## Gaussian beam propagation: comparison of the analytical closed-form Fresnel integral solution to the simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations |

JOSA A, Vol. 30, Issue 4, pp. 640-644 (2013)

http://dx.doi.org/10.1364/JOSAA.30.000640

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### Abstract

Simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations in the case of free-space propagation of a Gaussian beam are compared with analytical solutions. The most accurate results were obtained by the Rayleigh–Sommerfeld I approximation. The study reveals that the approximations are not uniform throughout the propagation region. While the accuracies of the Huygens and Fresnel methods generally increase as the propagation distance increases, the accuracy of the Rayleigh–Sommerfeld I approximation at first starts to diminish and later recovers as the propagation distance is further increased.

© 2013 Optical Society of America

## 1. INTRODUCTION

## 2. SPHERICAL WAVES, HUYGENS’ PRINCIPLE, AND THE FRESNEL APPROXIMATION

## 3. RAYLEIGH–SOMMERFELD I DIFFRACTION FORMULA

5. M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A **8**, 27–32 (1991). [CrossRef]

6. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. **17**, 35–100 (1954). [CrossRef]

5. M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A **8**, 27–32 (1991). [CrossRef]

## 4. PROPAGATION OF A GAUSSIAN BEAM

## 5. RESULTS AND DISCUSSION

## ACKNOWLEDGMENTS

## REFERENCES

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

2. | A. E. Siegman, |

3. | C. Huygens, |

4. | A. Sommerfeld, |

5. | M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A |

6. | C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. |

7. | L. Eyges, |

8. | M. Abramowitz and I. A. Stegun, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.1960) Diffraction and gratings : Diffraction theory

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 17, 2013

Manuscript Accepted: February 2, 2013

Published: March 18, 2013

**Citation**

Seyed M. Azmayesh-Fard, "Gaussian beam propagation: comparison of the analytical closed-form Fresnel integral solution to the simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations," J. Opt. Soc. Am. A **30**, 640-644 (2013)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-4-640

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### References

- M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
- A. E. Siegman, Lasers (University Science, 1986).
- C. Huygens, Treatise on Light (Dover, 1962).
- A. Sommerfeld, Optics, Vol. 3 of Lectures on Theoretical Physics (Academic, 1964).
- M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991). [CrossRef]
- C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954). [CrossRef]
- L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, 1972), p. 263.
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 7th ed. (National Bureau of Standards, 1968).

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