Editing Time–frequency analysis (section)

===Electromagnetic wave propagation===

We can represent an electromagnetic wave in the form of a 2 by 1 matrix

: <math>\begin{bmatrix}
  x  \\
  y
\end{bmatrix},</math>

which is similar to the time–frequency plane. When electromagnetic wave propagates through free-space, the [[Fresnel diffraction]] occurs. We can operate with the 2 by 1 matrix

: <math>\begin{bmatrix}
  x  \\
  y
\end{bmatrix}</math>

by [[Linear canonical transformation#Electromagnetic wave propagation|LCT]] with parameter matrix

: <math>\begin{bmatrix}
    a & b \\
    c & d
  \end{bmatrix} =
  \begin{bmatrix}
    1 & \lambda z \\
    0 & 1
  \end{bmatrix},
</math>

where ''z'' is the propagation distance and <math>\lambda </math> is the wavelength. When electromagnetic wave pass through a spherical lens or be reflected by a disk, the parameter matrix should be

: <math>\begin{bmatrix}
    a & b \\
    c & d
  \end{bmatrix} =
  \begin{bmatrix}
    1 & 0 \\
    -\frac{1}{\lambda f} & 1
  \end{bmatrix}
</math>

and

: <math>\begin{bmatrix}
    a & b \\
    c & d
  \end{bmatrix} =
  \begin{bmatrix}
    1 & 0 \\
    \frac{1}{\lambda R} & 1
  \end{bmatrix}
</math>

respectively, where ƒ is the focal length of the lens and ''R'' is the radius of the disk. These corresponding results can be obtained from

: <math>\begin{bmatrix}
  a & b \\
  c & d
\end{bmatrix}
\begin{bmatrix}
  x \\
  y
\end{bmatrix}.
</math>