Abstract

The polarization in isotropic radially symmetric lens and mirror systems in the paraxial approximation is examined. Polarized aberrations are variations in the phase, amplitude, and polarization state of the electromagnetic field across the exit pupil. Some are dependent on the incident polarization state and some are not. Expressions through fourth order for phase, amplitude, linear diattenuation, and linear retardance aberrations are derived in terms of the chief and marginal ray angles of incidence and the Taylor series expansion coefficients of the Fresnel equations for reflection and transmission at uncoated and thin-film-coated interfaces. Applications to polarization ray tracing are discussed.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Inoué, “Studies on depolarization of light at microscope lens surfaces. I: The origin of stray light by rotation at lens surfaces,” Exp. Cell Res. 3, 199–208 (1951).
    [CrossRef]
  2. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  3. E. W. Hansen, “Overcoming polarization aberrations in microscopy,” in Polarization Considerations in Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.891, 190–197 (1988).
  4. R. A. Chipman, “Polarization analysis of optical systems II,” in Polarization Considerations in Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 79–94 (1989).
  5. J. D. Mangus, J. Alonso, “Image sensitivity anomalies of glancing incidence telescopes,” in Proceedings of the X-ray Optics Symposium, P. W. Sanford, ed. (Mullard Space Sciences Laboratory of University College, London, 1973), pp. 244–275.
  6. 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]
  7. J. P. McGuire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations II: Amplitude response matrices for rotationally symmetric systems,” J. Opt. Soc. Am. A 8, 833–840 (1991).
    [CrossRef]
  8. J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
    [CrossRef]
  9. R. A. Chipman, J. E. Stacy, “pmap: polarization matrix analysis program,” cosmic program NPO-17273 (University of Georgia, Athens, Ga., 1983).
  10. synopsys, Breault Research Organization, Inc., Tucson Ariz.
  11. codev, Optical Research Associates, Pasadena, Calif.
  12. glad, Tucson, Ariz.
  13. S. Inoué, W. L. Hyde, “Studies on depolarization of light at microscope lens surfaces. II: The simultaneous realization of high resolution and high sensitivity with the polarization microscope,” J. Biophys. Biochem. Cyto. 3, 831–838 (1957).
    [CrossRef]
  14. J. P. McGuire, R. A. Chipman, “Polarization aberrations in the Solar Activity Measurement Experiments (SAMEX) solar vector magnetograph,” Opt. Eng. 28, 141–147 (1989).
  15. R. A. Chipman, “Polarization aberrations of lenses,” in International Lens Design Conference, W. H. Taylor, D. Moore, eds., Proc. Soc. Photo-Opt. Instrum. Eng.554, 82–87 (1985).
  16. R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (Optical Sciences, University of Arizona, Tucson, Ariz., 1987).
  17. R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).
  18. J. P. McGuire, R. A. Chipman, “Polarization aberrations in optical systems,” Current Developments in Optical Engineering II, R. E. Fisher, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 240–257 (1987). This is an earlier version of this paper, which lacked exponential form and several other refinements, which resulted in increased complexity of computation and interpretation.
  19. J. P. McGuire, “Diffraction image formation and analysis in optical systems with polarization aberrations,” Ph.D. dissertation (Department of Physics, University of Alabama in Huntsville, Huntsville, Ala., 1990).
  20. W. T. Welford, Aberrations of the Optical Systems (Hilger, Bristol, 1986).
  21. G. H. Spencer, M. V. R. K. Murty, “General ray-tracing procedure,” J. Opt. Soc. Am. 52, 672–678 (1962).
    [CrossRef]
  22. S. C. McClain, L. W. Hillman, R. A. Chipman, “Polarization in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
    [CrossRef]
  23. S. C. McClain, L. W. Hillman, R. A. Chipman, “Polarization in anisotropic optically active media. II. Theory and physics,” J. Opt. Soc. Am. A 10, 2383–2393 (1993).
    [CrossRef]
  24. R. C. Jones, “A new calculus for the treatment of optical systems. I: Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941). This and the other seven papers in the series are conveniently collected in Polarized Light, W. Swindell, ed. (Dowden, Hutchinson, & Ross, Stroudsburg, Pa., 1975).
    [CrossRef]
  25. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  26. R. C. Jones, “A new calculus for the treatment of optical systems VII: Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685(1948). In this paper the Pauli spin matrices and the identity matrix are used as a basis for the Jones matrices without identification of the Pauli spin matrices by name.
    [CrossRef]
  27. R. A. Chipman, “Polarization ray tracing,” in Recent Trends in Optical System Design; Computer Lens Design Workshop, C. Londoõn, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.776, 61–68 (1987).
  28. E. Waluschka, “Polarization ray trace,” Opt. Eng. 28, 86–89 (1989).
  29. J. Sánchez Mandragón, K. B. Wolf, eds., Lie Methods in Optics, Lecture Notes in Physics (Springer-Verlag, Berlin, 1986).
    [CrossRef]
  30. P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2, pp. 29–186.
    [CrossRef]
  31. C. Whitney, “Pauli-algebraic operators in polarization optics,” J. Opt. Soc. Am. 61, 1207–1213 (1971).
    [CrossRef]

1993 (2)

1991 (1)

1990 (2)

1989 (4)

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

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

J. P. McGuire, R. A. Chipman, “Polarization aberrations in the Solar Activity Measurement Experiments (SAMEX) solar vector magnetograph,” Opt. Eng. 28, 141–147 (1989).

R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).

1971 (1)

1962 (1)

1957 (1)

S. Inoué, W. L. Hyde, “Studies on depolarization of light at microscope lens surfaces. II: The simultaneous realization of high resolution and high sensitivity with the polarization microscope,” J. Biophys. Biochem. Cyto. 3, 831–838 (1957).
[CrossRef]

1951 (1)

S. Inoué, “Studies on depolarization of light at microscope lens surfaces. I: The origin of stray light by rotation at lens surfaces,” Exp. Cell Res. 3, 199–208 (1951).
[CrossRef]

1948 (1)

1941 (1)

Alonso, J.

J. D. Mangus, J. Alonso, “Image sensitivity anomalies of glancing incidence telescopes,” in Proceedings of the X-ray Optics Symposium, P. W. Sanford, ed. (Mullard Space Sciences Laboratory of University College, London, 1973), pp. 244–275.

Azzam, R. M. A.

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

Bashara, N. M.

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

Chipman, L. J.

R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).

Chipman, R. A.

S. C. McClain, L. W. Hillman, R. A. Chipman, “Polarization in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
[CrossRef]

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

J. P. McGuire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations II: Amplitude response matrices for rotationally symmetric systems,” J. Opt. Soc. Am. A 8, 833–840 (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. P. McGuire, R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[CrossRef]

J. P. McGuire, R. A. Chipman, “Polarization aberrations in the Solar Activity Measurement Experiments (SAMEX) solar vector magnetograph,” Opt. Eng. 28, 141–147 (1989).

R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).

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

J. P. McGuire, R. A. Chipman, “Polarization aberrations in optical systems,” Current Developments in Optical Engineering II, R. E. Fisher, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 240–257 (1987). This is an earlier version of this paper, which lacked exponential form and several other refinements, which resulted in increased complexity of computation and interpretation.

R. A. Chipman, “Polarization aberrations of lenses,” in International Lens Design Conference, W. H. Taylor, D. Moore, eds., Proc. Soc. Photo-Opt. Instrum. Eng.554, 82–87 (1985).

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

R. A. Chipman, J. E. Stacy, “pmap: polarization matrix analysis program,” cosmic program NPO-17273 (University of Georgia, Athens, Ga., 1983).

R. A. Chipman, “Polarization ray tracing,” in Recent Trends in Optical System Design; Computer Lens Design Workshop, C. Londoõn, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.776, 61–68 (1987).

R. A. Chipman, “Polarization analysis of optical systems II,” in Polarization Considerations in Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 79–94 (1989).

Hansen, E. W.

E. W. Hansen, “Overcoming polarization aberrations in microscopy,” in Polarization Considerations in Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.891, 190–197 (1988).

Hillman, L. W.

Hyde, W. L.

S. Inoué, W. L. Hyde, “Studies on depolarization of light at microscope lens surfaces. II: The simultaneous realization of high resolution and high sensitivity with the polarization microscope,” J. Biophys. Biochem. Cyto. 3, 831–838 (1957).
[CrossRef]

Inoué, S.

S. Inoué, W. L. Hyde, “Studies on depolarization of light at microscope lens surfaces. II: The simultaneous realization of high resolution and high sensitivity with the polarization microscope,” J. Biophys. Biochem. Cyto. 3, 831–838 (1957).
[CrossRef]

S. Inoué, “Studies on depolarization of light at microscope lens surfaces. I: The origin of stray light by rotation at lens surfaces,” Exp. Cell Res. 3, 199–208 (1951).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2, pp. 29–186.
[CrossRef]

Jones, R. C.

Mangus, J. D.

J. D. Mangus, J. Alonso, “Image sensitivity anomalies of glancing incidence telescopes,” in Proceedings of the X-ray Optics Symposium, P. W. Sanford, ed. (Mullard Space Sciences Laboratory of University College, London, 1973), pp. 244–275.

McClain, S. C.

McGuire, J. P.

J. P. McGuire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations II: Amplitude response matrices for rotationally symmetric systems,” J. Opt. Soc. Am. A 8, 833–840 (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. P. McGuire, R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” Opt. Eng. 29, 1478–1484 (1990).
[CrossRef]

J. P. McGuire, R. A. Chipman, “Polarization aberrations in the Solar Activity Measurement Experiments (SAMEX) solar vector magnetograph,” Opt. Eng. 28, 141–147 (1989).

J. P. McGuire, “Diffraction image formation and analysis in optical systems with polarization aberrations,” Ph.D. dissertation (Department of Physics, University of Alabama in Huntsville, Huntsville, Ala., 1990).

J. P. McGuire, R. A. Chipman, “Polarization aberrations in optical systems,” Current Developments in Optical Engineering II, R. E. Fisher, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 240–257 (1987). This is an earlier version of this paper, which lacked exponential form and several other refinements, which resulted in increased complexity of computation and interpretation.

Murty, M. V. R. K.

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2, pp. 29–186.
[CrossRef]

Spencer, G. H.

Stacy, J. E.

R. A. Chipman, J. E. Stacy, “pmap: polarization matrix analysis program,” cosmic program NPO-17273 (University of Georgia, Athens, Ga., 1983).

Waluschka, E.

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

Welford, W. T.

W. T. Welford, Aberrations of the Optical Systems (Hilger, Bristol, 1986).

Whitney, C.

Exp. Cell Res. (1)

S. Inoué, “Studies on depolarization of light at microscope lens surfaces. I: The origin of stray light by rotation at lens surfaces,” Exp. Cell Res. 3, 199–208 (1951).
[CrossRef]

J. Biophys. Biochem. Cyto. (1)

S. Inoué, W. L. Hyde, “Studies on depolarization of light at microscope lens surfaces. II: The simultaneous realization of high resolution and high sensitivity with the polarization microscope,” J. Biophys. Biochem. Cyto. 3, 831–838 (1957).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

Opt. Eng. (5)

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

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

J. P. McGuire, R. A. Chipman, “Polarization aberrations in the Solar Activity Measurement Experiments (SAMEX) solar vector magnetograph,” Opt. Eng. 28, 141–147 (1989).

R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).

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

Other (16)

J. Sánchez Mandragón, K. B. Wolf, eds., Lie Methods in Optics, Lecture Notes in Physics (Springer-Verlag, Berlin, 1986).
[CrossRef]

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, Chap. 2, pp. 29–186.
[CrossRef]

R. A. Chipman, “Polarization ray tracing,” in Recent Trends in Optical System Design; Computer Lens Design Workshop, C. Londoõn, R. E. Fischer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.776, 61–68 (1987).

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

J. P. McGuire, R. A. Chipman, “Polarization aberrations in optical systems,” Current Developments in Optical Engineering II, R. E. Fisher, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 240–257 (1987). This is an earlier version of this paper, which lacked exponential form and several other refinements, which resulted in increased complexity of computation and interpretation.

J. P. McGuire, “Diffraction image formation and analysis in optical systems with polarization aberrations,” Ph.D. dissertation (Department of Physics, University of Alabama in Huntsville, Huntsville, Ala., 1990).

W. T. Welford, Aberrations of the Optical Systems (Hilger, Bristol, 1986).

R. A. Chipman, “Polarization aberrations of lenses,” in International Lens Design Conference, W. H. Taylor, D. Moore, eds., Proc. Soc. Photo-Opt. Instrum. Eng.554, 82–87 (1985).

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

E. W. Hansen, “Overcoming polarization aberrations in microscopy,” in Polarization Considerations in Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.891, 190–197 (1988).

R. A. Chipman, “Polarization analysis of optical systems II,” in Polarization Considerations in Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 79–94 (1989).

J. D. Mangus, J. Alonso, “Image sensitivity anomalies of glancing incidence telescopes,” in Proceedings of the X-ray Optics Symposium, P. W. Sanford, ed. (Mullard Space Sciences Laboratory of University College, London, 1973), pp. 244–275.

R. A. Chipman, J. E. Stacy, “pmap: polarization matrix analysis program,” cosmic program NPO-17273 (University of Georgia, Athens, Ga., 1983).

synopsys, Breault Research Organization, Inc., Tucson Ariz.

codev, Optical Research Associates, Pasadena, Calif.

glad, Tucson, Ariz.

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 (19)

Fig. 1
Fig. 1

Calculation of the optical path length W and the polarization J along a ray. In classical aberration calculations (a) the optical path length is the sum of the contributions between each interface W nm and the interfaces are described by a scalar transmittance τ n . In the polarization calculations in this paper (b) the polarization-dependent transmission at each interface is described with a Jones matrix J n and the optical path length is calculated as in (a).

Fig. 2
Fig. 2

Normalized coordinates, chief ray, and marginal ray. The center of the pupil coordinates, (x, y) or (ρ, ϕ), coincides with the center of the pupil. The optic axis lies along the z axis. The point object lies along a line parallel to they axis at height H.

Fig. 3
Fig. 3

Comparison of approximations with the angle of incidence for a spherical mirror. Collimated light incident is normally incident. ρ/R i is the ratio of the pupil coordinate to the radius of curvature for the sphere.

Fig. 4
Fig. 4

Contours of constant aberration in the pupil for the scalar aberrations through fourth order.

Fig. 5
Fig. 5

Paraxial vector-polarization-aberration patterns across the pupil through fourth order.

Fig. 6
Fig. 6

Effect of linear diattenuation aberrations on linearly polarized light: (a) uniform linearly polarized light in the entrance pupil; (b), (c), and (d) the polarization state across the exit pupil if the system has only the aberrations P 10222 = e −1, P 11111 = e −1, and P 12000 = e −1.

Fig. 7
Fig. 7

Effect of linear retardance aberrations on linearly polarized light: (a) uniform linearly polarized light in the entrance pupil; (b), (c), (d) the polarization state across the exit pupil if the system has only the aberrations P 10222 = jπ/2, P 11111 = jπ/2, and P 12000 = jπ/2.

Fig. 8
Fig. 8

Effect of linear diattenuation aberrations on circularly polarized light: (a) uniform circularly polarized light in the entrance pupil; (b), (c), (d) polarization state across the exit pupil if the system has only the aberrations P 10222 = e −1, P 11111 = e −1, and P 12000 = e −1.

Fig. 9
Fig. 9

Effect of linear retardance aberrations on circularly polarized light: (a) uniform circularly polarized light in the entrance pupil; (b), (c), (d) the polarization state across the exit pupil if the system has only the aberration P 10222 = jπ/2, P 11111 = jπ/2, and P 12000 = jπ/2.

Fig. 10
Fig. 10

Paraxial angle of incidence at a spherical surface for three field angles. The length of the lines denotes the magnitude of the angle of incidence. The orientation of the lines denotes the orientation of the plane of incidence.

Fig. 11
Fig. 11

First three principal vector-aberration patterns for isotropic rotationally symmetric interfaces. The P 10222 pattern is quadratic in radius. The P 10422 pattern is quartic in radius. Finally the P 10622 pattern varies with radius to the sixth power.

Fig. 12
Fig. 12

Change in the single-surface vector aberrations with the object height. The principal patterns are decentered as the object height increases.

Fig. 13
Fig. 13

Addition of second-order aberration patterns to give a decentered view of the second-order principal aberration pattern.

Fig. 14
Fig. 14

Exit-pupil-polarization-aberration maps for the fourth-order circular aberrations.

Fig. 15
Fig. 15

Example of a pupil-aberration section for an on-axis object. Exit-pupil-aberration maps with the tangential and sagittal sections highlighted are shown in (a) and (b), respectively. The tangential and sagittal sections are shown in (c) and (d), respectively. The sagittal section is rotated 90° for a more compact display. The aberration is P 10222P 10422 + P 33110.

Fig. 16
Fig. 16

Example of tangential- and sagittal-pupil-aberration plots for off-axis objects. The aberration is the same as in Fig. 15.

Fig. 17
Fig. 17

Tangential- and sagittal-pupil-aberration plot for a system with nth-order principal aberrations and piston.

Fig. 18
Fig. 18

(a) Amplitude reflectance and (b) phase change on reflection from a gold coating at 10.6 μm. Curves are shown for both the s and p components of the incident light. Values were computed with a refractive index, n = 8.8 − 64j.

Fig. 19
Fig. 19

Comparison of the exact and second-order polarization of gold at 10.6 μm. The diattenuation (a) is the difference between the amplitude reflectance of s and p light. The retardance (b) is the difference between the reflected phases. Values were computed with a refractive index, n = 8.8 − 64j.

Tables (6)

Tables Icon

Table 1 Polarization Terminology

Tables Icon

Table 2 Identity Matrix, Pauli Spin Matrices, and Their Eigenstates

Tables Icon

Table 3 Physical Significance of the Exponential Polarization Coefficients

Tables Icon

Table 4 Commutation Relations for the Unit and Pauli Spin Matrices

Tables Icon

Table 5 Physical Significance of the Aberration Coefficients

Tables Icon

Table 6 Symmetry of Pupil Aberration Sections of Rotationally Symmetric Systemsa

Equations (114)

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

U ( t ) = [ U q ( t ) U r ( t ) ] ,
U = J U ,
J = [ j 11 j 12 j 21 j 22 ] .
J = exp ( a 0 σ 0 + a 1 σ 1 + a 2 σ 1 + a 2 σ 2 + a 3 σ 3 ) ,
U x p ( h , ρ , λ ) = J sys ( h , ρ , λ ) U n p ( h , ρ , λ ) ,
J sys ( h , ρ , λ ) = [ j sys , 11 ( h , ρ , λ ) j sys , 12 ( h , ρ , λ ) j sys , 21 ( h , ρ , λ ) j sys , 22 ( h , ρ , λ ) ] = exp ( a sys , 0 σ 0 + α sys , 1 σ 1 + a sys , 2 σ 2 + σ sys , 3 σ 3 ) ,
J sys ( h , ρ , λ ) = J ( h , ρ , λ ) exp ( - j k ρ · ρ 2 f ) ,
J ( h , ρ , λ ) = P ( h , ρ ) exp [ j k W ( h , ρ , λ ) ] [ 1 0 0 1 ] ,
S ^ q = R ^ q × N ^ q R ^ q × N ^ q ,
P ^ q = R ^ q × S ^ q ,
U q = ( U s q U p q ) ,
( U s , q + 1 U p , q + 1 ) = [ S ^ q + 1 · S ^ q S ^ q + 1 · P ^ q S ^ q + 1 · P ^ q S ^ q + 1 · S ^ q ] ( U s , q U p , q ) ,
U q + 1 = R ( θ q ) U q
R ( θ q ) = [ cos θ q sin θ q sin θ q cos θ q ] ,
θ q = arctan ( S ^ q + 1 · P ^ q S ^ q + 1 · S ^ q ) .
U q = J q ( i q ) U q ,
i q = arcsin R ^ q × N ^ q ,
J q ( i q ) = [ a s q ( i q ) 0 0 a p q ( i q ) ] ,
J q ( i q , θ q ) = J q ( i q ) R ( θ q ) = [ a s q ( i q ) cos θ q a s q ( i q ) sin θ q a p q ( i q ) sin θ q a p q ( i q ) cos θ q ] .
J ( R ^ 1 ) = q = Q 1 J q ( i q , θ q ) .
x = ρ sin ϕ ,
y = ρ cos ϕ .
( U x , q + 1 U y , q + 1 ) = R ( - θ q ) J q ( i q ) R ( θ q ) ( U x q U y q ) ,
θ q = arctan ( S ^ q · y ^ S ^ q · x ^ ) ,
J q ( i q , θ q ) = R ( - θ q ) J q ( i q ) R ( θ q ) = exp [ a 0 q ( i q ) σ 0 + a 1 q ( i q ) ( cos 2 θ q σ 1 + sin 2 θ q σ 2 ) ] ,
a 0 q ( i q ) = [ ln a s q ( i q ) + ln a p q ( i q ) ] / 2 ,
a 1 q ( i q ) = [ ln a s q ( i q ) + ln a p q ( i q ) ] / 2.
i q ( H , ρ , ϕ ) = arcsin S q = S q + [ S q ] 3 + = ( H 2 i c q 2 + 2 H ρ cos ϕ i c q i m q + ρ 2 i m q 2 ) 1 / 2 ,
cos [ 2 θ q ( H , ρ , ϕ ) ] = H 2 i c q 2 + 2 H ρ cos ϕ i c q i m q + ρ 2 i m q 2 cos 2 ϕ i q 2 ,
sin [ 2 θ q ( H , ρ , ϕ ) ] = - 2 H ρ sin ϕ i c q i m q - ρ 2 i m q 2 sin 2 ϕ i q 2 .
I x y ( ϑ ) = U x 2 + U y 2 U 2 = cos 2 ϑ ,
N ^ = { - x R i , y R i , 1 R i [ R i 2 - ( x 2 + y 2 ) ] 1 / 2 } ,
i = arcsin ( ρ R i ) = ρ R i + 1 6 ( ρ R i ) 3 + 3 40 ( ρ R i ) 5 + ,
a s q ( i q ) = ( r s 0 q + r s 2 q i q 2 + r s 4 q i q 4 + ) × exp [ j ( ψ s 0 q + ψ s 2 q i q 2 + ψ s 4 q i q 4 + ) ] ,
a p q ( i q ) = ( r p 0 q + r p 2 q i q 2 + r p 4 q i q 4 + ) × exp [ j ( ψ p 0 q + ψ p 2 q i q 2 + ψ p 4 q i q 4 + ) ] ,
r s 0 q = r q 0 q = r 0 q ,             ψ s 0 q = ψ p 0 q = ψ 0 q
ln a a q ( i ) = r 0 q + r a 2 q i q 2 + r a 4 q i q 4 + + j ( ψ 0 q + ψ a 2 q i q 2 + ψ a 4 q i q 4 + ) ,
r 0 = ln r 0 ,
r a 2 = r a 2 / r 0 ,
r a 4 = r a 4 / r 0 - 0.5 ( r a 2 / r 0 ) 2 .
J q ( H , ρ , ϕ ) = exp [ ( a 00 q + a 02 q i q 2 + a 04 q i q 4 + ) σ 0 + ( a 12 q i q 2 + a 14 q i q 4 + ) × ( cos 2 θ q σ 1 + sin 2 θ q σ 2 ) ] ,
a 0 , 2 k , q = [ r s , 2 k , q + r p , 2 k , q + j ( ψ s , 2 k , q + ψ p , 2 k , q ) ] / 2 ,
a 1 , 2 k , q = [ r s , 2 k , q - r p , 2 k , q + j ( ψ s , 2 k , q - ψ p , 2 k , q ) ] / 2.
exp A exp B = exp ( A + B + C 2 + C 3 + ) ,
C 2 = ½ [ A , B ] ,
C 3 = / 12 1 [ [ A , [ A , B ] ] - [ B , [ A , B ] ] ] .
J = J Q J Q - 1 J 2 J 1 ,
J = exp ( a 0 σ 0 + a 1 σ 1 + a 2 σ 2 + a 3 σ 3 ) ,
a 0 = q = 1 Q ( a 00 q + a 02 q i q 2 + a 04 q i q 4 ) ,
a 1 = q = 1 Q ( a 12 q i q 2 + a 14 q i q 4 ) cos 2 θ q ,
a 2 = q = 1 Q ( a 12 q i q 2 + a 14 q i q 4 ) sin 2 θ q ,
a 3 = j q = 1 Q ( a 12 q cos 2 θ q i q 2 p = 1 q - 1 a 12 p sin 2 θ p i p 2 - a 12 q sin 2 θ q i q 2 p = 1 q - 1 a 12 p cos 2 θ p i p 2 ) .
J ( H , ρ , ϕ ) = exp [ P 0000 σ 0 + constant piston P 02000 H 2 σ 0 + quadratic piston P 01110 H ρ cos ϕ σ 0 + tilt P 00220 ρ 2 σ 0 + defocus P 00400 ρ 4 σ 0 + spherical aberration P 01310 H ρ 3 cos ϕ σ 0 + coma P 02200 H 2 ρ 2 σ 0 + field curvature P 02220 H 2 ρ 2 cos 2 ϕ σ 0 + astigmatism P 03110 H 3 ρ cos ϕ σ 0 + distortion P 04000 H 4 σ 0 + quartic piston P 12000 H 2 A 0 ( ϕ ) + vector quadratic piston P 11111 H ρ A 1 ( ϕ ) + vector tilt P 10222 ρ 2 A 2 ( ϕ ) + vector defocus P 14000 H 4 A 0 ( ϕ ) + vector quartic piston P 13110 H 3 ρ cos ϕ A 0 ( ϕ ) + vector distortion type 0 P 13111 H 3 ρ A 1 ( ϕ ) + vector distortion type 1 P 12200 H 2 ρ 2 A 0 ( ϕ ) + vector fast type 0 P 12221 H 2 ρ 2 cos ϕ A 1 ( ϕ ) + vector fast type 1 P 12222 H 2 ρ 2 A 2 ( ϕ ) + vector fast type 2 P 11311 H ρ 3 A 1 ( ϕ ) + vector coma type 1 P 11332 H ρ 3 cos ϕ A 2 ( ϕ ) + vector coma type 2 P 10422 ρ 4 A 2 ( ϕ ) + vector spherical aberration P 31310 H ρ 3 sin ϕ σ 3 + circular polarization coma P 33110 H 3 ρ sin ϕ σ 3 + circular polarization distortion sixth - order terms ]
J ( H , x , y ) = exp [ P 00000 σ 0 + constant piston P 02000 H 2 σ 0 + quadratic piston P 01110 H y σ 0 + tilt P 00220 ( x 2 + y 2 ) σ 0 + defocus P 00400 ( x 2 + y 2 ) σ 0 + spherical aberration P 01310 H y ( x 2 + y 2 ) σ 0 + coma P 02200 H 2 ( x 2 + y 2 ) σ 0 + field curvature P 02220 H 2 y 2 σ 0 + astigmatism P 03110 H 3 y σ 0 + distortion P 04000 H 4 σ 0 + quartic piston P 12000 H 2 σ 1 + vector quadratic piston P 11111 H ( y σ 1 + x σ 2 ) + vector tilt P 10222 [ ( y 2 - x 2 ) σ 1 - 2 x y σ 2 ] + vector defocus P 14000 H 4 σ 1 + vector quartic piston P 13110 H 3 y σ 1 + vector distortion type 0 P 13111 H 3 ( y σ 1 - x σ 2 ) + vector distortion type 1 P 12200 H 2 ( x 2 + y 2 ) σ 1 + vector fast type 0 P 12221 H 2 y ( y σ 1 - x σ 2 ) + vector fast type 1 P 12222 H 2 [ ( y 2 - x 2 ) σ 1 - 2 x y σ 2 ] + vector fast type 2 P 11311 H ( x 2 + y 2 ) ( y σ 1 - x σ 2 ) + vector coma type 1 P 11332 H y [ ( y 2 - x 2 ) σ 1 - 2 x y σ 2 ] + vector coma type 2 P 10422 [ ( y 4 - x 4 ) σ 1 - 2 x y ( x 2 + y 2 ) σ 2 ] + vector spherical aberration P 31310 H x ( x 2 + y 2 ) σ 3 + circular polarization coma P 33110 H 3 x σ 3 + circular polarization distortion sixth - order terms ]
A 0 ( ϕ ) = σ 1 ,
A 1 ( ϕ ) = ( cos ϕ σ 1 - sin ϕ σ 2 ) ,
A 2 ( ϕ ) = ( cos 2 ϕ σ 1 - sin 2 ϕ σ 2 ) .
ρ ( i = 0 ) = - H i c i m .
P 10 n 22 ( ρ ) = ρ n ( cos 2 ϕ σ 1 - sin 2 ϕ σ 2 ) ,
R ^ q ( H , x , y , z ) = S w q ( H , x , y , z ) S w q ( H , x , y , z ) ,
N ^ q ( H , x , y , z ) = S i q ( H , x , y , z ) S i q ( H , x , y , z ) ,
= ( x , y , z )
S w q ( H , x , y , z ) = z w q - R w q ( 1 - { x w q 2 + [ y w q + R w q tan ( u c q H ) ] 2 } R w q - 2 ) 1 / 2 + W q ( H , x , y ) ,
S i q ( H , x , y , z ) = z i q - R i q [ 1 - ( x i q 2 + y i q 2 ) R i q - 2 ] 1 / 2 + A i q ( x i q 2 + y i q 2 ) .
( x w q , y w q , z w q ) = ( r w q x , r w q y , r w q z ) ,
( x i q , y i q ) = ( y m q x , y c q H + y m q y ) ,
N ^ q = ( - y m q x / R i q , - [ y c q H + y m q y ] / R i q , 1 ) ,
R ^ q = ( u m q x , u c q H + u m q y , 1 ) .
S q = ( i c q H + i m q y , - i m q x , 0 ) ,
P q = ( 0 , 0 , 1 ) × S q ,
i c q = u c q + y c q / R i q ,
i m q = u m q + y m q / R i q ,
r s = sin ( i - i ) sin ( i + i ) = N - 1 N + 1 ( 1 + 1 N i 2 - N 2 - 6 N - 3 12 N 3 i 4 + ) ,
r p = tan ( i - i ) tan ( i + i ) = N - 1 N + 1 ( 1 - 1 N i 2 - 5 N 2 - 6 N + 9 12 N 3 i 4 + ) ,
t s = 2 cos i sin i sin ( i + i ) = 2 N + 1 ( 1 - N - 1 2 N i 2 + 3 - 3 N - 7 N 2 + N 3 24 N 3 i 4 + ) ,
t p = 2 cos i sin i sin ( i + i ) cos ( i - i ) = 2 N + 1 [ 1 - N - 1 2 N 2 i 2 - ( N - 1 ) ( 9 - 6 N + 5 N 2 ) 24 N 4 i 4 + ] ,
n sin i = n sin i ,
r s ( i ) = exp ( r 0 + r 2 i 2 + ) ,
r p ( i ) = exp ( r 0 - r 2 i 2 + ) ,
t s ( i ) = exp ( t 0 + t s 2 i 2 + ) ,
t p ( i ) = exp ( t 0 + t p 2 i 2 + ) ,
r 0 = log ( N - 1 N + 1 ) ,
r 2 = ( n + j k ) N 2 ,
t 0 = log ( 2 N + 1 ) ,
t s 2 = - N - 1 2 N ,
t p 2 = - N ( N - 1 ) 2 N 2 ,
P 0000 = q = 1 Q a 00 q ,
P 02000 = q = 1 Q a 02 q i c q 2 ,
P 01110 = 2 q = 1 Q a 02 q i c q i m q ,
P 00220 = q = 1 Q a 02 q i m q 2 ,
P 04000 = q = 1 Q a 04 q i c q 4 ,
P 03110 = 4 q = 1 Q a 04 q i c q 3 i m q ,
P 02200 = 2 q = 1 Q a 04 q i c q 2 i m q 2 ,
P 02220 = 2 P 02200 ,
P 01310 = 4 q = 1 Q a 04 q i c q i m q 3 ,
P 00400 = q = 1 Q a 04 q i m q 4 ,
P 12000 = q = 1 Q a 12 q i c q 2 ,
P 11111 = 2 q = 1 Q a 12 q i c q i m q ,
P 10222 = q = 1 Q a 12 q i m q 2 ,
P 14000 = q = 1 Q a 14 q i c q 4 ,
P 13111 = 2 q = 1 Q a 14 q i c q 3 i m q ,
P 13112 = P 13111 ,
P 12200 = 0 ,
P 12221 = 3 q = 1 Q a 14 q i c q 2 i m q 2 ,
P 12222 = 0 ,
P 11331 = 2 q = 1 Q a 14 q i c q i m q 3 ,
P 11332 = P 11331 ,
P 10422 = q = 1 Q a 14 q i m q 4 ,
P 31310 = j q = 1 Q ( a 12 q i m q 2 p = 1 q - 1 a 12 p i c p i m p - a 12 q i c q i m q p = 1 q - 1 a 12 p i m p 2 ) ,
P 33310 = j q = 1 Q ( a 12 q i c q i m q p = 1 q - 1 a 12 p i c p 2 - a 12 q i c q 2 p = 1 q - 1 a 12 p i c p i m p ) .
W ( H , ρ , ϕ ) = W ( H 2 , H ρ cos ϕ , ρ 2 ) = u , v , w P 0 u v w 0 H u ρ v cos w ϕ ,
H 2 ,             H ρ cos ϕ ,             ρ 2 .
J ( H , ρ , ϕ ) = x = 0 , 1 , 2 P x ( H 2 , H ρ cos ϕ , ρ 2 ) H 2 - x ρ x A x ( ϕ ) = u , v , w , x P 1 u v w x H u ρ v cos n ϕ A x ( ϕ ) ,
H 2 A 0 ( ϕ ) ,             H ρ A 1 ( ϕ ) ,             ρ 2 A 2 ( ϕ ) ,

Metrics