Page 47 - Phase Space Optics Fundamentals and Applications

28 Chapter One

It is not difficult to show now that M can be represented in the general
form

10 cos ϑ exp(i ) sin ϑ

M = Q 0 + Q (1.67)
01 exp(−i ) sin ϑ − cos ϑ
where the angles ϑ and follow from the relations Q cos ϑ = Q 1 (with
0 ≤ ϑ ≤ ) and Q exp(i ) sin ϑ = Q 2 + iQ 3 .
A phase-space rotator will only change the values of the angles ϑ
and , but does not change the invariants Q 0 and Q. To transform a
diagonal matrix M with diagonal entries Q 0 + Q and Q 0 − Q into

the general form (1.67), we can use, for instance, the phase-space ro-
1
1
1
1
1
tating system 81 F( , − ) R(− ϑ) F(− , ); see also Sec. 1.6.2
2 2 2 2 2
and Eq. (1.48). Moreover, from Eq. (1.65), we easily derive 80 that
for a separable fractional Fourier transformer F( x , y ), Q 1 is an
invariant and Q 2 + iQ 3 undergoes a rotation-type transformation:
(Q 2 + iQ 3 ) o = exp[i( x − y )] (Q 2 + iQ 3 ) i . Similar properties hold
for a gyrator G(
), for which Q 2 is an invariant and (Q 3 + iQ 1 ) o =
exp(i2
)(Q 3 + iQ 1 ) i , and for a rotator R(−
), for which Q 3 is an
invariant and (Q 1 + iQ 2 ) o = exp(i2
)(Q 1 + iQ 2 ) i .
1.7.3 Symplectic Moment Matrix—The
Bilinear ABCD Law
If the moment matrix M is proportional to a symplectic matrix, it can
be expressed in the form 77
−1 −1
G G H
M = m −1 −1 (1.68)
HG G + HG H
with m a positive scalar, G and H real symmetric 2 × 2 matrices, and
G positive definite; the two positive eigenvalues of MJ are now equal
to +m, and the two negative eigenvalues are equal to −m.
We recall that for a symplectic moment matrix, the input-output
t
relation M o = TM i T can be expressed equivalently in the form of the
bilinear relationship
H o ± iG o = [C + D(H i ± iG i )][A + B(H i ± iG i )] −1 (1.69)

This bilinear relationship, together with the invariance of det M, com-
pletely describes the propagation of a symplectic matrix M through
a first-order optical system. Note that the bilinear relationship (1.69)
is identical to the ABCD law for spherical waves: for spherical waves
−1
we have H o = [C + DH i ][A + BH i ] , and we have only replaced the
(real) curvature matrix H by the (generally complex) matrix H±iG.We
are thus led to the important result that if matrix M of second-order