Page 102 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 83

+
LG m,n

θ
HG n,m

Ψ

HG m,n

–
LG m,n

FIGURE 3.4 (m, n)-Poincar´e sphere for orthosymplectic mode presentation.

U
The orthosymplectic mode H m,n (r), Eq. (3.68), can be obtained from
theHGone,H m,n (r), byapplicationoftheRCTassociatedwiththema-
U
U l
trix U or alternatively from the LG mode as H m,n (r) = R [L ± (r i )](r),
m,n
where U l = U × U g (± /4) [see Eq. (3.73)]. Since U l can be written as
U l = U r ( )U f ( , − )U r ( )U f (
,
) and the LG modes are eigen-
functions for the symmetrical fractional FT and the signal rotator,
U
we conclude that all different orthosymplectic modes H (r) can be
m,n
generated from the LGs by two-parameter RCTs described by matrix
U r ( )U f ( , − ).
It has been proposed to present all different orthosymplectic modes
U
H (r) (here a constant phase factor of the mode is ignored) for
m,n
fixed indices m and n on the sphere called the orbital (m, n)-Poincar´e
sphere, 43−45 which is similar to the one used for characterization of
polarized light (see Fig. 3.4).
(0,.)
+
For example, by starting from the LG mode L m,n (r) = L m,n (r),
living on the north pole of the (m, n)-Poincar´e sphere, and ap-
plying the RCT associated with two-parameter matrix U( , ) =
U r (− /4 + /2) U f ( /2, − /2) U r ( /4 − /2) to this mode, the en-
tire sphere can be populated by the different orthosymplectic modes
( , ) U( , ) +
L m,n (r) = R [L m,n (r i )](r), where the parameters ∈ [0, ] and
∈ [− , ] indicate the colatitude of a parallel and the longitude of
a meridian on the sphere, respectively. The HG modes H m,n (r) and
H n,m (r) are located at the intersection of the main meridian and equa-
tor at points ( , ) = ( /2, 0) and ( /2, ), respectively. Moreover,
it has been shown 43 that the transformation along the main meridian