Page 69 - Phase Space Optics Fundamentals and Applications

50 Chapter Two

After insertion of this expression in (2.2), it can be seen that the inte-
gration over x results in the delta function D ( − + a − Df ). The
multiple integral is thus reduced to a single integral

a − Df a − Df
A D ( f, a) = d exp (−i2 f ) + , −
2 2
= A 0 ( f, a − Df ) (2.18)
This result can be more readily obtained by recalling that the Fresnel
diffraction integral is the convolution of the input function T(x) by the
√
2
propagator G(x) = exp (i x / D − i /4)/ D; the output spectrum
is therefore the product of the input spectrum ˜ T( f ) by the spectrum
2
˜ G( f ) = exp (−i Df )of G(x); this is translated in terms of mutual
intensity as
$ %
2 2
f f f f
˜ D m + ,m − = exp −i D m + − m −
2 2 2 2

f f
× ˜ 0 m + ,m −
2 2

f f
= exp (−i2 m Df )˜ 0 m + ,m − (2.19)
2 2
Inserting this expression in (2.3), we indeed obtain directly 2,11

f f
A D ( f, a) = dm exp [i2 m(a − Df )]˜ 0 m + ,m −
2 2
= A 0 ( f, a − Df ) (2.20)
As shown in Ref. 3, a similar formula also exists for the WDF:

W D (x, g) = W 0 (x − Dg, g) (2.21)
According to Eqs. (2.18) and (2.21), the AF and the WDF propagate in a
uniform medium without a change of their functional forms; only the
variables are linearly transformed. This is an elegant representation
of Fresnel diffraction phenomena.

2.3.2 Transmission through a Thin Object
In this case, the incident mutual intensity inc (x, x ) is multiplied by

∗
T(x)T (x ), where T(x) is the object transmittance. The AF of the in-

cident beam is then to be convoluted with the object AF A T ( f, a)as
follows:

A( f, a) = dh A inc (h, a)A T ( f − h, a) (2.22)