Page 118 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 99

and as well as the HG mode can be associated with the point
( /2, 0) on the orbital Poincar´e sphere. By performing the appro-
priate phase-space rotations, the entire sphere can be populated.
The signal f , at point ( , ) is characterized by the vector Q =
(m xx −m yy )(sin cos , sin sin , cos ),whichdefinesthesignalsym-
metry. Thus signal with Q = (Q 1 , 0, 0) is better described in the Carte-
sian coordinates; meanwhile for Q = (0, 0,Q 3 ) polar coordinates are
the best choice for signal analysis. Moreover, in the last case the signal
f 0, possesses a longitudinal component of the OAM (m xx − m yy ).
Let us consider a signal f (x, y) presented as a superposition of the
HG modes
, ,
2
f (x, y) = a mn H mn (x, y) |a mn | = 1 (3.98)
m,n m,n
Based on the expressions for the signal second-order moments, 75 we
derive its Q components.
, ,
2 2
Q 0 = |a mn | (m + n + 1) = 1 + |a mn | (m + n)
m,n m,n
,
2
Q 1 = |a mn | (m − n)
m,n

,
Q 2 = 2Re a m,n+1 a ∗ (m + 1)(n + 1)
m+1,n
m,n

,
Q 3 = 2Im a m,n+1 a ∗ (m + 1)(n + 1) (3.99)
m+1,n
m,n
We observe that Q 2 = Q 3 = 0 if in the signal decomposition Eq. (3.98)
there are no subsequent terms a m,n+1 and a m+1,n . Moreover Q 3 = 0if
the signal is real.
The stable beams that do not change their form under the propaga-
tion in free space contain in the decomposition Eq. (3.98) only the HG
modes with the same index sum m + n. Then since the signal decom-
position is normalized, they have integer parameter Q 0 = m + n + 1,
which is related to the Gouy phase of the beam.
If a signal corresponds to a circular harmonic c l (r,
) [see Eq. ( 3.90],
l
it has the same parameter Q = (0, 0,l) as the LG mode L .
p
Representing signals on the orbital Poincar´e spheres, we simplify
the analysis and processing as well as the comparison with other sig-
nals. Since the search of the CT T c is related to the diagonalization of
the moment matrix, the signal presentation on the Poincar´e sphere is
valid for coherent as well as partially coherent beams.