ABSTRACT
Consider the calculation of the refractive index distribution that makes a given set of light rays on a plane possible. If we could calculate the optical path length along the rays, we could use the eikonal equation
n 2 = ∇S 2 (9.1)
to calculate the refractive index n, except that to calculate S, we need to know n. Although we cannot calculate S, we can defi ne a set of curves on the plane which are perpendicular to the light rays. These are given by i(x1, x2) = C, where C is a constant. For different values of C, different perpendicular lines to the rays are obtained. We then have S(x1, x2) = S(i(x1, x2)) or S = S(i) and
∇S = dS___di ∇i (9.2)
∇i is known because the curves i(x1, x2) = C were defi ned. Giving the function dS/di, n can be calculated using the eikonal equation. Replacing Equation 9.2 in Equation 9.1 we get:
n 2 = (dS___di) 2 ∇i 2 (9.3)
1 and S2 = constant propagating in a given medium of refractive index n(x1, x2). For these two wave fronts, we can write
p1 = ∇S1(9.4) p2 = ∇S2
or p 1 + p 2 = ∇( S 1 + S 2 )
(9.5) p 1 − p 2 = ∇( S 1 − S 2 )
If the refractive index is the same for both wave fronts, then p1 = p2 = n and the vector p1 + p2 points in the direction of the bisector of the two sets of rays.
