Page 369 - Phase Space Optics Fundamentals and Applications

350 Chapter Eleven

with det(T) = 1. This matrix relates the input and output properties
of the ray trajectories, i.e., the ray height y with respect to the principal
ray of a bundle and its angle u with respect to this ray. The refrac-
†
tive index of the medium n in which the ray propagates is usually
appended to the ray angle, so that the specification of the ray is the
column vector

y
Y = (11.29)
u
The output and input rays are related by the equation

Y out = T Y in (11.30)

This relation may be represented in the phase space consisting of a
transverse coordinate (the ray height) and the corresponding trans-
verse wave vector (proportional to the ray angle). A single ray is a
point in this space, a ray bundle emanating from a single point occu-
pies a region of constant height, and a plane wave occupies a region
of the phase space of constant angle (Fig. 11.3).
The elements of the transfer matrix may be derived by using Hamil-
ton’s characteristic function in the paraxial approximation, i.e., using
the Fresnel approximation to the propagation kernel in the Kirchhoff
formula. The output and input scalar electric fields for such a system
are related by

E out (x) = dx K(x, x )E in (x ) (11.31)
The most general form of the Fresnel kernel K is (for light of wave
number k 0 = 2 / )

ik 0 ik 0
2 2
K(x, x ) = exp − (Ax − 2xx + Dx ) (11.32)
2 B 2B
The parameters of the transfer matrix determine the action of the op-
tical system on the input field through this kernel. This may be illus-
trated by some important simple paraxial optical elements.

† The notation used for specifying the phase-space coordinates of a ray in ge-
ometrical optics is conventionally in terms of the ray height y and the ray angle
u with respect to the principal ray of the bundle. This notation is used, e.g., in
paraxial ray tracing for first-order system design, which is sometimes known as
ynu tracing. The corresponding coordinates used for specifying the electric field
are more usually the transverse coordinate x = y and the transverse wave vector
k x = k 0 tan u ∼ k 0 u with k 0 = 2 / , where is the optical wavelength.
=