Non-iterative method for designing super-resolving pupil filters

Noé Alcalá Ochoa; J. E. A. Landgrave

doi:10.1364/OE.19.023613

Abstract

We propose a method of designing pupil filters for transverse super-resolution without making use of recursive algorithms or the parabolic approximation for the point spread function (PSF). We represent the amplitude of the PSF as an expansion of orthogonal functions from the Fourier-Bessel transform of a Dini series. Their coefficients are related with desired features of the PSF, such as the transversal super-resolution gain and the intensity of the secondary maxima. We show the possibility to derive closed formulas to obtain large super-resolution gains with tolerable side-lobe intensities, at the expense of increasing the intensity of a chosen secondary lobe.

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References

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Publication

M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett. 32(15), 2239–2241 (2007).
[Crossref] [PubMed]
D. M. de Juana, J. E. Otti, V. F. Canales, and M. P. Cagigal, “Tranverse or axial superresolution in a 4Pi-confocal microscope by phase only filters,” J. Opt. Soc. Am. A 20(11), 2172–2178 (2003).
[Crossref]
V. F. Canales, P. J. Valle, J. E. Oti, and M. P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. 7, 720–723 (2009).
[Crossref]
T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. 81(18), 3844–3847 (1998).
[Crossref]
G. Toraldo di Francia, “Super-gain antennas and optical resolving Power,” Nuovo Cim. 9(S3Suppl.), 426–438 (1952).
[Crossref]
C. J. R. Sheppard and Z. S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am. A 5(5), 643–647 (1988).
[Crossref]
T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14(7), 1637–1646 (1997).
[Crossref]
D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003).
[Crossref] [PubMed]
M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” PASP 116(824), 965–970 (2004).
[Crossref]
V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analitically designed phase masks,” Opt. Commun. 247(1-3), 11–18 (2005).
[Crossref]
P. N. Gundu, E. Hack, and P. Rastogi, “High efficient superresolution combination filter with twin LCD spatial light modulators,” Opt. Express 13(8), 2835–2842 (2005).
[Crossref] [PubMed]
V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express 14(22), 10393–10402 (2006).
[Crossref] [PubMed]
P. N. Gundu, E. Hack, and P. Rastogi, “Apodized superresolution – concept and simulations,” Opt. Commun. 249(1-3), 101–107 (2005).
[Crossref]
N. A. Ochoa, J. García-Márquez, and A. González-Vega, “Hybrid pupil filter design using Bessel series,” Opt. Commun. (To be published).
J. E. A. Landgrave and L. R. Berriel-Valdos, “Sampling expansions for three-dimensional light amplitude distribution in the vicinity of an axial image point,” J. Opt. Soc. Am. A 14(11), 2962–2976 (1997).
[Crossref]

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analitically designed phase masks,” Opt. Commun. 247(1-3), 11–18 (2005).
[Crossref]

P. N. Gundu, E. Hack, and P. Rastogi, “Apodized superresolution – concept and simulations,” Opt. Commun. 249(1-3), 101–107 (2005).
[Crossref]

2004 (1)

M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” PASP 116(824), 965–970 (2004).
[Crossref]

2003 (2)

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003).
[Crossref] [PubMed]

D. M. de Juana, J. E. Otti, V. F. Canales, and M. P. Cagigal, “Tranverse or axial superresolution in a 4Pi-confocal microscope by phase only filters,” J. Opt. Soc. Am. A 20(11), 2172–2178 (2003).
[Crossref]

1998 (1)

T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. 81(18), 3844–3847 (1998).
[Crossref]

1997 (2)

T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14(7), 1637–1646 (1997).
[Crossref]

J. E. A. Landgrave and L. R. Berriel-Valdos, “Sampling expansions for three-dimensional light amplitude distribution in the vicinity of an axial image point,” J. Opt. Soc. Am. A 14(11), 2962–2976 (1997).
[Crossref]

1988 (1)

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

1952 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving Power,” Nuovo Cim. 9(S3Suppl.), 426–438 (1952).
[Crossref]

Berriel-Valdos, L. R.

J. E. A. Landgrave and L. R. Berriel-Valdos, “Sampling expansions for three-dimensional light amplitude distribution in the vicinity of an axial image point,” J. Opt. Soc. Am. A 14(11), 2962–2976 (1997).
[Crossref]

Boyer, G.

M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett. 32(15), 2239–2241 (2007).
[Crossref] [PubMed]

Cagigal, M. P.

V. F. Canales, P. J. Valle, J. E. Oti, and M. P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. 7, 720–723 (2009).
[Crossref]

V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express 14(22), 10393–10402 (2006).
[Crossref] [PubMed]

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analitically designed phase masks,” Opt. Commun. 247(1-3), 11–18 (2005).
[Crossref]

M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” PASP 116(824), 965–970 (2004).
[Crossref]

D. M. de Juana, J. E. Otti, V. F. Canales, and M. P. Cagigal, “Tranverse or axial superresolution in a 4Pi-confocal microscope by phase only filters,” J. Opt. Soc. Am. A 20(11), 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(8), 607–609 (2003).
[Crossref] [PubMed]

Canales, V. F.

V. F. Canales, P. J. Valle, J. E. Oti, and M. P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. 7, 720–723 (2009).
[Crossref]

V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express 14(22), 10393–10402 (2006).
[Crossref] [PubMed]

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analitically designed phase masks,” Opt. Commun. 247(1-3), 11–18 (2005).
[Crossref]

M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” PASP 116(824), 965–970 (2004).
[Crossref]

D. M. de Juana, J. E. Otti, V. F. Canales, and M. P. Cagigal, “Tranverse or axial superresolution in a 4Pi-confocal microscope by phase only filters,” J. Opt. Soc. Am. A 20(11), 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(8), 607–609 (2003).
[Crossref] [PubMed]

Chériaux, G.

M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett. 32(15), 2239–2241 (2007).
[Crossref] [PubMed]

de Juana, D. M.

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003).
[Crossref] [PubMed]

D. M. de Juana, J. E. Otti, V. F. Canales, and M. P. Cagigal, “Tranverse or axial superresolution in a 4Pi-confocal microscope by phase only filters,” J. Opt. Soc. Am. A 20(11), 2172–2178 (2003).
[Crossref]

García-Márquez, J.

N. A. Ochoa, J. García-Márquez, and A. González-Vega, “Hybrid pupil filter design using Bessel series,” Opt. Commun. (To be published).

González-Vega, A.

N. A. Ochoa, J. García-Márquez, and A. González-Vega, “Hybrid pupil filter design using Bessel series,” Opt. Commun. (To be published).

Gundu, P. N.

P. N. Gundu, E. Hack, and P. Rastogi, “High efficient superresolution combination filter with twin LCD spatial light modulators,” Opt. Express 13(8), 2835–2842 (2005).
[Crossref] [PubMed]

P. N. Gundu, E. Hack, and P. Rastogi, “Apodized superresolution – concept and simulations,” Opt. Commun. 249(1-3), 101–107 (2005).
[Crossref]

Hack, E.

P. N. Gundu, E. Hack, and P. Rastogi, “Apodized superresolution – concept and simulations,” Opt. Commun. 249(1-3), 101–107 (2005).
[Crossref]

P. N. Gundu, E. Hack, and P. Rastogi, “High efficient superresolution combination filter with twin LCD spatial light modulators,” Opt. Express 13(8), 2835–2842 (2005).
[Crossref] [PubMed]

Hegedus, Z. S.

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

Landgrave, J. E. A.

J. E. A. Landgrave and L. R. Berriel-Valdos, “Sampling expansions for three-dimensional light amplitude distribution in the vicinity of an axial image point,” J. Opt. Soc. Am. A 14(11), 2962–2976 (1997).
[Crossref]

Merano, M.

M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett. 32(15), 2239–2241 (2007).
[Crossref] [PubMed]

Morris, G. M.

T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14(7), 1637–1646 (1997).
[Crossref]

Mourou, G.

M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett. 32(15), 2239–2241 (2007).
[Crossref] [PubMed]

Ochoa, N. A.

N. A. Ochoa, J. García-Márquez, and A. González-Vega, “Hybrid pupil filter design using Bessel series,” Opt. Commun. (To be published).

Oti, J. E.

V. F. Canales, P. J. Valle, J. E. Oti, and M. P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. 7, 720–723 (2009).
[Crossref]

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analitically designed phase masks,” Opt. Commun. 247(1-3), 11–18 (2005).
[Crossref]

M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” PASP 116(824), 965–970 (2004).
[Crossref]

D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. 28(8), 607–609 (2003).
[Crossref] [PubMed]

Otti, J. E.

D. M. de Juana, J. E. Otti, V. F. Canales, and M. P. Cagigal, “Tranverse or axial superresolution in a 4Pi-confocal microscope by phase only filters,” J. Opt. Soc. Am. A 20(11), 2172–2178 (2003).
[Crossref]

Rastogi, P.

P. N. Gundu, E. Hack, and P. Rastogi, “Apodized superresolution – concept and simulations,” Opt. Commun. 249(1-3), 101–107 (2005).
[Crossref]

P. N. Gundu, E. Hack, and P. Rastogi, “High efficient superresolution combination filter with twin LCD spatial light modulators,” Opt. Express 13(8), 2835–2842 (2005).
[Crossref] [PubMed]

Sales, T. R. M.

T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. 81(18), 3844–3847 (1998).
[Crossref]

T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14(7), 1637–1646 (1997).
[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(5), 643–647 (1988).
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving Power,” Nuovo Cim. 9(S3Suppl.), 426–438 (1952).
[Crossref]

Trisorio, A.

M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett. 32(15), 2239–2241 (2007).
[Crossref] [PubMed]

Valle, P. J.

V. F. Canales, P. J. Valle, J. E. Oti, and M. P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. 7, 720–723 (2009).
[Crossref]

Chin. Opt. Lett. (1)

V. F. Canales, P. J. Valle, J. E. Oti, and M. P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. 7, 720–723 (2009).
[Crossref]

J. Opt. Soc. Am. A (4)

D. M. de Juana, J. E. Otti, V. F. Canales, and M. P. Cagigal, “Tranverse or axial superresolution in a 4Pi-confocal microscope by phase only filters,” J. Opt. Soc. Am. A 20(11), 2172–2178 (2003).
[Crossref]

T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A 14(7), 1637–1646 (1997).
[Crossref]

J. E. A. Landgrave and L. R. Berriel-Valdos, “Sampling expansions for three-dimensional light amplitude distribution in the vicinity of an axial image point,” J. Opt. Soc. Am. A 14(11), 2962–2976 (1997).
[Crossref]

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

Nuovo Cim. (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving Power,” Nuovo Cim. 9(S3Suppl.), 426–438 (1952).
[Crossref]

Opt. Commun. (3)

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analitically designed phase masks,” Opt. Commun. 247(1-3), 11–18 (2005).
[Crossref]

P. N. Gundu, E. Hack, and P. Rastogi, “Apodized superresolution – concept and simulations,” Opt. Commun. 249(1-3), 101–107 (2005).
[Crossref]

N. A. Ochoa, J. García-Márquez, and A. González-Vega, “Hybrid pupil filter design using Bessel series,” Opt. Commun. (To be published).

Opt. Express (2)

P. N. Gundu, E. Hack, and P. Rastogi, “High efficient superresolution combination filter with twin LCD spatial light modulators,” Opt. Express 13(8), 2835–2842 (2005).
[Crossref] [PubMed]

V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express 14(22), 10393–10402 (2006).
[Crossref] [PubMed]

Opt. Lett. (2)

M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett. 32(15), 2239–2241 (2007).
[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(8), 607–609 (2003).
[Crossref] [PubMed]

PASP (1)

M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” PASP 116(824), 965–970 (2004).
[Crossref]

Phys. Rev. Lett. (1)

T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. 81(18), 3844–3847 (1998).
[Crossref]

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Fig. 1 PSFs obtained with our method for different values of the super-resolution gain parameter:

ε = 0.80, 0.65, 0.63, 0.60

. Only two basis functions (K = 1) were employed in this example.

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Fig. 2 Pupil functions corresponding to the PSFs with

ε = 0.65

and

0.80

in Fig. 1.

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Fig. 3 PSFs obtained with our method for

Γ_{1} = 3.0, 3.5, 4.5

. In all cases K = 2 and

ε = 0.65

. Notice that when

Γ_{1} ≃ 4.5

,

Γ_{2} ≃ Γ_{1}

.

Download Full Size | PPT Slide | PDF

Fig. 4 Relative errors for various nominal values of the parameter

Γ_{1}

. In all cases

ε = 0.65

and K = 2.

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Fig. 5 PSF designed for a high super-resolving gain parameter,

ε = 0.50

, with

Γ_{1} = Γ_{2} = 2 .5

.

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Fig. 6 Comparison of the PSF obtained with our method and the best PSF presented in ref [12]. For the purposes of comparison, both PSFs have been normalized.

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Equations (14)

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G (δ, v) = 2 \int_{0}^{1} g (ρ) \exp (- i 2 π ​ δ ​ ρ^{2}) J_{0} (v ​ ρ) ρ d ​ ​ ρ,

g (ρ) = \sum_{n = 0}^{K} \frac{C_{n}}{J_{0}^{2} (α_{n})} J_{0} (α_{n} ρ),

ϕ (v) = G (0, v) = \frac{1}{J_{0}^{2} (α_{n})} \sum_{n = 0}^{K} C_{n} 2 \int_{0}^{1} J_{0} (α_{n} ρ) J_{0} (v ρ) ρ d ρ .

ϕ (v) = \sum_{n = 0}^{K} C_{n} ϕ_{n} (v),

ϕ_{n} (ν) = \frac{1}{J_{0} (α_{n})} \frac{2 ν J_{1} (ν)}{ν^{2} - α_{n}^{2}} .

ϕ (α_{k}) = C_{k} .

ε = \frac{D}{D_{c}},

S = \frac{{| ϕ (0) |}^{2}}{| ϕ_{c} (0) |^{2}},

Γ_{n} = \frac{{| ϕ (0) |}^{2}}{| ϕ (μ_{n}) |^{2}}, n = 0, ..., K,

\frac{| C_{0} |}{| C_{n} |} = \frac{| ϕ (α_{0}) |}{| ϕ (α_{n}) |} ≃ \frac{| ϕ (0) |}{| ϕ (μ_{n}) |} = \sqrt{Γ_{n}}, n = 0, ..., K,

| C_{n} | ≃ \sqrt{\frac{S}{Γ_{n}}}, n = 0, ..., K .

C_{n} ≃ \sqrt{\frac{S}{Γ_{n}}} \exp (i n π) = {(- 1)}^{n} \sqrt{\frac{S}{Γ_{n}}}, n = 0, ..., K .

ϕ (ε α_{1}) = \sum_{n = 0}^{K} C_{n} ϕ_{n} (ε α_{1}) = 0, ε < 1 .

C_{K} = \frac{\sqrt{S}}{ϕ_{K} (ε α_{1})} \sum_{n = 0}^{K - 1} \frac{{(- 1)}^{n}}{\sqrt{Γ_{n}}} ϕ_{n} (ε α_{1}) .

Abstract

References

2009 (1)

2007 (1)

2006 (1)

2005 (3)

2004 (1)

2003 (2)

1998 (1)

1997 (2)

1988 (1)

1952 (1)

Berriel-Valdos, L. R.

Boyer, G.

Cagigal, M. P.

Canales, V. F.

Chériaux, G.

de Juana, D. M.

García-Márquez, J.

González-Vega, A.

Gundu, P. N.

Hack, E.

Hegedus, Z. S.

Landgrave, J. E. A.

Merano, M.

Morris, G. M.

Mourou, G.

Ochoa, N. A.

Oti, J. E.

Otti, J. E.

Rastogi, P.

Sales, T. R. M.

Sheppard, C. J. R.

Toraldo di Francia, G.

Trisorio, A.

Valle, P. J.

Chin. Opt. Lett. (1)

J. Opt. Soc. Am. A (4)

Nuovo Cim. (1)

Opt. Commun. (3)

Opt. Express (2)

Opt. Lett. (2)

PASP (1)

Phys. Rev. Lett. (1)

Cited By

Figures (6)

Equations (14)

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