The Optical Lagrangian and the Ray Equation

  • Vasudevan Lakshminarayanan
  • Ajoy K. Ghatak
  • K. Thyagarajan


According to Hamilton’s principle in classical mechanics, the trajectory of a particle between times t 1 and t 2 is such that*
$$ \delta \int\limits_{{{t_1}}}^{{{t_2}}} {L({q_j},{{\dot{q}}_j}t)dt = 0} $$
where L is called the Lagrangian, the integration is over time, q j (j=,2,…) represent the generalized coordinates and dots represent differentiation with respect to time. Equation (1) is referred to as the Hamilton’s principle of least action. From {zyEq.(1)|33-1} it is possible to derive the Lagrange’s equations of motion [1]:
$$ \frac{d}{{dt}}\left( {\frac{{\partial L}}{{\partial {{\dot{q}}_j}}}} \right) = \frac{{\partial L}}{{\partial {q_j}}} $$
In this chapter we will write Fermat’s principle in the form of Eq.(1) and derive the ray equation using Cartesian coordinates. We will obtain explicit solutions of the ray equation. In the next chapter we will obtain the optical Lagrangian in cylindrical coordinates and derive ray equations valid for optical fibers which are characterized by cylindrically symmetric refractive index distribution.


Sound Propagation Refractive Index Profile Refractive Index Variation Sound Velocity Profile Launching Condition 
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    H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass. (1960).Google Scholar
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    M. Born and E. Wolf, Principles of optics, Pergamon Press, Oxford (1975).Google Scholar
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    Ghatak and K. Thyagarajan, Introduction to Fiber Optics, Cambridge University Press (1998).Google Scholar
  4. 4.
    W.S. Burdic, Underwater Acoustic System Analysis, Prentice Hall, Englewood Cliffs, N.J. (1984).Google Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Vasudevan Lakshminarayanan
    • 1
  • Ajoy K. Ghatak
    • 2
  • K. Thyagarajan
    • 2
  1. 1.University of Missouri-St. LouisSt. LouisUSA
  2. 2.Indian Institute of TechnologyNew DelhiIndia

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