Abstract

This paper is the first in a series that will examine image formation in optical systems with polarization aberrations. The present paper derives the point-spread function (PSF) and the optical transfer function for optical systems with polarization aberration and explores how image formation depends on the coherence and polarization state of the source. It is shown that the scalar PSF of Fourier optics can be generalized in the presence of polarization aberration to a 4 × 4 point-spread matrix (PSM) in Mueller matrix notation. A similar 4 × 4 optical transfer matrix (OTM) is shown to be an appropriate generalization of the optical transfer function. The PSM and the OTM are associated with the optical system and are independent of the incident polarization state but dependent on the coherence of the illumination. Since an optical system with polarization aberrations will have a different PSF and optical transfer function for different incident polarization states, the PSM and the OTM act as filters with regard to the incident polarization state. Example calculations are performed for a circularly retarding lens.

© 1990 Optical Society of America

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References

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  1. L. Seidel, Astron. Nachr. 43, 289, 305, 321 (1856).
    [CrossRef]
  2. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, London, 1950).
  3. E. Abbe, “Beiträge zur theorie des Mikroskops und der mikroskopischer Wahrnehmung,” Arch. Mikrosk. Anat. 9, 413 (1873).
    [CrossRef]
  4. P. M. Deffieux, L’Intégral de Fourier et ses Application à l’Optique, 2nd ed. (Masson, Paris, 1970).
  5. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165 (1959).
    [CrossRef]
  6. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
    [CrossRef]
  7. H. H. Hopkins, “Applications of coherence theory in microscopy and interferometry,” J. Opt. Soc. Am. 47, 508–526 (1956).
    [CrossRef]
  8. B. J. Thompson, “Image formation with partially coherent light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969).
    [CrossRef]
  9. H. Kubota, S. Inoué, “Diffraction images in the polarizing microscope,” J. Opt. Soc. Am. 49, 191–198 (1959).
    [CrossRef] [PubMed]
  10. W. Urbańczyk, “Optical imaging in polarized light,” Optik 63, 25–35 (1982).
  11. W. Urbańczyk, “Optical imaging systems changing the state of light polarization,” Optik 66, 301–309 (1984).
  12. W. Urbańczyk, “Optical transfer functions for imaging systems which change the state of light polarization,” Opt. Acta 33, 53–62 (1986).
    [CrossRef]
  13. B. Chakraborty, “Effect of polarizers used as masks on the perfect-lens aperture,” J. Opt. Soc. Am. A 2, 743–746 (1985).
    [CrossRef]
  14. A. Gosh, K. Murata, A. K. Chakraborty, “Frequency-response characteristics of a perfect lens masked by polarizing devices,” J. Opt. Soc. Am. A 5, 277–284 (1988).
    [CrossRef]
  15. B. Chakraborty, “Polychromatic optical transfer function of a perfect lens masked by birefringent plate devices,” J. Opt. Soc. Am. A 5, 1207–1212 (1988).
    [CrossRef]
  16. J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” in Polarization Considerations in Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 95–108 (1989).
    [CrossRef]
  17. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  18. R. A. Chipman, “Polarization aberrations,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1987).
  19. M. C. Simon, “Image formation through monoaxial plane parallel plates,” Appl. Opt. 27, 4176–4182 (1988).
    [CrossRef] [PubMed]
  20. J. E. Stewart, W. S. Gallaway, “Diffraction anomalies in grating spectrophotometers,” Appl. Opt. 1, 421–429 (1962).
    [CrossRef]
  21. J. B. Breckinridge, “Polarization properties of a grating spectrograph,” Appl. Opt. 10, 286–294 (1971).
    [CrossRef] [PubMed]
  22. R. R. A. Syms, “Vector effects in holographic optical elements,” Opt. Acta 32, 1413–1425 (1986).
    [CrossRef]
  23. J. P. McGuire, R. A. Chipman, “Polarization aberrations in optical systems,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 240–257 (1987).
  24. E. W. Hansen, “Overcoming polarization aberrations in microscopy,” in Polarization Consideration for Optical Systems, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.891, 190–197 (1988).
    [CrossRef]
  25. J. P. McGuire, R. A. Chipman, “Polarization aberrations in the Solar Activity Measurement Experiments (SAMEX) solar vector magnetograph,” Opt. Eng. 28, 141–147 (1989).
    [CrossRef]
  26. 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, and Ross, Stroudsburg, Pa., 1975).
    [CrossRef]
  27. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  28. R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).
    [CrossRef]
  29. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  30. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).
  31. S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986).
  32. An electromagnetic field is considered monochromatic or quasi-monochromatic if it satisfies the conditions Δν/ν¯≪1(the quasi-monochromatic condition) and 1/Δν≫ d/c, where Δνis the bandwidth of the signal, ν¯is the mean frequency of the signal, dis the longest optical path length, and cis the speed of light.28
  33. G. B. Parrent, “Imaging of extended polychromatic sources and generalized transfer functions,” J. Opt. Soc. Am. 51, 143–151 (1961).
    [CrossRef]
  34. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  35. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  36. K. D. Abhyankar, A. L. Fymat, “Relation between the elements of the phase matrix for scattering,” J. Math Phys. 10, 1935–1938 (1969).
    [CrossRef]
  37. Jones vectors are usually written asU=[UxUy], where Uxand Uyare the projections of field in the xand ydirections, respectively. We use arbitrary basis polarization states in this paper to simplify calculations. Additional information on the Jones algebra with arbitrary basis states (especially circular) is given in Ref. 26.
  38. E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  39. If Aand Bare n× mmatrices, then direct products, direct correlations, and direct convolutions are defined as A⊗ B= Cwith cαβ= aijbkl, AB= Cwith cαβ= aij★ bkl, and A⊛ B= Cwith cαβ= akj* bkl. Here Cis a nn× mmmatrix, ★ denotes correlation, * denotes convolution, and the indices assume the values α= n(i− 1) + kand β= m(j− 1) − l.
  40. Wolf’s mutual coherence function Γ is a linear combination of mutual coherence vector elements, Γ = (co,0+ co,3)/2.

1989 (3)

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).
[CrossRef]

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

1988 (3)

1986 (2)

W. Urbańczyk, “Optical transfer functions for imaging systems which change the state of light polarization,” Opt. Acta 33, 53–62 (1986).
[CrossRef]

R. R. A. Syms, “Vector effects in holographic optical elements,” Opt. Acta 32, 1413–1425 (1986).
[CrossRef]

1985 (1)

1984 (1)

W. Urbańczyk, “Optical imaging systems changing the state of light polarization,” Optik 66, 301–309 (1984).

1982 (1)

W. Urbańczyk, “Optical imaging in polarized light,” Optik 63, 25–35 (1982).

1971 (1)

1969 (1)

K. D. Abhyankar, A. L. Fymat, “Relation between the elements of the phase matrix for scattering,” J. Math Phys. 10, 1935–1938 (1969).
[CrossRef]

1962 (1)

1961 (1)

1959 (2)

H. Kubota, S. Inoué, “Diffraction images in the polarizing microscope,” J. Opt. Soc. Am. 49, 191–198 (1959).
[CrossRef] [PubMed]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165 (1959).
[CrossRef]

1956 (1)

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

1941 (1)

1873 (1)

E. Abbe, “Beiträge zur theorie des Mikroskops und der mikroskopischer Wahrnehmung,” Arch. Mikrosk. Anat. 9, 413 (1873).
[CrossRef]

1856 (1)

L. Seidel, Astron. Nachr. 43, 289, 305, 321 (1856).
[CrossRef]

Abbe, E.

E. Abbe, “Beiträge zur theorie des Mikroskops und der mikroskopischer Wahrnehmung,” Arch. Mikrosk. Anat. 9, 413 (1873).
[CrossRef]

Abhyankar, K. D.

K. D. Abhyankar, A. L. Fymat, “Relation between the elements of the phase matrix for scattering,” J. Math Phys. 10, 1935–1938 (1969).
[CrossRef]

Azzam, R. M. A.

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

Bashara, N. M.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Breckinridge, J. B.

Chakraborty, A. K.

Chakraborty, B.

Chipman, L. J.

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

Chipman, R. A.

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

J. P. McGuire, R. A. Chipman, “Polarization aberrations in optical systems,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 240–257 (1987).

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

J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” in Polarization Considerations in Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 95–108 (1989).
[CrossRef]

Crosignani, B.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986).

Deffieux, P. M.

P. M. Deffieux, L’Intégral de Fourier et ses Application à l’Optique, 2nd ed. (Masson, Paris, 1970).

DiPorto, P.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986).

Fymat, A. L.

K. D. Abhyankar, A. L. Fymat, “Relation between the elements of the phase matrix for scattering,” J. Math Phys. 10, 1935–1938 (1969).
[CrossRef]

Gallaway, W. S.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Gosh, A.

Hansen, E. W.

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

Hopkins, H. H.

H. H. Hopkins, “Applications of coherence theory in microscopy and interferometry,” J. Opt. Soc. Am. 47, 508–526 (1956).
[CrossRef]

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, London, 1950).

Inoué, S.

Jones, R. C.

Kubota, H.

McGuire, J. P.

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

J. P. McGuire, R. A. Chipman, “Polarization aberrations in optical systems,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 240–257 (1987).

J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” in Polarization Considerations in Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 95–108 (1989).
[CrossRef]

Murata, K.

O’Neil, E. L.

E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Parrent, G. B.

Seidel, L.

L. Seidel, Astron. Nachr. 43, 289, 305, 321 (1856).
[CrossRef]

Simon, M. C.

Solimeno, S.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986).

Stewart, J. E.

Syms, R. R. A.

R. R. A. Syms, “Vector effects in holographic optical elements,” Opt. Acta 32, 1413–1425 (1986).
[CrossRef]

Thompson, B. J.

B. J. Thompson, “Image formation with partially coherent light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969).
[CrossRef]

Urbanczyk, W.

W. Urbańczyk, “Optical transfer functions for imaging systems which change the state of light polarization,” Opt. Acta 33, 53–62 (1986).
[CrossRef]

W. Urbańczyk, “Optical imaging systems changing the state of light polarization,” Optik 66, 301–309 (1984).

W. Urbańczyk, “Optical imaging in polarized light,” Optik 63, 25–35 (1982).

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Appl. Opt. (3)

Arch. Mikrosk. Anat. (1)

E. Abbe, “Beiträge zur theorie des Mikroskops und der mikroskopischer Wahrnehmung,” Arch. Mikrosk. Anat. 9, 413 (1873).
[CrossRef]

Astron. Nachr. (1)

L. Seidel, Astron. Nachr. 43, 289, 305, 321 (1856).
[CrossRef]

J. Math Phys. (1)

K. D. Abhyankar, A. L. Fymat, “Relation between the elements of the phase matrix for scattering,” J. Math Phys. 10, 1935–1938 (1969).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165 (1959).
[CrossRef]

Opt. Acta (2)

W. Urbańczyk, “Optical transfer functions for imaging systems which change the state of light polarization,” Opt. Acta 33, 53–62 (1986).
[CrossRef]

R. R. A. Syms, “Vector effects in holographic optical elements,” Opt. Acta 32, 1413–1425 (1986).
[CrossRef]

Opt. Eng. (3)

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

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

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

Optik (2)

W. Urbańczyk, “Optical imaging in polarized light,” Optik 63, 25–35 (1982).

W. Urbańczyk, “Optical imaging systems changing the state of light polarization,” Optik 66, 301–309 (1984).

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

Other (18)

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, London, 1950).

P. M. Deffieux, L’Intégral de Fourier et ses Application à l’Optique, 2nd ed. (Masson, Paris, 1970).

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

B. J. Thompson, “Image formation with partially coherent light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969).
[CrossRef]

J. P. McGuire, R. A. Chipman, “Analysis of spatial pseudodepolarizers in imaging systems,” in Polarization Considerations in Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 95–108 (1989).
[CrossRef]

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

J. P. McGuire, R. A. Chipman, “Polarization aberrations in optical systems,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 240–257 (1987).

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986).

An electromagnetic field is considered monochromatic or quasi-monochromatic if it satisfies the conditions Δν/ν¯≪1(the quasi-monochromatic condition) and 1/Δν≫ d/c, where Δνis the bandwidth of the signal, ν¯is the mean frequency of the signal, dis the longest optical path length, and cis the speed of light.28

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Jones vectors are usually written asU=[UxUy], where Uxand Uyare the projections of field in the xand ydirections, respectively. We use arbitrary basis polarization states in this paper to simplify calculations. Additional information on the Jones algebra with arbitrary basis states (especially circular) is given in Ref. 26.

E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

If Aand Bare n× mmatrices, then direct products, direct correlations, and direct convolutions are defined as A⊗ B= Cwith cαβ= aijbkl, AB= Cwith cαβ= aij★ bkl, and A⊛ B= Cwith cαβ= akj* bkl. Here Cis a nn× mmmatrix, ★ denotes correlation, * denotes convolution, and the indices assume the values α= n(i− 1) + kand β= m(j− 1) − l.

Wolf’s mutual coherence function Γ is a linear combination of mutual coherence vector elements, Γ = (co,0+ co,3)/2.

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Figures (21)

Fig. 1
Fig. 1

Circularly retarding lens used as an example in this paper. The cavity between two plano-convex lenses is filled with a circularly retarding material (such as corn syrup) to give different focal lengths for right and left light circularly polarized light.

Fig. 2
Fig. 2

Coordinate system used for the object, the pupil, and the image.

Fig. 3
Fig. 3

ARM element |h11(r, α)| with the image plane midway between the foci for right and left circularly polarized light. Five different aberration strengths are shown. The curve for no CRD, α = 0 (an Airy disk), is shown for comparison.

Fig. 4
Fig. 4

ARM element ϑ(r, α). The parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

Amplitude and polarization distribution of light in the exit pupil of a circularly retarding lens when vertically polarized light is incident. The lens has α = 2π of CRD.

Fig. 6
Fig. 6

Amplitude distribution of light in the image plane of a circularly retarding lens. This is the far-field amplitude distribution produced by the light in Fig. 5.

Fig. 7
Fig. 7

Displacement of the best foci by a circularly retarding lens. The primary aberration is a polarization-dependent defocus that is even and distributes the multiple images along the optic axis.

Fig. 8
Fig. 8

Displacement of the best foci by a pair of birefringent lenses. The primary aberration is a polarization-dependent defocus that is even and distributes the multiple images along the optic axis. H, fast axis (horizontal); 45, fast axis at 45°.

Fig. 9
Fig. 9

Displacement of the best foci by a Cornu pseudodepolarizer. The primary aberration is a polarization-dependent tilt aberration that is odd and distributes the multiple images transverse to the optic axis. R, right circularly retarding wedge; L, left circularly retarding wedge.

Fig. 10
Fig. 10

Focal planes A and B at which the PSM and the OTM are calculated.

Fig. 11
Fig. 11

PSM element ps,11 = ps,44 with the image plane at the focus for right circularly polarized light. The curves are the average of the point-spread functions for a defocused image (left circular polarized) and a defocused image (right circular polarized).

Fig. 12
Fig. 12

PSM element ps,14 = −ps,41. The curves are the difference between the point-spread functions for a defocused image (left circular polarized) and a focused image (right circular polarized). The parameters are the same as in Fig. 11.

Fig. 13
Fig. 13

Intensity distribution of the linearly polarized component o(r, α). The parameters are the same as in Fig. 11.

Fig. 14
Fig. 14

Rotation ω(r, α) of the linearly polarized component. The parameters are the same as in Fig. 11.

Fig. 15
Fig. 15

OTM element a(ρ, α) versus the spatial frequency. The parameters are the same as in Fig. 3.

Fig. 16
Fig. 16

OTM element b(ρ, α) versus the spatial frequency. The parameters are the same as in Fig. 3.

Fig. 17
Fig. 17

OTM element β(ρ, α) versus the spatial frequency. The parameters are the same as in Fig. 3.

Fig. 18
Fig. 18

OTM element h ˜ s , 11 = h ˜ s , 44. The curves are the average of the optical-transfer functions for a defocused image (left circular polarized) and a focused image (right circular polarized). The parameters are the same as in Fig. 11.

Fig. 19
Fig. 19

OTM element h ˜ s , 14 = h ˜ s , 41. The curves are the difference of the optical-transfer functions for a defocused image (left circular polarized) and a focused image (right circular polarized). The parameters are the same as in Fig. 11.

Fig. 20
Fig. 20

OTM element O(ρ, α) showing the transfer of the spatial frequencies of the linearly polarized component. The parameters are the same as in Fig. 11.

Fig. 21
Fig. 21

Rotation of the linearly polarized component Ω(ρ, α) as a function of spatial frequency. The parameters are the same as in Fig. 11.

Tables (2)

Tables Icon

Table 1 Operators and Operands for Image Analysis in Optical Systems with Polarization Aberrations

Tables Icon

Table 2 Stokes Parameters and Their Significance

Equations (84)

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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 ) ]
J s y s ( 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 ] ,
j circ ( h , ρ ) = P ( h , ρ ) [ exp [ j k Δ t ( h , ρ ) / 2 ] 0 0 exp [ j k Δ t ( h , ρ ) / 2 ] ] ,
J cart ( h , ρ ) = P ( h , ρ ) [ cos [ k Δ t ( h , ρ ) / 2 ] sin [ k Δ t ( h , ρ ) / 2 ] sin [ k Δ t ( h , ρ ) / 2 ] cos [ k Δ t ( h , ρ ) / 2 ] ] ,
Δ = n r n l .
J cart = F 1 J circ F ,
F = 1 2 [ 1 1 j j ] .
t ( h , ρ ) = t 0 ρ 2 2 ( 1 R 1 1 R 2 ) ,
J circ ( ρ , α ) = [ j 11 0 0 j 11 * ] ,
j 11 = a P ( ρ ) exp ( j α ρ 2 2 )
P ( ρ ) = { 1 ρ < 1 0 ρ 1 ,
α = k Δ 2 ( 1 R 1 1 R 2 ) ,
a = exp ( j k Δ t 0 2 ) .
J sys , circ ( ρ , α ) = J circ ( ρ , α ) ( j k ρ 2 2 f ) ,
J sys , circ ( ρ , α ) = [ P ( ρ ) 0 0 P ( ρ ) exp ( j α ρ 2 ) ] exp ( j k ρ 2 2 f r ) ,
1 f r = 1 f Δ 2 ( 1 R 1 1 R 2 ) ,
M ( ρ , α ) = [ 1 0 0 0 0 cos α ρ 2 sin α ρ 2 0 0 sin α ρ 2 cos α ρ 2 0 0 0 0 1 ] ,
U l ( x , y ) = j A λ Λ 1 exp ( j k r 1 ) r 1 U o ( ξ , η ) d ξ d η ,
U l ( x , y ) = j A λ exp [ j k 2 f ( x 2 + y 2 ) ] × J ( ξ , η ; x , y ) Λ 1 exp ( j k r 1 ) r 1 U o ( ξ , η ) d ξ d η .
U i ( u , υ ) = A 2 4 λ 2 Λ 2 exp ( j k r 2 ) r 2 exp [ j k ( x 2 + y 2 ) 2 f ] × J ( ξ , η ; x , y ) Λ 1 exp ( j k r 1 ) r 1 U o ( ξ , η ) d ξ d η d x d y ,
r 1 = z 1 + ( x ξ ) 2 + ( y η ) 2 2 z 1 ,
r 2 = z 2 + ( x u ) 2 + ( y υ ) 2 2 z 2 ,
1 f = 1 z 1 + 1 z 2 .
U i ( u , υ ) = C exp [ j k 2 z 2 ( u 2 + υ 2 ) ] [ j k 2 z 1 ( ξ 2 + η 2 ) ] × exp [ j k x ( ξ z 1 + u z 2 ) j k y ( η z 1 + υ z 2 ) ] × J ( ξ , η ; x , y ) U o ( ξ , η ) d ξ d η d x d y ,
C = A 2 4 λ 2 z 1 z 2 exp [ j k ( z 1 + z 2 ) ] .
h ( ξ , η ; u , υ ) = C exp [ j k Ψ ( ξ , η ; u , υ ) ] × exp [ j k x z 2 ( M ξ + u ) j k y z 2 ( M η + υ ) ] × J ( ξ , η ; x , y ) d x d y = C exp [ j k Ψ ( ξ , η ; u , υ ) ] [ F { j 11 } F { j 12 } F { j 21 } F { j 22 } ] f u , f υ ,
f u = M ξ + u λ z 2 ,
f υ = M η + υ λ z 2 ,
Ψ ( ξ , η ; u , υ ) = ξ 2 + η 2 2 z 1 + u 2 + υ 2 2 z 2 .
U i ( u , υ ) = h ( ξ , η ; u , υ ) U o ( ξ , η ) d ξ d η .
U i ( u , υ , t ) = C h ( ξ , η ; u , υ ) U o ( ξ , η , t τ ) d ξ d η ,
f ( ξ , η ; u , υ ) = f ( ξ u , η υ ) .
U i = h * U o ,
h circ ( r , α ) = | h 11 ( r , α ) | [ exp [ j ϑ ( r , α ) / 2 ] 0 0 exp [ j ϑ ( r , α ) / 2 ] ] ,
h 11 ( r , α ) = F { j 11 } = 2 π a 0 1 exp ( j α ρ 2 / 2 ) j 0 ( 2 π ρ r ) ρ d ρ ,
ϑ ( r , α ) = 2 arg [ h 11 ( r , α ) ] = 2 tan 1 [ 0 1 sin ( α ρ 2 / 2 ) J 0 ( 2 π ρ r ) ρ d ρ 0 1 cos ( α ρ 2 / 2 ) J 0 ( 2 π ρ r ) ρ d ρ ] ,
U ( t ) = ( U q ( t ) U r ( t ) ) ,
c i ( u , υ ) = U i ( u , υ , t ) U i * ( u , υ , t ) = [ U i q ( u , υ , t ) U i q * ( u , υ , t ) U i q ( u , υ , t ) U i r * ( u , υ , t ) U i r ( u , υ , t ) U i q * ( u , υ , t ) U i r ( u , υ , t ) U i r * ( u , υ , t ) ] ,
f ( t ) lim T 1 2 T T T f ( t ) d t .
c i ( u , υ ) = h ( ξ 1 , η 1 ; u , υ ) U o ( ξ 1 , η 1 ; t ) h * ( ξ 2 , η 2 ; u , υ ) U o * ( ξ 2 , η 2 ; t ) d ξ 1 d η 1 d ξ 2 d η 2 = P ( ξ 1 , η 1 ; ξ 2 , η 2 ; u , υ ) × c 0 ( ξ 1 , η 1 ; ξ 2 , η 2 ) d ξ 1 d η 1 d ξ 2 d η 2 ,
AB A B = ( A A ) ( B B ) ,
c o ( ξ 1 , η 1 ; ξ 2 , η 2 ) = U o ( ξ 1 , η 1 , t ) U 0 * ( ξ 2 , η 2 , t ) ,
P ( ξ 1 , η 1 ; ξ 2 ; η 2 ; u , υ ) = h ( ξ 1 , η 1 ; u , υ ) h * ( ξ 2 , η 2 ; u , υ ) .
s = S c ,
S Cart = [ 1 0 0 1 1 0 0 1 0 1 1 0 0 j j 0 ] ,
S circ = [ 1 0 0 1 1 1 1 0 0 j j 0 1 0 0 1 ] .
P s = SPS 1 .
c o ( ξ 1 , η 1 ; ξ 2 , η 2 ) = c o ( ξ 1 , η 1 ) δ ( ξ 1 ξ 2 , η 1 η 2 ) ,
c i ( u , υ ) = P ( u ξ 1 , υ η 1 ) c o ( ξ 1 , η 1 ) d ξ 1 d η 1 = P * c o .
s i 0 = p s 11 * s o 0 + p s 12 * s o 1 + p s 13 * s o 2 + p s 14 * s o 4 ,
Det J = | t | 2 ,
Det J < 1 ,
s o ( ξ , η ) = [ | t ( ξ , η ) | 2 0 0 0 ] , c o ( ξ , η ) = 1 2 [ | t ( ξ , η ) | 2 0 0 | t ( ξ , η ) | 2 ] ,
s i ( ξ , η ) = [ p s 11 * | t | 2 p s 21 * | t | 2 p s 31 * | t | 2 p s 41 * | t | 2 ] , c i ( ξ , η ) = 1 2 [ ( p 11 + p 14 ) * | t | 2 ( p 21 + p 24 ) * | t | 2 ( p 31 + p 34 ) * | t | 2 ( p 41 + p 44 ) * | t | 2 ] ,
c i ( ξ , η ) = [ | h 11 | 2 + | h 12 | 2 h 11 h 21 * + h 12 h 22 * h 11 * h 21 + h 12 * h 22 | h 21 | 2 + | h 22 | 2 ] * | t | 2 ,
s i , Cart ( ξ , η ) = 1 2 [ | h 11 | 2 + | h 12 | 2 + | h 21 | 2 + | h 22 | 2 | h 11 | 2 + | h 12 | 2 | H 21 | 2 | h 22 | 2 2 Re [ h 11 * h 21 + h 12 * h 22 ] 2 Im [ h 11 * h 21 + h 12 * h 22 ] ] * | t | 2 ,
s i , circ ( ξ , η ) = 1 2 [ | h 11 | 2 + | h 12 | 2 + | h 21 | 2 + | h 22 | 2 2 Re [ h 11 * h 21 + h 12 * h 22 ] 2 Im [ h 11 * h 21 + h 12 * h 22 ] | h 11 | 2 + | h 12 | 2 | h 21 | 2 | h 22 | 2 ] * | t | 2 .
C i = F { c i } = F { P * c o } = H ˜ C o ,
H ˜ ( ν u , ν υ ) = F { P } N = F { h h * } N = H ( p , q ) H * ( p λ ¯ z 2 ν u , q λ ¯ z 2 ν υ ) d p d q N ,
N = 1 2 [ | H 11 ( p , q ) | 2 + | H 12 ( p , q ) | 2 + | H 21 ( p , q ) | 2 + | H 22 ( p , q ) | 2 ] d p d q ,
H ˜ ( ν u , ν υ ) = 2 J ( p , q ) J * ( p λ ¯ z 2 ν u , q λ ¯ z 2 ν υ ) d p d q N ,
N = [ | j 11 ( p , q ) | 2 + j 12 ( p , q ) | 2 + | j 21 ( p , q ) | 2 + | j 22 ( p , q ) | 2 ] d p d q .
H ˜ = J ( x , y ) J ( x , y ) N
H ˜ = J ( x , y ) J * ( x , y ) N ,
H ˜ ( ν u , ν υ ) = P ˜ ( p , q ) P ˜ * ( p λ ¯ z 2 υ u , q λ ¯ z 2 ν υ ) d p d q | P ˜ ( p , q ) | 2 d p d q × [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
P = h h * = | h 11 ( r , α ) | 2 [ 1 0 0 0 0 exp [ j ϑ ( r , α ) ] 0 1 0 0 exp [ j ϑ ( r , α ) ] 0 0 0 0 1 ] ,
P s = S circ PS circ 1 = | h 11 ( r , α ) | 2 [ 1 0 0 0 0 cos ϑ ( r , α ) sin ϑ ( r , α ) 0 0 sin ϑ ( r , α ) cos ϑ ( r , α ) 0 0 0 0 1 ] .
s i 0 ( r , α ) = | h 11 ( r , α ) | 2 * s o 0 ( r , α ) ,
P s = 1 2 [ | h 11 ( r , 0 ) | 2 + | h 11 ( r , 2 α ) | 2 0 0 | h 11 ( r , 0 ) | 2 + | h 11 ( r , 2 α ) | 2 0 o ( r , α ) cos ω ( r , α ) o ( r , α ) sin ( r , α ) 0 0 o ( r , α ) sin ω ( r , α ) o ( r , α ) cos ω ( r , α ) 0 | h 11 ( r , 0 ) | 2 | h 11 ( r , 2 α ) | 2 0 0 | h 11 ( r , 0 ) | 2 + | h 11 ( r , 2 α ) | 2 ] ,
o ( r , α ) = 2 π | F { exp ( j α ρ 2 ) P ( ρ ) } F { P ( ρ ) } | ,
ω ( r , α ) = arg [ F { exp ( j α ρ 2 ) P ( ρ ) } F { P ( ρ ) } ] .
s i 0 ( r , α ) = 1 2 [ | h 11 ( r , 0 ) | 2 * ( s o 0 + s o 3 ) + | h 11 ( r , 2 α ) | 2 * ( s o 0 + s o 3 ) ] ,
H ˜ = 1 2 [ | j 11 ( p , q ) | 2 d p d q × [ j 11 * j 11 * 0 0 0 j 11 * j 11 0 0 0 0 0 0 j 11 * * j 11 ] .
H ˜ s = S circ H ˜ S circ 1 = [ a ( ρ , α ) 0 0 0 0 b ( ρ , α ) cos β ( ρ , α ) b ( ρ , α ) sin β ( ρ , α ) 0 0 b ( ρ , α ) sin β ( ρ , α ) b ( ρ , α ) cos β ( ρ , α ) 0 0 0 0 a ( ρ , α ) ] ,
a ( ρ , α ) = j 11 * j 11 * π ,
b ( ρ , α ) = | j 11 * j 11 | π ,
β ( ρ , ρ ) = tan 1 ( Im j 11 * j 11 Re j 11 * j 11 ) ,
S i 0 = a ( ρ , α ) S o 0 ,
H ˜ s = 1 2 [ a ( ρ , 0 ) + a ( ρ , 2 α ) 0 0 a ( ρ , 0 ) a ( ρ , 2 α ) 0 O ( ρ , α ) cos Ω ( ρ , α ) O ( ρ , α ) sin Ω ( ρ , α ) 0 0 O ( ρ , α ) sin Ω ( ρ , α ) O ( ρ , α ) cos Ω ( ρ , α ) 0 a ( ρ , 0 ) a ( ρ , 2 α ) 0 0 a ( ρ , 0 ) + a ( ρ , 2 α ) ] ,
O ( ρ , α ) = 2 | exp ( j α ρ 2 ) P ( ρ ) * P ( ρ ) | π ,
Ω ( ρ , α ) = arg [ exp ( j α ρ 2 ) P ( ρ ) * P ( ρ ) ] .
U=[UxUy],

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