Page 83 - Phase Space Optics Fundamentals and Applications

64 Chapter Three

thin convergent spherical lens. It will be shown that all other phase-
space rotators also can be realized experimentally by using thin lenses,
but in this case some of them must be cylindrical.
In this chapter, we briefly summarize the main properties of the
two-dimensional CTs, corresponding to rotations in four-dimensional
phase space; we also consider in detail the basic phase-space rotators:
symmetric and antisymmetric fractional FTs, signal (image) rotator
and gyrator, as well as the first-order optical systems performing these
transforms. Finally we discuss the applications of the phase-space
rotators.

3.2 First-Order Optical Systems and
Canonical Integral Transforms
3.2.1 Canonical Integral Transforms and Ray
Transformation Matrix Formalism
In paraxial approximation of the scalar diffraction theory, the propa-
gation of a coherent monochromatic light through a first-order system
is described by a canonical integral transform. 1,2,4 Thus starting from
the complex field amplitude f i (r i ) at the input plane of the system,
we have its CT at the output plane f o (r o )

∞
T
f o (r o ) = f i (r i )K (r i , r o ) dr i (3.1)
−∞
T
The kernel K (r i , r o ) is parameterized by the wavelength and the
real symplectic ray transformation 4 × 4 matrix T that relates the po-
sition r i and direction p i of an incoming ray to the position r o and
direction p o of the outgoing ray

r o A B r i r i
= = T (3.2)
p o C D p i p i
t
t
where r = (x, y) and p = ( p x ,p y ) . The superscript t denotes trans-
position. Note that the term related to the time dependence and the
phase accumulation exp(i2 z/ ) due to propagation at distance z will
be omitted. Here and further in this chapter we use the normalized
dimensionless variables and the matrix parameters. The normalized
variable p can also be interpreted as spatial frequency or ray mo-
mentum. To convert them to real position r and ray direction p, the
√ √
following relations have to be used: r = r w, p = p /w, a = A,
b = Bw, c = Cw, and d = D, where w is some length factor defined
by the used optical system and the beam width.