Page 85 - Phase Space Optics Fundamentals and Applications

66 Chapter Three

is given by
t t
T −1 = D t −B t (3.9)
−C A

3.2.2 Modified Iwasawa Decomposition
of Ray Transformation Matrix
Any proper normalized symplectic ray transformation matrix can be
decomposed in the modified Iwasawa form as 20−22

A B I 0 S 0 X Y
T = = −1 = T L T S T O (3.10)
C D −G I 0 S −Y X
in which the first matrix represents a lens transform described by the
symmetric matrix

t −1
t
t
t
G =−(CA + DB )(AA + BB ) = G t (3.11)
The second matrix corresponds to a scaler described by the positive
definite symmetric matrix
t 1/2
t
S = (AA + BB ) = S t (3.12)
and the third matrix, T O , is orthogonal 21,22 and due its symmetry can
be shortly represented by the unitary 2 × 2 matrix

t −1/2
t
U = X + iY = (AA + BB ) (A + iB) (3.13)
TheactionoftheCTsdescribedbythefirsttwomatricesisobvious.The
lens transform produces the second-order polynomial phase modu-
lation of the signal, and the scaler is responsible for the magnification
of the signal. Therefore the most significant signal transformations
are related to the last orthosymplectic matrix T O . They correspond
to phase-space rotators and will be denoted as rotational canonical
(integral) transforms (RCTs). The phase-space rotators include the sig-
nal rotator, separable fractional FT, and gyrator among others.
The signal rotator [ray transformation matrix T r ( )] can be ex-
pressed by the unitary matrix, Eq. (3.13),

cos sin
U r ( )= (3.14)
− sin cos
associated with the clockwise rotation in the xy and p x p y planes at
angle .