Abstract

Complex pupil filters are introduced to improve the three-dimensional resolving power of an optical imaging system. Through the design of the essential parameters of such filters, the transmittance and radius of the first zone, three-dimensional superresolution is realized. The Strehl ratio and the transverse and axial gains of such filters are analyzed in detail. A series of simulation examples of such filters are also presented that prove that three-dimensional superresolution can be realized. The advantage of such filters is that it is easy to realize three-dimensional superresolution, and the disadvantage is that the sidelobes of the axial intensity distribution are too high. But this can be overcome by the application of a confocal system.

© 2005 Optical Society of America

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References

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  1. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).
    [CrossRef]
  2. G. Boyer, “New class of axially apodizing filters for confocal scanning microscopy,” J. Opt. Soc. Am. A 19, 584–589 (2002).
    [CrossRef]
  3. M. Gu, T. Tannous, C. J. R. Sheppard, “Effect of an annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1995).
    [CrossRef]
  4. C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).
  5. C. J. R. Sheppard, Z. S. Hegedus, “Axial behavior of pupil plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
    [CrossRef]
  6. D. M. de Juana, J. E. Oti, V. F. Canales, 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]
  7. T. R. M. Sales, G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14, 1637–1646 (1997).
    [CrossRef]
  8. T. R. M. Sales, G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. 156, 227–230 (1998).
    [CrossRef]
  9. H. Liu, Y. Yan, Q. Tan, G. Jin, “Theories for the design of diffractive superresolution elements and limits of optical superresolution,” J. Opt. Soc. Am. A 19, 2185–2193 (2002).
    [CrossRef]
  10. M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).
  11. M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
    [CrossRef]
  12. H. Liu, Y. Yan, D. Yi, G. Jin, “Design of three-dimensional superresolution filters and limits of axial optical superresolution,” Appl. Opt. 42, 1463–1476 (2003).
    [CrossRef] [PubMed]
  13. X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).
  14. M. Martinez-Corral, V. F. Canales, P. J. Valle, M. P. Cagigal, “Focusing property of annular binary phase filters,” Opt. Commun. 229, 71–77 (2004).
    [CrossRef]
  15. A. I. Whiting, A. F. Abouraddy, B. E. A. Saleh, M. C. Teich, J. T. Fourkas, “Polarization-assisted transverse and axial optical superresolution,” Opt. Express 11, 1714–1723 (2003).
    [CrossRef] [PubMed]
  16. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, UK, 1999).
  17. D. M. de Juan, J. E. Oti, V. F. Canales, M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28, 607–609 (2003).
    [CrossRef]
  18. M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 643–647 (1992).
    [CrossRef]

2004 (1)

M. Martinez-Corral, V. F. Canales, P. J. Valle, M. P. Cagigal, “Focusing property of annular binary phase filters,” Opt. Commun. 229, 71–77 (2004).
[CrossRef]

2003 (4)

2002 (2)

2001 (1)

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

1999 (1)

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

1998 (2)

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

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

1997 (1)

1995 (2)

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

M. Gu, T. Tannous, C. J. R. Sheppard, “Effect of an annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1995).
[CrossRef]

1992 (1)

1988 (1)

1952 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).
[CrossRef]

Abouraddy, A. F.

Andres, P.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, UK, 1999).

Boyer, G.

Cagigal, M. P.

Canales, V. F.

de Juan, D. M.

de Juana, D. M.

Deng, X.

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

Fourkas, J. T.

Gu, M.

Hegedus, Z. S.

Jin, G.

Kowalczyk, M.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

Liu, H.

Martinez-Corral, M.

M. Martinez-Corral, V. F. Canales, P. J. Valle, M. P. Cagigal, “Focusing property of annular binary phase filters,” Opt. Commun. 229, 71–77 (2004).
[CrossRef]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

Morris, G. M.

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

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

Oti, J. E.

Saleh, B. E. A.

Sales, T. R. M.

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

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

Sheppard, C. J. R.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

M. Gu, T. Tannous, C. J. R. Sheppard, “Effect of an annular pupil on confocal imaging through highly scattering media,” Opt. Lett. 21, 312–314 (1995).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9, 643–647 (1992).
[CrossRef]

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

Tan, Q.

Tannous, T.

Teich, M. C.

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).
[CrossRef]

Valle, P. J.

M. Martinez-Corral, V. F. Canales, P. J. Valle, M. P. Cagigal, “Focusing property of annular binary phase filters,” Opt. Commun. 229, 71–77 (2004).
[CrossRef]

Wang, G.

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

Whiting, A. I.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, UK, 1999).

Xu, Z.

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

Yan, Y.

Yi, D.

Zapata-Rodriguez, C. J.

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

Appl. Opt. (1)

Chin. J. Lasers (1)

X. Deng, G. Wang, Z. Xu, “3-D superresolution pupil filter,” Chin. J. Lasers A28, 459–462 (2001).

J. Opt. Soc. Am. A (6)

Nuovo Cimento, Suppl. (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426–435 (1952).
[CrossRef]

Opt. Commun. (3)

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

M. Martinez-Corral, V. F. Canales, P. J. Valle, M. P. Cagigal, “Focusing property of annular binary phase filters,” Opt. Commun. 229, 71–77 (2004).
[CrossRef]

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, M. Kowalczyk, “Three-dimensional superresolution by annular binary filters,” Opt. Commun. 165, 267–278 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Optik (Stuttgart) (2)

M. Martinez-Corral, P. Andres, C. J. Zapata-Rodriguez, C. J. R. Sheppard, “Improvement of three-dimensional resolution in confocal scanning microscopy by combination of two pupil filters,” Optik (Stuttgart) 107, 145–148 (1998).

C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik (Stuttgart) 99, 32–34 (1995).

Other (1)

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, UK, 1999).

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Figures (9)

Fig. 1
Fig. 1

Structure of the three-zone complex pupil filter.

Fig. 2
Fig. 2

Relation among the transverse gain (solid curve), the axial gain (dashed curve), and the transmittance of the first zone.

Fig. 3
Fig. 3

Relation between the Strehl ratio and the transmittance of the first zone.

Fig. 4
Fig. 4

For a given value a=0.60, the (a) transverse and (b) axial intensity distributions for the complex filter with t=0.55 (dashed curve), t=0.65 (dotted curve), t=0.75 (dashed–dotted curve) compared with the clear pupil (solid curve).

Fig. 5
Fig. 5

Relation between the influence of the sidelobes and the transmittance: (a) effect of transverse sidelobes and (b) effect of axial sidelobes.

Fig. 6
Fig. 6

Relation among the transverse gain (solid curve), the axial gain (dashed curve), and the radius of the first zone.

Fig. 7
Fig. 7

Relation between the Strehl ratio and the radius of the first zone.

Fig. 8
Fig. 8

For a given value t=0.80, the (a) transverse and (b) axial intensity distributions for the complex filter with a=0.55 (dashed curve), a=0.60 (dotted curve), and a=0.65 (dashed–dotted curve) compared with the clear pupil (solid curve).

Fig. 9
Fig. 9

Relation between the effects of the sidelobes and the radius: (a) effect of transverse sidelobes and (b) effect of axial sidelobes.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

U(v, u)=201P(ρ)J0(vρ)exp-iuρ22ρdρ.
v=kr sin α,
u=4kz sin2α2,
U(v, 0)=201P(ρ)J0(vρ)ρdρ,
U(0, u)=201P(ρ)exp-iuρ22ρdρ.
U(0, u)=01Q(t)exp-iut2dt.
I(ν, 0)=|I0|2-12 Re(I0I1*)ν2,
I(0, u)=|I0|2-Im(I0*I1)u-14[Re(I2*I0-|I1|2)]u2,
In=201P(ρ)ρ2n+1dρ.
uF=-2 Im(I0*I1)Re(I2*I0)-|I1|2,
S=|I0|2-uF Im(I0*I1).
GT=2Re(I0I1*)-uF Im(I0*I2)S,
GA=12Re(I0I2*)-|I1|2S.
P(ρ)=T(ρ)exp[iφ(ρ)]=texp(0.6πi)0ρa1a<ρbexp(0.6πi)b<ρ1.
I0=b2-a2+exp(iϕ)(ta2-b2+1)=a0+ib0,
I1=12[b4-a4+exp(iϕ)(ta4-b4+1)]=a1+ib1,
I2=13[b6-a6+exp(iϕ)(ta6-b6+1)]=a2+ib2,
uF=a1b0-a0b113(a2a0+b2b0)-14(a12+b12),
S=a02+b02-12uF(a0b1-a1b0),
GT=212(a1a0+b0b1)-13uF(a2b0-a0b2)S,
GA=1213(a2a0+b0b2)-14uF(a12+b12)S.
U(ν, 0)=2ν{exp(0.6πi)J1(ν)+a[t exp(0.6πi)-1]J1(aν)+b[1-exp(0.6πi)]J1(bν)},
U(0, u)=i2u{t exp(0.6πi)[exp(-iuρ2/2)-1]+exp(-iub2/2)-exp(-iua2/2)+exp(0.6πi)[exp(-iu/2)-exp(-iub2/2)]}.
I(ν, 0)=U(ν, 0)U*(ν, 0),I(0, u)=U(0, u)U*(0, u),

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