Abstract

We report the frequency response characteristics of an optical system consisting of a lens made of a uniaxial birefringent crystal sandwiched between two linear polarizers; the lens has prespecified off-axis aberrations such as primary astigmatism and primary coma. An analytical expression is obtained for the optical transfer function of the proposed system by use of the autocorrelation of the pupil function over the lens aperture. Some specific cases are computed and illustrated graphically. It has been shown that the proposed system has imaging characteristics distinctly different from those of an ordinary glass lens, and these may be advantageous for better balancing of aberrations in conventional imaging systems.

© 2004 Optical Society of America

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  1. J. Tsujiuchi, “A density filter improving aberrant optical image,” J. Phys. Soc. Jpn. 12, 744–000 (1957).
    [CrossRef]
  2. 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]
  3. S. C. Biswas, A. Boivin, “Influence of primary astigmatism on the performance of optimum apodizers,” J. Opt. (Paris) 4, 1–00 (1975).
  4. S. C. Biswas, A. Boivin, “Influence of spherical aberration on the performance of optimum apodizers,” Opt. Acta 23, 569–588 (1976).
    [CrossRef]
  5. S. C. Biswas, A. Boivin, “Performance of optimum apodizers in the presence of primary coma,” Can. J. Phys. 57, 1388 (1979).
    [CrossRef]
  6. 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]
  7. 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]
  8. 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]
  9. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. Montes, “Bessel annular apodizers: imaging characteristics,” Appl. Opt. 26, 2770–2772 (1987).
    [CrossRef] [PubMed]
  10. J. Tsujiuchi, “Correction of optical images by compensation of aberrations and by spatial frequency filtering,” in Progress in Optics, Vol. II, E. Wolf, ed. (North Holland, Amsterdam, 1963), pp. 131–180.
    [CrossRef]
  11. S. Mezouari, A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28, 771–773 (2003).
    [CrossRef] [PubMed]
  12. A. K. Chakraborty, S. Das, D. K. Basu, A. Ghosh, “Imaging characteristics of a birefringent lens,” Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 130–134 (1990).
    [CrossRef]
  13. S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
    [CrossRef]
  14. S. Sanyal, A. Ghosh, “Imaging characteristics of birefringent lenses under focused and defocused condition,” Optik 110, 513–520 (1999).
  15. S. Sanyal, A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000).
    [CrossRef]
  16. S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. 41, 592–597 (2002).
    [CrossRef]
  17. S. Sanyal, A. Ghosh, “Image assessment of a birefringent lens based on the factor of encircled energy,” Opt. Eng. 42, 1058–1064 (2003).
    [CrossRef]
  18. S. Sanyal, A. Ghosh, “High tolerance to spherical aberrations and defects of focus using a birefringent lens,” Appl. Opt. 41, 4611–4619 (2002).
    [CrossRef] [PubMed]
  19. S. Sanyal, A. Ghosh, “Light throughput of a birefringent lens—a comparative study,” J. Opt. (India) 30(2), 85–93 (2001).
  20. S. Sanyal, A. Ghosh, “Imaging behaviour of a birefringent lens suffering from primary coma,” J. Opt. (India) 29(1), 15–23 (2000).
  21. S. Sanyal, Y. Kawata, S. Mandal, A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng. (to be published).
  22. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London A 231, 91–103 (1955).
    [CrossRef]
  23. M. De, “The influence of astigmatism on the response function of an optical system,” Proc. R. Soc. London A 233, 91–104 (1956).
  24. M. De, B. K. Nath, “Response of optical systems suffering from primary coma,” Optik 15, 739–750 (1958).
  25. F. Ratajczyk, “A method of calculation of permissible birefringence in lenses of the optical instruments,” Optik (Stuttgart) 68, 61–68 (1984).
  26. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  27. R. A. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).
    [CrossRef]
  28. J. P. Mcguire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I. Formulation and examples,” J. Opt. Soc. Am. A 7, 1614–1626 (1990).
    [CrossRef]
  29. J. P. Mcguire, R. A. Chipman, “Polarization aberrations. I. Rotationally symmetric systems,” Appl. Opt. 33, 5080–5100 (1994).
    [CrossRef] [PubMed]
  30. H. Kikuta, K. Iwata, H. Shimomura, “First-order aberration of a double-focus lens made of a uniaxial crystal,” J. Opt. Soc. Am. A 9, 814–819 (1992).
    [CrossRef]
  31. Y. Unno, “Distorted wave front produced by a high-resolution projection optical system having rotationally symmetric birefringence,” Appl. Opt. 37, 7241–7247 (1998).
    [CrossRef]
  32. J. Lesso, A. Duncan, W. Sibbett, M. Padgett, “Aberrations introduced by a lens made from a birefringent material,” Appl. Opt. 39, 592–598 (2000).
    [CrossRef]
  33. Y. Unno, “Influence of birefringence on the image formation of high-resolution projection optics,” Appl. Opt. 39, 3243–3252 (2000).
    [CrossRef]
  34. Y. Unno, A. Suzuki, “Analyses of imaging performance degradation caused by birefringence residual in lens materials,” in Optical Microlithography XIV, C. J. Progler, ed., Proc. SPIE4346, 1306–1317 (2001).
    [CrossRef]
  35. I. Abdulhalim, C. N. Pannell, D. N. Payne, “Fiber compatible fast acousto-optic modulator using a gradient index lens as the interaction medium,” Appl. Phys. Lett. 62, 3402–3404 (1993).
    [CrossRef]
  36. I. Abdulhalim, C. N. Pannell, “Photoelastically induced light modulation in gradient-index lenses,” Opt. Lett. 18, 1274–1276 (1993).
    [CrossRef] [PubMed]
  37. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1985).
  38. H. H. Hopkins, Wave Theory of Aberration (Clarendon, Oxford, 1950).
  39. V. N. Mahajan, Aberration Theory Made Simple (SPIE, Bellingham, Wash., 1991).
    [CrossRef]
  40. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1966).
  41. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

2003 (2)

S. Sanyal, A. Ghosh, “Image assessment of a birefringent lens based on the factor of encircled energy,” Opt. Eng. 42, 1058–1064 (2003).
[CrossRef]

S. Mezouari, A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28, 771–773 (2003).
[CrossRef] [PubMed]

2002 (2)

S. Sanyal, A. Ghosh, “High tolerance to spherical aberrations and defects of focus using a birefringent lens,” Appl. Opt. 41, 4611–4619 (2002).
[CrossRef] [PubMed]

S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. 41, 592–597 (2002).
[CrossRef]

2001 (1)

S. Sanyal, A. Ghosh, “Light throughput of a birefringent lens—a comparative study,” J. Opt. (India) 30(2), 85–93 (2001).

2000 (4)

1999 (1)

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

1998 (2)

1994 (1)

1993 (2)

I. Abdulhalim, C. N. Pannell, “Photoelastically induced light modulation in gradient-index lenses,” Opt. Lett. 18, 1274–1276 (1993).
[CrossRef] [PubMed]

I. Abdulhalim, C. N. Pannell, D. N. Payne, “Fiber compatible fast acousto-optic modulator using a gradient index lens as the interaction medium,” Appl. Phys. Lett. 62, 3402–3404 (1993).
[CrossRef]

1992 (1)

1990 (1)

1989 (2)

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

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

1987 (1)

1986 (1)

1984 (1)

F. Ratajczyk, “A method of calculation of permissible birefringence in lenses of the optical instruments,” Optik (Stuttgart) 68, 61–68 (1984).

1983 (1)

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]

1979 (2)

S. C. Biswas, A. Boivin, “Performance of optimum apodizers in the presence of primary coma,” Can. J. Phys. 57, 1388 (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, 1–00 (1975).

1971 (1)

1958 (1)

M. De, B. K. Nath, “Response of optical systems suffering from primary coma,” Optik 15, 739–750 (1958).

1957 (1)

J. Tsujiuchi, “A density filter improving aberrant optical image,” J. Phys. Soc. Jpn. 12, 744–000 (1957).
[CrossRef]

1956 (1)

M. De, “The influence of astigmatism on the response function of an optical system,” Proc. R. Soc. London A 233, 91–104 (1956).

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London A 231, 91–103 (1955).
[CrossRef]

Abdulhalim, I.

I. Abdulhalim, C. N. Pannell, D. N. Payne, “Fiber compatible fast acousto-optic modulator using a gradient index lens as the interaction medium,” Appl. Phys. Lett. 62, 3402–3404 (1993).
[CrossRef]

I. Abdulhalim, C. N. Pannell, “Photoelastically induced light modulation in gradient-index lenses,” Opt. Lett. 18, 1274–1276 (1993).
[CrossRef] [PubMed]

Abramowitz, M.

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

Andres, P.

Bandyopadhyay, P.

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

Basu, D. K.

A. K. Chakraborty, S. Das, D. K. Basu, A. Ghosh, “Imaging characteristics of a birefringent lens,” 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 (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, 1–00 (1975).

Boivin, A.

S. C. Biswas, A. Boivin, “Performance of optimum apodizers in the presence of primary coma,” Can. J. Phys. 57, 1388 (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, 1–00 (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,” Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 130–134 (1990).
[CrossRef]

Chipman, R. A.

Das, S.

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

De, M.

M. De, B. K. Nath, “Response of optical systems suffering from primary coma,” Optik 15, 739–750 (1958).

M. De, “The influence of astigmatism on the response function of an optical system,” Proc. R. Soc. London A 233, 91–104 (1956).

Diaz, A.

Duncan, A.

Ghosh, A.

S. Sanyal, A. Ghosh, “Image assessment of a birefringent lens based on the factor of encircled energy,” Opt. Eng. 42, 1058–1064 (2003).
[CrossRef]

S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. 41, 592–597 (2002).
[CrossRef]

S. Sanyal, A. Ghosh, “High tolerance to spherical aberrations and defects of focus using a birefringent lens,” Appl. Opt. 41, 4611–4619 (2002).
[CrossRef] [PubMed]

S. Sanyal, A. Ghosh, “Light throughput of a birefringent lens—a comparative study,” J. Opt. (India) 30(2), 85–93 (2001).

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(1), 15–23 (2000).

S. Sanyal, A. Ghosh, “Imaging characteristics of birefringent lenses under focused and defocused condition,” 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]

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

S. Sanyal, Y. Kawata, S. Mandal, A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng. (to be published).

Harvey, A. R.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London A 231, 91–103 (1955).
[CrossRef]

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

Iwata, K.

Kawata, Y.

S. Sanyal, Y. Kawata, S. Mandal, A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng. (to be published).

Kikuta, H.

Lesso, J.

Mahajan, V. N.

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

Mandal, S.

S. Sanyal, Y. Kawata, S. Mandal, A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng. (to be published).

Mcguire, J. P.

Mezouari, S.

Mino, M.

Montes, E.

Nath, B. K.

M. De, B. K. Nath, “Response of optical systems suffering from primary coma,” Optik 15, 739–750 (1958).

Ojeda-Castaneda, J.

Okano, Y.

Padgett, M.

Pannell, C. N.

I. Abdulhalim, C. N. Pannell, “Photoelastically induced light modulation in gradient-index lenses,” Opt. Lett. 18, 1274–1276 (1993).
[CrossRef] [PubMed]

I. Abdulhalim, C. N. Pannell, D. N. Payne, “Fiber compatible fast acousto-optic modulator using a gradient index lens as the interaction medium,” Appl. Phys. Lett. 62, 3402–3404 (1993).
[CrossRef]

Payne, D. N.

I. Abdulhalim, C. N. Pannell, D. N. Payne, “Fiber compatible fast acousto-optic modulator using a gradient index lens as the interaction medium,” Appl. Phys. Lett. 62, 3402–3404 (1993).
[CrossRef]

Ratajczyk, F.

F. Ratajczyk, “A method of calculation of permissible birefringence in lenses of the optical instruments,” Optik (Stuttgart) 68, 61–68 (1984).

Sanyal, S.

S. Sanyal, A. Ghosh, “Image assessment of a birefringent lens based on the factor of encircled energy,” Opt. Eng. 42, 1058–1064 (2003).
[CrossRef]

S. Sanyal, A. Ghosh, “High tolerance to spherical aberrations and defects of focus using a birefringent lens,” Appl. Opt. 41, 4611–4619 (2002).
[CrossRef] [PubMed]

S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. 41, 592–597 (2002).
[CrossRef]

S. Sanyal, A. Ghosh, “Light throughput of a birefringent lens—a comparative study,” J. Opt. (India) 30(2), 85–93 (2001).

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(1), 15–23 (2000).

S. Sanyal, A. Ghosh, “Imaging characteristics of birefringent lenses under focused and defocused condition,” 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, Y. Kawata, S. Mandal, A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng. (to be published).

Shimomura, H.

Sibbett, W.

Stegun, I. A.

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

Suzuki, A.

Y. Unno, A. Suzuki, “Analyses of imaging performance degradation caused by birefringence residual in lens materials,” in Optical Microlithography XIV, C. J. Progler, ed., Proc. SPIE4346, 1306–1317 (2001).
[CrossRef]

Tsujiuchi, J.

J. Tsujiuchi, “A density filter improving aberrant optical image,” J. Phys. Soc. Jpn. 12, 744–000 (1957).
[CrossRef]

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

Unno, Y.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1966).

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. (8)

Appl. Phys. Lett. (1)

I. Abdulhalim, C. N. Pannell, D. N. Payne, “Fiber compatible fast acousto-optic modulator using a gradient index lens as the interaction medium,” Appl. Phys. Lett. 62, 3402–3404 (1993).
[CrossRef]

Can. J. Phys. (1)

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

J. Opt. (India) (2)

S. Sanyal, A. Ghosh, “Light throughput of a birefringent lens—a comparative study,” J. Opt. (India) 30(2), 85–93 (2001).

S. Sanyal, A. Ghosh, “Imaging behaviour of a birefringent lens suffering from primary coma,” J. Opt. (India) 29(1), 15–23 (2000).

J. Opt. (Paris) (1)

S. C. Biswas, A. Boivin, “Influence of primary astigmatism on the performance of optimum apodizers,” J. Opt. (Paris) 4, 1–00 (1975).

J. Opt. Soc. Am. A (2)

J. Phys. Soc. Jpn. (1)

J. Tsujiuchi, “A density filter improving aberrant optical image,” J. Phys. Soc. Jpn. 12, 744–000 (1957).
[CrossRef]

Opt. Acta (2)

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

Opt. Acta. (1)

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]

Opt. Eng. (5)

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

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

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

S. Sanyal, A. Ghosh, “Frequency response characteristics of a birefringent lens,” Opt. Eng. 41, 592–597 (2002).
[CrossRef]

S. Sanyal, A. Ghosh, “Image assessment of a birefringent lens based on the factor of encircled energy,” Opt. Eng. 42, 1058–1064 (2003).
[CrossRef]

Opt. Lett. (3)

Optik (2)

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

M. De, B. K. Nath, “Response of optical systems suffering from primary coma,” Optik 15, 739–750 (1958).

Optik (Stuttgart) (1)

F. Ratajczyk, “A method of calculation of permissible birefringence in lenses of the optical instruments,” Optik (Stuttgart) 68, 61–68 (1984).

Proc. R. Soc. London A (2)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London A 231, 91–103 (1955).
[CrossRef]

M. De, “The influence of astigmatism on the response function of an optical system,” Proc. R. Soc. London A 233, 91–104 (1956).

Other (9)

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

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

S. Sanyal, Y. Kawata, S. Mandal, A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng. (to be published).

Y. Unno, A. Suzuki, “Analyses of imaging performance degradation caused by birefringence residual in lens materials,” in Optical Microlithography XIV, C. J. Progler, ed., Proc. SPIE4346, 1306–1317 (2001).
[CrossRef]

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

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

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

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1966).

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

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Figures (11)

Fig. 1
Fig. 1

Proposed system.

Fig. 2
Fig. 2

OTF of the proposed system along the 0° azimuth of the image plane for various birefringent lenses under the parallel polarizer configuration. Each lens contains the prespecified astigmatism W 22 = 1λ. The image plane under consideration is midway between sagittal and tangential foci (W 20 = -W 22/2).

Fig. 3
Fig. 3

Same as Fig. 2 for a 30° azimuth of the image plane.

Fig. 4
Fig. 4

Same as Fig. 2 for a 60° azimuth of the image plane.

Fig. 5
Fig. 5

OTF of the proposed system along various azimuths of the image plane for the birefringent lens with α = 0.6λ, W 22 = 1λ in the parallel polarizer configuration. The image plane under consideration is midway between sagittal and tangential foci (W 20 = -W 22/2).

Fig. 6
Fig. 6

MTF and PTF of a birefringent lens with α = 0.55λ, W 31 = 2λ in the parallel polarizer configuration at the image plane W 20 = ±0.3λ for the 0° azimuth of the sinusoidal grating test object. Dashed curves, MTF; solid curves, PTF; 1, proposed system; 2, similar ordinary lens.

Fig. 7
Fig. 7

Same as Fig. 6 for a 30° azimuth of the sinusoidal grating test object.

Fig. 8
Fig. 8

Same as Fig. 6 for a 60° azimuth of the sinusoidal grating test object.

Fig. 9
Fig. 9

Same as Fig. 6 for a 75° azimuth of the sinusoidal grating test object.

Fig. 10
Fig. 10

Same as Fig. 6 for a 90° azimuth of the sinusoidal grating test object.

Fig. 11
Fig. 11

PTF of a birefringent lens with α = 0.55λ, W 31 = 2λ in the parallel polarizer configuration at the image plane W 20 = ±0.3λ along various azimuths.

Equations (15)

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f-+x0, y0, α, W22, W20=cossinkα1-x02+y02 ×expikWx0, y0 when x02+y021 =0 otherwise,
x0=x cos φ-y sin φ, y0=x sin φ+y cos φ,
f-+x, y, α, W22, W20=cossinkα1-x2+y2 ×expikWx, ywhen x2+y21 =0 otherwise,
Wx, y=W22x2 sin2 φ+y2 cos2 φ+xy sin 2φ+W20x2+y2
OTF-+s, φ, α, W22, W20=1N-+αx=0ay=0bcosks2W20-α+W22 sin2 φx+W22y sin 2φdxdy+x=0ay=0bcosks2W20+α+W22 sin2 φx+W22y sin 2φdxdy±x=0ay=0bcosks2W20+W22 sin2 φx+W22y sin 2φ+2kα1-s24-x2+y2dxdy±x=0ay=0bcosks2W20+W22 sin2 φx+W22y sin 2φ-2kα1-s24-x2+y2dxdy,
a=1-s2,b=1-x+s221/2,N-+α=π21±sinc4αλ
OTF-+s, φ, α, W22, W20=1N-+αI11+I12±I21±I22.
I2221=x=0ay=0bcosPx+QyRx2+y2±Cdxdy,
I2221=θ1=0βθ2=0θ1cosP cosθ1+Q sinθ2±R cos2θ1±R cos2θ2+Csinθ1cosθ2dθ1dθ2,
P=P±Rs,R=R2,C=±C-Ps2Rs241,
cosϕ cos δ=n=0-1nεnJ2nϕcos2nδ,sinϕ cos δ=2 n=0-1nJ2n+1ϕcos2n+1δ,cosϕ sin δ=n=0 εnJ2nϕcos2nδ,sinϕ sin δ=2 n=0 J2n+1ϕsin2n+1δ,
I2221=βcossinCl=0m=0p=0q=0 Al,m,p,qJlP×JmQJpRJqRi=132sincDl,m,p,qiβEm,qi.
f-+x0, y0, α, W31, W20 =cossinkα1-x02+y02expikWx0, y0 when x02+y021 =0 otherwise,
OTF-+s, φ, α, W31, W20=12N-+αx=0ay=0b×Ωx, y, α, W31, W20×dxdy+x=0-ay=0b¯×Ωx, y, α, W31, W20×dxdy,
b¯=1-x-s221/2,Ωx, y, α, W31, W20=expiks2W20-αx+W31cosφ3x2+y2+s24-2W31sin φxy+expiks2W20+αx+W31cos φ3x2+y2+s24-2W31sin φxy±expik2sxW20+W31x cos φ-y sin φ+W31s cos φ-2α×x2+y2+s24+2α±expik2sxW20+W31x cos φ-y sin φ+W31s cos φ+2αx2+y2+s24-2α.

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