Statistical Dispersion Measure

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A Statistical Dispersion Measure is variability measure of a probability distribution.



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


where [math]\displaystyle{ \bar{u} }[/math] is the average of {[math]\displaystyle{ u_i }[/math]}.