Page 89 - Phase Space Optics Fundamentals and Applications

70 Chapter Three

(a) (b)

FIGURE 3.1 (a) The test real positive image and (b) its transformation after
the rotation at angle = /4.

diagonal block matrices X and Y, they lead to the relations
2
2
X + Y = 1
11
11
2
2
X + Y = 1 (3.25)
22 22
which are satisfied only if X 11,22 = cos x,y and Y 11,22 = sin x,y . Thus
the associated unitary matrix corresponds to the separable fractional
FT one, Eq. (3.15).
The kernel of the two-dimensional separable fractional FT is a prod-
uct of two one-dimensional fractional FT kernels, K U f ( x , y ) (r i , r o ) =
y
x
x
K (x i ,x o )K (y i ,y o ) , with K (x i ,x o ) given by
f f f

2 2
x + x o cos x − 2x o x i
−1/2
i
K x (x i ,x o ) = (i sin x ) exp i (3.26)
f
sin x
There are two main definitions of the fractional FT kernel which differ
by the phase factor exp(i x /2); see, for example, Refs. 4 and 5. Indeed,
to obtain the ordinary FT for x = /2 and rigorously satisfy the angle
x
additivity, the kernel exp(i x /2)K (x i ,x o ) has to be used. Neverthe-
f
less here we consider one, Eq. (3.26), that describes the complex field
amplitude propagation through the related first-order optical systems
as well as time evolution of the harmonic oscillator. In general the dif-
ference in the kernel definition is not important for the applications of
the RCTs, except in such particular cases as the definition of the Gouy
phase, 27 where Eq. (3.26) is preferable. Moreover, the matrix formal-
ism widely used for the description of phase-space rotators permits
one to avoid the differences in the fractional FT kernel definition.