Page 43 - Phase Space Optics Fundamentals and Applications

24 Chapter One

the Wigner distribution has a constant value. When we eliminate the
frequency variable q from Eqs. (1.60), we are immediately led to

d dr ∂k d dz ∂k
k = k = (1.61)
ds ds ∂r ds ds ∂z
which are the equations for an optical ray in geometrical optics. 71 We
are thus led to the general conclusion that in the geometric-optical ap-
proximation, the Wigner distribution has a constant value along the
geometric-optical ray paths, which conforms to our conclusions in
Sec. 1.6.4: W o (r, q) W i [g (r, q), g (r, q)]. For a more detailed treat-
x u
ment of rays, see Chap. 8 by Miguel Alonso.

1.7 Wigner Distribution Moments in
First-Order Optical Systems
The Wigner distribution moments provide valuable tools for the char-
acterization of optical beams (see, for instance, Ref. 37). First-order
moments, defined as

1
[m x ,m y ,m u ,m v ] = [x, y, u, v] W(x, y, u, v) dx dy du dv
E
(1.62)
yield the position of the beam (m x and m y ) and its direction (m u
and m v ). Second-order moments, defined by Eq. (1.25), give infor-
mation about the spatial width of the beam (the shape m xx and
m yy of the spatial ellipse and its orientation m xy ) and the angu-
lar width in which the beam is radiating (the shape m uu and m vv
of the spatial-frequency ellipse and its orientation m uv ). Moreover,
they provide information about its curvature (m xu and m yv ) and its
twist (m xv and m yu ), with a possible definition of the twistedness as 46
m yy m xv − m xx m yu + m xy (m xu − m yv ). Many important beam charac-
terizers, such as the overall beam quality 72
2 2
m xx m uu − m + m yy m vv − m + 2(m xy m uv − m xv m yu )
xu yv
(see also Sec. 1.7.1), are based on second-order moments. Also the
longitudinal component of the orbital angular momentum = a +
v ∝ (m xv − m yu ) [see Eq. (3) in Ref. 73] and its antisymmetrical part
a and vortex part v ,
(m xx − m yy )(m xv + m yu ) − 2m xy (m xu − m yv )
a ∝
m xx + m yy
m yy m xv − m xx m yu + m xy (m xu − m yv )
v ∝ 2
m xx + m yy