# Statistical Dispersion Measure

(Redirected from Statistical Variability)

## References

### 2016

• (Shan et al., 2016) ⇒ Shan, M., Nastasa, V., & Popescu, G. (2016). “Statistical dispersion relation for spatially broadband fields". Optics letters, 41(11), 2490-2492. DOI:10.1364/OL.41.002490 [1]
• Let us consider first the Helmholtz equation:
$\displaystyle{ \nabla^2 U(\mathbf{r},\omega)+n^2\beta_0^2U(\mathbf{r},\omega)=0\quad\quad(1) }$
where $\displaystyle{ U }$ is the field in a medium, $\displaystyle{ n }$ is the refractive index of the medium, and $\displaystyle{ \beta_0 }$ is the wavenumber in vacuum, $\displaystyle{ \beta_0 =\omega/c }$. Note that, if the medium is homogeneous, i.e., $\displaystyle{ n }$ is independent of $\displaystyle{ \mathbf{r} }$ (...)
Finally, we obtain the statistical dispersion relation for a field in weakly scattering medium, namely,
$\displaystyle{ \langle \kappa^2 \rangle =n^2_0\beta_0^2 \left(1+\frac{\sigma^2_n}{n_0^2}\right)\quad\quad (13) }$
Equation (13) represents the main result of this Letter. It establishes the relationship between the second-order moment of the k-vector, $\displaystyle{ \langle \kappa^2\rangle=\langle \kappa_x^2\rangle\langle \kappa_y^2\rangle\langle \kappa_z^2\rangle }$ , and the statistics of the refractive index fluctuations. Clearly, when $\displaystyle{ \sigma_n \rightarrow 0 }$, we recover the homogeneous dispersion relation, $\displaystyle{ \langle \kappa^2\rangle=n_0^2\beta_0^2 }$.

### 2016

where $\displaystyle{ \bar{u} }$ is the average of {$\displaystyle{ u_i }$}.