Page 46 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 27

Another way to find moment invariants for phase-space rotators is
to consider the Hermitian matrix

1
†
M = (r − iq)(r − iq) W(r, q) dr dq
E
= Mrr + Mqq + i(Mrq − M t rq )

m xx + m uu m xy + m uv + i(m xv − m yu )
=
m xy + m uv − i(m xv − m yu ) m yy + m vv

Q 0 + Q 1 Q 2 + iQ 3
= (1.64)
Q 2 − iQ 3 Q 0 − Q 1
and to use Eq. (1.50) to get the relation

M = UM U = UM U −1 (1.65)
†
i
o
i
which is again a similarity transformation. Note that the moments
m xu and m yv , i.e., the diagonal entries of submatrix Mrq , do not enter
matrixM andthatwehaveintroducedthefourmomentcombinations

Q j ( j = 0, 1, 2, 3) as
1
Q 0 = [(m xx + m uu ) + (m yy + m vv )] (1.66a)
2
1
Q 1 = [(m xx + m uu ) − (m yy + m vv )] (1.66b)
2
Q 2 = m xy + m uv (1.66c)
Q 3 = m xv − m yu (1.66d)

The characteristic equation with which the eigenvalues of M can be
determined reads
2
2
2
2
2
det(M − I) = 0 = − 2Q 0 + Q − Q = ( − Q 0 ) − Q ,

0
where we have also introduced

2
2
Q = Q + Q + Q 2 (1.66e)
1 2 3
The eigenvalues are real and we can write 1,2 = Q 0 ± Q. Since the
eigenvalues are invariant, we immediately get that 1 − 2 = 2Q is
an invariant, 80 and we also get the invariants 1 + 2 = 2Q 0 = b 1 ,
2
2
which is the trace of M and of M, and 1 2 = Q − Q = b 2 − a 2 ,

0

which is the determinant of M . We remark that Q 3 corresponds to
the longitudinal component of the orbital angular momentum of a
paraxial beam propagating in the z direction. From the invariance
of Q, we conclude that the three-dimensional vector (Q 1 ,Q 2 ,Q 3 ) =
(Q cos ϑ,Q sin ϑ cos ,Q sin ϑ sin ) lives on a sphere with radius Q.