Statistical Dispersion Measure

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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:
$\nabla^2 U(\mathbf{r},\omega)+n^2\beta_0^2U(\mathbf{r},\omega)=0\quad\quad(1)$
where $U$ is the field in a medium, $n$ is the refractive index of the medium, and $\beta_0$ is the wavenumber in vacuum, $\beta_0 =\omega/c$. Note that, if the medium is homogeneous, i.e., $n$ is independent of $\mathbf{r}$ (...)
Finally, we obtain the statistical dispersion relation for a field in weakly scattering medium, namely,
$\langle \kappa^2 \rangle =n^2_0\beta_0^2 \left(1+\frac{\sigma^2_n}{n_0^2}\right)\quad\quad (13)$
Equation (13) represents the main result of this Letter. It establishes the relationship between the second-order moment of the k-vector, $\langle \kappa^2\rangle=\langle \kappa_x^2\rangle\langle \kappa_y^2\rangle\langle \kappa_z^2\rangle$ , and the statistics of the refractive index fluctuations. Clearly, when $\sigma_n \rightarrow 0$, we recover the homogeneous dispersion relation, $\langle \kappa^2\rangle=n_0^2\beta_0^2$.

2016

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