## A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling |

Optics Express, Vol. 18, Issue 18, pp. 19141-19155 (2010)

http://dx.doi.org/10.1364/OE.18.019141

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### Abstract

We present a technique to solve numerically the Fresnel diffraction integral by representing a given complex function as a finite superposition of complex Gaussians. Once an accurate representation of these functions is attained, it is possible to find analytically its diffraction pattern. There are two useful consequences of this representation: first, the analytical results may be used for further theoretical studies and second, it may be used as a versatile and accurate numerical diffraction technique. The use of the technique is illustrated by calculating the intensity distribution in a vicinity of the focal region of an aberrated converging spherical wave emerging from a circular aperture.

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## 1. Introduction

7. M. Sypek, “Ligth propagation in the Fresnel region. new numerical approach,” Opt. Commun. **116**(1-3), 43–48 (1995). [CrossRef]

10. M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. **195**(5-6), 351–359 (2001). [CrossRef]

## 2. Analytical description

*N*large, and

*u*bounded, Eq. (6) can be written in a good approximation as,Aside of the term

*π*, Eq. (7) is the exact Fourier transform of our rectangle function with arbitrary width

*A*.

*f*exhibiting a maximum frequency

*g*as,where

*Δ*is the sampling interval.

*K*a real number.

*K*can be chosen giving other characteristics to the superposition of Gaussians but the Rayleigh-like criterion is the most straightforward way to implement.

## 3. Amplitude distribution in a vicinity of a focal region

*a*, centered at the origin of a plane

*λ*is the wavelength of the illuminating beam and

*a*centered at the origin.

*z*of the incident plane. In order to compare our results with [1,2] we will limit ourselves to the case

*φ*and

*θ*are defined as usual (Fig. 1). Thus, Eq. (16) can be rewritten as,

*s*. Figure 8 bellow shows a plot of one of these functions. We have found that 50 terms are sufficient to make the maximum error less than 2/1000. For an even better accuracy we are using 60 Gaussians in our calculations. Clearly, a much greater number of pixels would be required for sampling the quadratic phase in Eq. (20).

*n*in the above equations can be truncated to moderate values due to the properties of the error function. To compare the results obtained with the above equations, and only for comparative purposes, we verified the accuracy of the results with those obtained with Wolfram’s Mathematica®.

### 3.1 Numerical examples

*a*and

*b*stand for a tilted beam in the

*x*and

*y*axes respectively and

*x*and

*y*respectively. Note however that our method may be applied to a far more general aberrated and/or apodizated wave-front. Our results shown below are normalized in intensity.

*x*axis, with corresponding values,

*x*axis. The corresponding values are,

7. M. Sypek, “Ligth propagation in the Fresnel region. new numerical approach,” Opt. Commun. **116**(1-3), 43–48 (1995). [CrossRef]

## 4. Conclusions

## References and links

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

2. | E. H. Linfoot, |

3. | 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 |

4. | J. J. Stamnes and H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. |

5. | Y. Li, “Expansions for irradiance distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A |

6. | J. J. Stamnes, “Waves in focal regions,” The Adam Hilger series on Optics and Optoelectronics, (1986). |

7. | M. Sypek, “Ligth propagation in the Fresnel region. new numerical approach,” Opt. Commun. |

8. | A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE |

9. | M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion |

10. | M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. |

11. | H. P. Hsu, |

12. | R. W. Southworth, and S. L. Deleeuw, |

13. | M. Abramovitz, and I. Stegun, |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(080.1010) Geometric optics : Aberrations (global)

(080.1510) Geometric optics : Propagation methods

(080.1753) Geometric optics : Computation methods

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 7, 2010

Revised Manuscript: July 21, 2010

Manuscript Accepted: August 4, 2010

Published: August 25, 2010

**Citation**

Moisés Cywiak, Arquímedes Morales, Manuel Servín, and Rafael Gómez-Medina, "A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling," Opt. Express **18**, 19141-19155 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19141

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### References

- M. Born, and E. Wolf, Principles of Optics 7th ed. (Pergamon, New York, 1980), Chap. 8.
- E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, 1955), Chap. 3.
- 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]
- J. J. Stamnes and H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. 37(17), 3612–3622 (1998). [CrossRef]
- Y. Li, “Expansions for irradiance distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 23(3), 730–740 (2006). [CrossRef]
- J. J. Stamnes, “Waves in focal regions,” The Adam Hilger series on Optics and Optoelectronics, (1986).
- M. Sypek, “Ligth propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995). [CrossRef]
- A. W. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).
- M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4(1), 85–97 (1982). [CrossRef]
- M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Commun. 195(5-6), 351–359 (2001). [CrossRef]
- H. P. Hsu, Fourier Analysis (Simon & Schuster, Inc. New York, 1970), Chap. 9.
- R. W. Southworth, and S. L. Deleeuw, Digital and computation and numerical methods (McGraw-Hill, N.Y., 1965) Chap. 9.
- M. Abramovitz, and I. Stegun, Handbook of mathematical functions in Applied Mathematics series-55, 299 (1972).

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