## Tailoring the axial shape of the point spread function using the Toraldo concept

Optics Express, Vol. 10, Issue 1, pp. 98-103 (2002)

http://dx.doi.org/10.1364/OE.10.000098

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### Abstract

A novel procedure for shaping the axial component of the point spread function of nonparaxial focusing systems by use of phase-only pupil filters is presented. The procedure is based on the Toraldo technique for tailoring focused fields. The resulting pupil filters consist of a number of concentric annular zones with constant real transmittance. The number of zones and their widths can be adapted according to the shape requirements. Our method is applied to design filters that produce axial superresolution in confocal scanning systems.

© Optical Society of America

## 1. Introduction

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

13. Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A **3**, 1892–1896 (1986). [CrossRef]

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

## 2. The axial response of apodized focusing systems

*A*(θ) represents the apodization function, i.e. the amplitude transmittance of the exit pupil, θ being the angular coordinate at the exit pupil plane as seeing from the focal point.

*z*is the axial coordinate as measured from the focus of the system, α the semiaperture angle,

*λ*the wavelength, and

*n*the refractive index of the medium.

*q*(

*ζ*) such that its 1D Fourier transform approximates the desired form. The optical implementation of a filter whose transmittance maps to such

*q*(

*ζ*) is in general a non-trivial task. An alternative approach is based on the Toraldo concept. As we show below it is possible to design ring-shaped pupil filters to control the position of the zeros of the axial-amplitude PSF. In the shaping procedure the influence of the angular aperture is two-fold. On one hand, for a given function

*q*(

*ζ*) the actual form of the apodization function explicitly depends on the value of α through the mapping of Eq. (2). On the other hand, the axial extent of the focal spot is determined by the value of α through the scale factor of Eq. (4). This means that, in general, the optimal pupil transmittance for a given purpose does not scale linearly with α, and then must be calculated explicitly for the NA value of interest.

## 3. The Toraldo concept

13. Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A **3**, 1892–1896 (1986). [CrossRef]

16. C. J. R. Sheppard, G. Calvert, and M. Wheatland, “Focal distribution for superresolving toraldo filters,” J. Opt. Soc. Am. A 1**5**, 849–856 (1998). [CrossRef]

*t*(

*r*), is subdivided into

*m*concentric annular zones to control the radii of

*m*-1 rings of zero intensity. In our approach the function

*q*(

*ζ*) is also subdivided into constant-transmittance zones. Here it is important to take into account that if one wants to tailor the axial PSF with the constraint that the transverse PSF should remain almost invariant, the resulting function

*q*(

*ζ*) must be centrosymmetric [3

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

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

*m*-1 axial zeros, the interval [-0.5, 0.5] should be divided into 2

*m*-1 subintervals such that in each subinterval the function

*q*(

*ζ*) is constant. On the basis of the above reasoning, the procedure for shaping the axial intensity of nonparaxially focused scalar fields is as follows:

*q*(

*ζ*) is decomposed as

*q*(

*ζ*), that is

*m*= 2 , which allows one to control the position of one zero, the axial amplitude PSF is given by

*h*(0) = 1, we obtain

*z*=

_{N}*z*

_{1}. Then from Eq. (7)

*q*(

*ζ*), and therefore the coefficients

*k*, which produces an axial intensity that is symmetric around the focus, is real. If, additionally, one wants to maximize the filter throughput, the zones should have no absorption and have opposite phases, that is,

_{i}*k*

_{2}= -

*k*

_{1}. The choice of a phase-only filter yields a design that can be manufactured with relative ease. From Eq. (9) we find

*y*= sinc (

*z*

_{1})/ sinc (Δ

*z*

_{1}) and the straight line

*y*= 2Δ.

*q*(

*ζ*) = rect(

*ζ*), the first axial zero appears at

*z*= 1, then the value of the constraint parameter should be

_{N}*z*

_{1}= 0.7 . The solution of the transcendental Eq. (10) is then Δ = 0.19. In Fig. 1 we compare the normalized axial intensity PSF provided by the Toraldo-like filter with that corresponding to the nonapodized circular pupil. In addition, we have represented, in contour plots, the 3D intensity distribution in the meridian plane for both cases. These are of interest because they show that there are no significant off-axis lobes, which was not guaranteed by the design. Due to the centrosymmetry of the Toraldo filter the central lobe in the transverse direction remains almost invariant.

*q*(

*ζ*) together with the 2D representation of the actual amplitude transmittance of the filter for two different values of α. Note that a similar effect, but in a different context, was found in [5–9].

## 4. Application to Confocal Microscopy

*z*= 4 thick can be imaged without bleaching by the sidelobes. Of course, if one wants to send the huge axial sidelobe further away it is necessary to design a Toraldo filter composed by a higher number of zones. A practical rule for the design procedure is that the position of the huge side-lobe is approximately

_{N}*z*=

_{N}*m*+1.

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

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

## 5. Conclusions

## Acknowledgements

## References and links

1. | P. Jacquinot and B. Rozien-Dossier, “Apodisation” in |

2. | A. Boivin, |

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

4. | M. Martínez-Corral, P. Andrés, J. Ojeda-Castan͂eda, and G. Saavedra, “Tunable axial superresolution by annular binary filters. Application to confocal microscopy,” Opt. Commun. |

5. | J. Campos, J. C. Escalera, C. J. R. Sheppard, and M. J. Yzuel, “Axially invariant pupil filters,” J. Mod. Opt. |

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

7. | C. J. R. Sheppard, “Leaky annular pupils for improved axial imaging,” Optik |

8. | M. Martínez-Corral, P. Andrés, C. J. Zapata-Rodríguez, and M. Kowalczyk, “Three-dimensional superresolu-tion by annular binary filters,” Opt. Commun. |

9. | S. Grill and E. H. K. Stelzer, “Method to calculate lateral and axial gain factors of optical setups with a large solid angle,” J. Opt. Soc. Am. A |

10. | M. A. A. Neil, R. Juskaitis, T. Wilson, Z. J. Laczik, and V. Sarafis, “Optimized pupil-plane filters for confocal microscope point-spread function engineering,” Opt. Lett. |

11. | M. Martínez-Corral, L. Mun͂oz-Escrivá, M. Kowalczyk, and T. Cichocki, “One-dimensional iterative algorithm for three-dimensional point-spread function engineering,” Opt. Lett. |

12. | G. Francia Toraldo di, “Nuovo pupille superresolventi,” Atti Fond. Giorgio Ronchi |

13. | Z. S. Hegedus and V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A |

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

15. | T. R. M. Sales and G. M. Morris, “Axial superresolution with phase-only pupil filters,” Opt. Commun. |

16. | C. J. R. Sheppard, G. Calvert, and M. Wheatland, “Focal distribution for superresolving toraldo filters,” J. Opt. Soc. Am. A 1 |

17. | H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. |

18. | M. Gu, |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(110.1220) Imaging systems : Apertures

(140.3300) Lasers and laser optics : Laser beam shaping

(170.1790) Medical optics and biotechnology : Confocal microscopy

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 4, 2001

Published: January 14, 2002

**Citation**

Manuel Martinez-Corral, M. Caballero, Ernst H. K. Stelzer, and Jim Swoger, "Tailoring the axial shape of the point spread function using the Toraldo concept," Opt. Express **10**, 98-103 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-98

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### References

- P. Jacquinot and B. Rozien-Dossier, "Apodisation" in Progress in Optics, E. Wolf, ed., Vol III (North-Holland, Amsterdam, 1964).
- A. Boivin, Th?orie el Calcul des Figures de Diffraction de R?volution (Universit? de Laval, Quebec, 1964).
- C. J. R. Sheppard and Z. S. Hegedus, "Axial behavior of pupil-plane filters," J. Opt. Soc. Am. A 5, 643-647 (1988). [CrossRef]
- M. Mart?nez-Corral, P. Andr?s, J. Ojeda-Casta?eda and G. Saavedra, "Tunable axial superresolution by annular binary filters. Application to confocal microscopy," Opt. Commun. 119, 491-498 (1995). [CrossRef]
- J. Campos, J. C. Escalera, C. J. R. Sheppard and M. J. Yzuel, "Axially invariant pupil filters," J. Mod. Opt. 47, 57-68 (2000).
- 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, "Leaky annular pupils for improved axial imaging," Optik 99, 32-34 (1995).
- M. Mart?nez-Corral, P. Andr?s, C. J. Zapata-Rodr?guez and M. Kowalczyk, "Three-dimensional superresolution by annular binary filters," Opt. Commun. 165, 267-278 (1999). [CrossRef]
- S. Grill and E. H. K. Stelzer, "Method to calculate lateral and axial gain factors of optical setups with a large solid angle," J. Opt. Soc. Am. A 16, 2658-2665 (1999). [CrossRef]
- M. A. A. Neil, R. Juskaitis, T. Wilson, Z. J. Laczik and V. Sarafis, "Optimized pupil-plane filters for confocal microscope point-spread function engineering," Opt. Lett. 25, 245-247 (2000). [CrossRef]
- M. Mart?nez-Corral, L. Mu?oz-Escriv?, M. Kowalczyk and T. Cichocki, "One-dimensional iterative algorithm for three-dimensional point-spread function engineering," Opt. Lett. 26, 1861-1863 (2001). [CrossRef]
- G. Toraldo di Francia, "Nuovo pupille superresolventi," Atti Fond. Giorgio Ronchi 7, 366-372 (1952).
- Z. S. Hegedus and V. Sarafis, "Superresolving filters in confocally scanned imaging systems," J. Opt. Soc. Am. A 3, 1892-1896 (1986). [CrossRef]
- T. R. M. Sales and G. M. Morris, "Diffractive superresolution elements," J. Opt. Soc. Am. A 14, 1637-1646 (1997). [CrossRef]
- T. R. M. Sales and G. M. Morris, "Axial superresolution with phase-only pupil filters," Opt. Commun. 156, 227-230 (1998). [CrossRef]
- C. J. R. Sheppard, G. Calvert and M. Wheatland, "Focal distribution for superresolving toraldo filters," J. Opt. Soc. Am. A 15, 849-856 (1998). [CrossRef]
- H. Wang and F. Gan, "High focal depth with a pure-phase apodizer," Appl. Opt. 40, 5658-562 (2001). [CrossRef]
- M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, Berlin, 2000).

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