Page 45 - Phase Space Optics Fundamentals and Applications

26 Chapter One

with a 4 = det M and

2 2
a 2 = m xx m uu − m xu + m yy m vv − m yv + 2(m xy m uv − m xv m yu )

Since the eigenvalues of MJ are invariants, the same holds for the co-
efficients of the characteristic equation. And since the characteristic
2
equation is an equation in , we have only two such independent
eigenvalues (± x and ± y , say) and thus only two independent in-
variants (such as x and y ,or a 2 and a 4 ).
An interesting property follows from Williamson’s theorem: 78,79
For any real, positive definite symmetric matrix M, there exists a
t
real symplectic matrix T o such that M = T o Δ o T , where Δ o =
o
−1 t
−1
T M(T ) takes the normal form
o
o

Λ o 0 x 0
Δ o = with Λ o = and x , y > 0
0 Λ o 0 y
(1.63)
−1
From the similarity transformation MJ = T o (Δ o J)T , we conclude
o
that Δ o follows directly from the eigenvalues ± x and ± y of MJ and
2
2
2
that T o follows from the eigenvectors of (MJ) :(MJ) T o = T o Δ . Any
o
moment matrix M can thus be brought into the diagonal form Δ o by
means of a realizable first-order optical system with ray transforma-
−1
tion matrix T .
o
1.7.2 Moment Invariants for Phase-Space
Rotators
In the special case that we are dealing with a phase-space rota-
tor, for which the ray transformation matrix satisfies the orthogo-
t
nality relation T −1 = T , we have not only the similarity trans-
formation M o J = T(M i J)T −1 but also the similarity transformation
−1
M o = TM i T . The eigenvalues of M are now also invariants, and
the same holds for the coefficients of the corresponding characteristic
equation

4 3 2
det(M − I) = 0 = − b 1 + b 2 − b 3 + b 4

Since b 4 = det M is already a known invariant (= a 4 ), this yields at
most three new independent invariants.