Page 193 - Phase Space Optics Fundamentals and Applications

174 Chapter Five

In Eq. (5.22) the kernel of the Volterra transformation, V(x; x 1 ,x 2 ), is
the bilinear impulse response. 13 The concept of bilinearity has found
practical applications in some optical problems. 14 In what follows
we show that bilinearity, in the above sense, is related to a phase-
space transformation. 15 To our end, we use the following center and
difference coordinates:

x 1 + x 2 y y
ˆ y = , y = x 1 − x 2 , B(x;ˆy, y) = V x, ˆy + , ˆy −
2 2 2
(5.23)

By employing Eq. (5.23), the definition of the product-space represen-
tation, and the definition of the WDF, we rewrite Eq. (5.22) as

∞
∞

2 y ∗ y
ˆ
|u(x)| = B(x;ˆy, y)u 0 y + u 0 ˆ y − d ˆydy
2 2
−∞ −∞
∞
∞
= B(x;ˆy, y) p 0 ( ˆy, y) d ˆydy
−∞ −∞
⎡ ⎤
∞ ∞
∞
= ⎣ B(x;ˆy, y) exp (i2 ˆ y) dy ⎦ W 0 (ˆy, ˆ )d ˆyd ˆ (5.24)
−∞ −∞ −∞
Equation (5.24) tells one how to tailor the output irradiance distri-
bution if one takes as input either the product-space representation,
p 0 (x, y) or the WDF W 0 (ˆy, ˆ ). Here it is convenient to recall the result
in Eq. (5.14).

∞

2
|u(x)| = W(x, ) d
−∞
∞ ∞
∞
= W 0 (ˆy, ˆ )W b (x, ;ˆy, ˆ )d ˆyd ˆ d
−∞ −∞ −∞
⎡ ⎤
∞ ∞
∞
= ⎣ W b (x, ; x 0 , 0 ) d ⎦ W 0 (ˆy, ˆ ) d ˆyd ˆ (5.25)
−∞ −∞ −∞