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Wigner Distribution in Optics 25

74
[seeEqs.(22)and(21)inRef.73]arebasedonthesemoments. Higher-
order moments are used, for instance, to characterize the beam’s sym-
metry and its sharpness. 37
Because the Wigner distribution of a two-dimensional signal is a
function of four variables, it is difficult to analyze. Therefore, the signal
is often represented not by the Wigner distribution itself, but by its
moments. Beam characterization based on the second-order moments
of the Wigner distribution thus became the basis of an International
Organization for Standardization standard. 75
In this section we restrict ourselves mainly to second-order mo-
ments. The propagation of the matrix M of second-order moments
of the Wigner distribution through a first-order optical system with
ray transformation matrix T can be described by the input-output
t
relationship 9,76 M o = TM i T . This relationship can be readily derived
by combining the input-output relationship (1.39) of the first-order op-
tical system with the definition (1.25) of the moment matrices of the
input and the output signal. Since the ray transformation matrix T is
symplectic, we immediately conclude that a possible symplecticity of
the moment matrix (to be discussed later) is preserved in a first-order
optical system: if M i is proportional to a symplectic matrix, then M o
is proportional to a symplectic matrix as well, with the same propor-
tionality factor.

1.7.1 Moment Invariants
t
If we multiply the moment relation M o = TM i T from the right by J,
and use the symplecticity property (1.41) and the properties of J, the
−1
77
input-output relationship can be written as M o J = T(M i J)T .From
the latter relationship we conclude that the matrices M i J and M o J are
related to each other by a similarity transformation. As a consequence
of this similarity transformation, and writing the matrix MJ in terms
−1
of its eigenvalues and eigenvectors according to MJ = SΛS ,we
can formulate the relationships Λ o = Λ i and S o = TS i . We are thus
led to the important property 77 that the eigenvalues of the matrix MJ
(and any combination of these eigenvalues) remain invariant under
propagation through a first-order optical system, while the matrix of
t
t t
eigenvectors S transforms in the same way as the ray vector [r , q ]
does.
It can be shown 77 that the eigenvalues of MJ are real. Moreover,
if is an eigenvalue of MJ, then − is an eigenvalue, too; this im-
plies that the characteristic polynomial det(MJ − I), with the help of
2
which we determine the eigenvalues, is a polynomial of . Indeed,
the characteristic equation takes the form

4 2
det(MJ − I) = 0 = − a 2 + a 4