Abstract

Many applications can benefit from the use of pupil filters for controlling the light intensity distribution near the focus of an optical system. Most of the design methods for such filters are based on a second-order expansion of the Point Spread Function (PSF). Here, we present a new procedure for designing radially-symmetric pupil filters. It is more precise than previous procedures as it considers the exact expression of the PSF, expanded as a function of first-order Bessel functions. Furthermore, this new method presents other advantages: the height of the side lobes can be easily controlled, it allows the design of amplitude-only, phase-only or hybrid filters, and the coefficients of the PSF expansion can be directly related to filter parameters. Finally, our procedure allows the design of filters with very different behaviours and optimal performance.

© 2006 Optical Society of America

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References

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  1. G. Boyer, “New class of axially apodizing filters for confocal scanning microscopy,” J. Opt. Soc. Am A 19584–589 (2002).
    [Crossref]
  2. M. Martínez-Corral, P. Andrés, C.J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filtres,” Opt.Commun. 165, 267–278 (1999).
    [Crossref]
  3. M.T. Caballero, P. Andrés, A. Pons, J. Lancis, and M. Martínez-Corral, “Axial resolution in two-color excitation fluorescence microscopy by phase-only apodization,” Opt.Commun. 246, 313–321 (2005).
    [Crossref]
  4. D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Transverse or axial superresolution in a 4Pi-confocal microscope by phase-only filters,” J. Opt. Soc. Am A 20, 2172–2178 (2003).
    [Crossref]
  5. X. Zhao, C. Li, and H. Ruan, “A new approach for improving transverse superresolution in optical data storage,” Opt. Eng. 44, 125202 1–4 (2005).
    [Crossref]
  6. S. Pereira and A.S. van de Nes , “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt.Commun. 234, 119–124 (2004).
    [Crossref]
  7. V.F. Canales, D.M. de Juana, and M.P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29 (2004) 935–937
    [Crossref] [PubMed]
  8. M.P. Cagigal, V.F. Canales, and J.E. Oti, “Design of continuous superresolving masks for ground-based telescopes”, Publ. Astron. Soc. Pac. 116, 965–970 (2004).
    [Crossref]
  9. J. Jia, C, Zhou, and L. Liu, “Superresolution technology for reduction of the far-field diffraction spot size in the laser free-space communication system,” Opt.Commun. 228, 271–278 (2003).
    [Crossref]
  10. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9426>–435 (1952).
    [Crossref]
  11. C.J.R. Sheppard and Z.S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am A 5, 643>–647 (1988).
    [Crossref]
  12. D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607>–609 (2003).
    [Crossref] [PubMed]
  13. M. Yun, L. Liu, J. Sun, and D. Liu, “Three-dimensional superresolution by three-zone complex pupil filters,” J. Opt. Soc. Am A 22 (2005) 272>–277.
    [Crossref]
  14. X. Zhao, C. Li, and H. Ruan, “Improvement of three-dimensional resolution in optical data storage by combination of two annular binary phase filters,” Chin. Phys. Lett. 21, 1515>–1517 (2004).
    [Crossref]
  15. M. Yun, L. Liu, J. Sun, and D. Liu, “Transverse or axial superresolution with radial birefringent filter,” J. Opt. Soc. Am A 21, 1869>–1874 (2004).
    [Crossref]
  16. S. Zhou and C. Zhou, “Discrete continuous-phase superresolving filters,” Opt. Lett. 23, 2746>–2748 (2004).
    [Crossref]
  17. H. Ding, Q. Li, and W. Zou, “Design and comparison of amplitude-type and phase-only transverse super-resolving pupil filters,” Opt. Commun. 229, 117>–122 (2004).
    [Crossref]
  18. S. Ledesma, J.C. Escalera, J. Campos, and M.J. Yzuel, “Evolution of the transverse response of an optical system with complex filtres,” Opt. Commun. 249, 183>–192 (2005).
    [Crossref]
  19. H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Appl. Opt. 43, 6242>–6247 (2004).
    [Crossref] [PubMed]
  20. P. Narayan, E. Hack, and P. Rastogi, “High efficient superesolution combination filter with twin LCD spatial light modulators,” Opt. Express 13, 2835>–2842 (2005).
    [Crossref]
  21. J. Jia, C. Zhou, X. Sun, and L. Liu, “Superresolution laser beam shaping,” Appl. Opt. 43, 2112>–2117 (2004).
    [Crossref] [PubMed]
  22. T.R.M. Sales and G.M. Morris, “Axial superresolution with phase-only pùpil filters,” Opt.Commun. 156, 227>–230 (1998).
    [Crossref]
  23. M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Aunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt.Commun. 119, 491>–498 (1995).
    [Crossref]
  24. V.F. Canales, J.E. Oti, and M.P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks” Opt. Commun. 247, 11>–18 (2005).
    [Crossref]
  25. H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl.Opt. 41, 5263>–5266 (2002).
    [Crossref] [PubMed]
  26. V.F. Canales, J.E. Oti, P.J. Valle, and M.P. Cagigal, “Reduction of the diffraction pattern in segmented apertures”, Opt. Eng. (to be published).
  27. M.P. Cagigal, J.E. Oti, V.F. Canales, and P.J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241, 249>–253 (2004).
    [Crossref]
  28. C.J.R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” Optica Acta 24, 1051>–1073 (1977).
    [Crossref]

2005 (6)

M.T. Caballero, P. Andrés, A. Pons, J. Lancis, and M. Martínez-Corral, “Axial resolution in two-color excitation fluorescence microscopy by phase-only apodization,” Opt.Commun. 246, 313–321 (2005).
[Crossref]

X. Zhao, C. Li, and H. Ruan, “A new approach for improving transverse superresolution in optical data storage,” Opt. Eng. 44, 125202 1–4 (2005).
[Crossref]

M. Yun, L. Liu, J. Sun, and D. Liu, “Three-dimensional superresolution by three-zone complex pupil filters,” J. Opt. Soc. Am A 22 (2005) 272>–277.
[Crossref]

S. Ledesma, J.C. Escalera, J. Campos, and M.J. Yzuel, “Evolution of the transverse response of an optical system with complex filtres,” Opt. Commun. 249, 183>–192 (2005).
[Crossref]

P. Narayan, E. Hack, and P. Rastogi, “High efficient superesolution combination filter with twin LCD spatial light modulators,” Opt. Express 13, 2835>–2842 (2005).
[Crossref]

V.F. Canales, J.E. Oti, and M.P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks” Opt. Commun. 247, 11>–18 (2005).
[Crossref]

2004 (10)

M.P. Cagigal, J.E. Oti, V.F. Canales, and P.J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241, 249>–253 (2004).
[Crossref]

J. Jia, C. Zhou, X. Sun, and L. Liu, “Superresolution laser beam shaping,” Appl. Opt. 43, 2112>–2117 (2004).
[Crossref] [PubMed]

H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Appl. Opt. 43, 6242>–6247 (2004).
[Crossref] [PubMed]

X. Zhao, C. Li, and H. Ruan, “Improvement of three-dimensional resolution in optical data storage by combination of two annular binary phase filters,” Chin. Phys. Lett. 21, 1515>–1517 (2004).
[Crossref]

M. Yun, L. Liu, J. Sun, and D. Liu, “Transverse or axial superresolution with radial birefringent filter,” J. Opt. Soc. Am A 21, 1869>–1874 (2004).
[Crossref]

S. Zhou and C. Zhou, “Discrete continuous-phase superresolving filters,” Opt. Lett. 23, 2746>–2748 (2004).
[Crossref]

H. Ding, Q. Li, and W. Zou, “Design and comparison of amplitude-type and phase-only transverse super-resolving pupil filters,” Opt. Commun. 229, 117>–122 (2004).
[Crossref]

S. Pereira and A.S. van de Nes , “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt.Commun. 234, 119–124 (2004).
[Crossref]

V.F. Canales, D.M. de Juana, and M.P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29 (2004) 935–937
[Crossref] [PubMed]

M.P. Cagigal, V.F. Canales, and J.E. Oti, “Design of continuous superresolving masks for ground-based telescopes”, Publ. Astron. Soc. Pac. 116, 965–970 (2004).
[Crossref]

2003 (3)

J. Jia, C, Zhou, and L. Liu, “Superresolution technology for reduction of the far-field diffraction spot size in the laser free-space communication system,” Opt.Commun. 228, 271–278 (2003).
[Crossref]

D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Transverse or axial superresolution in a 4Pi-confocal microscope by phase-only filters,” J. Opt. Soc. Am A 20, 2172–2178 (2003).
[Crossref]

D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607>–609 (2003).
[Crossref] [PubMed]

2002 (2)

G. Boyer, “New class of axially apodizing filters for confocal scanning microscopy,” J. Opt. Soc. Am A 19584–589 (2002).
[Crossref]

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl.Opt. 41, 5263>–5266 (2002).
[Crossref] [PubMed]

1999 (1)

M. Martínez-Corral, P. Andrés, C.J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filtres,” Opt.Commun. 165, 267–278 (1999).
[Crossref]

1998 (1)

T.R.M. Sales and G.M. Morris, “Axial superresolution with phase-only pùpil filters,” Opt.Commun. 156, 227>–230 (1998).
[Crossref]

1995 (1)

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Aunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt.Commun. 119, 491>–498 (1995).
[Crossref]

1988 (1)

C.J.R. Sheppard and Z.S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am A 5, 643>–647 (1988).
[Crossref]

1977 (1)

C.J.R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” Optica Acta 24, 1051>–1073 (1977).
[Crossref]

1952 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9426>–435 (1952).
[Crossref]

Andrés, P.

M.T. Caballero, P. Andrés, A. Pons, J. Lancis, and M. Martínez-Corral, “Axial resolution in two-color excitation fluorescence microscopy by phase-only apodization,” Opt.Commun. 246, 313–321 (2005).
[Crossref]

M. Martínez-Corral, P. Andrés, C.J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filtres,” Opt.Commun. 165, 267–278 (1999).
[Crossref]

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Aunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt.Commun. 119, 491>–498 (1995).
[Crossref]

Boyer, G.

G. Boyer, “New class of axially apodizing filters for confocal scanning microscopy,” J. Opt. Soc. Am A 19584–589 (2002).
[Crossref]

Caballero, M.T.

M.T. Caballero, P. Andrés, A. Pons, J. Lancis, and M. Martínez-Corral, “Axial resolution in two-color excitation fluorescence microscopy by phase-only apodization,” Opt.Commun. 246, 313–321 (2005).
[Crossref]

Cagigal, M.P.

V.F. Canales, J.E. Oti, and M.P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks” Opt. Commun. 247, 11>–18 (2005).
[Crossref]

M.P. Cagigal, J.E. Oti, V.F. Canales, and P.J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241, 249>–253 (2004).
[Crossref]

M.P. Cagigal, V.F. Canales, and J.E. Oti, “Design of continuous superresolving masks for ground-based telescopes”, Publ. Astron. Soc. Pac. 116, 965–970 (2004).
[Crossref]

V.F. Canales, D.M. de Juana, and M.P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29 (2004) 935–937
[Crossref] [PubMed]

D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607>–609 (2003).
[Crossref] [PubMed]

D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Transverse or axial superresolution in a 4Pi-confocal microscope by phase-only filters,” J. Opt. Soc. Am A 20, 2172–2178 (2003).
[Crossref]

V.F. Canales, J.E. Oti, P.J. Valle, and M.P. Cagigal, “Reduction of the diffraction pattern in segmented apertures”, Opt. Eng. (to be published).

Campos, J.

S. Ledesma, J.C. Escalera, J. Campos, and M.J. Yzuel, “Evolution of the transverse response of an optical system with complex filtres,” Opt. Commun. 249, 183>–192 (2005).
[Crossref]

Canales, V.F.

V.F. Canales, J.E. Oti, and M.P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks” Opt. Commun. 247, 11>–18 (2005).
[Crossref]

M.P. Cagigal, J.E. Oti, V.F. Canales, and P.J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241, 249>–253 (2004).
[Crossref]

M.P. Cagigal, V.F. Canales, and J.E. Oti, “Design of continuous superresolving masks for ground-based telescopes”, Publ. Astron. Soc. Pac. 116, 965–970 (2004).
[Crossref]

V.F. Canales, D.M. de Juana, and M.P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29 (2004) 935–937
[Crossref] [PubMed]

D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607>–609 (2003).
[Crossref] [PubMed]

D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Transverse or axial superresolution in a 4Pi-confocal microscope by phase-only filters,” J. Opt. Soc. Am A 20, 2172–2178 (2003).
[Crossref]

V.F. Canales, J.E. Oti, P.J. Valle, and M.P. Cagigal, “Reduction of the diffraction pattern in segmented apertures”, Opt. Eng. (to be published).

Choudhury, A.

C.J.R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” Optica Acta 24, 1051>–1073 (1977).
[Crossref]

de Juana, D.M.

de Nes, A.S. van

S. Pereira and A.S. van de Nes , “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt.Commun. 234, 119–124 (2004).
[Crossref]

Ding, H.

H. Ding, Q. Li, and W. Zou, “Design and comparison of amplitude-type and phase-only transverse super-resolving pupil filters,” Opt. Commun. 229, 117>–122 (2004).
[Crossref]

Escalera, J.C.

S. Ledesma, J.C. Escalera, J. Campos, and M.J. Yzuel, “Evolution of the transverse response of an optical system with complex filtres,” Opt. Commun. 249, 183>–192 (2005).
[Crossref]

Francia, G. Toraldo di

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9426>–435 (1952).
[Crossref]

Gan, F.

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl.Opt. 41, 5263>–5266 (2002).
[Crossref] [PubMed]

Hack, E.

Hegedus, Z.S.

C.J.R. Sheppard and Z.S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am A 5, 643>–647 (1988).
[Crossref]

Jia, J.

J. Jia, C. Zhou, X. Sun, and L. Liu, “Superresolution laser beam shaping,” Appl. Opt. 43, 2112>–2117 (2004).
[Crossref] [PubMed]

J. Jia, C, Zhou, and L. Liu, “Superresolution technology for reduction of the far-field diffraction spot size in the laser free-space communication system,” Opt.Commun. 228, 271–278 (2003).
[Crossref]

Kowalczyk, M.

M. Martínez-Corral, P. Andrés, C.J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filtres,” Opt.Commun. 165, 267–278 (1999).
[Crossref]

Lancis, J.

M.T. Caballero, P. Andrés, A. Pons, J. Lancis, and M. Martínez-Corral, “Axial resolution in two-color excitation fluorescence microscopy by phase-only apodization,” Opt.Commun. 246, 313–321 (2005).
[Crossref]

Ledesma, S.

S. Ledesma, J.C. Escalera, J. Campos, and M.J. Yzuel, “Evolution of the transverse response of an optical system with complex filtres,” Opt. Commun. 249, 183>–192 (2005).
[Crossref]

Li, C.

X. Zhao, C. Li, and H. Ruan, “A new approach for improving transverse superresolution in optical data storage,” Opt. Eng. 44, 125202 1–4 (2005).
[Crossref]

X. Zhao, C. Li, and H. Ruan, “Improvement of three-dimensional resolution in optical data storage by combination of two annular binary phase filters,” Chin. Phys. Lett. 21, 1515>–1517 (2004).
[Crossref]

Li, Q.

H. Ding, Q. Li, and W. Zou, “Design and comparison of amplitude-type and phase-only transverse super-resolving pupil filters,” Opt. Commun. 229, 117>–122 (2004).
[Crossref]

Liu, D.

M. Yun, L. Liu, J. Sun, and D. Liu, “Three-dimensional superresolution by three-zone complex pupil filters,” J. Opt. Soc. Am A 22 (2005) 272>–277.
[Crossref]

M. Yun, L. Liu, J. Sun, and D. Liu, “Transverse or axial superresolution with radial birefringent filter,” J. Opt. Soc. Am A 21, 1869>–1874 (2004).
[Crossref]

Liu, L.

M. Yun, L. Liu, J. Sun, and D. Liu, “Three-dimensional superresolution by three-zone complex pupil filters,” J. Opt. Soc. Am A 22 (2005) 272>–277.
[Crossref]

M. Yun, L. Liu, J. Sun, and D. Liu, “Transverse or axial superresolution with radial birefringent filter,” J. Opt. Soc. Am A 21, 1869>–1874 (2004).
[Crossref]

J. Jia, C. Zhou, X. Sun, and L. Liu, “Superresolution laser beam shaping,” Appl. Opt. 43, 2112>–2117 (2004).
[Crossref] [PubMed]

J. Jia, C, Zhou, and L. Liu, “Superresolution technology for reduction of the far-field diffraction spot size in the laser free-space communication system,” Opt.Commun. 228, 271–278 (2003).
[Crossref]

Luo, H.

Martínez-Corral, M.

M.T. Caballero, P. Andrés, A. Pons, J. Lancis, and M. Martínez-Corral, “Axial resolution in two-color excitation fluorescence microscopy by phase-only apodization,” Opt.Commun. 246, 313–321 (2005).
[Crossref]

M. Martínez-Corral, P. Andrés, C.J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolution by annular binary filtres,” Opt.Commun. 165, 267–278 (1999).
[Crossref]

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Aunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt.Commun. 119, 491>–498 (1995).
[Crossref]

Morris, G.M.

T.R.M. Sales and G.M. Morris, “Axial superresolution with phase-only pùpil filters,” Opt.Commun. 156, 227>–230 (1998).
[Crossref]

Narayan, P.

Ojeda-Castañeda, J.

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Aunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt.Commun. 119, 491>–498 (1995).
[Crossref]

Oti, J.E.

V.F. Canales, J.E. Oti, and M.P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically-designed phase masks” Opt. Commun. 247, 11>–18 (2005).
[Crossref]

M.P. Cagigal, V.F. Canales, and J.E. Oti, “Design of continuous superresolving masks for ground-based telescopes”, Publ. Astron. Soc. Pac. 116, 965–970 (2004).
[Crossref]

M.P. Cagigal, J.E. Oti, V.F. Canales, and P.J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241, 249>–253 (2004).
[Crossref]

D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Transverse or axial superresolution in a 4Pi-confocal microscope by phase-only filters,” J. Opt. Soc. Am A 20, 2172–2178 (2003).
[Crossref]

D.M. de Juana, J.E. Oti, V.F. Canales, and M.P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607>–609 (2003).
[Crossref] [PubMed]

V.F. Canales, J.E. Oti, P.J. Valle, and M.P. Cagigal, “Reduction of the diffraction pattern in segmented apertures”, Opt. Eng. (to be published).

Pereira, S.

S. Pereira and A.S. van de Nes , “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt.Commun. 234, 119–124 (2004).
[Crossref]

Pons, A.

M.T. Caballero, P. Andrés, A. Pons, J. Lancis, and M. Martínez-Corral, “Axial resolution in two-color excitation fluorescence microscopy by phase-only apodization,” Opt.Commun. 246, 313–321 (2005).
[Crossref]

Rastogi, P.

Ruan, H.

X. Zhao, C. Li, and H. Ruan, “A new approach for improving transverse superresolution in optical data storage,” Opt. Eng. 44, 125202 1–4 (2005).
[Crossref]

X. Zhao, C. Li, and H. Ruan, “Improvement of three-dimensional resolution in optical data storage by combination of two annular binary phase filters,” Chin. Phys. Lett. 21, 1515>–1517 (2004).
[Crossref]

Saavedra, G.

M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Aunable axial superresolution by annular binary filtres. Application to confocal microscopy,” Opt.Commun. 119, 491>–498 (1995).
[Crossref]

Sales, T.R.M.

T.R.M. Sales and G.M. Morris, “Axial superresolution with phase-only pùpil filters,” Opt.Commun. 156, 227>–230 (1998).
[Crossref]

Sheppard, C.J.R.

C.J.R. Sheppard and Z.S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am A 5, 643>–647 (1988).
[Crossref]

C.J.R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” Optica Acta 24, 1051>–1073 (1977).
[Crossref]

Sun, J.

M. Yun, L. Liu, J. Sun, and D. Liu, “Three-dimensional superresolution by three-zone complex pupil filters,” J. Opt. Soc. Am A 22 (2005) 272>–277.
[Crossref]

M. Yun, L. Liu, J. Sun, and D. Liu, “Transverse or axial superresolution with radial birefringent filter,” J. Opt. Soc. Am A 21, 1869>–1874 (2004).
[Crossref]

Sun, X.

Valle, P.J.

M.P. Cagigal, J.E. Oti, V.F. Canales, and P.J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241, 249>–253 (2004).
[Crossref]

V.F. Canales, J.E. Oti, P.J. Valle, and M.P. Cagigal, “Reduction of the diffraction pattern in segmented apertures”, Opt. Eng. (to be published).

Wang, H.

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl.Opt. 41, 5263>–5266 (2002).
[Crossref] [PubMed]

Yun, M.

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Figures (4)

Fig. 1.
Fig. 1.

Transverse PSF designed with our procedure to obtain a G value of 0.64 with S = 0.14 (solid curve) in comparison with the result found by Narayan et al. (dotted curve). For comparison the PSFs for an unobstructed pupil (dashed double-dotted curve) is also shown. v is normalized, so that the first zero of the PSF for the unobstructed pupil is unity.

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Fig. 2.
Fig. 2.

Axial PSFs corresponding to the filters in figure 1: filter designed with our procedure (solid curve), the one designed by Narayan et al. (dotted curve) and the unobstructed pupil (dashed double-dotted curve). u is normalized, so that the first zero of the PSF for the unobstructed pupil is unity.

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Fig. 3.
Fig. 3.

Transverse PSF designed with our procedure to obtain a G value of 0.7 (dotted-dashed curve). The result for a two-zones phase-only filter (dashed curve) is very similar. For comparison the PSFs for an unobstructed pupil (dashed double-dotted curve) and for the filters in figure 1, the filter designed with our procedure for G=0.64 (solid curve) and the one designed by Narayan et al. (dotted curve), are also shown. v is normalized, so that the first zero of the PSF for the unobstructed pupil is unity.

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Fig. 4.
Fig. 4.

Axial PSFs corresponding to the filters in figure 3: filter designed with our procedure to obtain a G value of 0.7 (dotted-dashed curve), the two-zones phase-only filter (dashed curve), an unobstructed pupil (dashed double-dotted curve), filter designed with our procedure for G=0.64 (solid curve) and the one designed by Narayan et al. (dotted curve). u is normalized, so that the first zero of the PSF for the unobstructed pupil is unity.

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Equations (15)

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U ( v , u ) = 2 ∫ 0 1 P ( ρ ) J 0 ( vρ ) exp ( ju ρ 2 / 2 ) ρ d ρ
P ( ρ ) = { t 1 e j π if ρ 0 ≤ ρ < ρ 1 t 2 e j 0 if ρ 1 ≤ ρ < ρ 2 ⋯ t i ( e j π ) i if ρ i − 1 ≤ ρ < ρ i ⋯ t N ( e j π ) N if ρ N − 1 ≤ ρ ≤ ρ N 0 if ρ > 1
U ( v , u = 0 ) = ∑ i = 1 N 2 t i ( − 1 ) i ( ρ i J 1 ( ρ i v ) v − ρ i − 1 J 1 ( ρ i − 1 v ) v ) =
= 2 t N ( − 1 ) N ρ N J 1 ( ρ N v ) v + ∑ i = 1 N − 1 2 ( t i + 1 + t i ) ( − 1 ) i ρ i J 1 ( ρ i v ) v
J 0 ( ρ n + ρ m 1 v ) = lim ρ n − ρ m → 0 ( 2 ρ n J 1 ( ρ n v ) v − 2 ρ m J 1 ( ρ m v ) v ) 1 ρ n 2 − ρ m 2
S = PSE ( 0,0 ) PSF clear pupil ( 0,0 )
Γ = PSE ( 0,0 ) PSF ( v 1 st sidelobe , 0 )
d U d v ( v = v 1 st sidelobe , u = 0 ) = 2 t N ( − 1 ) N ρ N 2 J 2 ( ρ N v ) v + ∑ i = 1 N − 1 2 ( t i + 1 + t i ) ( − 1 ) i ρ i 2 J 2 ( ρ i v ) v = 0
Resolution condition : PSF ( G· v 1 zA , 0 ) < ε Energy loss : S = PSF ( 0,0 ) > S 0 Side lobe height : Γ = PSF ( 0,0 ) PSF ( G· v 1 s , 0 ) > Γ 0
U ( v = 0 , u ) = ∑ i = 1 N t i ( − 1 ) N − i sin [ πu ( ρ i 2 − ρ i − 1 2 ) ] πu exp [ πju ( ρ i 2 + ρ i − 1 2 ) ]
d d u PSF ( v = 0 , u ) ∣ u = 0 = 0
d d u PSF ( v = 0 , u ) = [ ∫ 0 1 2 ρP ( ρ ) d ρ exp ( juρ 2 2 ) ( jρ 2 2 ) ] [ ∫ 0 1 2 ρP ( ρ ) d ρ exp ( juρ 2 2 ) ] * +
[ ∫ 0 1 2 ρP ( ρ ) d ρ exp ( juρ 2 2 ) ( jρ 2 2 ) ] * [ ∫ 0 1 2 ρP ( ρ ) d ρ exp ( juρ 2 2 ) ]
d d u PSF ( v = 0 , u ) ∣ u = 0 = ( ∑ i = 1 N t i 4 j exp ( j ϕ i ) ( ρ i 4 − ρ i − 1 4 ) ) ( ∑ i = 1 N exp ( − j ϕ i ) ( ρ i 2 − ρ i − 1 2 ) ) +
( ∑ i = 1 N t i 4 ( − j ) exp ( − j ϕ i ) ( ρ i 4 − ρ i − 1 4 ) ) ( ∑ i = 1 N t i exp ( j ϕ i ) ( ρ i 2 − ρ i − 1 2 ) )

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