The frequency of light scattered by particles undergoing electrophoresis is shifted by the amount of the Doppler effect, \upsilon_D\, from that of the incident light, :\upsilon\, . 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, g(\tau) \,, can be approximately described by the following damped cosine function [7]. : g(\tau)=A+B \exp(-\Gamma\tau)\cos(2\pi\upsilon_o)+C \exp(-2\Gamma \tau)\,\qquad (1) where \Gamma\, is a decay constant and A, B, and C are positive constants dependent on the optical system. Damping frequency \upsilon_o\, is an observed frequency, and is the frequency difference between scattered and reference light. : \upsilon_o =| \upsilon_s - \upsilon_r | = | (\upsilon_i+\upsilon_D)-(\upsilon_i+\upsilon_M ) |\qquad(2) where \upsilon_s\, is the frequency of scattered light, \upsilon_r\, the frequency of the reference light, \upsilon_i\, the frequency of incident light (laser light), and \upsilon_M\, the modulation frequency. The power spectrum of mixed light, namely the Fourier transform of g(\tau) \,, gives a couple of Lorenz functions at \pm\Delta\upsilon \, having a half-width of \Gamma/2\pi\, at the half maximum. In addition to these two, the last term in equation (1) gives another Lorenz function at \upsilon = 0\, 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. : \upsilon_D = \frac{Vq}{2\pi} \qquad(3) and :\Gamma = D|q|^2 \qquad(4) where \ V \, is velocity of the particles, \ q \, is the amplitude of the scattering vector, and \ D \, is the
translational diffusion constant of particles. The amplitude of the scattering vector \ q \, is given by the equation : \ |q| = \frac{4 \pi n}{\lambda_0 }\sin\left( \frac{\theta}{2}\right) \qquad(5) Since velocity \ V \, is proportional to the applied electric field, \ E \,, the apparent electrophoretic mobility \ \mu_{obs} \, is define by the equation : \ \vec{V} = \mu_{obs} \vec{E} \qquad(6) 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 : \upsilon_D = \mu_{obs} \frac{n E}{\lambda_0} \sin \theta \qquad(7) where \ E \, is the strength of the electric field, \ n\, the
refractive index of the medium, \ \lambda_0 \,, the wavelength of the incident light in vacuum, and \ \theta \, the scattering angle. The sign of \ v_D \, 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 \ | \upsilon_M | > | \upsilon _D | \,, Eq. (2) is modified and expressed as : \upsilon_p = \upsilon_o = \pm( \upsilon _D -| \upsilon _M | ) \qquad(8) The modulation frequency \upsilon _M \, 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, \ d_H\, . : \ d_H = \frac{k_BT}{3 \pi \eta D} \qquad(10) where \ k_B \, is
Boltzmann coefficient, \ T \, is the absolute temperature, and \ \eta \, the
dynamic viscosity of the surrounding fluid. ==Profile of electro-osmotic flow==