Page 39 - Phase Space Optics Fundamentals and Applications

20 Chapter One

and hence
W o (Ar + Bq, −Br + Aq) = W i (r, q) (1.46)

In the one-dimensional case, such a system reduces to a fractional
Fourier transformer (A = cos , B = sin ); the extension to a higher-
dimensional separable fractional Fourier transformer (with diagonal
matrices A and B, and different fractional angles for the different co-
ordinates) is straightforward.
In the two-dimensional case, the three basic systems with an orthog-
onal ray transformation matrix are (1) the separable fractional Fourier
transformer F( x , y ), (2) the rotator R(
), and (3) the gyrator G(
),
with unitary representations U = A + iB equal to

exp(i x ) 0 cos
sin
cos
i sin
and
0 exp(i y ) − sin
cos
i sin
cos
(1.47)
respectively. All three systems correspond to rotations in phase space,
which justifies the name phase-space rotators!
From the many decompositions of a general phase-space rotator
1
1
1
1
into the more basic ones, we mention F( , − ) R(
) F(− , )
2 2 2 2
F( x , y ), which follows directly if we represent the unitary matrix as

exp(i x ) cos
exp[i( y + )] sin
U = (1.48)
− exp[i( x − )] sin
exp(i y ) cos
1 1 1 1
Note that we have the relationship F( , − ) R(
) F(− , ) =
4 4 4 4
G(
), which is just one of the many similarity-type relationships that
exist between a rotator R( ), a gyrator G( ), and an antisymmetric
fractional Fourier transformer F( , − ):
1 1 1 1
F ± , ∓ G(±
) F ∓ , ± = R(−
) (1.49a)
4 4 4 4
1 1 1 1
F ± , ∓ R(±
) F ∓ , ± = G(
) (1.49b)
4 4 4 4
1 1
R ± F(±
, ∓
) R ∓ = G(−
) (1.49c)
4 4

1
1
G ± F(±
, ∓
) G ∓ = R(
) (1.49d)
4 4
1 1
R ± G(±
) R ∓ = F(
, −
) (1.49e)
4 4
1 1
G ± R(±
) G ∓ = F(−
,
) (1.49f)
4 4
If we separate from U the scalar matrix U f (ϑ, ϑ) = exp(iϑ) I with
exp(2iϑ) = det U, which matrix corresponds to a symmetric frac-
tional Fourier transformer F(ϑ, ϑ), the remaining matrix is a so-called
quaternion, and thus a 2 × 2 unitary matrix with unit determinant;