Abstract

A three-by-three polarization ray-tracing matrix method for polarization ray tracing in optical systems is presented for calculating the polarization transformations associated with ray paths through optical systems. The method is a three-dimensional generalization of the Jones calculus. Reflection and refraction algorithms are provided. Diattenuation of the optical system is calculated via singular value decomposition. Two numerical examples, a three fold-mirror system and a hollow corner cube, demonstrate the method.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. E. Knowlden, “Wavefront errors produced by multilayer thin film optical coatings,” Ph.D. dissertation (University of Arizona, 1981).
  2. R. A. Chipman, “Polarization ray tracing,” Proc. SPIE 766, 61–68 (1987).
  3. E. Waluschka, “A polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
  4. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  5. J. Wesner, F. Eisenkramer, J. Heil, and T. Sure, “Improved polarization ray tracing of thin-film optical coatings,” Proc. SPIE 5524, 261–272 (2004).
    [CrossRef]
  6. R. A. Chipman, “Mechanics of polarization ray tracing,” Opt. Eng. 34, 1636–1645 (1995).
    [CrossRef]
  7. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  8. P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
    [CrossRef]
  9. P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
    [CrossRef]
  10. R. A. Chipman, “Challenges for polarization ray tracing,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA4.
  11. W. S. T. Lam, S. McClain, G. Smith, and R. Chipman, “Ray tracing in biaxial materials,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA1.
  12. S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).
  13. G. Yun, S. C. McClain, and R. A. Chipman, “Three-dimensional polarization ray tracing calculus II: retardance,” Appl. Opt. 50, 2866–2874 (2011).
    [CrossRef] [PubMed]
  14. R. C. Jones, “A new calculus for the treatment of optical systems I,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  15. R. C. Jones, “A new calculus for the treatment of optical systems II,” J. Opt. Soc. Am. 31, 493–499 (1941).
    [CrossRef]
  16. R. C. Jones, “A new calculus for the treatment of optical systems III,” J. Opt. Soc. Am. 31, 500–503 (1941).
    [CrossRef]
  17. R. C. Jones, “A new calculus for the treatment of optical systems. IV,” J. Opt. Soc. Am. 32, 486–493 (1942).
    [CrossRef]
  18. R. C. Jones, “A new calculus for the treatment of optical systems. V. A more general formulation, and description of another calculus,” J. Opt. Soc. Am. 37, 107–110 (1947).
    [CrossRef]
  19. R. C. Jones, “A new calculus for the treatment of optical systems. VI. Experimental determination of the matrix,” J. Opt. Soc. Am. 37, 110–112 (1947).
    [CrossRef]
  20. R. C. Jones, “A new calculus for the treatment of optical systems. VII. Properties of the N matrices,” J. Opt. Soc. Am. 38, 671–683 (1948).
    [CrossRef]
  21. R. C. Jones, “A new calculus for the treatment of optical systems. VIII. Electromagnetic theory,” J. Opt. Soc. Am. 46, 126–131 (1956).
    [CrossRef]
  22. W. Singer, M. Totzeck, and H. Gross, “Physical Image Formation,” in Handbook of Optical Systems (Wiley, New York, 2005), pp. 613–620.
  23. S. G. Krantz, “The index or winding number of a curve about a point,” in Handbook of Complex Variables (Birkhauser, 1999), pp. 49–50.
  24. H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, 1986), pp. 179–209.
  25. P. H. Berning, “Theory and calculations of optical thin films,” Physics of thin films: advances in research and development 1, 69–121 (1963).
  26. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), pp. 63–74.
  27. S. Lu and R. A. Chipman, “Homogeneous and inhomogeneous jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  28. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977), pp. 67–84.
  29. R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (University of Arizona, 1987).
  30. S. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
    [CrossRef]
  31. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
    [CrossRef]
  32. P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985), pp. 1535–1544.
  33. E. R. Peck, “Polarization properties of corner reflectors and cavities,” J. Opt. Soc. Am. 52, 253–257 (1962).
    [CrossRef]
  34. M. A. Acharekar, “Derivation of internal incidence angles and coordinate transformations between internal reflections for corner reflectors at normal incidence,” Opt. Eng. 23, 669–674 (1984).
  35. R. Kalibjian, “Stokes polarization vector and Mueller matrix for a corner-cube reflector,” Opt. Commun. 240, 39–68 (2004).
    [CrossRef]
  36. J. Liu and R. M. A. Azzam, “Polarization properties of corner-cube retroreflectors: theory and experiment,” Appl. Opt. 36(1997).
    [CrossRef] [PubMed]

2011 (1)

2004 (2)

J. Wesner, F. Eisenkramer, J. Heil, and T. Sure, “Improved polarization ray tracing of thin-film optical coatings,” Proc. SPIE 5524, 261–272 (2004).
[CrossRef]

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

1998 (1)

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

1997 (2)

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[CrossRef]

J. Liu and R. M. A. Azzam, “Polarization properties of corner-cube retroreflectors: theory and experiment,” Appl. Opt. 36(1997).
[CrossRef] [PubMed]

1996 (1)

1995 (2)

1994 (1)

1989 (2)

E. Waluschka, “A polarization ray trace,” Opt. Eng. 28, 86–89 (1989).

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

1987 (2)

R. A. Chipman, “Polarization ray tracing,” Proc. SPIE 766, 61–68 (1987).

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

1984 (1)

M. A. Acharekar, “Derivation of internal incidence angles and coordinate transformations between internal reflections for corner reflectors at normal incidence,” Opt. Eng. 23, 669–674 (1984).

1963 (1)

P. H. Berning, “Theory and calculations of optical thin films,” Physics of thin films: advances in research and development 1, 69–121 (1963).

1962 (1)

1956 (2)

R. C. Jones, “A new calculus for the treatment of optical systems. VIII. Electromagnetic theory,” J. Opt. Soc. Am. 46, 126–131 (1956).
[CrossRef]

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

1948 (1)

1947 (2)

1942 (1)

1941 (3)

Acharekar, M. A.

M. A. Acharekar, “Derivation of internal incidence angles and coordinate transformations between internal reflections for corner reflectors at normal incidence,” Opt. Eng. 23, 669–674 (1984).

Azzam, R. M. A.

J. Liu and R. M. A. Azzam, “Polarization properties of corner-cube retroreflectors: theory and experiment,” Appl. Opt. 36(1997).
[CrossRef] [PubMed]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977), pp. 67–84.

Barakat, R.

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977), pp. 67–84.

Berning, P. H.

P. H. Berning, “Theory and calculations of optical thin films,” Physics of thin films: advances in research and development 1, 69–121 (1963).

Booker, G. R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), pp. 63–74.

Chipman, R.

W. S. T. Lam, S. McClain, G. Smith, and R. Chipman, “Ray tracing in biaxial materials,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA1.

Chipman, R. A.

G. Yun, S. C. McClain, and R. A. Chipman, “Three-dimensional polarization ray tracing calculus II: retardance,” Appl. Opt. 50, 2866–2874 (2011).
[CrossRef] [PubMed]

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

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

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

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

R. A. Chipman, “Polarization ray tracing,” Proc. SPIE 766, 61–68 (1987).

R. A. Chipman, “Challenges for polarization ray tracing,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA4.

R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (University of Arizona, 1987).

Eisenkramer, F.

J. Wesner, F. Eisenkramer, J. Heil, and T. Sure, “Improved polarization ray tracing of thin-film optical coatings,” Proc. SPIE 5524, 261–272 (2004).
[CrossRef]

Gross, H.

W. Singer, M. Totzeck, and H. Gross, “Physical Image Formation,” in Handbook of Optical Systems (Wiley, New York, 2005), pp. 613–620.

Heil, J.

J. Wesner, F. Eisenkramer, J. Heil, and T. Sure, “Improved polarization ray tracing of thin-film optical coatings,” Proc. SPIE 5524, 261–272 (2004).
[CrossRef]

Higdon, P. D.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

Jones, R. C.

Kalibjian, R.

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

Knowlden, R. E.

R. E. Knowlden, “Wavefront errors produced by multilayer thin film optical coatings,” Ph.D. dissertation (University of Arizona, 1981).

Krantz, S. G.

S. G. Krantz, “The index or winding number of a curve about a point,” in Handbook of Complex Variables (Birkhauser, 1999), pp. 49–50.

Laczik, Z.

Lam, W. S. T.

W. S. T. Lam, S. McClain, G. Smith, and R. Chipman, “Ray tracing in biaxial materials,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA1.

Lancaster, P.

P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985), pp. 1535–1544.

Liu, J.

J. Liu and R. M. A. Azzam, “Polarization properties of corner-cube retroreflectors: theory and experiment,” Appl. Opt. 36(1997).
[CrossRef] [PubMed]

Lu, S.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, 1986), pp. 179–209.

McClain, S.

W. S. T. Lam, S. McClain, G. Smith, and R. Chipman, “Ray tracing in biaxial materials,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA1.

McClain, S. C.

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Peck, E. R.

Singer, W.

W. Singer, M. Totzeck, and H. Gross, “Physical Image Formation,” in Handbook of Optical Systems (Wiley, New York, 2005), pp. 613–620.

Smith, G.

W. S. T. Lam, S. McClain, G. Smith, and R. Chipman, “Ray tracing in biaxial materials,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA1.

Sure, T.

J. Wesner, F. Eisenkramer, J. Heil, and T. Sure, “Improved polarization ray tracing of thin-film optical coatings,” Proc. SPIE 5524, 261–272 (2004).
[CrossRef]

Tismenetsky, M.

P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985), pp. 1535–1544.

Török, P.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[CrossRef]

P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

Totzeck, M.

W. Singer, M. Totzeck, and H. Gross, “Physical Image Formation,” in Handbook of Optical Systems (Wiley, New York, 2005), pp. 613–620.

Varga, P.

Waluschka, E.

E. Waluschka, “A polarization ray trace,” Opt. Eng. 28, 86–89 (1989).

Wesner, J.

J. Wesner, F. Eisenkramer, J. Heil, and T. Sure, “Improved polarization ray tracing of thin-film optical coatings,” Proc. SPIE 5524, 261–272 (2004).
[CrossRef]

Wilson, T.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), pp. 63–74.

Yun, G.

Appl. Opt. (2)

G. Yun, S. C. McClain, and R. A. Chipman, “Three-dimensional polarization ray tracing calculus II: retardance,” Appl. Opt. 50, 2866–2874 (2011).
[CrossRef] [PubMed]

J. Liu and R. M. A. Azzam, “Polarization properties of corner-cube retroreflectors: theory and experiment,” Appl. Opt. 36(1997).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. 34, 1535–1544 (1987).
[CrossRef]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

J. Opt. Soc. Am. (9)

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

Opt. Commun. (2)

P. Török and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. 137, 127–135 (1997).
[CrossRef]

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

Opt. Eng. (4)

M. A. Acharekar, “Derivation of internal incidence angles and coordinate transformations between internal reflections for corner reflectors at normal incidence,” Opt. Eng. 23, 669–674 (1984).

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

E. Waluschka, “A polarization ray trace,” Opt. Eng. 28, 86–89 (1989).

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

Physics of thin films: advances in research and development (1)

P. H. Berning, “Theory and calculations of optical thin films,” Physics of thin films: advances in research and development 1, 69–121 (1963).

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Proc. SPIE (2)

J. Wesner, F. Eisenkramer, J. Heil, and T. Sure, “Improved polarization ray tracing of thin-film optical coatings,” Proc. SPIE 5524, 261–272 (2004).
[CrossRef]

R. A. Chipman, “Polarization ray tracing,” Proc. SPIE 766, 61–68 (1987).

Other (10)

R. E. Knowlden, “Wavefront errors produced by multilayer thin film optical coatings,” Ph.D. dissertation (University of Arizona, 1981).

R. A. Chipman, “Challenges for polarization ray tracing,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA4.

W. S. T. Lam, S. McClain, G. Smith, and R. Chipman, “Ray tracing in biaxial materials,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWA1.

W. Singer, M. Totzeck, and H. Gross, “Physical Image Formation,” in Handbook of Optical Systems (Wiley, New York, 2005), pp. 613–620.

S. G. Krantz, “The index or winding number of a curve about a point,” in Handbook of Complex Variables (Birkhauser, 1999), pp. 49–50.

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, 1986), pp. 179–209.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999), pp. 63–74.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1977), pp. 67–84.

R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (University of Arizona, 1987).

P. Lancaster and M. Tismenetsky, The Theory of Matrices (Academic, 1985), pp. 1535–1544.

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 (5)

Fig. 1
Fig. 1

(a) Longitude and latitudes lines as the local x and y bases. (b) Singularity at the north pole.

Fig. 2
Fig. 2

The exiting polarization state E q at the qth optical interface is calculated from the matrix P q for a given incident polarization state E q 1 .

Fig. 3
Fig. 3

Sequence of three fold-mirrors. The transformation of the incident x-polarized light (solid red) and y- polarized light (dashed blue) are shown through the system.

Fig. 4
Fig. 4

Top view of a hollow aluminum-coated corner cube. Three surfaces of the corner cube are perpendicular to each other.

Fig. 5
Fig. 5

Top and side views of a corner cube. Propagation vectors are shown in solid black, local s coordinate vectors in solid red, and local p coordinate vectors in dashed blue.

Tables (4)

Tables Icon

Table 1 Polarization Ray-Tracing Matrix for a Horizontal Fast Axis Linear Quarter-Wave Retarder Without Beam Deviation for Three Different Propagation Directions, along Z Axis, Y Axis, and X Axis a

Tables Icon

Table 2 Polarization Ray-Tracing Matrices for Each Ray Intercept for a Ray Propagating On-Axis through a Three Fold-Mirror System

Tables Icon

Table 3 Propagation Vectors, Local Coordinate Basis Vectors, Surface Normal Vectors, and Polarization Ray-Tracing Matrices Associated with a Ray Path Through an Aluminum-Coated Hollow Corner Cube

Tables Icon

Table 4 Maximum Intensity of Transmitted Electric Field and Associated Incident Electric Field, the Minimum Intensity of Transmitted Electric Field and Associated Incident Electric Field, and the Diattenuation from the Ray through the Corner Cube System

Equations (43)

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

J = ( E x E y ) ,
e ( r , t ) = Re { E e i ( k k ^ r ω t ) } ,
E k ^ = 0 .
Re { E } Im { E } = 0 , | Re { E } | = | Im { E } | .
E q = ( E x , q E y , q E z , q ) = P q E q 1 .
P Total = P Q P Q 1 P q P 2 P 1 = q = Q , 1 1 P q .
P Total = P Q A Q , Q 1 P Q 1 A 3 , 2 P 2 A 2 , 1 P 1 = P Q q = Q 1 , 1 1 A q + 1 , q P q .
w q = ( E x , q E y , q ) = J q w q 1 ,
E a = P q E a , E b = P q E b .
P q k ^ q 1 = γ k ^ q .
P q k ^ q 1 = k ^ q .
s ^ q = k ^ q 1 × k ^ q | k ^ q 1 × k ^ q | , p ^ q = k ^ q 1 × s ^ q , s ^ q = s ^ q , p ^ q = k ^ q × s ^ q .
O in , q 1 = ( s ^ x , q s ^ y , q s ^ z , q p ^ x , q p ^ y , q p ^ z , q k ^ x , q 1 k ^ y , q 1 k ^ z , q 1 ) , O out , q = ( s ^ x , q p ^ x , q k ^ x , q s ^ y , q p ^ y , q k ^ y , q s ^ z , q p ^ z , q k ^ z , q ) .
( E s , q 1 E p , q 1 0 ) ,
J t , q = ( α s , t , q 0 0 0 α p , t , q 0 0 0 1 ) , J r , q = ( α s , r , q 0 0 0 α p , r , q 0 0 0 1 ) .
P q = O out , q J q O in , q 1 ,
P q = O out , q J q O in , q 1 where     J q = ( j 11 j 12 0 j 21 j 22 0 0 0 1 ) ,
P q = O in , q J q O in , q 1 .
( e i π / 4 0 0 e i π / 4 ) .
( e i π / 4 0 0 0 e i π / 4 cos 2 θ + sin 2 θ cos θ sin θ ( 1 e i π / 4 0 cos θ sin θ ( 1 e i π / 4 ) cos 2 θ + e i π / 4 sin 2 θ ) ) .
D = I max I min I max + I min , 0 D 1 .
P = U D V = ( k ^ x , Q u x , 1 u x , 2 k ^ y , Q u y , 1 u y , 2 k ^ z , Q u z , 1 u z , 2 ) ( 1 0 0 0 Λ 1 0 0 0 Λ 2 ) ( k ^ x , 0 * k ^ y , 0 * k ^ z , 0 * v x , 1 * v y , 1 * v z , 1 * v x , 2 * v y , 2 * v z , 2 * ) ,
Λ 1 Λ 2 0 ,
V = ( V * ) T .
P v 1 = Λ 1 u 1 , P v 2 = Λ 2 u 2 , P k ^ 0 = k ^ Q .
E = α v 1 + β v 2 ,
I Trans = | P E | 2 = E P P E .
I Trans = | α | 2 Λ 1 2 + | β | 2 Λ 2 2 = | α | 2 ( Λ 1 2 Λ 2 2 ) + Λ 2 2 .
I Trans = { I max = Λ 1 2 if     | α | 2 = 1 I min = Λ 2 2 if     | α | 2 = 0 ,
D = Λ 1 2 Λ 2 2 Λ 1 2 + Λ 2 2 ,
P 1 = ( 0 0 1 0 1 0 1 0 0 ) .
P Total = P 3 P 2 P 1 = ( 0 1 0 1 0 0 0 0 1 ) .
P Total ( 1 0 0 ) = ( 0 1 0 1 0 0 0 0 1 ) ( 1 0 0 ) = ( 0 1 0 ) .
P cc = P 3 P 2 P 1 = ( 0.39 + 0.78 i 0.01 + 0.02 i 0 0.02 i 0.40 + 0.78 i 0 0 0 1 ) .
U cc = ( 0 0.63 + 0.15 i 0.74 + 0.17 i 0 0.37 0.66 i 0.32 + 0.57 i 1 0 0 ) , D cc = ( 1 0 0 0 0.88 0 0 0 0.87 ) , V cc = ( 0 0.43 0.49 i 0.47 0.6 i 0 0.41 0.64 i 0.38 + 0.52 i 1 0 0 ) .
E ( r ) = P ( r ) E 0 ( r ) exp ( i 2 π / λ OPL ( r ) ) .
k ^ s = ( k ^ 1 , 1 k ^ 1 , 2 k ^ 1 , M k ^ L , 1 k ^ L , 2 k ^ L , M ) , L × M   propagation vector grid r s = ( r 1 , 1 r 1 , 2 r 1 , M r L , 1 r L , 2 r L , M ) , L × M   exit pupil position grid OPL s = ( OPL 1 , 1 OPL 1 , 2 OPL 1 , M OPL L , 1 OPL L , 2 OPL L , M ) , L × M   optical path length grid ,
P Total , s = ( P 1 , 1 P 1 , 2 P 1 , M P 2 , 1 P 2 , 2 P 2 , M P L , 1 P L , 2 P L , M ) ,
P Total , s = ( 0 0 0 0 0 0 0 0 0 0 0 0 P 2 , 2 P 2 , M 1 0 0 0 0 0 P 3 , 2 P 3 , M 1 0 0 0 0 P N 2 , 2 P N 2 , M 1 0 0 0 0 0 P N 1 , 2 P N 1 , M 1 0 0 0 0 0 0 0 0 0 0 0 0 ) .
P combined = ( P 1 k D ) exp ( i 2 π / λ OPL 1 ) + ( P 2 k D ) exp ( i 2 π / λ OPL 2 ) + k D ,
e ( r = 0 , t = π / 2 ω ) = Re { E e i π / 2 } = Im { E } ,
e ( 0 , 0 ) × e ( 0 , π / 2 ω ) = z ^ .
{ e ( 0 , 0 ) × e ( 0 , π / 2 ω ) } k ^ .

Metrics