Page 291 - Phase Space Optics Fundamentals and Applications

272 Chapter Eight

simplified model for the square of the refractive index
$ 2
n x ≥ x 0
2 0
n (x) = (8.103)
2
2
2
n [1 − (x − x 0 ) ]
0 x < x 0
where x 0 is the height at which the refractive index becomes constant
and determines the speed of the variation of the refractive index
near x = 0. Consider a point source located at (x s , 0), where x s >
x 0 , emitting light uniformly in all directions. Let us choose the ray
parameter as the angle (with respect to the z axis) at which each ray
leaves the source. Since the refractive index is independent of z, H is
invariant under propagation, according to Eq. (8.19c). From geometry,
its value is found to be
H( ) = n 0 cos (8.104)
The solutions to Eqs. (8.19a), (8.19b), and (8.20) take different forms in
three regions:

1. If the ray has not entered the inhomogeneous region x < x 0 ,
then it is a straight line defined by

X(z, ) = x s + z tan (8.105a)
P(z, ) = n 0 sin (8.105b)
L(z, ) = n 0 z sec (8.105c)

2. If the ray is inside the inhomogeneous region, then
sin
X(z, ) = x 0 + sin{ [z − Z 1 ( )] sec } (8.106a)

P(z, ) = n 0 sin cos{ [z − Z 1 ( )] sec } (8.106b)

sin
L(z, ) = n 0 sec Z 1 ( ) + 1 + [z − Z 1 ( )]
2
n 0
+ sin sin{2 [z − Z 1 ( )] sec } (8.106c)
4
where Z 1 ( ) = (x s − x 0 ) cot is the value of z at which the ray
enters the inhomogeneous region.
3. If the ray has entered and exited the inhomogeneous region,
then
X(z, ) = x 0 − [z − Z 2 ( )] tan (8.107a)

P(z, ) =−n 0 sin (8.107b)

n 0 sin
L(z, ) = n 0 [z + Z 1 ( ) − Z 2 ( )] sec + 1 +
2
(8.107c)