Page 119 - Phase Space Optics Fundamentals and Applications

100 Chapter Three

3.7.7 Gouy Phase Accumulation
It was found by Gouy 76 more than 100 years ago that the Gaussian
beam accumulates the additional constant phase during its free-space
propagation. Now it is known that other transversal modes with
a Gaussian envelope, undergoing a cycle of transformations while
propagating through a paraxial optical system, accumulate Gouy
phase, 77,78 usually divided into two parts: a dynamic part and a geo-
metric part. 44,45,79,80 The identification of the Gouy phase is important
in resonator theory, 81 in optical trapping, 82 and in possible applica-
tions of its geometric part for quantum computation. 83,84 A simple
method for the determination of the Gouy phase—and in particu-
lar its dynamic and geometric parts—accumulated by an appropriate
Gaussian-type mode during its propagation through a first-order op-
tical system has been proposed in Ref. 27. It is based on the analysis of
the eigenvalues and eigenvectors of the ray transformation matrix as-
sociated with the first-order optical system as it is briefly summarized
below.
Strictly speaking, a beam of light (r) propagating through an op-
tical system, described by operator R, accumulates a phase shift only
if it is an eigenfunction of R with eigenvalue in the form of complex
exponent: R[ (r i )](r o ) = exp(i ) (r o ). We recall (see Sec. 3.5.1), that
U
an orthosymplectic mode H m,n (r i ) is an eigenfunction for the phase-
space rotator associated with unitary matrix U s = UU f ( x , y )U −1
1
1
with eigenvalue exp[−i(m + ) x −i(n+ ) y ], which corresponds to
2 2
the Gouy phase. The decomposition Eq. (3.29) is crucial for the identifi-
cation of the dynamic and geometric parts of the Gouy phase. Indeed,
during the propagation through the symmetric fractional FT system,
U
the mode H (r i ) acquires the dynamic phase d =−(m + n + 1)
,
m,n
defined by the sum of mode indices; meanwhile in the case of a sys-
tem similar to the antisymmetric fractional FT, the accumulated phase,
known as the geometric one, is proportional to the index difference
g =−(m−n) . Note that the dynamic and geometric phases are also
U
defined by the second-order moments of H (r i ) through parameters
m,n
Q 0 =− d and Q =− g . This emphasizes that the geometric phase
accumulation is related to the orbital angular momentum operators
defined in phase space.
If the eigenvalues of the ray transformation matrix are not unimod-
ular, we can speak about phase accumulation only in a wide sense,
where scaling and quadratic-phase modulation of the field amplitude
at the output system plane are present. In this case we permit the
beam to be an eigenfunction of the transformation described by the
orthosymplectic matrix in the Iwasawa decomposition Eq. (3.10).
We conclude that the Gouy phase accumulation of Gaussian-type
beams is associated with rotations in phase space. Dynamic phase is
acquired in symmetric, rotationally invariant systems, whose T O in
the decomposition Eq. (3.10) corresponds to the symmetric fractional