Page 38 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 19

−1
coordinates, W r =: r and Wq =: q, we have

r o A B r i
= (1.39)
q C D q
o i
We recall that the ray transformation matrix is symplectic. Using the
matrix J,

0 −I −1 t
†
J = i = J = J =−J (1.40)
I 0
∗ t
t
−1
†
where J , J = (J ) , and J are the inverse, the adjoint, and the trans-
pose of J, respectively, symplecticity can be elegantly expressed as
t
T −1 = JT J. In detail we have

−1 t t
A B D −B t
−1
T = = t t = JT J (1.41)
C D −C A
If det B = 0, the coherent point-spread function of the first-order
optical system reads
h(r o , r i ) = (det iB) −1/2 t −1 t −1 t −1
exp i r DB r o − 2r B r o + r B Ar i
o i i
(1.42)
see also Refs. 60 and 61. In the limiting case that B → 0, we have
−1/2 t −1 −1
h(r o , r i ) =| det A| exp i r CA r o r i − A r o (1.43)
o
In the degenerate case det B = 0 but B = 0, a representation in terms
of the coherent point-spread function can also be formulated. 62 The
relationship between the input Wigner distribution W i (r, q) and the
output Wigner distribution W o (r, q) takes the simple form
W o (Ar + Bq, Cr + Dq) = W i (r, q) (1.44)
and this is independent of the possible degeneracy of submatrix B.

1.6.2 Phase-Space Rotators—More Rotations
in Phase Space
If the ray transformation matrix is not only symplectic but also orthog-
t
onal, T −1 = T , the system acts as a general phase-space rotator, 53 as
we will see shortly. We then have A = D and B =−C, and U = A+iB
−1
is a unitary matrix: U = U . We thus have
†

A B
t
†
T = and (A − iB) = U = U −1 = (A + iB) −1
−BA
(1.45)