Contrast

For a fully developed speckle pattern which formed when the complete coherent and polarized light illuminate a static medium, the contrast (K) range from 0 to 1 is defined by the ratio between the standard deviation and mean intensity:^[6][7]

${\displaystyle K={\frac {\sigma }{\langle I\rangle }}}$

The intensity distribution of the speckle pattern will be used to compute the contrast value.

Autocorrelation functions

Autocorrelation functions of electric field are used to measure the relationship between contrast and the motion of scatterers because the intensity fluctuations are produced by electric field changes of scatterers. E(t) is the electric field over time, E* is the complex conjugate of electric field and ${\displaystyle \tau }$ is the autocorrelation delay time.^[7][6]

${\displaystyle g_{1}(\tau )={\frac {\langle E(t)\cdot E^{*}(t+\tau )\rangle }{\langle E(t)\cdot E^{*}(t)\rangle }}}$

Bandyopadhyay et al. showed that the reduced intensity variances of speckle pattern are related to ${\displaystyle g_{1}}$. Therefore, the contrast can be written as^[7][8][9]

${\displaystyle K(T)^{2}={\frac {2\beta }{T}}\int \limits _{0}^{T}|g_{1}(\tau )|^{2}\left(1-{\frac {\tau }{T}}\right)\mathrm {d} \tau }$

where T is the exposure time. The normalization constant ${\displaystyle \beta }$ takes into account the loss of correlation due to the detector pixel size, and depolarization of the light through the medium.

Motion Distributions

Dynamic scatterers' motion can be classified into two categories, one is the ordered motion and the other one is disordered motion. The ordered motion is the ordered flow of scattered while the disordered motion is caused by the temperature effects. The total dynamic scatterers' motions were thought of as Brownian motion historically, the approximate velocity distribution of Brownian motion can be considered as the Lorentzian profile. However, the ordered motion in dynamic scatterers follows Gaussian distribution. When considering the motion distribution, the contrast equation related to the autocorrelation can be updated. The updated equations are as follows, ${\displaystyle K^{L}}$ is the contrast equation function in Lorentzian profile and ${\displaystyle K^{G}}$ is the contrast equation function in Gaussian profile. ${\displaystyle \tau }$ is the decorrelation time. Both equations can be used in contrast measurement, some scientists also use contrast equations with the combination of them. However, what the correct theoretical contrast equation should be is still under investigation.^[7][10][11]

${\displaystyle K^{L}(T)^{2}=\beta {\begin{bmatrix}{\frac {\tau _{cl}}{T}}+{\frac {(\tau _{cl})^{2}}{2T^{2}}}(e^{-2T/\tau _{cl}}-1)\end{bmatrix}}}$

${\displaystyle K^{G}(T)^{2}=\beta {\begin{bmatrix}{\frac {\tau _{cg}}{T}}{\sqrt {\frac {\pi }{2}}}\mathrm {erf} ({\sqrt {2}}T/\tau _{cg})+{\frac {\tau _{cg}^{2}}{2T^{2}}}(e^{-2T^{2}/\tau _{cg}^{2}}-1)\end{bmatrix}}}$

Normalization constants

${\displaystyle \beta }$ is the normalization constants that vary in different LSCI systems, the value of it is ${\displaystyle \leq }$ 1, the most common method to determine the value of it is using the following equation. ${\displaystyle \beta }$ is account for the instability and maximum contrast of each LSCI system.

${\displaystyle \beta =\lim _{T\to 0}K(T)}$

Effect of static scatterers

Static scatterers are present in the assessed sample, speckle contrast produced by static scatterers remains constant. By adding statics scatterers, the contrast equation can be updated again.

${\displaystyle K(T)^{2}={\frac {2\beta }{T}}\int \limits _{0}^{T}P_{1}^{2}|g_{1}(\tau )|^{2}\left(1-{\frac {\tau }{T}}\right)P_{2}^{2}\mathrm {d} \tau }$

*The above equation did not account for the motion distributions.

P1 and P2 are two constants that range from 0 to 1, they are determined by fitting this equation to the actual experimental data.^[7]

Scatterers velocity determination

The relationship between the velocity of scatterers and decorrelation time is as follows, velocity of scatterers such as the blood flow is proportional to the decorrelation time, ${\displaystyle \lambda }$ is the laser light wavelength.^[7][11]

${\displaystyle V={\frac {\lambda }{2\pi \tau _{c}}}}$