Page 99 - Phase Space Optics Fundamentals and Applications

80 Chapter Three

It includes a rotation, scaling of the coordinates described by the
−1
matrix X , and phase modulation associated with the matrix
−1
product −YX .
Phase-space rotation of the Dirac function leads to the generalized
chirp function, which corresponds to the point-spread function of the
related first-order optical system.

U
R [ (r i − v)](r o )
1 t −1 t −1 t −1
= √ exp i v Y Xv − 2v Y r o + r XY r o (3.65)
o
det iY
Correspondingly applying the inverse transform parameterized by
the matrix U −1 to this chirp function, we obtain (r i − v). Therefore
the phase-space rotations can be used for the localization of certain
chirp signals, as will be discussed further.

3.5 Eigenfunctions for Phase-Space
Rotators
3.5.1 Some Relations for the Eigenfunctions

It is known that the Hermite-Gaussian (HG) functions H m,n (r) =
√
2
n
H m (x) H n (y), where H n (x) = 2 1/4 (2 n!) −1/2 H n ( 2 x) exp(− x )
and H n (·) denotes the Hermite polynomials, are eigenfunctions for
the separable fractional FT for any angles x and y with eigenvalues
1
1
exp[−i(m + ) x − i(n + ) y ] (see, for example, Ref. 4).
2 2
To find the eigenfunctions for the RCT parameterized by the
unitary matrix U s , first we perform the similarity decomposition
−1
U s = UU f ( x , y )U , where x and y and the matrix U are de-
fined from the eigenvalues and eigenvectors of U s correspondingly.
Then it is clear that 39 the functions obtained from H m,n (r i ) by the
U U
RCT parameterized by U: H m,n (r) = R [H m,n (r i )](r) are eigenfunc-
tions for the RCT described by the matrix U s with eigenvalues
1
1
exp[−i(m + ) x − i(n + ) y ].
2 2
U
There are various names for H m,n (r) modes: two-dimensional
Hermite-Gaussian functions, 40 Hermite-Laguerre Gaussian func-
tions, 35 and orthosymplectic modes. 41 Here we will use the last one
U
since H m,n (r) is an eigenfunction for the RCT parameterized by the
orthosymplectic ray transformation matrix associated with U s .
U
As well as the HG functions, the modes H m,n (r) for the same U
and different indices m, n ∈ [0, ∞) form a complete orthonormal
set, and therefore, any function can be represented as their linear
superposition.