The Optical Lagrangian and the Ray Equation

Abstract

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.