## Exact ray tracing formulas based on a nontrigonometric alternative to Snell’s law |

JOSA A, Vol. 29, Issue 12, pp. 2679-2687 (2012)

http://dx.doi.org/10.1364/JOSAA.29.002679

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### Abstract

In this work, Fermat’s principle is applied to derive a simple exact formula for refraction (reflection) in terms of the lengths of the incident and refracted rays. This formula is a nontrigonometric alternative to Snell’s law and is general for all optical surfaces. It is used to derive the paraxial optics equations in a more simple and direct way than that often used in the literature. It’s also applied to derive a new single, exact ray tracing formula for the nonparaxial refraction (reflection) at a single optical surface. The obtained formulas are used to develop a simple ray tracing procedure for meridional refraction through systems of spherical surfaces without the need to use any form of Snell’s law. Numerical examples are provided and discussed.

© 2012 Optical Society of America

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(120.5700) Instrumentation, measurement, and metrology : Reflection

(120.5710) Instrumentation, measurement, and metrology : Refraction

(080.1753) Geometric optics : Computation methods

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: September 21, 2012

Revised Manuscript: November 2, 2012

Manuscript Accepted: November 4, 2012

Published: November 29, 2012

**Citation**

Hassan A. Elagha, "Exact ray tracing formulas based on a nontrigonometric alternative to Snell’s law," J. Opt. Soc. Am. A **29**, 2679-2687 (2012)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-12-2679

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### References

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- P. Mouroulis and J. Macdonald, “Rays and foundations of geometrical optics,” in Geometrical Optics and Optical Design (Oxford University, 1997), pp. 11–13.
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- D. Malacara and Z. Malacara, Handbook of Optical Design(Marcel Dekker, 2004).
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- J. E. Harvey, “Exact ray trace procedure,” http://ebookbrowse.com/3-5-exact-ray-trace-procedure-pdf-d354859582 .

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