Page 114 - Phase Space Optics Fundamentals and Applications

Rotations in Phase Space 95

can be summarized as
U 2
F = exp(i N )R U N [··· [exp(i 2 )R [exp(i 1 )R [ f ]]]] (3.91)
U 1
The decryption procedure is written correspondingly as

1 [exp(−i 1 ) ··· [R
N−1 [exp(−i N−1 )R
f = R U −1 U −1 U −1
N [exp(−i N )F]]]]
(3.92)
The randomness of the phase masks together with a large number
of encryption parameters U n provides a high security of the encryp-
tion procedure. More sophisticated algorithms for signal encryption
applying phase-space rotators have been developed in Refs. 17 and 18.

3.7.5 Mode Converters
The Hermite-Gaussian and the helical Laguerre-Gaussian (LG) modes
are probably the best-known functions used in optics. Indeed, the
transversal field distributions for widely applied laser cavities are de-
scribed by these modes. Moreover, they, as well as all orthosymplectic
modes, are structurally stable which means that, ignoring the scaling,
their intensity profiles remain the same during the propagation in ho-
mogeneous medium. This is a consequence of the fact that the modes
U
H m,n (r) are eigenfunctions for the symmetric fractional FT, which ap-
pear in the Iwasawa decomposition of the Fresnel ray transformation
matrix, Eq. (3.10), T O = T f (
,
).
Although LG and HG modes can be produced directly from laser
cavities, it is often needed to switch from one type of mode to another.
The simplest and cheapest way to do it is based on cylindrical lens
application. Since a Laguerre-Gaussian beam is rotationally symmet-
ric and is an eigenfunction of a symmetric fractional Fourier trans-
former, there are several first-order optical systems which produce
this operation. 25,35,42,52,61,62 Any of the phase-space rotators associ-
ated with the matrix 42

1 exp(i 1 ) ±i exp(i 2 )
U = √ (3.93)
2 ±i exp(i 1 ) exp(i 2 )
can serve as HG-to-LG mode converter. The LG beams at the output
of these systems differ one from another by only a constant phase
shift. The special case 1 = 0, 2 = /2 has been considered in Ref. 35;
meanwhile for 1 = 2 = 0, matrix U reduces to the gyrator ma-
trix U g (± /4). Moreover, the gyrator transform of HG mode at other
angles ϑ generates all possible orthosymplectic modes, beside their
rotated replicas.
Therefore the flexible scheme proposed for the gyrator imple-
mentation can serve as a tunable mode converter. In Fig. 3.10 the