Page 260 - Phase Space Optics Fundamentals and Applications

Rays and Waves 241

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(Recall that v =| R| is the speed of the parameterization.) The first two
of these equations rule the propagation of each ray, and the third im-
pliesthattheeikonalcorrespondstotheopticalpathlengthalongtheray.
Equation (8.12a) is the geometrical definition of the optical momen-
tum as a vector that is locally tangent to the ray and whose magnitude
is given by the local refractive index (see Fig. 8.1), while Eq. (8.12b)
states that local changes in the refractive index modify the direction
of propagation of the ray. For a homogeneous medium, Eq. (8.12b)
implies that P remains constant, so rays are straight lines and, due to
Eq. (8.12c), L increases linearly with the propagation length. At the
interface between two homogeneous media, on the other hand, ∇n
is deltalike, pointing in the direction of the interface’s normal. In this
case, Eq. (8.12b) states that the component of P locally parallel to the
interface remains constant. This fact, combined with the requirement
that | P|= nat either side of the interface, leads to Snell’s law. When the
refractive index changes continuously and smoothly, the rays change
direction gradually and become curved.
While varying the parameter causes R to move along a ray, vari-
ations of 1 or 2 make R move from one ray to another. That is,
the above set of equations describes the evolution of a two-parameter
family of rays, with each ray corresponding to a set of values of the pa-
rameters = ( 1 , 2 ). However, the evolution of each ray is completely
autonomous,ascanbeappreciatedfromEqs.(8.12a)and(8.12b),which
involve no operation on the parameters 1 , 2 . Nevertheless, the opti-
cal path lengths of the different rays in the family are interconnected,
as one can see from considering the derivative of Eq. (8.10) with re-
spect to one of these parameters,

∂L ∂ R ∂ R
=∇ · = P · (8.13)
∂ j ∂ j ∂ j
This relation is a consequence of the fact that in Eq. (8.5) the rays
were chosen to be perpendicular to the surfaces of constant (and
therefore L). That is, all along their propagation, the rays constitute
what is known as a normal congruence, which means that there exists
a continuous set of surfaces that intersect perpendicularly all rays in
the family. Then L corresponds to the optical path length along the
rays measured from one of these normal surfaces. For example, for a

P
R
n(R)

FIGURE 8.1 Definition of the optical momentum.