Page 101 - Phase Space Optics Fundamentals and Applications

82 Chapter Three

and the recurrence relations
! √ √ " t
√ t U U U
2 [x, y] H (r) = U m + 1 H (r), n + 1 H (r)
m,n m+1,n m,n+1
√ U √ U t
+ U ∗ mH m−1,n (r), n H m,n−1 (r) (3.71)
Based on Eqs. (3.70) and (3.71), we can determine 41 the z-OAM
U
component for the orthosymplectic mode H m,n (r).

∞ ∂ ∂
L m,n = Im H U (r) ∗ x − y H U (r) dx dy
m,n
m,n
z
−∞ ∂y ∂x

∗
= 2Im mU 11 U − nU 22 U 12 (3.72)
∗
21
As an example, let us find the eigenfunctions for the signal rota-
tor. Using the similarity transformation Eq. (3.48), we observe that
U g (∓ /4+ k)
the orthosymplectic mode H m,n (r) is an eigenfunction for the
U g (∓ /4+ k)
signal rotator for any angle. Note that H m,n with integer k
±
corresponds to the helicoidal Laguerre-Gaussian (LG) modes L (r)
m,n
apart from the constant phase factor

1/2
U g (∓ /4+ k) 1/2 (min{m, n})! √
H m,n (r) ∝ L ± (r) = 2 ( 2 r) |m−n|
m,n
(max{m, n})!
(|m−n|) 2
× exp[±i(m − n) ] L (2 r )
min{m,n}
2
× exp(− r ) (3.73)
( )
where L n (·) denotes the generalized Laguerre polynomials, and spa-
tial coordinates are represented by the two-dimensional column vec-
t t
tor r = (x, y) = (r cos ,r sin ) . Therefore, the LG mode L ± (r)is
m,n
an eigenfunction for the signal rotator. From Eq. (3.72) it follows that
± m,n
L (r) possesses the integer OAM projection L =±(m − n), also
m,n z
known as a topological charge.
Correspondingly using the similarity transformation Eq. (3.47) for
the gyrator, we conclude that its eigenfunctions are the HG ones
rotated at ∓ /4 + k.
3.5.2 Mode Presentation on Orbital
Poincar´e Sphere
Wehaveemphasizedthataphase-spacerotatorisdescribedbytheuni-
U
tary matrix U, which has 4 degrees of freedom. Nevertheless H (r)
m,n
is characterized by only two parameters because it is an eigenfunction
for the symmetric fractional FT R U f (
,
) and for the RCT associated
with matrix U s = UU f ( , − )U −1 for any
and . Let us demonstrate
this statement with the following example.