176 (1989).
    [CrossRef]
  16. S. C. McClain, L. W. Hillman, R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
    [CrossRef]
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  18. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).
  19. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 69–102.
  20. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  21. D. E. Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972), p. 6–248.
  22. R. C. Weast, ed., CRC Handbook of Chemistry and Physics, 66th ed. (CRC, Boca Raton, Fla., 1985), p. E-409.
  23. G. Szvivessy, C. Muenster, “Lattice optics of active crystals,” Ann. Phys. (Leipzig), 20, 703–736 (1934).
  24. E. E. Wahlstrom, Optical Crystallography, 3rd ed. (Wiley, New York, 1960), pp. 155–156.
  25. F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), p. 552.
  26. M. C. Simon, “Internal total reflection in monoaxial crystals,” Appl. Opt. 26, 3878–3883 (1987).
    [CrossRef] [PubMed]
  27. M. C. Simon, R. M. Echarri, “Inhibited reflection in uniaxial crystals,” Opt. Lett. 14, 257–259 (1989).
    [CrossRef] [PubMed]
  28. G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973), pp. 317–320.
  29. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md.1983), pp. 293–295.
  30. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), pp. 45, 60–72.
  31. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 52.
  32. W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York, 1990), pp. 288–297.
  33. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 19–38.
  34. J. P. McGuire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I: Formulation and example,” J. Opt. Soc. Am. A 7, 1614–1626 (1990).
    [CrossRef]
  35. S. C. McClain, R. A. Chipman, L. W. Hillman, “Aberrations of a horizontal–vertical depolarizer,” Appl. Opt. 31, 2326–2331 (1992).
    [CrossRef] [PubMed]
  36. R. V. Shack, Optical Sciences Center, University of Arizona, Tuscon, Ariz. 85721 (personal communication, 1991).
  37. H. A. Macleod, “Microstructure of optical thin films,” in Optical Thin Films, R. I. Seddon, ed., Proc. Soc. Photo-Opt. Instrum. Eng.325, 21–28 (1982).
    [CrossRef]

1993

1992

1991

J. D. Trolinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466 (1991).
[CrossRef]

1990

1989

M. C. Simon, R. M. Echarri, “Inhibited reflection in uniaxial crystals,” Opt. Lett. 14, 257–259 (1989).
[CrossRef] [PubMed]

E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
[CrossRef]

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

1987

1983

1979

1975

1962

1948

1937

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

1934

G. Szvivessy, C. Muenster, “Lattice optics of active crystals,” Ann. Phys. (Leipzig), 20, 703–736 (1934).

Bennett, H. E.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, ed. (McGraw-Hill, New York, 1978).

Bennett, J. M.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, ed. (McGraw-Hill, New York, 1978).

Billings, B. H.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Bruegge, T. J.

T. J. Bruegge, “Analysis of polarization in optical systems,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 165–176 (1989).
[CrossRef]

Chipman, R. A.

S. C. McClain, L. W. Hillman, R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
[CrossRef]

S. C. McClain, R. A. Chipman, L. W. Hillman, “Aberrations of a horizontal–vertical depolarizer,” Appl. Opt. 31, 2326–2331 (1992).
[CrossRef] [PubMed]

J. D. Trolinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466 (1991).
[CrossRef]

J. P. McGuire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I: Formulation and example,” J. Opt. Soc. Am. A 7, 1614–1626 (1990).
[CrossRef]

J. D. McGuire, R. A. Chipman, “Analysis of spatial pseudo-depolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[CrossRef]

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

R. A. Chipman, “Polarization ray tracing,” in Recent Trends in Optical Design; Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 61–68 (1987).
[CrossRef]

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Echarri, R. M.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), pp. 45, 60–72.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md.1983), pp. 293–295.

Greivenkamp, J. E.

Hillman, L. W.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), p. 552.

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 19–38.

Liang, Q.-T.

Macleod, H. A.

H. A. Macleod, “Microstructure of optical thin films,” in Optical Thin Films, R. I. Seddon, ed., Proc. Soc. Photo-Opt. Instrum. Eng.325, 21–28 (1982).
[CrossRef]

McClain, S. C.

McGuire, J. D.

J. D. McGuire, R. A. Chipman, “Analysis of spatial pseudo-depolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[CrossRef]

McGuire, J. P.

Muenster, C.

G. Szvivessy, C. Muenster, “Lattice optics of active crystals,” Ann. Phys. (Leipzig), 20, 703–736 (1934).

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), pp. 45, 60–72.

Rosenberg, W. J.

Shack, R. V.

R. V. Shack, Optical Sciences Center, University of Arizona, Tuscon, Ariz. 85721 (personal communication, 1991).

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1966), pp. 71, 87–108.

Simon, M.

Simon, M. C.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York, 1990), pp. 288–297.

Stavroudis, O. N.

Stewart, G. W.

G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973), pp. 317–320.

Swindell, W.

Szvivessy, G.

G. Szvivessy, C. Muenster, “Lattice optics of active crystals,” Ann. Phys. (Leipzig), 20, 703–736 (1934).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), pp. 45, 60–72.

Title, A. M.

Trolinger, J. D.

J. D. Trolinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466 (1991).
[CrossRef]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md.1983), pp. 293–295.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), pp. 45, 60–72.

Wahlstrom, E. E.

E. E. Wahlstrom, Optical Crystallography, 3rd ed. (Wiley, New York, 1960), pp. 155–156.

Waluschka, E.

E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 52.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), p. 552.

Wilson, D. K.

J. D. Trolinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466 (1991).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 69–102.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 69–102.

Ann. Phys. (Leipzig)

G. Szvivessy, C. Muenster, “Lattice optics of active crystals,” Ann. Phys. (Leipzig), 20, 703–736 (1934).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

J. D. Trolinger, R. A. Chipman, D. K. Wilson, “Polarization ray tracing in birefringent media,” Opt. Eng. 30, 461–466 (1991).
[CrossRef]

J. D. McGuire, R. A. Chipman, “Analysis of spatial pseudo-depolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[CrossRef]

E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
[CrossRef]

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

Opt. Lett.

Rev. Mod. Phys.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Other

D. E. Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972), p. 6–248.

R. C. Weast, ed., CRC Handbook of Chemistry and Physics, 66th ed. (CRC, Boca Raton, Fla., 1985), p. E-409.

R. V. Shack, Optical Sciences Center, University of Arizona, Tuscon, Ariz. 85721 (personal communication, 1991).

H. A. Macleod, “Microstructure of optical thin films,” in Optical Thin Films, R. I. Seddon, ed., Proc. Soc. Photo-Opt. Instrum. Eng.325, 21–28 (1982).
[CrossRef]

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1966), pp. 71, 87–108.

R. A. Chipman, “Polarization ray tracing,” in Recent Trends in Optical Design; Computer Lens Design Workshop, C. Londono, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.766, 61–68 (1987).
[CrossRef]

T. J. Bruegge, “Analysis of polarization in optical systems,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 165–176 (1989).
[CrossRef]

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, ed. (McGraw-Hill, New York, 1978).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 69–102.

E. E. Wahlstrom, Optical Crystallography, 3rd ed. (Wiley, New York, 1960), pp. 155–156.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), p. 552.

G. W. Stewart, Introduction to Matrix Computations (Academic, New York, 1973), pp. 317–320.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md.1983), pp. 293–295.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988), pp. 45, 60–72.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 52.

W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York, 1990), pp. 288–297.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), pp. 19–38.

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 (3)

Fig. 1
Fig. 1

(a) Geometry for refraction at a birefringent-to-birefringent interface. A ray with wave vector k ^ i is incident upon surface q with surface normal η ^. Two rays are transmitted, with wave vectors k ^ to and k ^ te; two rays are reflected, with wave vectors k ^ ro and k ^ re. (b) Geometry for refraction at nonbirefringent-to-birefringent interface. A ray with wave vector k ^ i is incident upon surface q with surface normal η ^. Two rays are transmitted, with wave vectors k ^ to and k ^ te; one ray is reflected, with wave vector k ^ r. (c) Geometry for refraction at a birefringent-to-nonbirefringent interface. A ray with wave vector k ^ i is incident upon surface q with surface normal η ^. One ray is transmitted, with wave vector k ^ t; two rays are reflected, with wave vectors k ^ ro and k ^ re. (d) Geometry for refraction at a nonbirefringent-to-nonbirefringent interface. A ray with wave vector ki is incident upon surface q with surface normal η ^. One ray is transmitted, with wave vector k ^ t; one ray is reflected, with wave vector k ^ r.

Fig. 2
Fig. 2

Ray incident with wave-vector direction k ^ i onto a quartz rotator. The ray divides into ordinary and extraordinary modes with ray vectors ρ ^ o and ρ ^ e. The rays exit parallel to but offset from each other.

Fig. 3
Fig. 3

Ordinary and extraordinary rays interfere at point O. The ordinary ray originated from a lower portion of the incident plane wave than did the extraordinary ray, resulting in a phase lag that is due to the shear distance, in addition to the difference in optical path lengths between the two sides of the quartz block.

Tables (2)

Tables Icon

Table 1 Quartz Parametersa

Tables Icon

Table 2 Numerical Values for Refraction into a Quartz Rotator at 762 nma

Equations (70)

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

D = [ o 0 0 0 o 0 0 0 e ] .
G D = [ g o 0 0 0 g o 0 0 0 g e ] .
o = n o 2 ,
e = n e 2 ,
g o = ½ ( n r - n l ) ,
ρ = π λ ( n r - n l ) = 2 π λ g o
g o = λ ρ 2 π .
g e = - 1.92 g o .
= R - 1 D R ,
G = R - 1 G D R ,
R = [ cos α 0 - sin α 0 1 0 sin α 0 cos α ] [ cos β sin β 0 - sin β cos β 0 0 0 1 ] .
Γ t = - n i k ^ i · η ^ + [ n i 2 ( k ^ i · η ^ ) 2 + ( n t 2 - n i 2 ) ] 1 / 2
Γ r = - n i k ^ i · η ^ - [ n i 2 ( k ^ i · η ^ ) 2 + ( n r 2 - n i 2 ) ] 1 / 2
k ^ = n i k ^ i + Γ η ^ n i k ^ i + Γ η ^ .
θ = arccos ( k ^ · c ^ ) .
o = o q - g o q 2 ,
e = e q - g e q 2 ,
e = cos 2 θ e + sin 2 θ o ,
v = 4 e g o q 2 cos 2 θ + 0 ( g o q + g e q ) 2 sin 2 θ ,
w = ( g o q - g e q ) 2 sin 2 θ cos 2 θ .
n = { o ( o + e ) + v ± [ o 2 ( e + e ) 2 + 2 o v ( e + e ) + v 2 - 4 ( e + w ) o 2 e ] 1 / 2 2 ( e + w ) } 1 / 2 .
K μ ν = [ 0 - k μ ν z k μ ν y k μ ν z 0 - k μ ν x - k μ ν y k μ ν x 0 ] ,
M μ ν = q + ( n μ ν K μ ν + i G q ) 2 ,
M μ ν · E ^ μ ν = 0 ,
E ^ μ ν * · E ^ μ ν = 1.
M μ ν = U μ ν S μ ν V μ ν t ,
M μ ν · E μ ν = 0.
E ^ μ ν = E μ ν E μ ν .
H ^ μ ν = ( n μ ν K μ ν + i G q ) E ^ μ ν ,
H ^ i = ( n i K i + i G q - 1 ) E ^ i .
ρ ^ μ ν = R e { E ^ μ ν × H ^ μ ν * } R e { E ^ μ ν × H ^ μ ν * } ,
s 1 = k ^ i × η ^ ,
s 2 = η ^ × s 1 .
F = [ s 1 · E ^ t o s 1 · E ^ t e - s 1 · E ^ r o - s 1 · E ^ r e s 2 · E ^ t o s 2 · E ^ t e - s 2 · E ^ r o - s 2 · E ^ r e s 1 · H ^ t o s 1 · H ^ t e - s 1 · H ^ r o - s 1 · H ^ r e s 2 · H ^ t o s 2 · H ^ t e - s 2 · H ^ r o - s 2 · H ^ r e ] .
[ a t o a t e a r o a r e ] = F - 1 · [ s 1 · E ^ i s 2 · E ^ i s 1 · H ^ i s 2 · H ^ i ] .
E t ( r , t ) = R e { E i [ a t o E ^ t o exp ( i n t o k ^ t o · r ) + a t e E ^ t e exp ( i n t e k ^ t e · r ) ] exp ( - i ω t ) } ,
E r ( r , t ) = R e { E i [ a r o E ^ r o exp ( i n r o k ^ r o · r ) + a r e E ^ r e exp ( i n r e k ^ r e · r ) ] exp ( - i ω t ) } .
k ^ r = n i k ^ i - 2 n i ( k ^ i · η ^ ) η ^ n i k ^ i - 2 n i ( k ^ i · η ^ ) η ^ ,
E ^ rs = k ^ r × η ^ k ^ r × η ^ ,
E ^ rp = k ^ r × E ^ r s .
H ^ rs = n i k ^ r × E ^ rs ,
H ^ rp = n i k ^ r × E ^ rp .
ρ ^ r = k ^ r .
F = [ s 1 · E ^ to s 1 · E ^ te - s 1 · E ^ rs 0 s 2 · E ^ to s 2 · E ^ te 0 - s 2 · E ^ rp s 1 · H ^ to s 1 · H ^ te 0 - s 1 · H ^ rp s 2 · H ^ to s 2 · H ^ te - s 2 · H ^ rs 0 ] .
[ a to a te a rs a rp ] = F - 1 · [ s 1 · E ^ i s 2 · E ^ i s 1 · H ^ i s 2 · H ^ i ] .
E r ( r , t ) = R e { E i ( a rs E ^ rs + a rp E ^ rp ) exp ( i n r k ^ r · r ) exp ( - i ω t ) } ,
Γ t = - n i k ^ i · η ^ + [ n i 2 ( k ^ t · η ^ ) 2 + ( n q 2 - n i 2 ) ] 1 / 2 ,
k ^ t = n i k ^ i + Γ t η ^ n i k ^ i + Γ t η ^ .
E ^ ts = k ^ t × η ^ k ^ t × η ^ ,
E ^ tp = k ^ t × E ^ t s .
H ^ ts = n i k ^ t × E ^ ts ,
H ^ tp = n i k ^ t × E ^ tp .
ρ ^ t = k ^ t .
F = [ s 1 · E ^ ts 0 - s 1 · E ^ ro - s 1 · E ^ re 0 s 2 · E ^ tp - s 2 · E ^ ro - s 2 · E ^ re 0 s 1 · H ^ tp - s 1 · H ^ ro - s 1 · H ^ re s 2 · H ^ ts 0 - s 2 · H ^ ro - s 2 · H ^ re ] .
[ a ts a tp a ro a re ] = F - 1 · [ s 1 · E ^ i s 2 · E ^ i s 1 · H ^ i s 2 · H ^ i ] .
E t ( r , t ) = R e { E i ( a ts E ^ ts + a tp E ^ tp ) exp ( i n t k ^ t · r ) exp ( - i ω t ) } ,
[ a ts a tp a rs a rp ] = F - 1 · [ s 1 · E ^ i s 2 · E ^ i s 1 · H ^ i s 2 · H ^ i ] ,
F = [ s 1 · E ^ ts 0 - s 1 · E ^ rs 0 0 s 2 · E ^ tp 0 - s 2 · E ^ rp 0 s 1 · H ^ tp 0 - s 1 · H ^ rp s 2 · H ^ ts 0 - s 2 · H ^ rs 0 ] .
l = [ ( x q + 1 - x q ) 2 + ( y q + 1 - y q ) 2 + ( z q + 1 - z q ) 2 ] 1 / 2 .
OPL = n l k ^ · ρ ^ .
E ( r ) = s E s ( r ) ,
PSF ( u ) = F { E ( r ) } ,
δ = π ( n r - n l ) l λ 0 ,
E upper = [ ( a 1 t o s · a 2 t s o ) s ^ + ( a 1 t o s · a 2 t p o ) p ^ ] e i ϕ o × exp [ i ( k ^ i · r - ω t ) ] ,
E lower ( r , t ) = [ ( a 1 t e s · a 2 t s e ) s ^ + ( a 1 t e s · a 2 t p e ) p ^ ] e i ϕ e × exp [ i ( k ^ i · r - ω t ) ] ,
OPD = OPL e - OPL o + ( y o - y e ) sin θ i .
Δ ϕ = OPD λ 0 2 π = 3.46 rad .
E out = { [ a 1 t o s · a 2 t s o + a 1 t e s · a 2 t s e exp ( - i Δ ϕ ) ] s ^ + [ a 1 t o s · a 2 t p o + a 1 t e p · a 2 t p e exp ( - i Δ ϕ ) ] p ^ } × exp [ i ( k ^ t · r - ω t ) ] .
E out = [ 0.433 s ^ + 0.851 exp ( - 1.97 i ) p ^ ] exp [ i ( k ^ t · r - ω t ) ] .
D = [ x 0 0 0 y 0 0 0 z ] .

Metrics