It's pretty simple. The physics is that (in the 1D approximation), in order to move the edge of the depletion zone from z to z+dz, you have to create a sheet of charge
d sigma = rho dz
and the capacitance change is
dC = A epsilon(1/(z+dz) - 1/z) ~ -A epsilon/z**2 dz. Here epsilon is the dielectric constant of fully depleted silicon.
The E field goes as the integral of the sheet charge elements
E = 1/epsilon integral (0 to z) rho(z) dz
and V is the line integral of the E field.
V = integral (0 to z) E(z) dz
So you get z from the capacitance, and then rho from the second derivative of V with respect to z. Haven't got time to do the derivation properly, but that's more or less how it goes.
There are fine points having to do with where you take the origin of voltage (probably the contact potential of the junction) and Debye shielding, so some care is needed.
What I found was that a lot of my favourite high-speed and high-linearity PDs have a buried layer of higher doping, like an APD.
Boonton 72BDs are all the go.
Cheers
Phil Hobbs