Page 95 - Phase Space Optics Fundamentals and Applications

76 Chapter Three

which, in particular, yields the energy conservation law

∞ ∞
2 U 2
| f (r i ) | dr i = |R [ f (r i )](r o )| dr o (3.44)
−∞ −∞
We also remind (see Sec. 1.6) that the Wigner distribution is rotated
in phase space under the RCT
U
W f (r, p) = W R [ f ] (Xr + Yp, −Yr + Xp) (3.45)
Moreover its projection corresponds to the squared modulus of the ap-
propriated RCT, which can be registered experimentally. These prop-
erties are crucial for phase-space tomography, which permits one to
reconstruct the Wigner distribution from its projections. The details
of this method for the case of the fractional FT are clarified in Chap. 4.

3.4.2 Similarity to the Fractional
Fourier Transform
It has been shown 36,37 that any unitary matrix U s is similar to one
U f associated with the fractional FT. Indeed, the unitary matrix has
unimodular eigenvalues and can be diagonalized. The diagonal uni-
tary matrix corresponds to the fractional FT, Eq. (3.15). Moreover, the
matrix that diagonalizes the matrix is also unitary, and therefore we
can write
U s = UU f ( x , y )U −1 (3.46)

where x and y and the matrix U are defined from the eigenvalues
and eigenvectors of U s , correspondingly. Then we can conclude that
any phase-space rotator is similar to the fractional FT. For example, the
signal rotator and gyrator are similar to the antisymmetric fractional
FT because

U r ( ) = U g U f ( , − ) U g −
4 4

U g ( ) = U r − U f ( , − ) U r (3.47)
4 4
Note that due to the symmetry of the phase-space rotator matrices,
such as U g ( ± /4) =−U g (± /4), there exist various similarity
relations (see Sec. 1.6.2). For example, we can also write for the signal
rotator

U r (± ) = U g ± U f ( , − ) U g ∓
4 4

= U g ± U f ( , − ) U g ∓ (3.48)
4 4