## Non-iterative method for designing super-resolving pupil filters |

Optics Express, Vol. 19, Issue 23, pp. 23613-23620 (2011)

http://dx.doi.org/10.1364/OE.19.023613

Acrobat PDF (1354 KB)

### 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

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]

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]

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

## 2. Methodology

*NA*is the numerical aperture,

*r*is the radial distance and

*C*are the, possibly complex, coefficients to be calculated, and

_{n}*K*+ 1 the number of basis functions that we adopt. We must recall here that the pupil function must satisfy the condition

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]

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]

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]

*S*, and the relative intensity of the first lobe

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]

*D*is the diameter of the central lobe of the PSF, the sub index

*c*stands for clear aperture, and

*n*th

*-*secondary maximum. Note that we have extended the definition of the parameter

*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

*S*was its intended value. Thus, with our method the parameters

*S*and

*S*must be used to ensure that

*S*= 1 to start the design of any pupil filter.

## 3. Results and discussion

*K*= 1) and

*k*, and from them the correct values for

*S.*For example, in the case of

*S*= 0.174, in agreement with the corresponding plot in Fig. 1 . From the heights of the central and the first peaks in this plot we find

*K*= 2, to show how can we manipulate the intensity of one of the side lobes. Using again

*S*and

*S =*0.131, 0.155, 0.100 (Fig. 3 ). Note that a higher 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

*S*= 0.100 but

*n*th maximum of the PSF. In practice, the tails of the basis functions adjacent to the

*n*th basis function add small contributions to the PSF in the vicinity of

*n*th basis function, to

*n*th maximum of the PSF. We believe that this is not a serious drawback, however, giving that the desired value of

*K*= 2. Notice that the percentual error is under 20% in the range

*K*= 3). We set

## 4. Conclusions

## Acknowledgements

## References and links

1. | M. Merano, G. Boyer, A. Trisorio, G. Chériaux, and G. Mourou, “Superresolved femtosecond laser ablation,” Opt. Lett. |

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 |

3. | V. F. Canales, P. J. Valle, J. E. Oti, and M. P. Cagigal, “Pupil apodization for increasing data storage density,” Chin. Opt. Lett. |

4. | T. R. M. Sales, “Smallest focal spot,” Phys. Rev. Lett. |

5. | G. Toraldo di Francia, “Super-gain antennas and optical resolving Power,” Nuovo Cim. |

6. | C. J. R. Sheppard and Z. S. Hegedus, “Axial behavior of pupil-plane filters,” J. Opt. Soc. Am. A |

7. | T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A |

8. | D. M. de Juana, J. E. Oti, V. F. Canales, and M. P. Cagigal, “Design of superresolving continuous phase filters,” Opt. Lett. |

9. | M. P. Cagigal, V. F. Canales, and J. E. Oti, “Design of continuous superresolving masks for ground-based telescopes,” PASP |

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. |

11. | P. N. Gundu, E. Hack, and P. Rastogi, “High efficient superresolution combination filter with twin LCD spatial light modulators,” Opt. Express |

12. | V. F. Canales and M. P. Cagigal, “Pupil filter design by using a Bessel functions basis at the image plane,” Opt. Express |

13. | P. N. Gundu, E. Hack, and P. Rastogi, “Apodized superresolution – concept and simulations,” Opt. Commun. |

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 |

**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/vjbo/abstract.cfm?URI=oe-19-23-23613

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### References

- 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. A20(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. A5(5), 643–647 (1988). [CrossRef]
- T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” J. Opt. Soc. Am. A14(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,” PASP116(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. Express13(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. Express14(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. A14(11), 2962–2976 (1997). [CrossRef]

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