In order to reconstruct the electron number density by the measured phase difference data
of the microwave, I tried several algorithms. The first algorithm is the simple factor
multiplication to the electron number density derived from the equation (1) and the
simulation result. If the phase difference result in a position is twice of the measured
phase difference, we can assume the electron number density corresponding to the
position is a half of the previous assumed one. However, because the range in which the
electron in a point have a influence on the phase difference is not a point as the equation
(1) indicates, the iteration time to find the right electron number density profile can be too
long.
So, the second method is to use the spatial transfer function between a point of electron
integrated number density and the phase difference distribution in xz plane. It seems like
radially propagating wave from a center. Using the convolution of the transfer function
with the line integrated electron density profile, the phase difference distribution can be
achieved without the FDTD simulation. And inversely, using the deconvolution of
measured phase difference with the transfer function, I can find the real 3-D electron
number density profile. However, in complicate electron distribution case, the transfer
functions in other positions are not same because electric fields in these positions that are
changed by the electrons influence each other and they are convoluted very well. So it
also takes too much iteration to obtain the precise electron profile corresponding to the
measured phase difference.
The last algorithm I suggest is to use several variables that determine the radially
symmetric electron profile. I can set the height, width, unsymmetrical factor and the
position of the electron profile in a z position as variables. So, there are hundreds
variables in the electron distribution. The optimization to find the best electron profile
requires a lot of time.