Page 145 - Phase Space Optics Fundamentals and Applications

126 Chapter Four

can be rewritten as
1
¯ I(r N , ,z) =
2
( f + z) 2
0.5
1
u u

× q t + ,r N , ,z q ∗ t − ,r N , ,z
2 2
−1 −0.5

i2
× exp [W 40 + W 20 (z) + 2 W 40] u dt du (4.61)

The above integration over the variable u can be clearly identified as
the WDF of q (s, r N , ,z) with respect to the first variable, as stated in
Eq. (4.1). Thus, it is straightforward to show that
0.5

1 W 40 W 40 + W 20 (z)
¯ I(r N , ,z) = 2 2 W q t, −2 t − dt
( f + z)
−0.5
(4.62)
This expression relates the irradiance at any observation point to the
line integral of the function W q (x, ) along a straight line in phase
space described by the equation

W 40 W 40 + W 20 (z)
=−2 x − (4.63)

as depicted in Fig. 4.8. One can identify this integration as a projection
of the WDF at an angle given by [see Eq. (4.10)]

tan =− (4.64)
2W 40

FIGURE 4.8 Integration line in phase space.