Time delay in 1D disordered media with high transmission

Abstract

We study the time delay of reflected and transmitted waves in 1D disordered media with high transmission. Highly transparent and translucent random media are found in nature or can be synthetically produced. We perform numerical simulations of microwaves propagating in disordered waveguides to show that reflection amplitudes are described by complex Gaussian random variables with the remarkable consequence that the time-delay statistics in reflection of 1D disordered media are described as in random media in the diffusive regime. For transmitted waves, we show numerically that the time delay is an additive quantity and its fluctuations thus follow a Gaussian distribution. Ultimately, the distributions of the time delay in reflection and transmission are physical illustrations of the central limit theorem at work.

Graphic abstract

Appendix A Deviations of the reflection amplitudes from the Gaussian circular ensemble

Fig. 9

Average of the real and imaginary parts of the reflection at two different frequencies ($\omega _1=50.262$ rad ns$^{-1}$ and $\omega _2=50.268$ rad ns$^{-1}$) as a function of the ratio $L/\ell$. Blue and orange dots correspond to $\ell =57634$ cm and 214082 cm, respectively

Fig. 10

Chi-square test values as a function of the ratio $L/\ell$ to test the Gaussian distribution of the real and imaginary parts of the reflection at two frequency values. Blue and orange dots correspond to $\ell = 57634$ cm and 214082 cm, respectively. The dashed line indicate the critical value $\chi ^2_c =31.41$ which corresponds to the level of significance $\alpha$ = 0.05 with 20 degrees of freedom

In the main text we have assumed that the reflection amplitudes are complex Gaussian random variables in the ballistic regime. Here we quantify the range of validity of this assumption as a function of the ratio $L/\ell$. We perform chi-square goodness of fit tests to determine whether the real and imaginary parts of the reflection amplitudes follow a Gaussian distribution and the distribution of $\tau _r$ is described by Eq. (3), in the main text.

First, we have found that the expectation values that define the circular symmetry, $E[\textrm{Re}(r_1)]=E[\textrm{Im}(r_1)]=E[\textrm{Re}(r_1 r_2)]=E[\textrm{Im}(r_1 r_2)]=0$, of the reflections are preserved even for values of $L \sim \ell$, as it can be seen in Fig. 9.

However, the chi-square goodness of fit reveals that the reflection amplitudes are no longer described by a Gaussian distribution as L approaches $\ell$. In Fig. 10, we show the values of $\chi ^2$ of the real and imaginary parts of the reflection at two different frequencies and two values of the mean free path as function of the system length. The $\chi ^2$ values are obtained from 1000 samples and the data is grouped into 21 histogram classes. We obtained 1000 values of $\chi ^2$ for each value of $L/\ell$. We thus plot the average value $\langle \chi ^2 \rangle$. The horizontal dashed lines in Fig. 10 indicate the $\chi ^2$ critical value with 20 degrees of freedom and level of significance $\alpha$ = 0.05. As it can be seen, the $\chi ^2$ values are within the acceptance level up to $L/\ell \sim 10^{-1}$.

It is thus expected that for systems with $L \sim \ell$, the distribution of the time delay in reflection would not be well described by Eq. (3) in the main text. Indeed, we have performed chi-square tests for the distribution of $\tau _r$. In Fig. 11, the values of the chi-square goodness of fit indicate that the numerical distributions of $\tau _r$ are well described by Eq. (3) in the main text up to $L/\ell \sim 10^{-1}$.

Fig. 11

Chi-square test values as a function of the ratio $L/\ell$ to test the agreement of the numerical and the expected (Eq. (3)) distribution of $\tau _r$. Blue and orange dots correspond to $\ell =57634$ cm and 214082 cm, respectively. The dashed line indicates the critical value $\chi ^2_c =31.41$ which corresponds to the level of significance $\alpha$ = 0.05 with 20 degrees of freedom