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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 23613–23620
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Non-iterative method for designing super-resolving pupil filters

Noé Alcalá Ochoa and J. E. A. Landgrave  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 23613-23620 (2011)
http://dx.doi.org/10.1364/OE.19.023613


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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.

© 2011 OSA

1. Introduction

There are fields where it is crucial to improve the resolution limit of an optical system. For example, in laser ablation the PSF diameter needs to be reduced for femtosecond laser machining [1

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

]; in microscopy, such improvement is required in confocal microscopes [2

2. 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]

]; in high density data storage, super-resolving confocal readout systems are needed to retrieve the data [3

3. 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]

], etc. In general, super-resolution involves a transversal or a longitudinal reduction of the PSF, although these are not independent [4

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

]. In particular, the design of super-resolving pupils demands the reduction of the PSF radius, the optimization of the Strehl ratio and the brightness of its rings. Toraldo Di Francia [5

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

] proposed an interpolating method to design pupils with equally spaced rings, and proved the possibility of reducing the size of the PSF as much as desired by displacing the brighter secondary lobes farther from the central lobe. The cost was a drastic reduction of the Strehl ratio. Later, Sheppard and Hegedus [6

6. 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]

] introduced the parabolic approximation of the PSF, which has been used to design pupils with iterative methods [7

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

9

9. 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]

], or to derive closed expressions in the case of pupils with three zones [10

10. 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]

]. More recently, the use of parabolic approximations has been abandoned, and instead accurate merit functions have been constructed to use in conjunction with iterative methods [11

11. 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]

13

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

]. In any case, these methods require an initial guess, which in most cases determines the solution that we obtain.

The aim of this work is to show the possibility of designing useful super-resolving pupil filters, without resorting to iterative schemes and second order approximations for the PSF (parabolic approximation). The desired pupil function is expanded into a Dini series, and its Fourier-Bessel transform is found analytically. This also leads to a series in terms of a set of orthogonal functions, but now representing the amplitude of the PSF. Previously, one of the authors used this kind of series to find super-resolving pupils with iterative methods [14

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

]. What we want now is to avoid the iterative scheme, by relating the coefficient of the amplitude PSF series to parameters such as the super-resolving gain and the maximum intensity of the secondary lobes. In this form, we can readily obtain new super-resolving pupils by the simple expedient of providing the coefficients of the Dini series, which are found from simple algebraic expressions involving the parameters just mentioned above. Although the method becomes less accurate outside the interval 0.55<ε<0.75, where 0<ε<1is the super-resolving gain parameter, we can still achieve with relative ease large reductions of the central lobe of the PSF, with tolerable side-lobe intensity ratios. For this we have to accept a drastic drop of the Strehl ratio, arising from channeling a large portion of the energy in the PSF to a predefined secondary lobe. This resembles, of course, the result obtained by Toraldo di Francia with binary pupils [5

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

]. In the last part of the paper we present some examples that clearly show this kind of results.

2. Methodology

We shall start with a derivation of the expansion for the PSF, and its connection with the pupil function. Let G(δ,v) be the normalized complex amplitude distribution of an axially symmetric, complex pupil function g(ρ). Then
G(δ,v)=201g(ρ)exp(i2πδρ2)J0(vρ)ρdρ,
(1)
where ρ is the normalized radial coordinate and (δ,v) are the axial and the transverse dimensionless optical coordinates, respectively, defined by the equations δ=ξ(NA)2/2λ and v=2πrNA/λ, where ξ=zf is the axial distance from the focus, NA is the numerical aperture, r is the radial distance and λ the wavelength. Let us now suppose that we can describe the pupil function as a truncated Dini series:
g(ρ)=n=0KCnJ02(αn)J0(αnρ),
(2)
where J0(x) is the Bessel function of the first kind and zero order, αn are the roots of J1(x), Cn are the, possibly complex, coefficients to be calculated, and K + 1 the number of basis functions that we adopt. We must recall here that the pupil function must satisfy the condition |g(ρ)|1 in the interval [0, 1].

To obtain the corresponding transverse amplitude of the PSF we use Eq. (2) in Eq. (1) and set δ=0. We obtain:

ϕ(v)=G(0,v)=1J02(αn)n=0KCn201J0(αnρ)J0(vρ)ρdρ.
(3)

Evaluating the integrals as shown in [15

15. 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]

], we can rewrite Eq. (3) as
ϕ(v)=n=0KCnϕn(v),
(4)
where

ϕn(ν)=1J0(αn)2νJ1(ν)ν2αn2.
(5)

From this equation it can be readily shown thatϕn(αk)=δnk, where δnk is the kronecker delta; in other words, every basis function ϕn(ν) nulls at the location of the maxima of the others [15

15. 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]

]. Using this property in Eq. (4) we find

ϕ(αk)=Ck.
(6)

Therefore, the coefficients that we seek are the amplitudes of ϕ(ν) evaluated at the roots of J1(x). In general the location of the peaks of the PSF, |ϕ(ν)|2, it is not at the roots αn, but it is close to them if 0.55<ε<0.75, given that the basis functions ϕn(ν)have their principal maximum precisely at αn [15

15. 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]

].

We shall now proceed to adjust the shape of the function |ϕ(v)|2, taking into account certain parameters. These are the super-resolving gain ε, the Strehl ratio S, and the relative intensity of the first lobe Γ1 [12

12. 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]

]. They are defined as follows:
ε=DDc,
(7)
S=|ϕ(0)|2|ϕc(0)|2,
(8)
Γn=|ϕ(0)|2|ϕ(μn)|2,n=0,...,K,
(9)
where D is the diameter of the central lobe of the PSF, the sub index c stands for clear aperture, and μn is the “optical” radius of its nth-secondary maximum. Note that we have extended the definition of the parameter Γ1 to all side-lobe intensities.

We will now relate the coefficients of Eq. (4) with these parameters. Since μnαn, recalling that α0=0, from Eqs. (6) and (9) we obtain:
|C0||Cn|=|ϕ(α0)||ϕ(αn)||ϕ(0)||ϕ(μn)|=Γn,n=0,...,K,
(10)
with Γ0=1. Assuming |ϕc(0)|2=1, from Eq. (6) and (8) we have |C0|=|ϕ(0)|=S. Therefore

|Cn|SΓn,n=0,...,K.
(11)

Since Eq. (11) only involves the modulus of the coefficients, in particular we can choose

CnSΓnexp(inπ)=(1)nSΓn,n=0,...,K.
(12)

ϕ(εα1)=n=0KCnϕn(εα1)=0,ε<1.
(13)

In order to satisfy Eqs. (13) and (12), one of the coefficients in Eq. (13) must be calculated from the rest. For example, the last one:

CK=SϕK(εα1)n=0K1(1)nΓnϕn(εα1).
(14)

With this, we also fix the value of ΓK [Eqs. (14) and (12)]. At this point we have obtained the coefficients Cnthat, substituted in Eq. (4), will give us the PSF with the desired modifications, namely, a reduced central lobe, and known side-lobe relative intensities. However, these coefficients must be scaled to ensure that the condition |g(ρ)|1 will be satisfied in the interval [0, 1]. The new coefficients, therefore, will be kCn, with k=1/max[|g(ρ)|], where max[|g(ρ)|] is computed from Eq. (2) and the original coefficients; typically we obtain k < 1. This normalization procedure does not affect the relative intensities of the secondary lobes, which are given by ratios of the coefficients, nor the value of the super-resolution gain. But the actual Strehl ratio becomes k2|C0|2=k2S, where S was its intended value. Thus, with our method the parameters ε, S and Γ1 of Eqs. (7), (8) and (9) cannot be satisfied independently, since the value of S must be used to ensure that |g(ρ)|1 in the interval [0, 1] [Eqs. (12), (14) and (2)]. For simplicity, therefore, we will set S = 1 to start the design of any pupil filter.

3. Results and discussion

In the second example we choose K = 2, to show how can we manipulate the intensity of one of the side lobes. Using again ε=0.65, we assigned different values to Γ1, obtaining in this form PSFs with different values of S and CK=2 - and thus of Γ2. For example, setting Γ1=3.0,3.5and4.5, after scaling the coefficients in each case we found that S = 0.131, 0.155, 0.100 (Fig. 3
Fig. 3 PSFs obtained with our method for Γ1=3.0,3.5,4.5. In all cases K = 2 and ε=0.65. Notice that when Γ14.5, Γ2Γ1.
). Note that a higher value of Γ1 does not necessarily mean a lower value of S. In particular, the three coefficients which yielded S = 0.100 were (0.3147, −0.1484, 0.0951). Substituting them in Eq. (4) and plotting the resulting PSF we can see that in fact ε=0.65, S = 0.100 but Γ1=4.2 (curve with a dotted line). The value that we expected for Γ1was C02/C12=[0.3147/(0.1484)]24.5. As we mentioned before, the discrepancy between the expected and the actual value of Γ1 arises from the fact that we assumed that Γn could be obtained from the values of |ϕ(αn)|2, instead of those of |ϕ(μn)|2, where μn is the location of the nth maximum of the PSF. In practice, the tails of the basis functions adjacent to the nth basis function add small contributions to the PSF in the vicinity of αn, shifting the position of the local maximum of the PSF from αn, the position of the maximum of the nth basis function, to μn, the actual location of the nth maximum of the PSF. We believe that this is not a serious drawback, however, giving that the desired value of Γn may be obtained by increasing slightly its intended value. There is no such problem with the super-resolving gain parameter ε, which is predicted accurately. Figure 4
Fig. 4 Relative errors for various nominal values of the parameter Γ1. In all cases ε=0.65 and K = 2.
shows the relative errors of Γ1 for various nominal values of this parameter. In all cases ε=0.65 and K = 2. Notice that the percentual error is under 20% in the range0.55<ε<0.75, for all the values of Γ1 that we chose.

The third example shows the possibility of further reductions of the super-resolution gain parameter, at the expense of a large increment in the intensity of a predefined side-lobe. Figure 5
Fig. 5 PSF designed for a high super-resolving gain parameter, ε=0.50, with Γ1=Γ2=2.5.
shows a PSF with ε=0.50, obtained with only four coefficients of the Dini expansion (K = 3). We set Γ1=Γ2=2.5 and calculated C3 from Eq. (14). After scaling the resulting coefficients we had Cn=(0.0340,-0.0215,0.0215,-0.0711). Plotting the corresponding PSF we can see that in fact ε=0.50, S=(0.0340)2=0.0012, but Γ12.2. The reason for the slight discrepancy in the value of Γ1 was given above.

Finally, in Fig. 6
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.
we compare our results (the curve with a dotted line of Fig. 3), with those of Canales and Cagigal (the curve with an interrumped line in Fig. 6) [12

12. 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]

]. Both results were normalized to show the behavior of the secondary lobes. Our PSF shows a higher reduction of the secondary lobes, but this was achieved with a lower value of the Strehl ratio, S. Similar results were obtained by one of the authors when using an iterative method for the same design problem [14

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

].

4. Conclusions

We have presented a non-iterative method to design super-resolving pupils based on Dini series, obtaining formulas which relate the coefficients of the series with relevant design parameters, like the super-resolution gain and the relative intensities of the side lobes of the PSF. Both, the pupil function and its corresponding PSF can be readily computed from these coefficients.

Acknowledgements

Noé Alcala Ochoa would like to thank the support of CONACyT through project 133495.

References and links

1.

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

2.

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]

3.

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]

4.

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

5.

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

6.

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]

7.

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

8.

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]

9.

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]

10.

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]

11.

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]

12.

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]

13.

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

14.

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

15.

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]

OCIS Codes
(110.0180) Imaging systems : Microscopy
(110.1220) Imaging systems : Apertures
(170.1790) Medical optics and biotechnology : Confocal microscopy

ToC Category:
Imaging Systems

History
Original Manuscript: July 26, 2011
Revised Manuscript: October 8, 2011
Manuscript Accepted: October 13, 2011
Published: November 4, 2011

Virtual Issues
Vol. 7, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Noé Alcalá Ochoa and J. E. A. Landgrave, "Non-iterative method for designing super-resolving pupil filters," Opt. Express 19, 23613-23620 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23613


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References

  1. M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett.32(15), 2239–2241 (2007). [CrossRef] [PubMed]
  2. 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. A20(11), 2172–2178 (2003). [CrossRef]
  3. 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]
  4. T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett.81(18), 3844–3847 (1998). [CrossRef]
  5. G. Toraldo di Francia, “Super-gain antennas and optical resolving Power,” Nuovo Cim.9(S3Suppl.), 426–438 (1952). [CrossRef]
  6. C. J. R. Sheppard and Z. S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am. A5(5), 643–647 (1988). [CrossRef]
  7. T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A14(7), 1637–1646 (1997). [CrossRef]
  8. 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]
  9. M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” PASP116(824), 965–970 (2004). [CrossRef]
  10. 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]
  11. P. N. Gundu, E. Hack, and P. Rastogi, “High efficient superresolution combination filter with twin LCD spatial light modulators,” Opt. Express13(8), 2835–2842 (2005). [CrossRef] [PubMed]
  12. V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express14(22), 10393–10402 (2006). [CrossRef] [PubMed]
  13. P. N. Gundu, E. Hack, and P. Rastogi, “Apodized superresolution – concept and simulations,” Opt. Commun.249(1-3), 101–107 (2005). [CrossRef]
  14. N. A. Ochoa, J. García-Márquez, and A. González-Vega, “Hybrid pupil filter design using Bessel series,” Opt. Commun. (To be published).
  15. 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. A14(11), 2962–2976 (1997). [CrossRef]

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