## Equivalent functions for the Fresnel integral

Optics Express, Vol. 13, Issue 21, pp. 8469-8482 (2005)

http://dx.doi.org/10.1364/OPEX.13.008469

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### Abstract

Fresnel integral is modeled with three equivalent functions. The first function is derived by considering the sum of the first term of the Fresnel integral�??s asymptotic expansion {F(x)} and an exponential function which approaches to infinity at the zero of the Fresnel function�??s argument and has the properties of a unit step function. The second one is the sum of a unit step function and the transition function defined for the simplified uniform theory of diffraction. The third function considers directly eliminating the infinity coming from F(x). The amplitude and the phase of Fresnel integral and its equivalent functions are compared numerically. The result is applied to the modified theory of physical optics solution of the diffraction of edge waves from a half plane problem.

© 2005 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(080.0080) Geometric optics : Geometric optics

(080.2720) Geometric optics : Mathematical methods (general)

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 14, 2005

Revised Manuscript: October 3, 2005

Published: October 17, 2005

**Citation**

Yusuf Umul, "Equivalent functions for the Fresnel integral," Opt. Express **13**, 8469-8482 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8469

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### References

- A. Sommerfeld, Optics (Academic Press, New York, 1954).
- D. S. Ahluwalia, R. M. Lewis and J. Boersma, �??Uniform asymptotic theory of diffraction by a plane screen,�?? SIAM J. Appl. Math. 16, 783-807 (1968). [CrossRef]
- R. M. Lewis and J. Boersma, �??Uniform asymptotic theory of edge diffraction theory,�?? J. Math. Physics 10, 2291-2305 (1969). [CrossRef]
- R. G. Kouyoumjian and P. H. Pathak, �??A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,�?? Proceedings of IEEE 62, 1448-1461 (1974).
- Y. Z. Umul, �??Simplified uniform theory of diffraction,�?? Opt. Lett. 30, 1614-1616 (2005). [CrossRef]
- Y. Z. Umul, �??Modified theory of physical optics,�?? Opt. Express 12, 4959-4972 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4959.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4959.</a> [CrossRef]

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