Page 104 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 85

φ 1

O x
φ 2

FIGURE 3.5 Generalized lens constructed from two cylindrical lenses
rotated at different angles.

with the block matrix

g xx g xy
G = (3.74)
g xy g yy
produces the quadratic phase modulation of the input wavefront

2
2
f o (x, y) = exp[−i (g xx x + 2g xy xy + g yy y )] f i (x, y) (3.75)
In practice, the generalized lens can be implemented by a spatial
light modulator (SLM) that allows one to change the lens parameters
almost in real time. Also it can be constructed as a combination
of n aligned cylindrical lenses of power p j (p j > 0 for conver-
gent lens), which are attached one to another and rotated coun-
terclockwise with respect to the transversal OX axis at angles j .
- n 2 - n
Then g xx = j=1 p j cos j , g xy =− j=1 p j (sin 2 j )/2, and g yy =
n 2
-
j=1 p j sin j . Depending on the angles and the powers of the
cylindrical lenses, we obtain the elliptic (including spherical), hyper-
bolic, or parabolic phase modulations. In Fig. 3.5 the generalized lens
that contains only two cylindrical lenses is displayed.
Below we will consider flexible optical schemes with fixed loca-
tion of the generalized lenses which implement the basic phase-space
rotators.

3.6.1 Flexible Optical Setups for Fractional
FT and Gyrator
Based on the matrix formalism, flexible optical setups, which per-
U f ( x , y ) U g (ϑ)
form the fractional FT R , and the gyrator R have been
designed. 21,33,54 These optical schemes contain three generalized
lenses L 1 , L 2 , and L 3 ; the last is identical to L 1 , with fixed equal