Page 41 - Phase Space Optics Fundamentals and Applications

22 Chapter One

1.6.4 Geometric-Optical Systems
Let us start by studying a modulator, described—in the case of par-
tially coherent light—by the input-output relationship o (r 1 , r 2 ) =
∗
m(r 1 ) i (r 1 , r 2 ) m (r 2 ). The input and output Wigner distributions are
related by

W o (r, q) = W m (r, q − q ) W i (r, q ) dq i (1.53)
i
i
where W m (r, q) is the Wigner distribution of the modulation function
m(r).
We now confine ourselves to the case of a pure phase modulation
function m(r) = exp[i2 (r)]. We then get

1
∗
1
1
1
m r + r m r − r = exp i2 r + r − r − r
2 2 2 2
t
= exp{i2 [(d /dr) r + higher-order terms]}
(1.54)
If we consider only the first-order derivative in relation (1.54), we get
W m (r, q) (q − d /dr), and the input-output relationship of the
pure phase modulator becomes W o (r, q) W i (r, q − d /dr), which
is a mere coordinate transformation. We conclude that a single input
ray yields a single output ray.
The ideas described above have been applied to the design of optical
coordinate transformers 63,64 and to the theory of aberrations. 65 Now,
if the first-order approximation is not sufficiently accurate, i.e., if we
have to take into account higher-order derivatives in relation (1.54),
the Wigner distribution allows us to overcome this problem. Indeed,
we still have the exact input-output relationship (1.53), and we can
take into account as many derivatives in relation (1.54) as necessary.
We thus end up with a more general form 66 than W o (r, q) W i (r, q −
d /dr). This will yield an Airy function instead of a Dirac function, for
instance, when we take not only the first but also the third derivative
into account.
We concluded that a single input ray yields a single output
ray. This may also happen in more general—not just modulation-
type—systems; we call such systems geometric-optical systems.
These systems have the simple input-output relationship W o (r, q)
W i [g (r, q), g (r, q)], where the sign becomes an = sign in the case
x u
of linear functions g and g , i.e., in the case of Luneburg’s first-order
x u
optical systems. There appears to be a close relationship to the de-
scription of such geometric-optical systems by means of the Hamilton
characteristics. 6