Page 103 - Phase Space Optics Fundamentals and Applications

84 Chapter Three

= 0 corresponds to the gyrator transform Eq. (3.16), along the merid-
ian with = /2—to the antisymmetric fractional FT—and along the
equator—to the signal rotator transform Eq. (3.14). Thus the HG mode
H m,n (r) rotated counterclockwise at angle /2 lives on the equator at
longitude .
It is easy to see from Eq. (3.72) that the modes from the same co-
latitude have the same projection of the OAM along the propagation
direction L m,n = (m−n) cos . Then for the LG mode L ± (r) the value
m,n
z
of the projection L m,n =±(m − n) is an integer; meanwhile L m,n = 0
z
z
( /2, )
for HG mode L m,n (r).
U
Correspondingly, any orthonormal set {H m,n (r)} with integer m, n ∈
[0, ∞) is characterized by two parameters and can be associated with
a certain direction ( , ) in three-dimensional parametric space.
3.6 Optical Setups for Basic
Phase-Space Rotators
It is well known (e.g., see Ref. 10) that in paraxial optics the Fourier
transform can be performed using a convergent thin lens. Thus the
complex field amplitude at the back focal plane of the lens corre-
sponds to the FT of one at the front focal plane. As derived in Ref. 28
and discussed in Sec. 1.5, the symmetric fractional FT at angle
can be also performed by the same scheme if the distance z be-
tween the input/output plane and the lens of focal distance f equals
2
z = 2 f sin (
/2). Another proposed scheme 28 consists of two identi-
cal spherical convergent lenses of focal distance f located at the input
2
and output system planes with the distance z = 2 f sin (
/2) between
them. Moreover, the propagation of the optical beam through the op-
tical fiber with a quadratic refractive index profile also produces the
symmetric fractional FT at angles defined by the propagation distance
and the refractive index gradient. 14,29
To perform the separable fractional FT, as well as signal rotator
and gyrator, the cylindrical lenses are needed. Several setups for sep-
arable fractional FT, 46−50 antisymmetric fractional FT, 51 and signal
rotator 21,22 have been proposed. Nevertheless most are difficult to
adapt to the often needed change of transformation parameter. More-
over, the parameter turning is usually accompanied by additional scal-
ing which depends on the transformation parameter.
The main objective of the system design is to find a minimal lens–
free-space configuration that is flexible for transformation parameter
changing. A setup with fixed free-space intervals between the gener-
alized lenses is a promising candidate for this task.
The generalized lens, 52,53 which can be mathematically described
by the CT parameterized by ray transformation matrix T L , Eq. (3.10),