Page 194 - Phase Space Optics Fundamentals and Applications

Imaging Systems: Phase-Space Representations 175

Mutual spectrum
J(m, n) n
Mutual intensity
m
j(x, y) y

f

f

f

FIGURE 5.6 Same as Fig. 5.1, but with a rotating scatter plate for reducing
the degree of spatial coherence.

Comparing Eqs. (5.24) and (5.25) and using Eq. (5.15), we obtain

∞
∞
B(x;ˆy, y) = W b (x, ;ˆy, ˆ ) exp (−i2 yˆ ) d ˆ d
−∞ −∞
∞

= W q (x − ˆy, ) exp [i2 (−y) ] d
−∞
! y " ! y "
ˆ
ˆ
= q x − y + q ∗ x − y −
2 2
= p q (x − ˆy, −y) (5.26)
And therefore, in suitable coordinates, the Volterra kernel is related
to the double WDF. Equivalently, one can say that phase-space repre-
sentations are bilinear transformations. For space-invariant systems,
the Volterra kernel is the product-space representation of the im-
pulse response. Here it is relevant to note that for optical systems
working with partially coherent illumination, one needs to substitute
the product-space representation for the mutual coherence function
(the mutual intensity for monochromatic illumination), as depicted in
Fig. 5.6. Equivalently, the product spectrum representation should be
substituted by the cross-spectral density (the mutual power spectrum
or the mutual spectrum for monochromatic illumination). Thus, the
road map changes as depicted in Fig. 5.7.