Abstract

The resolving power of an optical imaging system is limited by residual aberrations and diffraction effects. The Rayleigh–Abbe diffraction limit of resolution corresponds to radius of the central lobe of the point spread function of an aberration free diffraction limited system. An attempt to circumvent this limitation was proposed by Toraldo di Francia, who showed that suitable pupil plane filtering can overcome this resolution limit, albeit over a restricted field. This paper reports results of our investigations on the use of evolutionary programming to obtain globally or quasi-globally optimum solutions in synthesis of lossless Toraldo filters consisting of concentric unequal area zones of fixed phase.

© 2013 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  3. G. Toraldo di Francia, La Diffrazione Delle Luce (Edizioni Scientifiche Einaudi, 1958).
  4. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
    [CrossRef]
  5. G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi. 7, 366–372 (1952).
  6. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9, 204–206 (1960).
  7. W. Lukosz, “Optical systems with resolving power exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  8. B. R. Frieden, “On arbitrary perfact imaging with a finite aperture,” Opt. Acta 16, 795–807 (1969).
    [CrossRef]
  9. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North Holland, 1971), Vol. IX, pp. 311–407.
  10. G. R. Boyer and M. Sechaud, “Superresolution by Taylor filters,” Appl. Opt. 12, 893–894 (1973).
    [CrossRef]
  11. G. R. Boyer, “Pupil filters for moderate superresolution,” Appl. Opt. 15, 3089–3093 (1976).
    [CrossRef]
  12. G. R. Boyer, “Realisation d’un filtrage super-resolvant,” Opt. Acta 30, 807–816 (1983).
    [CrossRef]
  13. R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
    [CrossRef]
  14. R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. II. Application of the impulse generating filter,” Opt. Acta 27, 1641–1670 (1980).
    [CrossRef]
  15. R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. III. Effects due to variation of the field extent,” Opt. Acta 30, 681–688 (1983).
    [CrossRef]
  16. I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. 72, 1287–1291 (1982).
    [CrossRef]
  17. Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [CrossRef]
  18. T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646 (1997).
    [CrossRef]
  19. C. J. R. Sheppard and Z. S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
    [CrossRef]
  20. M. P. Cagigal, J. E. Oti, V. F. Canals, and P. J. Valle, “Analytical design of superresolving phase filters,” Opt. Commun. 241, 249–253 (2004).
    [CrossRef]
  21. C. J. R. Sheppard, M. D. Sharma, and A. Arbouet, “Axial apodizing filters for confocal imaging,” Optik 111, 347–354 (2000).
  22. M. Yun, L. Liu, J. Sun, and D. Liu, “Three-dimensional superresolution by three-zone complex pupil filters,” J. Opt. Soc. Am. A 22, 272–277 (2005).
    [CrossRef]
  23. T. R. M. Sales and G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. 156, 227–230 (1998).
    [CrossRef]
  24. M. Martínez-Corral, M. T. Caballero, E. H. K. Stelzer, and J. Swoger, “Tailoring the axial shape of the point spread function using the Toraldo concept,” Opt. Express 10, 98–103 (2002).
    [CrossRef]
  25. H. Luo and C. Zhou, “Comparison of superresolution effects with annular phase and amplitude filters,” Appl. Opt. 43, 6242–6247 (2004).
    [CrossRef]
  26. X. Liu, L. Liu, D. Liu, and L. Bai, “Design and application of three-zone annular filters,” Optik 117, 453–461 (2006).
    [CrossRef]
  27. C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281, 913–922 (2008).
    [CrossRef]
  28. C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun. 281, 3623–3630 (2008).
    [CrossRef]
  29. M. Martínez-Corral, P. Andrés, and J. Ojeda-Castaneda, “On-axis diffractional behavior of two dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
    [CrossRef]
  30. M. Martínez-Corral, P. Andrés, J. Ojeda-Castañeda, and G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. 119, 491–498 (1995).
    [CrossRef]
  31. S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters, with applications to super-Gaussian phase filters,” Opt. Lett. 29, 932–934 (2004).
    [CrossRef]
  32. S. Ledesma, J. C. Escalera, J. Campos, and M. J. Yzuel, “Evolution of the transverse response of an optical system with complex filters,” Opt. Commun. 249, 183–192 (2005).
    [CrossRef]
  33. T. G. Jabbour, M. Petrovich, and S. M. Kuebler, “Design of axially super resolving phase filters using the method of generalized projection,” Opt. Commun. 281, 2002–2011 (2008).
    [CrossRef]
  34. L. N. Hazra, “Walsh filters for tailoring of resolution in microscopic imaging,” Micron 38, 129–135 (2007).
    [CrossRef]
  35. M. Martínez-Corral and G. Saavedra, “The resolution challenge in 3D optical microscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 1–67.
  36. Y. Tan, R. Guo, S. Xiao, G. Chang, and W. Huang, “Design of superresolved phase plates,” J. Laser Micro/Nanoeng. 1, 281–287 (2006).
    [CrossRef]
  37. L. Liu and G. Wang, “Designing superresolution optical pupil filter with constrained global optimization algorithm,” Optik 119, 481–484 (2008).
    [CrossRef]
  38. L. N. Hazra and N. Reza, “Optimal design of Toraldo super resolving filters,” Proc. SPIE 7787, 77870D (2010).
    [CrossRef]
  39. L. N. Hazra and N. Reza, “Superresolution by pupil plane phase filtering,” Pramana 75, 855–867 (2010).
    [CrossRef]
  40. I. Rechenberg, Evolutionsstrategie: Optimeirung Technischer System nach Prinzipien der Biologischen Evolution (Frommen–Holzboog Verlag, 1943).
  41. D. E. Goldberg, Genetic Algorithm in Search, Optimization and Machine Learning (Addison-Wesley, 1989).
  42. S. Banerjee and L. N. Hazra, “Structural design of broken contact doublets with prespecified aberration targets using genetic algorithm,” J. Mod. Opt. 49, 1111–1123 (2002).
    [CrossRef]

2010 (2)

L. N. Hazra and N. Reza, “Optimal design of Toraldo super resolving filters,” Proc. SPIE 7787, 77870D (2010).
[CrossRef]

L. N. Hazra and N. Reza, “Superresolution by pupil plane phase filtering,” Pramana 75, 855–867 (2010).
[CrossRef]

2008 (4)

L. Liu and G. Wang, “Designing superresolution optical pupil filter with constrained global optimization algorithm,” Optik 119, 481–484 (2008).
[CrossRef]

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, “Design of axially super resolving phase filters using the method of generalized projection,” Opt. Commun. 281, 2002–2011 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281, 913–922 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun. 281, 3623–3630 (2008).
[CrossRef]

2007 (1)

L. N. Hazra, “Walsh filters for tailoring of resolution in microscopic imaging,” Micron 38, 129–135 (2007).
[CrossRef]

2006 (2)

Y. Tan, R. Guo, S. Xiao, G. Chang, and W. Huang, “Design of superresolved phase plates,” J. Laser Micro/Nanoeng. 1, 281–287 (2006).
[CrossRef]

X. Liu, L. Liu, D. Liu, and L. Bai, “Design and application of three-zone annular filters,” Optik 117, 453–461 (2006).
[CrossRef]

2005 (2)

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

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

2004 (3)

2002 (2)

M. Martínez-Corral, M. T. Caballero, E. H. K. Stelzer, and J. Swoger, “Tailoring the axial shape of the point spread function using the Toraldo concept,” Opt. Express 10, 98–103 (2002).
[CrossRef]

S. Banerjee and L. N. Hazra, “Structural design of broken contact doublets with prespecified aberration targets using genetic algorithm,” J. Mod. Opt. 49, 1111–1123 (2002).
[CrossRef]

2000 (1)

C. J. R. Sheppard, M. D. Sharma, and A. Arbouet, “Axial apodizing filters for confocal imaging,” Optik 111, 347–354 (2000).

1998 (1)

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

1997 (1)

1995 (1)

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

1994 (1)

1988 (1)

1986 (1)

1983 (2)

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. III. Effects due to variation of the field extent,” Opt. Acta 30, 681–688 (1983).
[CrossRef]

G. R. Boyer, “Realisation d’un filtrage super-resolvant,” Opt. Acta 30, 807–816 (1983).
[CrossRef]

1982 (1)

1980 (2)

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. II. Application of the impulse generating filter,” Opt. Acta 27, 1641–1670 (1980).
[CrossRef]

1976 (1)

1973 (1)

1969 (1)

B. R. Frieden, “On arbitrary perfact imaging with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

1966 (1)

1960 (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9, 204–206 (1960).

1952 (2)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
[CrossRef]

G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi. 7, 366–372 (1952).

Andrés, P.

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

M. Martínez-Corral, P. Andrés, and J. Ojeda-Castaneda, “On-axis diffractional behavior of two dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
[CrossRef]

Arbouet, A.

C. J. R. Sheppard, M. D. Sharma, and A. Arbouet, “Axial apodizing filters for confocal imaging,” Optik 111, 347–354 (2000).

Bai, L.

X. Liu, L. Liu, D. Liu, and L. Bai, “Design and application of three-zone annular filters,” Optik 117, 453–461 (2006).
[CrossRef]

Banerjee, S.

S. Banerjee and L. N. Hazra, “Structural design of broken contact doublets with prespecified aberration targets using genetic algorithm,” J. Mod. Opt. 49, 1111–1123 (2002).
[CrossRef]

Boivin, A.

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. III. Effects due to variation of the field extent,” Opt. Acta 30, 681–688 (1983).
[CrossRef]

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. II. Application of the impulse generating filter,” Opt. Acta 27, 1641–1670 (1980).
[CrossRef]

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Boivin, R.

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. III. Effects due to variation of the field extent,” Opt. Acta 30, 681–688 (1983).
[CrossRef]

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field. II. Application of the impulse generating filter,” Opt. Acta 27, 1641–1670 (1980).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Boyer, G. R.

Caballero, M. T.

Cagigal, M. P.

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

Campos, J.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281, 913–922 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun. 281, 3623–3630 (2008).
[CrossRef]

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

S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters, with applications to super-Gaussian phase filters,” Opt. Lett. 29, 932–934 (2004).
[CrossRef]

Canals, V. F.

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

Chang, G.

Y. Tan, R. Guo, S. Xiao, G. Chang, and W. Huang, “Design of superresolved phase plates,” J. Laser Micro/Nanoeng. 1, 281–287 (2006).
[CrossRef]

Cox, I. J.

Escalera, J. C.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun. 281, 3623–3630 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281, 913–922 (2008).
[CrossRef]

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

S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters, with applications to super-Gaussian phase filters,” Opt. Lett. 29, 932–934 (2004).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “On arbitrary perfact imaging with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North Holland, 1971), Vol. IX, pp. 311–407.

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithm in Search, Optimization and Machine Learning (Addison-Wesley, 1989).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Guo, R.

Y. Tan, R. Guo, S. Xiao, G. Chang, and W. Huang, “Design of superresolved phase plates,” J. Laser Micro/Nanoeng. 1, 281–287 (2006).
[CrossRef]

Hazra, L. N.

L. N. Hazra and N. Reza, “Optimal design of Toraldo super resolving filters,” Proc. SPIE 7787, 77870D (2010).
[CrossRef]

L. N. Hazra and N. Reza, “Superresolution by pupil plane phase filtering,” Pramana 75, 855–867 (2010).
[CrossRef]

L. N. Hazra, “Walsh filters for tailoring of resolution in microscopic imaging,” Micron 38, 129–135 (2007).
[CrossRef]

S. Banerjee and L. N. Hazra, “Structural design of broken contact doublets with prespecified aberration targets using genetic algorithm,” J. Mod. Opt. 49, 1111–1123 (2002).
[CrossRef]

Hegedus, Z. S.

Huang, W.

Y. Tan, R. Guo, S. Xiao, G. Chang, and W. Huang, “Design of superresolved phase plates,” J. Laser Micro/Nanoeng. 1, 281–287 (2006).
[CrossRef]

Jabbour, T. G.

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, “Design of axially super resolving phase filters using the method of generalized projection,” Opt. Commun. 281, 2002–2011 (2008).
[CrossRef]

Kartashev, A. I.

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectrosc. 9, 204–206 (1960).

Kuebler, S. M.

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, “Design of axially super resolving phase filters using the method of generalized projection,” Opt. Commun. 281, 2002–2011 (2008).
[CrossRef]

Ledesma, S.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281, 913–922 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun. 281, 3623–3630 (2008).
[CrossRef]

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

S. Ledesma, J. Campos, J. C. Escalera, and M. J. Yzuel, “Simple expressions for performance parameters of complex filters, with applications to super-Gaussian phase filters,” Opt. Lett. 29, 932–934 (2004).
[CrossRef]

Liu, D.

X. Liu, L. Liu, D. Liu, and L. Bai, “Design and application of three-zone annular filters,” Optik 117, 453–461 (2006).
[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, 272–277 (2005).
[CrossRef]

Liu, L.

L. Liu and G. Wang, “Designing superresolution optical pupil filter with constrained global optimization algorithm,” Optik 119, 481–484 (2008).
[CrossRef]

X. Liu, L. Liu, D. Liu, and L. Bai, “Design and application of three-zone annular filters,” Optik 117, 453–461 (2006).
[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, 272–277 (2005).
[CrossRef]

Liu, X.

X. Liu, L. Liu, D. Liu, and L. Bai, “Design and application of three-zone annular filters,” Optik 117, 453–461 (2006).
[CrossRef]

Lukosz, W.

Luo, H.

Martínez-Corral, M.

M. Martínez-Corral, M. T. Caballero, E. H. K. Stelzer, and J. Swoger, “Tailoring the axial shape of the point spread function using the Toraldo concept,” Opt. Express 10, 98–103 (2002).
[CrossRef]

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

M. Martínez-Corral, P. Andrés, and J. Ojeda-Castaneda, “On-axis diffractional behavior of two dimensional pupils,” Appl. Opt. 33, 2223–2229 (1994).
[CrossRef]

M. Martínez-Corral and G. Saavedra, “The resolution challenge in 3D optical microscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 1–67.

Morris, G. M.

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

T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646 (1997).
[CrossRef]

Ojeda-Castaneda, J.

Ojeda-Castañeda, J.

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

Oti, J. E.

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

Petrovich, M.

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, “Design of axially super resolving phase filters using the method of generalized projection,” Opt. Commun. 281, 2002–2011 (2008).
[CrossRef]

Rechenberg, I.

I. Rechenberg, Evolutionsstrategie: Optimeirung Technischer System nach Prinzipien der Biologischen Evolution (Frommen–Holzboog Verlag, 1943).

Reza, N.

L. N. Hazra and N. Reza, “Superresolution by pupil plane phase filtering,” Pramana 75, 855–867 (2010).
[CrossRef]

L. N. Hazra and N. Reza, “Optimal design of Toraldo super resolving filters,” Proc. SPIE 7787, 77870D (2010).
[CrossRef]

Saavedra, G.

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

M. Martínez-Corral and G. Saavedra, “The resolution challenge in 3D optical microscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 53, pp. 1–67.

Sales, T. R. M.

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

T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646 (1997).
[CrossRef]

Sarafis, V.

Sechaud, M.

Sharma, M. D.

C. J. R. Sheppard, M. D. Sharma, and A. Arbouet, “Axial apodizing filters for confocal imaging,” Optik 111, 347–354 (2000).

Sheppard, C. J. R.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Two-zone pupil filters,” Opt. Commun. 281, 913–922 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun. 281, 3623–3630 (2008).
[CrossRef]

C. J. R. Sheppard, M. D. Sharma, and A. Arbouet, “Axial apodizing filters for confocal imaging,” Optik 111, 347–354 (2000).

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

I. J. Cox, C. J. R. Sheppard, and T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. 72, 1287–1291 (1982).
[CrossRef]

Stelzer, E. H. K.

Sun, J.

Swoger, J.

Tan, Y.

Y. Tan, R. Guo, S. Xiao, G. Chang, and W. Huang, “Design of superresolved phase plates,” J. Laser Micro/Nanoeng. 1, 281–287 (2006).
[CrossRef]

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–438 (1952).
[CrossRef]

G. Toraldo di Francia, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi. 7, 366–372 (1952).

G. Toraldo di Francia, La Diffrazione Delle Luce (Edizioni Scientifiche Einaudi, 1958).

Valle, P. J.

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

Wang, G.

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Figures (8)

Fig. 1.
Fig. 1.

Limited field of view of a Toraldo filter. Dotted curve: point spread function of airy pupil. Solid curve: point spread function of a pupil with Toraldo filter.

Fig. 2.
Fig. 2.

Transverse section in the image space. EO, optical axis; E, exit pupil; O, image plane; α, semi-angular aperture; n, refractive index of the image space.

Fig. 3.
Fig. 3.

mth zero-one function Wm(r).

Fig. 4.
Fig. 4.

Flowchart of the evolutionary programming.

Fig. 5.
Fig. 5.

Illustration for different parameters of the merit function in the point spread function of an arbitrary phase filter.

Fig. 6.
Fig. 6.

Point spread functions of optimal phase filters of the first kind. Phase at a zone=mπ. (a) 2-zone: p1=2.99, p2=3.97; (b) 3-zone: p1=2.83, p2=5.95; (c) 4-zone: p1=2.69, p2=4.98; (d) 5-zone: p1=2.59, p2=5.01; and (e) 6-zone: p1=2.55, p2=5.19. In each case ε=0.02; Strehl ratio S and filter parameters are given in inset.

Fig. 7.
Fig. 7.

Point spread functions of optimal phase filters of the second kind. Phase at a zone=mπ. (a) 2-zone: p1=3.25, p2=5.03; (b) 3-zone: p1=3.15, p2=7.50; (c) 4-zone: p1=3.17, p2=6.80; (d) 5-zone: p1=3.09, p2=7.58; and (e) 6-zone: p1=3.27, p2=9.96. In each case ε=0.02; Strehl ratio S and filter parameters are given in inset.

Fig. 8.
Fig. 8.

Superresolving two-zone, two-phase filters with narrow central lobes. First zero occurs at (a) p=2.77, (b) p=2.41, (c) p=2.02, and (d) p=1.26. Strehl ratio S and filter parameters are given in inset.

Equations (18)

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F(p)=01f(r)J0(pr)rdr,
p=2πλ(nsinα)χ,
f(r)=1,for0r1=0,otherwise.
F0(p)=01J0(pr)rdr.
F0(0)=01rdr=12.
FN(p)=F(p)F0(0)=2F(p).
f(r)=m=1MfmWm(r),
Wm(r)=1,forrm1rrm=0,otherwise.
FN(p)=2m=1Mfm01Wm(r)J0(pr)rdr=2m=1Mfmrm1rmJ0(pr)rdr.
FN(p)=2m=1MfmIm(p),
Im(p)=[rmJ1(prm)rm1J1(prm1)p]=[rm2J1(prm)(prm)rm12J1(prm1)(prm1)].
fm=eiαm.
FN(p)=2m=1MeiαmIm(p).
IN(p)=|FN(p)|2=4m=1Mn=1M[cos(αmαn)]Im(p)In(p).
Φ=11+ψ.
ψ=ω1(p1Tp1)2+ω2(p2Tp2)2,
p1T=p1Bestδpp2T=p2Best+δp,
Φ=0ifIN(0)<SC,

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