## Fast vectorial calculation of the volumetric focused field distribution by using a three-dimensional Fourier transform |

Optics Express, Vol. 20, Issue 2, pp. 1060-1069 (2012)

http://dx.doi.org/10.1364/OE.20.001060

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### Abstract

We show that the volumetric field distribution in the focal region of a high numerical aperture focusing system can be efficiently calculated with a three-dimensional Fourier transform. In addition to focusing in a single medium, the method is able to calculate the more complex case of focusing through a planar interface between two media of mismatched refractive indices. The use of the chirp z-transform in our numerical implementation of the method allows us to perform fast calculations of the three-dimensional focused field distribution with good accuracy.

© 2012 OSA

## 1. Introduction

4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

*et al.*demonstrated that the 3D vectorial field distribution at the focus can be computed plane by plane under a proper transformation of the Debye-Wolf integral [5

5. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express **14**(23), 11277–11291 (2006). [CrossRef] [PubMed]

7. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. **54**(2), 240–244 (1964). [CrossRef]

8. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A **19**(8), 1721–1721 (2002). [CrossRef]

9. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. **271**(1), 40–47 (2007). [CrossRef]

## 2. Light focused into a homogenous medium

4. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. **253**(1274), 358–379 (1959). [CrossRef]

*x-*,

*y-*, and

*z-*direction) is governed by the individual scalar Debye integralwhere λ is the wavelength of the monochromatic light, in the corresponding medium, and

7. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. **54**(2), 240–244 (1964). [CrossRef]

10. J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. **36**(8), 1341–1343 (2011). [CrossRef] [PubMed]

**253**(1274), 358–379 (1959). [CrossRef]

*E*component in the focal region of an aplanatic system, with NA=0.95, for incident light linearly polarized in the

_{x}*x*-direction, λ

_{0}=532nm, as obtained by direct evaluation of the Debye-Wolf integral and the 3D-FT method. The relative error in the focal volume between the two methods was calculated as [11

11. P. Török, P. R. T. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A **23**(3), 713–722 (2006). [CrossRef] [PubMed]

*z*. This figure shows that the accuracy of the 3D-FT method is best at the focal plane, reduces as we move away in the

*z*-direction, and slightly improves at the edge of the volume analyzed. As can be expected, the accuracy is worse at planes close to the minima in the focused field axial distribution. This behavior reflects the role of computational precision rather than a limitation of the 3D-FT method.

*x*- and

*y*-direction in the focal plane for

^{−3}, which is enough for a large number of applications.

12. D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express **11**(21), 2747–2752 (2003). [CrossRef] [PubMed]

*x*-direction, with unity topological charge, has a space-variant phase distribution, in the transverse plane, and can be expressed as

13. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Variable-radius focused optical vortex with suppressed sidelobes,” Opt. Lett. **31**(11), 1600–1602 (2006). [CrossRef] [PubMed]

14. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. **1**(1), 1–57 (2009). [CrossRef]

## 3. Light focused through an interface

## 4. Conclusion

18. P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. **36**(11), 2305–2312 (1997). [CrossRef] [PubMed]

*x*,

*y*, and

*z*), which can speed up the calculation. However, even in these favorable conditions for direct integration of the Debye-Wolf integral, the 3D-FT method is significantly faster. We compared the speed using a volume of 4λ×4λ×4λ around the focal point, for cubic pixels with side λ/30 (121×121×121 points) using MATLAB on a PC with an Intel Core2 Duo 2.53 GHz processor and 4 GB of RAM. The typical calculation time for direct evaluation of the Debye-Wolf integral is around 23 minutes. The same result obtained with the 3D-FT method takes approximately 2 seconds, and the accuracy achieved, measured as the relative error, is in the order of 10

^{−3}. The maximum benefit of the method presented here is obtained when the field in the whole 3D focal volume is desired. If the focused field is required only in a small number of points (e.g. a single plane at the focus) direct integration of Debye-Wolf integral may be sufficiently fast.

## Acknowledgments

## References

1. | J. J. Stamnes, |

2. | J. W. Goodman, |

3. | M. Born and E. Wolf, |

4. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

5. | M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express |

6. | A. V. Oppenheim, R. W. Schafer, and J. R. Buck, |

7. | C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. |

8. | C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A |

9. | I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. |

10. | J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. |

11. | P. Török, P. R. T. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A |

12. | D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express |

13. | J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Variable-radius focused optical vortex with suppressed sidelobes,” Opt. Lett. |

14. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

15. | C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun. |

16. | P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A |

17. | S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A |

18. | P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(110.0180) Imaging systems : Microscopy

(180.6900) Microscopy : Three-dimensional microscopy

(260.1960) Physical optics : Diffraction theory

(260.5430) Physical optics : Polarization

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 23, 2011

Revised Manuscript: October 9, 2011

Manuscript Accepted: October 11, 2011

Published: January 4, 2012

**Virtual Issues**

Vol. 7, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

J. Lin, O. G. Rodríguez-Herrera, F. Kenny, D. Lara, and J. C. Dainty, "Fast vectorial calculation of the volumetric focused field distribution by using a three-dimensional Fourier transform," Opt. Express **20**, 1060-1069 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1060

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### References

- J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999).
- B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci.253(1274), 358–379 (1959). [CrossRef]
- M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express14(23), 11277–11291 (2006). [CrossRef] [PubMed]
- A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing, 2nd ed. (Prentice Hall, 1999).
- C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am.54(2), 240–244 (1964). [CrossRef]
- C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A19(8), 1721–1721 (2002). [CrossRef]
- I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun.271(1), 40–47 (2007). [CrossRef]
- J. Lin, X.-C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodríguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett.36(8), 1341–1343 (2011). [CrossRef] [PubMed]
- P. Török, P. R. T. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A23(3), 713–722 (2006). [CrossRef] [PubMed]
- D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express11(21), 2747–2752 (2003). [CrossRef] [PubMed]
- J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Variable-radius focused optical vortex with suppressed sidelobes,” Opt. Lett.31(11), 1600–1602 (2006). [CrossRef] [PubMed]
- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009). [CrossRef]
- C. J. R. Sheppard and M. Gu, “Axial imaging through an aberration layer of water in confocal microscopy,” Opt. Commun.88(2-3), 180–190 (1992). [CrossRef]
- P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A12(2), 325–332 (1995). [CrossRef]
- S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A14(7), 1482–1490 (1997). [CrossRef]
- P. Török and P. Varga, “Electromagnetic diffraction of light focused through a stratified medium,” Appl. Opt.36(11), 2305–2312 (1997). [CrossRef] [PubMed]

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