The frequency of light scattered by particles undergoing electrophoresis is shifted by the amount of the Doppler effect,   from that of the incident light, : .
The shift can be detected by means of heterodyne optics in which the scattering light is mixed with the reference light.
The autocorrelation function of intensity of the mixed light, , can be approximately described by the following damped cosine function [7].
where     is a decay constant and A, B, and C are positive constants dependent on the optical system.
Damping frequency   is an observed frequency, and is the frequency difference between scattered and reference light.
where    is the frequency of scattered light,   the frequency of the reference light,
  the frequency of incident light (laser light),
and   the modulation frequency.
The power spectrum of mixed light, namely the Fourier transform of , gives a couple of Lorenz functions at   having a half-width of   at the half maximum.
In addition to these two, the last term in equation (1) gives another Lorenz function at
The Doppler shift of frequency and the decay constant are dependent on the geometry of the optical system and are expressed respectively by the equations.
and
where    is velocity of the particles,   is the amplitude of the scattering vector, and  is the translational diffusion constant of particles.
The amplitude of the scattering vector   is given by the equation
Since velocity       is proportional to the applied electric field, , the apparent electrophoretic mobility   is define by the equation
Finally, the relation between the Doppler shift frequency and mobility is given for the case of the optical configuration of Fig. 3 by the equation
where     is the strength of the electric field,  the refractive index of the medium, , the wavelength of the incident light in vacuum, and     the scattering angle.
The sign of   is a result of vector calculation and depends on the geometry of the optics.
The spectral frequency can be obtained according to Eq. (2).
When , Eq. (2) is modified and expressed as
The modulation frequency   can be obtained as the damping frequency without an electric field applied.
The particle diameter is obtained by assuming that the particle is spherical. This is called the hydrodynamic diameter,     .
where    is Boltzmann coefficient,   is the absolute temperature, and   the dynamic viscosity of the surrounding fluid.