Page 116 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 97

distributions.Notethatthismethodisnoninterferometricandnoniter-
ative. The reconstruction of the Wigner distribution from the separable
fractional FT power spectra, known as the Radon-Wigner transform,
is discussed in detail in Chap. 4. Here we only mentioned that the flex-
ible optical setup for the fractional FT considered in Sec. 3.6 permits
the almost real-time measurements of the Radon-Wigner transform.
It has been shown in Ref. 67 that for the reconstruction of the Wigner
distribution of the optical field separable in the Cartesian coordinates,
the setup performing the antisymmetric fractional FT can be used. The
numerical simulations show that a large number of the Wigner dis-
tribution projections, which can be acquired only by a flexible setup,
are required for the correct field identification. This stresses the im-
portance of the flexible phase-space rotator setups considered in the
previous section.
U
We mentioned that the orthosymplectic modes H m,n (r) for fixed U
form a complete orthonormal set that permits one to represent a signal
- U U
a
as their linear superposition f (r) = m,n mn H m,n (r). For the determi-
U 2
nation of the spatial mode spectrum |a mn | of a complex field ampli-
U
tude, the two uniparametric phase-space rotators for which H (r)
m,n
are eigenfunctions are needed. Thus in Ref. 68 the symmetric frac-
tional FT and the signal rotator were applied for the LG spectrum
measurements. Another scheme based on the symmetric and anti
symmetric fractional FT has been proposed for the determination of
the HG spectrum. 69
The optical field is often represented not by the Wigner distribu-
tion itself, which for the two-dimensional case is a function of four
variables, but by its global moments (see Sec. 1.7). Thus beam char-
acterization by means of 10 the second-order WD moments, 26,54,70,71
defined in Eq. (1.25), became the basis of the international standard.
All these moments can be found from the measurements of the four
2
fractional FT power spectra 72 |R U f ( x , y ) [ f (r i )](r)| with at least one
of them corresponding to the different angles x = y .
It has been shown in Ref. 73 [see also Eq. (1.66)] that the eight nor-
malized second-order moments can be combined into four linear su-
perpositions

1
Q 0 = [m xx + m uu + m yy + m vv ]
2
1
Q 1 = [m xx + m uu − (m yy + m vv )]
2
Q 2 = m xy + m uv
Q 3 = m xv − m yu (3.94)

which are related to the four basic phase space rotators: symmetric and
antisymmetric fractional FTs, gyrator, and signal rotator, respectively.