Abstract

We report a systematic investigation of the imaging behavior of an optical system consisting of a lens from a uniaxial birefringent crystal sandwiched between two linear polarizers into which primary spherical aberration has been introduced. The proposed system has higher tolerance to primary spherical aberration and has a larger depth of focus than an imaging system found with an isotropic lens. Some specific cases are computed and illustrated graphically.

© 2002 Optical Society of America

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  1. J. Tsujiuchi, “A density filter improving aberrant optical image,” J. Phys. Soc. Jpn. 12, 744–752 (1957).
    [CrossRef]
  2. T. Asakura, R. Barakat, “Annular and annulus apertures with spherical aberration and defocusing,” Oyo Butsuri 30, 728–735 (1961).
  3. T. Asakura, “Axial intensity distribution for an annular aperture with primary spherical aberration,” Oyo Butsuri 31, 243–244 (1962).
  4. T. Asakura, H. Mishina, “Irradiance distribution in the diffraction patterns of an annual aperture with spherical aberration and coma,” Jpn. J. Appl. Phys. 7, 751–758 (1968).
    [CrossRef]
  5. T. Asakura, H. Mishina, “Three dimensional distribution in the diffraction patterns of annular apertures with primary spherical aberration,” Oyo Butsuri 37, 805–809 (1968).
  6. M. Mino, Y. Okano, “Improvement in the OTF of a defocused optical system through the use of shaded apertures,” Appl. Opt. 10, 2219–2225 (1971).
    [CrossRef] [PubMed]
  7. S. C. Biswas, A. Boivin, “Influence of primary astigmatism on the performance of optimum apodizers,” J. Opt. (Paris) 4, 1–5 (1975).
  8. S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
    [CrossRef]
  9. S. C. Biswas, A. Boivin, “Performance of optimum apodizers in the presence of primary coma,” Can. J. Phys. 57, 1388–1396 (1979).
    [CrossRef]
  10. M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
    [CrossRef]
  11. M. J. Yzuel, F. Calvo, “Point-spread function calculation for optical systems with residual aberrations and a non-uniform transmission pupil,” Opt. Acta 30, 233–242 (1983).
    [CrossRef]
  12. J. Ojeda-Castaneda, P. Andres, A. Diaz, “Annular apodizers for low sensitivity to defocus and to spherical aberration,” Opt. Lett. 11, 487–489 (1986).
    [CrossRef] [PubMed]
  13. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Line-spread function relatively insensitive to defocus,” Opt. Lett. 8, 458–460 (1983).
    [CrossRef] [PubMed]
  14. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett. 10, 520–522 (1985).
    [CrossRef] [PubMed]
  15. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Bessel annular apodizers: imaging characteristics,” Appl. Opt. 26, 2770–2772 (1987).
    [CrossRef] [PubMed]
  16. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Arbitrarily high focal depth with finite apertures,” Opt. Lett. 13, 183–185 (1988).
    [CrossRef] [PubMed]
  17. J. Ojeda-Castaneda, A. Diaz, “High focal depth by quasibifocus,” Appl. Opt. 27, 4163–4165 (1988).
    [CrossRef]
  18. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790–795 (1988).
    [CrossRef] [PubMed]
  19. J. Ojeda-Castaneda, E. Tepichin, A. Pons, “Apodization of annual apertures: Strehl ratio,” Appl. Opt. 27, 5140–5145 (1988).
    [CrossRef] [PubMed]
  20. J. Ojeda-Castaneda, C. M. Gomez-Sarabia, “Focal depth: optimum annular apodizer,” Appl. Opt. 28, 4263–4264 (1989).
    [CrossRef] [PubMed]
  21. J. Ojeda-Castaneda, E. Tepichin, A. Diaz, “Arbitrarily high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
    [CrossRef] [PubMed]
  22. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 (1990).
    [CrossRef] [PubMed]
  23. J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 131–180.
    [CrossRef]
  24. P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 29–186.
    [CrossRef]
  25. A. K. Chakraborty, S. Das, D. K. Basu, A. Ghosh, “Imaging characteristics of a birefringent lens,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 130–134 (1990).
    [CrossRef]
  26. S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
    [CrossRef]
  27. S. Sanyal, A. Ghosh, “Simulation of complex masks on the lens aperture using a birefringent lens,” Opt. Optoelectron. 1, 656–661 (1998).
  28. S. Sanyal, A. Ghosh, “Imaging characteristics of birefringent lenses under focused and defocused conditions,” Optik 110, 513–520 (1999).
  29. S. Sanyal, A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000).
    [CrossRef]
  30. S. Sanyal, A. Ghosh, “Imaging behaviour of a birefringent lens suffering from primary coma,” J. Opt. (India) 29, 15–24 (2000).
  31. S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. (to be published).
  32. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).
  33. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, 1950).
  34. V. N. Mahajan, Aberration Theory Made Simple (SPIE Optical Eng. Press, Bellingham, Wash.1991).
    [CrossRef]
  35. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A231, 91–103 (1955).
    [CrossRef]
  36. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  37. R. Barakat, A. Houston, “Reciprocity relations between the transfer function and total illuminance,” J. Opt. Soc. Am. 53, 1244–1249 (1963).
    [CrossRef]
  38. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

2000 (2)

S. Sanyal, A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000).
[CrossRef]

S. Sanyal, A. Ghosh, “Imaging behaviour of a birefringent lens suffering from primary coma,” J. Opt. (India) 29, 15–24 (2000).

1999 (1)

S. Sanyal, A. Ghosh, “Imaging characteristics of birefringent lenses under focused and defocused conditions,” Optik 110, 513–520 (1999).

1998 (2)

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

S. Sanyal, A. Ghosh, “Simulation of complex masks on the lens aperture using a birefringent lens,” Opt. Optoelectron. 1, 656–661 (1998).

1990 (1)

1989 (2)

1988 (4)

1987 (1)

1986 (1)

1985 (1)

1983 (2)

M. J. Yzuel, F. Calvo, “Point-spread function calculation for optical systems with residual aberrations and a non-uniform transmission pupil,” Opt. Acta 30, 233–242 (1983).
[CrossRef]

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Line-spread function relatively insensitive to defocus,” Opt. Lett. 8, 458–460 (1983).
[CrossRef] [PubMed]

1979 (2)

S. C. Biswas, A. Boivin, “Performance of optimum apodizers in the presence of primary coma,” Can. J. Phys. 57, 1388–1396 (1979).
[CrossRef]

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

1976 (1)

S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
[CrossRef]

1975 (1)

S. C. Biswas, A. Boivin, “Influence of primary astigmatism on the performance of optimum apodizers,” J. Opt. (Paris) 4, 1–5 (1975).

1971 (1)

1968 (2)

T. Asakura, H. Mishina, “Irradiance distribution in the diffraction patterns of an annual aperture with spherical aberration and coma,” Jpn. J. Appl. Phys. 7, 751–758 (1968).
[CrossRef]

T. Asakura, H. Mishina, “Three dimensional distribution in the diffraction patterns of annular apertures with primary spherical aberration,” Oyo Butsuri 37, 805–809 (1968).

1963 (1)

1962 (1)

T. Asakura, “Axial intensity distribution for an annular aperture with primary spherical aberration,” Oyo Butsuri 31, 243–244 (1962).

1961 (1)

T. Asakura, R. Barakat, “Annular and annulus apertures with spherical aberration and defocusing,” Oyo Butsuri 30, 728–735 (1961).

1957 (1)

J. Tsujiuchi, “A density filter improving aberrant optical image,” J. Phys. Soc. Jpn. 12, 744–752 (1957).
[CrossRef]

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A231, 91–103 (1955).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Andres, P.

Asakura, T.

T. Asakura, H. Mishina, “Irradiance distribution in the diffraction patterns of an annual aperture with spherical aberration and coma,” Jpn. J. Appl. Phys. 7, 751–758 (1968).
[CrossRef]

T. Asakura, H. Mishina, “Three dimensional distribution in the diffraction patterns of annular apertures with primary spherical aberration,” Oyo Butsuri 37, 805–809 (1968).

T. Asakura, “Axial intensity distribution for an annular aperture with primary spherical aberration,” Oyo Butsuri 31, 243–244 (1962).

T. Asakura, R. Barakat, “Annular and annulus apertures with spherical aberration and defocusing,” Oyo Butsuri 30, 728–735 (1961).

Bandyopadhyay, P.

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

Barakat, R.

R. Barakat, A. Houston, “Reciprocity relations between the transfer function and total illuminance,” J. Opt. Soc. Am. 53, 1244–1249 (1963).
[CrossRef]

T. Asakura, R. Barakat, “Annular and annulus apertures with spherical aberration and defocusing,” Oyo Butsuri 30, 728–735 (1961).

Basu, D. K.

A. K. Chakraborty, S. Das, D. K. Basu, A. Ghosh, “Imaging characteristics of a birefringent lens,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 130–134 (1990).
[CrossRef]

Berriel-Valdos, L. R.

Biswas, S. C.

S. C. Biswas, A. Boivin, “Performance of optimum apodizers in the presence of primary coma,” Can. J. Phys. 57, 1388–1396 (1979).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of primary astigmatism on the performance of optimum apodizers,” J. Opt. (Paris) 4, 1–5 (1975).

Boivin, A.

S. C. Biswas, A. Boivin, “Performance of optimum apodizers in the presence of primary coma,” Can. J. Phys. 57, 1388–1396 (1979).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of primary astigmatism on the performance of optimum apodizers,” J. Opt. (Paris) 4, 1–5 (1975).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).

Calvo, F.

M. J. Yzuel, F. Calvo, “Point-spread function calculation for optical systems with residual aberrations and a non-uniform transmission pupil,” Opt. Acta 30, 233–242 (1983).
[CrossRef]

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

Chakraborty, A. K.

A. K. Chakraborty, S. Das, D. K. Basu, A. Ghosh, “Imaging characteristics of a birefringent lens,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 130–134 (1990).
[CrossRef]

Das, S.

A. K. Chakraborty, S. Das, D. K. Basu, A. Ghosh, “Imaging characteristics of a birefringent lens,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 130–134 (1990).
[CrossRef]

Diaz, A.

Ghosh, A.

S. Sanyal, A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000).
[CrossRef]

S. Sanyal, A. Ghosh, “Imaging behaviour of a birefringent lens suffering from primary coma,” J. Opt. (India) 29, 15–24 (2000).

S. Sanyal, A. Ghosh, “Imaging characteristics of birefringent lenses under focused and defocused conditions,” Optik 110, 513–520 (1999).

S. Sanyal, A. Ghosh, “Simulation of complex masks on the lens aperture using a birefringent lens,” Opt. Optoelectron. 1, 656–661 (1998).

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

A. K. Chakraborty, S. Das, D. K. Basu, A. Ghosh, “Imaging characteristics of a birefringent lens,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 130–134 (1990).
[CrossRef]

S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. (to be published).

Gomez-Sarabia, C. M.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A231, 91–103 (1955).
[CrossRef]

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

Houston, A.

Jacquinot, P.

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

Mahajan, V. N.

V. N. Mahajan, Aberration Theory Made Simple (SPIE Optical Eng. Press, Bellingham, Wash.1991).
[CrossRef]

Mino, M.

Mishina, H.

T. Asakura, H. Mishina, “Three dimensional distribution in the diffraction patterns of annular apertures with primary spherical aberration,” Oyo Butsuri 37, 805–809 (1968).

T. Asakura, H. Mishina, “Irradiance distribution in the diffraction patterns of an annual aperture with spherical aberration and coma,” Jpn. J. Appl. Phys. 7, 751–758 (1968).
[CrossRef]

Montes, E.

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 (1990).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, E. Tepichin, A. Diaz, “Arbitrarily high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. Opt. 28, 2666–2670 (1989).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, C. M. Gomez-Sarabia, “Focal depth: optimum annular apodizer,” Appl. Opt. 28, 4263–4264 (1989).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, A. Diaz, “High focal depth by quasibifocus,” Appl. Opt. 27, 4163–4165 (1988).
[CrossRef]

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790–795 (1988).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, E. Tepichin, A. Pons, “Apodization of annual apertures: Strehl ratio,” Appl. Opt. 27, 5140–5145 (1988).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Arbitrarily high focal depth with finite apertures,” Opt. Lett. 13, 183–185 (1988).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Bessel annular apodizers: imaging characteristics,” Appl. Opt. 26, 2770–2772 (1987).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, P. Andres, A. Diaz, “Annular apodizers for low sensitivity to defocus and to spherical aberration,” Opt. Lett. 11, 487–489 (1986).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett. 10, 520–522 (1985).
[CrossRef] [PubMed]

J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Line-spread function relatively insensitive to defocus,” Opt. Lett. 8, 458–460 (1983).
[CrossRef] [PubMed]

Okano, Y.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Pons, A.

Roizen-Dossier, B.

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

Sanyal, S.

S. Sanyal, A. Ghosh, “Imaging behaviour of a birefringent lens suffering from primary coma,” J. Opt. (India) 29, 15–24 (2000).

S. Sanyal, A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000).
[CrossRef]

S. Sanyal, A. Ghosh, “Imaging characteristics of birefringent lenses under focused and defocused conditions,” Optik 110, 513–520 (1999).

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

S. Sanyal, A. Ghosh, “Simulation of complex masks on the lens aperture using a birefringent lens,” Opt. Optoelectron. 1, 656–661 (1998).

S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. (to be published).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Tepichin, E.

Tsujiuchi, J.

J. Tsujiuchi, “A density filter improving aberrant optical image,” J. Phys. Soc. Jpn. 12, 744–752 (1957).
[CrossRef]

J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 131–180.
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).

Yzuel, M. J.

M. J. Yzuel, F. Calvo, “Point-spread function calculation for optical systems with residual aberrations and a non-uniform transmission pupil,” Opt. Acta 30, 233–242 (1983).
[CrossRef]

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

Appl. Opt. (9)

Can. J. Phys. (1)

S. C. Biswas, A. Boivin, “Performance of optimum apodizers in the presence of primary coma,” Can. J. Phys. 57, 1388–1396 (1979).
[CrossRef]

J. Opt. (India) (1)

S. Sanyal, A. Ghosh, “Imaging behaviour of a birefringent lens suffering from primary coma,” J. Opt. (India) 29, 15–24 (2000).

J. Opt. (Paris) (1)

S. C. Biswas, A. Boivin, “Influence of primary astigmatism on the performance of optimum apodizers,” J. Opt. (Paris) 4, 1–5 (1975).

J. Opt. Soc. Am. (1)

J. Phys. Soc. Jpn. (1)

J. Tsujiuchi, “A density filter improving aberrant optical image,” J. Phys. Soc. Jpn. 12, 744–752 (1957).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Asakura, H. Mishina, “Irradiance distribution in the diffraction patterns of an annual aperture with spherical aberration and coma,” Jpn. J. Appl. Phys. 7, 751–758 (1968).
[CrossRef]

Opt. Acta (3)

M. J. Yzuel, F. Calvo, “A study of the possibility of image optimization by apodization filters in optical systems with residual aberrations,” Opt. Acta 26, 1397–1406 (1979).
[CrossRef]

M. J. Yzuel, F. Calvo, “Point-spread function calculation for optical systems with residual aberrations and a non-uniform transmission pupil,” Opt. Acta 30, 233–242 (1983).
[CrossRef]

S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
[CrossRef]

Opt. Eng. (1)

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

Opt. Lett. (4)

Opt. Optoelectron. (1)

S. Sanyal, A. Ghosh, “Simulation of complex masks on the lens aperture using a birefringent lens,” Opt. Optoelectron. 1, 656–661 (1998).

Optik (1)

S. Sanyal, A. Ghosh, “Imaging characteristics of birefringent lenses under focused and defocused conditions,” Optik 110, 513–520 (1999).

Oyo Butsuri (3)

T. Asakura, H. Mishina, “Three dimensional distribution in the diffraction patterns of annular apertures with primary spherical aberration,” Oyo Butsuri 37, 805–809 (1968).

T. Asakura, R. Barakat, “Annular and annulus apertures with spherical aberration and defocusing,” Oyo Butsuri 30, 728–735 (1961).

T. Asakura, “Axial intensity distribution for an annular aperture with primary spherical aberration,” Oyo Butsuri 31, 243–244 (1962).

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

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A231, 91–103 (1955).
[CrossRef]

Other (9)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. (to be published).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).

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

V. N. Mahajan, Aberration Theory Made Simple (SPIE Optical Eng. Press, Bellingham, Wash.1991).
[CrossRef]

J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1963), Vol. II, pp. 131–180.
[CrossRef]

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

A. K. Chakraborty, S. Das, D. K. Basu, A. Ghosh, “Imaging characteristics of a birefringent lens,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 130–134 (1990).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

The proposed system.

Fig. 2
Fig. 2

Variation of SR with the primary spherical aberration for several birefringent lenses: 1, clear-aperture isotropic lens; 2, α = 0.285λ; 3, α = 0.3142λ; 4, α = 0.6044λ, 5, α = 0.9λ. 2a–5a, Parallel-polarizer configuration (solid curves); 2b–5b, crossed-polarizer configuration (dashed curves).

Fig. 3
Fig. 3

Imaging characteristics of the proposed system for α = 0.285λ with various amounts of primary spherical aberration [at the corresponding best focal plane (BFP)]. Dashed curves, W 40 = 0, solid curves, W 40 = λ. 1, Clear-aperture isotropic lens (BFP, Gaussian image plane for W 40 = 0 and W 20 = -λ for W 40 = λ). 2, Birefringent lens with parallel-polarizer configuration (BFP, Gaussian image plane for W 40 = 0 and W 20 = -1.35λ for W 40 = λ). 3, Birefringent lens with crossed-polarizer configuration (BFP, Gaussian image plane for W 40 = 0 and W 20 = -0.83λ for W 40 = λ). (a) IPSF and FEE, (b) OTF, (c) axial irradiance distribution.

Fig. 4
Fig. 4

Imaging characteristics of the proposed system for α = 0.3142λ with various amounts of primary spherical aberration (at the corresponding BFP). Dashed curves, W 40 = 0, solid curves, W 40 = 1.08λ (for this amount of aberration the SR is the maximum for this birefringent lens with parallel-polarizer configuration). 1, Clear-aperture isotropic lens (BFP, Gaussian image plane for W 40 = 0 and W 20 = -1.08λ for W 40 = 1.08λ). 2, Birefringent lens with parallel-polarizer configuration (BFP, Gaussian image plane for W 40 = 0 and W 20 = -1.55λ for W 40 = 1.08λ). 3, Birefringent lens with crossed-polarizer configuration (BFP, Gaussian image plane for W 40 = 0 and W 20 = -0.9λ for W 40 = 1.08λ). (a) IPSF and FEE, (b) OTF, (c) axial irradiance distribution.

Fig. 5
Fig. 5

Imaging characteristics of the proposed system for α = 0.6044λ with various amounts of primary spherical aberration (at the corresponding BFP). Dashed curves, W 40 = 0; and solid curves, W 40 = 1.38λ (for this amount of aberration the SR is the maximum for this birefringent lens with crossed-polarizer configuration). 1, Clear-aperture isotropic lens (BFP, Gaussian image plane for W 40 = 0 and W 20 = -1.38λ for W 40 = 1.38λ). 2, Birefringent lens with parallel-polarizer configuration (BFP, W 20 = ±0.69λ for W 40 = 0 and W 20 = -0.73λ for W 40 = 1.38λ). 3, Birefringent lens with crossed-polarizer configuration (BFP, Gaussian image plane for W 40 = 0 and W 20 = -1.75λ for W 40 = 1.38λ). (a) IPSF and FEE, (b) OTF, (c) axial irradiance distribution.

Fig. 6
Fig. 6

Imaging characteristics of the proposed system for α = 0.9λ with primary spherical aberration W 40 = 0 and W 40 = 2.5λ (at the corresponding BFP). Dashed curves, W 40 = 0; solid curves, W 40 = 2.5λ (for this amount of aberration the SR is the maximum for this birefringent lens with parallel-polarizer configuration). 1, Clear-aperture isotropic lens (BFP, Gaussian image plane for W 40 = 0 and W 20 = -3.5λ for W 40 = 2.5λ). 2, Birefringent lens with parallel-polarizer configuration (BFP, W 20 = 1.05λ for W 40 = 0 and W 20 = -2.35λ for W 40 = 2.5λ). (a) IPSF and FEE, (b) OTF, (c) axial irradiance distribution.

Equations (14)

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f-+r, α=cos/sinkα1-r2,
I-+ρ, α, W20, W40=401cos/sinkα 1-r2exp× kW20r2 +W40r4J0ρrr dr2,
OTFs=-- fx+s2, yf* x-s2, ydxdy-- fx, yf*x, ydxdy,
N-+α= θ=02πr=01 fr, θf*r, θ rdrdθ=π21±sinc4αλ,
OTFs, α, W20, W40=1N-+αx=01-s/2 1-x+s/221/2× y=0 Qx, y, α, W20, W40dxdy,
Qx, y, α, W20, W40= cos4kW40sx x2+y2+s24+2kW20+α sx+cos4kW40 sxx2+y2+s24+2kW20-αsx ±cos2kα+4kW40sx×x2+y2+s24+2kW20sx-2kα± cos2kα -4kW40sxx2+y2+s24-2kW20sx-2kα.
EW=W 0OTFsJ1Wsds, OTFs=0, s2.
I-+ρ=0, α, W20, W40=401cos/sinkα1-r2×expik W20r2+W40r4dr2.
I-+ρ=0, α, W20, W40 =expikα201expikW40y2expikW20-αydy±exp-ikα201expikW40y2 ×expik W20+αydy2,
I-+ρ=0, α, W20, W40 =expikα2-recty-12expikW40y2×expikW20-αydy±exp-ikα2 ×-recty-12expikW40y2×expikW20+αydy2,
recty=1-1/2<y<1/2=0otherwise.
I-+ρ=0, α, W20, W40 =14gW20-αλ2+gW20+αλ2±gW20-αλg*W20+αλexpi2kα±g*W20-αλgW20+αλexp-i2kα,
gp=sincpexpiπp exp-iπp22W40,
SRα, W40=Iρ=0, α, W¯20, W40Iρ=0, α, W¯20, W40=0,

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