Light propagation in optical waveguides: A dynamic programming approach

Thumbnail Image
Full text at PDC
Publication Date
Lakshminarayanan, Vasudevan
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Optical Society of America
Google Scholar
Research Projects
Organizational Units
Journal Issue
We apply techniques of optimal-control theory, namely, the methods of dynamic programming, to the problem of light propagation in optical waveguides. This formulation is equivalent to the resolution of an eikonal equation. We illustrate this optimization technique for the case of an ideal parabolic refractive-index-profile distribution. We discuss the possibility of extending this procedure to other types of optical waveguides and optical media.
© 1997 Optical Society of America. The Optical Society (OSA) Annual Meeting (1995. Portland, Oregón, EE.UU.). Congress of the International Comission for Optics (ICO) (17º. 1996. Taejon, Corea del Sur). This work was supported in part by the School of Optometry of the University of Missouri–St. Louis (UMSL). A UMSL research grant is also gratefully acknowledged. We also thank Jerry Christensen, Dean, School of Optometry, USML, for support during the initial stages of this research program. Additional financial assistance from the Spanish Ministry of Education and Science (DGICYT) under project PR94-362 to M. L. Calvo is also acknowledged. Partial results were presented at the OSA Annual Meeting (Portland, Oregon, October 1995) and at the ICO-XVII Congress (Taejon, Korea, August 1996). The authors also thank an anonymous referee for a careful review and R. F. Álvarez-Estrada and G. F. Calvo for helpful suggestions and discussions.
1. R. K. Lagu and R. V. Ramaswamy, “A variational finite difference method for analyzing channel waveguides with arbitrary index profiles”, IEEE J. Quantum Electron. QE-22, 968–978 (1986). 2. A. Sharma and P. Bindal, “Variational analysis of diffused planar and channel waveguides and directional couplers”, J. Opt. Soc. Am. A 11, 2244–2248 (1994). 3. A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method”, Appl. Opt. 21, 947–987 (1982). 4. B. D. Stone and G. W. Forbes, “Optimal interpolants for Runge–Kutta ray tracing in inhomogeneous media”, J. Opt. Soc. Am. A 7, 248–254 (1990). 5. J. Puchalsky, “Numerical determination of continuous ray tracing: the four-component method”, Appl. Opt. 33, 1900–1906 (1994). 6. K. B. Wolf and G. Krötzen, “Geometry and dynamics in refracting systems”, Eur. J. Phys. 16, 14–20 (1995). 7. A. Ghatak, E. Sharma, and J. Kompella, “Exact ray paths in bent waveguides”, Appl. Opt. 27, 3180–3184 (1988). 8. A. Ghatak and E. G. Sauter, “The harmonic oscillator problem and the parabolic index optical waveguide I. Classical and ray optics analysis”, Eur. J. Phys. 10, 136–143 (1989). 9. E. G. Sauter and A. K. Ghatak, “The harmonic oscillator problem and the parabolic index optical waveguide II.Quantum mechanical and wave optical analysis”, Eur. J. Phys. 10, 144–150 (1989). 10. R. Kalaba, “Dynamic programming, Fermat’s principle, and the eikonal equation”, J. Opt. Soc. Am. 51, 1150–1151 (1961). See also V. Lakshminarayanan and S. Varadharajan, “Dynamic programming, Fermat’s principle, and the eikonal equation—revisited”, J. Optimization Theor. Appl. (to be published). 11. R. E. Bellman, Dynamic Programming (Princeton U. Press, Princeton, N.J., 1957). 12. V. Lakshminarayanan, S. Varadharajan, and M. L. Calvo, “A note on the applicability of dynamic programming to waveguide problems”, in Photonics ’96: Proceedings of the International Conference on Fiber Optics and Photonics, J. P. Raina and P. R. Vaya, eds. (Tata McGraw-Hill, New Delhi, 1977), Vol. 1, pp. 209–214. 13. R. E. Bellman and R. Kalaba, “Dynamic programming, invariant imbedding and quasi-linearization: comparison and interconnections”, in Computing Methods in Optimization Problems, A. V. Balakrishnan and L. W. Neustadt, eds. (Academic, New York, 1964), pp. 135–145. 14. S. E. Dreyfus, Dynamic Programming and the Calculus of Variation (Academic, New York, 1965). 15. R. E. Bellman and R. Vasudevan, Wave Propagation. An Invariant Imbedding Approach (Reidel, Dordrecht, The Netherlands, 1986). 16. S. E. Dreyfus and A. M. Law, The Art and Theory of Dynamic Programming (Academic, New York, 1977). 17. R. E. Bellman, R. Kalaba, and G. M. Wing, “Invariant imbedding and mathematical physics I. Particle process”, J. Math. Phys. 1, 280–308 (1960). 18. R. E. Bellman, Introduction to the Mathematical Theory of Control Processes. Linear Equations and Quadratic Criteria (Academic, New York, 1967), Chaps. 3 and 4. 19. R. E. Bellman and S. E. Dreyfus, Applied Dynamic Programming (Princeton U. Press, Princeton, N.J. 1962), Chap. VIII. 20. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1984), Sec. 4.2. 21. Caratheodory’s theorem establishes the connection between Fermat’s principle and the calculus of variation defining precise extremals. See C. Caratheodory, Geometrische Optik (Springer, Berlin, 1937). See also Ref. 20, App. 1, pp. 719–737. 22. M. L. Calvo and V. Lakshminarayanan, “Dynamic programming: an alternative approach to light propagation in arbitrary optical media”, in 17th Congress of the International Commission for Optics: Optics for Science and New Technology, J. S. Chang, J. H. Lee, S. Y. Lee, and C. H. Nam, eds., Proc. SPIE, 2778, 294–295 (1996). 23. See, for example, Ref. 20, p. 111. 24. R. F. Alvarez-Estrada, Department of Theoretical Physics, Faculty of Physical Sciences, Universidad Complutense, 28040 Madrid, Spain, June 1996 (personal communication). 25. Toda’s potential, also called the Toda lattice, is a well-known solution of Eq. (12) in the field of quantum mechanics and nonlinear waves. See M. Toda, “Vibration of a chain with nonlinear interaction”, J. Phys. Soc. Jpn. 22, 431–436 (1967); Wave propagation in anharmonic lattices”, J. Phys. Soc. Jpn. 23, 501–506 (1967); A. Khare and R. K. Bhaduri, “Exactly solvable noncentral potentials in two and three dimensions”, Am. J. Phys. AJPIAS 62, 1008–1014 (1994). 26. R. K. Luneburg, The Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), p. 180. 27. S. Cornbleet, Microwave Optics, The Optics of Microwave Antenna Design (Academic, New York, 1976), pp. 127–132. See also Sec. 2.6. 28. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. (Cambridge U. Press, London, 1964), Chap. X, p. 281. 29. L. B. Felsen and M. Marcuwitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973). 30. Some interesting formalism related to the so-called homogeneous-wave solution can be found in S. Solimeno, B. Crosignani, and P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, New York, 1986), pp. 63–66. 31. J. Mathews and R. L. Walker, Mathematical Methods of Physics (W. A. Benjamin, New York, 1970), Sec. 8–2. 32. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Sec. 1–9. 33. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Sec. 1–8. 34. Although the Rayleigh–Ritz method is discussed in the literature in relation to variational principles, here we introduce applications in the context of dynamic programming. See, for example, Ref. 15, Chap. X, Sec. 3. 35. G. A. Bliss, Lectures on the Calculus of Variation (U. of Chicago Press, Chicago, Ill., 1959). 36. The constant of integration β¯ plays an interesting role in the dynamic-programming formulation. It ensures the similarity between the trajectory z(x) given in Eq. (24) and the ray propagation in graded-index-profile waveguides under the geometrical optics approximation. Even if this similarity were to be ignored, there should appear values of β¯ n_0, n_0 being the refractive index of the core, for which the integral in Eq. (24) diverges. This indicates that only bound trajectories (bound rays) are convergent solutions of Eq. (24). 37. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Sec. 3–2.