Page 90 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 71

If x = y =
, then the fractional FT is symmetric. The symmetric
fractional FT produces the rotation in phase planes xp x and yp y at the
same angle
. Its kernel is written as

t
2
1 r + r 2 o cos
− 2r r i
o
i
U f (
,
)
K (r i , r o ) = exp i (3.27)
i sin
sin
For
= 0 it corresponds to the identity transform K U f (0,0) (r i , r o ) =
(r i −r o );for
= /2tothecommonFouriertransform[Eq.(3.5)]apart
from constant −i; for
= , to the reverse transform K U f ( , ) (r i , r o ) =
− (r i + r o ), which coincides, except for the sign, with signal rotation
at angle ; and for
= 3 /2 to the inverse FT apart from constant i.
We observe that the symmetric fractional FT is periodic, in the strict
sense, with 4 and not with 2 as the rest of basic phase-space rotators:
signal rotator, antisymmetric fractional FT, and gyrator.
The unitary matrix associated with the symmetric fractional FT
is a scalar matrix U f (
,
) = exp(i
)I with determinant exp(i2
).
From the scalar form of U f (
,
), it is easy to see that the symmetric
fractional FT commutes with any phase-space rotator: U f (
,
)U =
UU f (
,
).
The signal transformation under the symmetric fractional FT, ob-
tained by numericalsimulations,is demonstrated inFig.3.2,where the
real Fig 3.2a and imaginary Fig.3.2b parts of the symmetric fractional
FT at angle
= /4 of the signal shown in Fig. 3.1a are displayed.
If x =− y = , then the kernel is given by

K U f ( ,− ) (r i , r o )

2
2
2
1 x + x − y − y 2 o cos − 2 (x o x i − y o y i )
i
o
i
= exp i (3.28)
sin sin
and corresponds to the antisymmetric fractional FT, which also pro-
duces the rotation in phase planes xp x and yp y but at the angles

(a) (b)

FIGURE 3.2 (a) Real and (b) imaginary parts of the symmetric fractional FT
at angle
= /4 of the test signal (Fig. 3.1a).