TECHNICAL NOTE Dynamic Programming, the

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 95, No. 3, pp. 713-716, DECEMBER 1997

TECHNICAL NOTE Dynamic Programming, the Fermat Principle, and the Eikonal Equation Revisited V. LAKSHMINARAYANAN1 AND S. VARADHARAJAN2 Communicated by R. E. Kalaba

Abstract. We derive the eikonal equation using the technique of dynamic programming. The formalism of dynamic programming is based on the procedure defining the minimum pathway, and an immediate application of the Fermat principle leads to the well-known eikonal equation of classical geometrical optics. Key Words. Eikonal equation, optics, optimality principle, Fermat principle, dynamic programming.

1. Introduction The eikonal equation plays a major role in geometrical optics and in fact can be considered as the basic equation of geometrical optics. The solution of the eikonal equation gives the geometrical wavefronts of the propagating electromagnetic wave. A full description and derivation of the eikonal equation from the Maxwell equations is given in Born and Wolf (Ref. 1). The eikonal equation can also be thought of as the HamiltonJacobi equation of the variational problem

the optical counterpart of which is nothing more than the Fermat principle. Assistant Professor, School of Optometry and Adjunct Associate Professor, Department of Physics and Astronomy, University of Missouri at St. Louis, St. Louis, Missouri. 2 Graduate Student, Department of Physics and Astronomy, University of Missouri at St. Louis, St. Louis, Missouri.

713 0022-3239/97/1200-0713$12.50/0 © 1997 Plenum Publishing Corporation

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Kalaba (Ref. 2), in an early example of the application of dynamic programming techniques (Ref. 3) to the solution of problems in mathematical physics, showed how the eikonal equation may be derived using the Bellman principle of optimality (Ref. 3). In this note, we rederive the eikonal equation, not in terms of time coordinates (as Kalaba did), but in terms of spatial coordinates. Hence, in this respect, this note should be considered as a companion to the earlier note of Kalaba. The Fermat principle states that the time taken for an actual path traced by a ray of light is an extremum. Caratheodory showed that it was a true minimum and established the connection between the Fermat principle and the calculus of variations defining precise extremals (Ref. 4). In simple terms, the Caratheodory theorem can be stated as follows: "The optical path length of any ray is a minimum." This is the fundamental principle of geometrical optics. Therefore, we can find the path of a ray by minimizing the optical path length, that is, the path of the wavefront in an optical medium of refractive index n. Traditionally, that is how the eikonal equation is derived using variational techniques. We can also derive the eikonal equation by applying the powerful technique of dynamic programming. 2. Derivation For the present case, let us deal with a medium with refractive index n(x, z), where z is along the direction of propagation of ray or alternatively the wavefront, since the ray is nothing but the perpendicular to the wavefront; the physical situation is shown in Fig 1. Out of the infinite number of possible paths available, only that for which the integral in Equation (1) is a minimum is taken by the light ray. That is, the path of the ray is such that

where the path is assumed to start at x = 0 and z = 0. To do this minimization, we apply the principle of optimality, which states: For a total optimal policy, regardless of the first (last) decision, the subsequent (earlier) decisions must form an optimal policy. Applied to our problem, this implies that we consider an infinitesimal path As from (x-sin 9As, z-cos OAs) to (x, z), namely, from the point P in the figure to the point Q. Then, by applying backward dynamic programming, we get

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Fig. 1. Physical situation under consideration.

Expanding the above in Taylor series, we get

which reduces to

When As is small, we get Minimizing (5) with respect to 0, we obtain where Sx and S: are the partial derivatives of S with respect to x and z. Therefore, for the optimal path we get Substituting (8) in (6), we get

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which is exactly the eikonal equation. The method is very elegant. This can be generalized easily to three dimensions, yielding the general eikonal equation. 3. Conclusions The application of dynamic programming to optical problems is as follows. The principle of optimality is applied to the Fermat principle. Coupled with the initial conditions for the trajectory and its slope, this leads directly to the eikonal equation, which on solution gives the optimized trajectory. The full potential of this technique can be appreciated when applied to waveguiding in optical fibers. We have not only found that the technique does give a much deeper insight into the physics of light guiding, but that it also makes the problem computationally simpler. The method can be used in the design of waveguide profiles optimized for specific parameters, such as transit time, ray half period, etc. (Refs. 5-8). It has also been suggested that it might be possible to apply the methodology to study nonlinear fibers. This will be the subject of future research. References 1. BORN, M., and WOLF, E., Principles of Optics, Pergamon Press, 6th Edition, Oxford, UK, 1980. 2. KALABA, R., Dynamic Programming, Fermat's Principle, and the Eikonal Equation, Journal of the Optical Society of America, Vol. 51, pp. 1150-1151, 1961. 3. BELLMAN, R., Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957. 4. CARATHEODORY, C., Geometrische Optik, Springer, Berlin, Germany, 1937. 5. CALVO, M. L., and LAKSHMINARAYANAN, V., Light Propagation in Optical Waveguides: A Dynamic Programming Approach, Journal of the Optical Society of America, Vol. 14A, pp. 872-881, 1997. 6. LAKSHMINARAYANAN, V., VARADHARAJAN, S., and CALVO, M. L., A Note on the Applicability of Dynamic Programming to Waveguide Problems, Photonics 96: Proceedings of the International Conference on Fiber Optics and Photonics, Edited by J. P. Raina and P. R. Vaya, Tata McGraw Hill, New Delhi, India, Vol. 1, pp. 209-214, 1997. 7. LAKSHMINARAYANAN, V., and CALVO, M. L., Application of Dynamic Programming to Waveguide Analysis: An Example. Optics and Photonics News (Supplement), Vol. 6, p. 127, 1995. 8. CALVO, M. L. and LAKSHMINARAYANAN, V., Grain-Guided Segmented Planar Waveguides: Optimal Design Using Dynamic Programming Technique. Optics and Photonics News (Supplement), Vol. 8, p. 88, 1997.