Page 117 - Phase Space Optics Fundamentals and Applications

98 Chapter Three

2
2
2
2
Parameters Q 0 and Q = Q + Q + Q are invariant under the phase-
1 2 3
space rotations. The components Q j ( j = 1, 2, 3), which can be or-
ganized as a vector Q = (Q 1 ,Q 2 ,Q 3 ), define the degree of vorticity
of the beam, 46 which can be demonstrated on the example of the or-
thosymplectic modes.
For the HG modes only the diagonal second-order moments [see
1
1
Eq. (1.25)] differ from zero: m xx = m uu = m+ and m yy = m vv = n+ ,
2 2
and therefore
Q 0 = m + n + 1
Q = (m − n, 0, 0) (3.95)
Using the transformation relations for Q j under the phase-space
rotators 73 (see also Sec. 1.7.2), we find that the orthosymplectic mode
( , )
L m,n (r) presented on the (m, n)-orbital Poincar´e sphere, defined in
Sec. 3.5.2, is characterized by the parameters

Q 0 = m + n + 1
Q = Q(sin cos , sin sin , cos ) (3.96)

±
where Q = m − n. Thus for the LG mode L (r) we obtain Q 1 =
m,n
Q 2 = 0, and Q 3 =±(m − n); meanwhile the HG modes rotated
counterclockwise at ± /4 are characterized by Q 1 = Q 3 = 0 and
Q 2 =±(m − n). It has been mentioned that Q 3 corresponds to the z
component of the OAM of the beam propagating in the z direction.
A beam with nonzero integer Q 3 is referred as a vortex beam. Among
U
the orthosymplectic modes H m,n (r)(m = n), only the LG modes are
2
usually mentioned as vortex beams. Nevertheless others with Q =
2
(m − n) = 0 can be considered as potential vortices, since they are
converted to the LG modes by phase-space rotations. Note that for the
modes with symmetric indices m = n, we obtain Q 1 = Q 2 = Q 3 = 0.
This is the case of the fundamental Gaussian mode, for which Q 0 takes
a minimal value Q 0 = 1.
The orbital Poincar´e sphere introduced for the presentation of the
orthosymplectic modes can also be used for the characterization of
other two-dimensional signals, 74 which may be coherent or partially
coherent. It has been shown in Ref. 20 that there exists such an op-
tical first-order optical system associated with ray transformation
matrix T c that brings the moment matrix M to the diagonal form
t
M c = T c MT , called canonical f c , where m xx = m uu , m yy = m vv (see
c
details in Sec. 1.7.1). The signal f c has the parameters
Q 0 = m xx + m yy
Q = (m xx − m yy , 0, 0) (3.97)