Page 67 - Phase Space Optics Fundamentals and Applications

48 Chapter Two

The corresponding intensity spectrum can be expressed by the multi-
ple integral

d d
˜ I D ( f ) = dx exp (−i2 xf )
D
2 2
(x − ) − (x − )
exp i T( )T ( ) (2.9)
∗

D
As the integration over x results in the following delta function

−

dx exp −i2 x f + = D ( Df + − ) (2.10)
D
the intensity spectrum can thus be reduced to a single integration. 6, 7

2
∗
˜ I D ( f ) = exp (−i Df ) d exp (−i2 f )T( )T ( + Df )

Df Df
= dx exp (−i2 fx)T x − T ∗ x + (2.11)
2 2
Similar expressions also exist in terms of ˜ T( f ):

2
∗
˜ I D ( f ) = exp (−i Df ) dh exp (−i2 Dh f ) ˜ T(h + f ) ˜ T (h)

f f
= dh exp (−i2 Dh f ) ˜ T h + ˜ T ∗ h − (2.12)
2 2
It is interesting to note that the AF associated with T(x) is apparent in
this formulation if the intensity spectrum is formally considered as a
function of f and a = Df .

2.2.2 Application to Simple Objects
This formulation can provide interesting results for some typical
Fresnel diffraction patterns. For instance, in the case of a slit of full
width w, we obtain 7
(w−| Df |)/2 $
sin[ f (w−| Df |)] for | f |≤ w
˜ I D (f) = dx e −i2 fx = f D
0 otherwise
−(w−| Df |)/2
(2.13)
which is analytically much simpler than the intensity distribution I (x)
in terms of Fresnel integrals represented geometrically by the Cornu
spiral.