Intensity of waves inside a strongly disordered medium

TL;DR

This study investigates wave intensity behavior inside strongly disordered media, revealing non-exponential decay, a universal intensity distribution at mobility edges, and challenging existing diffusion theories.

Contribution

It provides new insights into wave intensity distribution in disordered media and critically evaluates current local diffusion theories of Anderson localization.

Findings

Intensity remains roughly constant in the first half of the slab

Sharp intensity drop occurs in the middle of the sample

Universal intensity distribution observed at mobility edges

Abstract

Anderson localization does not lead to an exponential decay of intensity of an incident wave with the depth inside a strongly disordered three-dimensional medium. Instead, the average intensity is roughly constant in the first half of a disordered slab, sharply drops in a narrow region in the middle of the sample, and then remains low in the second half of the sample. A universal, scale-free spatial distribution of average intensity is found at mobility edges where the intensity exhibits strong sample-to-sample fluctuations. Our numerical simulations allow us to discriminate between two competing local diffusion theories of Anderson localization and to pinpoint a deficiency of the self-consistent theory.

Equations

$$\mathbf{\psi }={\mathbf{\psi }}_0+\alpha (\omega )\left[\hat{G}(\omega )-i1\right]\mathbf{\psi },$$

$$G_{mn}(\omega )=i{\delta }_{mn}+\left(1-{\delta }_{mn}\right)\frac{\exp (ik|{\mathbf{r}}_m-{\mathbf{r}}_n|)}{k|{\mathbf{r}}_m-{\mathbf{r}}_n|}.$$

$$\mathbf{\psi }={\left(1-\alpha (\omega )\left[\hat{G}(\omega )-i1\right]\right)}^{-1}{\mathbf{\psi }}_0.$$

$$-\nabla \cdot D(\mathbf{r})\nabla ⟨I(\mathbf{r})⟩=S(\mathbf{r}),$$

$$\frac{1}{D(\mathbf{r})}=\frac{1}{D_B}+\frac{12\pi }{K^2\ell }P(\mathbf{r},\mathbf{r}),$$

$$⟨I(z)⟩=I_0-J_{\mathrm{dif}}{\int }_0^z\frac{d{z}^{\prime }}{D(z^{\prime })}\text{for }z\gg \ell ,$$

$$⟨I(z){⟩}_{\mathrm{SC}}$$

$$⟨I(z){⟩}_{\mathrm{SUSY}}$$

$$\delta ={{\left[-\frac{1}{⟨I(z)⟩}\frac{\partial }{\partial z}⟨I(z)⟩\right]}^{-1}|}_{z=L/2}.$$

$$⟨I(z){⟩}_{\mathrm{SUSY}}$$

$$⟨I(z){⟩}_{\mathrm{SC}}=\frac{I_0}{2}\{1$$

$$+\frac{3\sqrt{2}\mathrm{arcsinh}(1-2\frac{z}{L})-2(1-2\frac{z}{L})\sqrt{1-2\frac{z}{L}(1-\frac{z}{L})}}{3\sqrt{2}\mathrm{arcsinh}(1)-2}\},$$

$$⟨I(z){⟩}_{\mathrm{SUSY}}=I_0{\left(1-\frac{z}{L}\right)}^2\left(1+2\frac{z}{L}\right).$$

$$-\nabla \cdot D(\mathbf{r})\nabla P(\mathbf{r},{\mathbf{r}}^{\prime })=\delta (\mathbf{r}-{\mathbf{r}}^{\prime }),$$

$$\frac{1}{D(\mathbf{r})}=\frac{1}{D_B}+\frac{12\pi }{K^2\ell }P(\mathbf{r},\mathbf{r}),$$

$$P(\mathbf{r},\mathbf{r})=\frac{1}{2\pi }{\int }_0^{q_{\perp }^{\max }}d{q}_{\perp }{q}_{\perp }\hat{P}(q_{\perp },z=z^{\prime }),$$

$$\hat{P}(q_{\perp },z=z^{\prime })$$

$$P(\mathbf{r},{\mathbf{r}}^{\prime })\mp z_0\frac{D(z)}{D_B}\frac{\partial }{\partial z}P(\mathbf{r},{\mathbf{r}}^{\prime })=0$$

$$D(z)={\left[D_{\infty }(z{)}^{1/2}+D_{\infty }(L-z{)}^{1/2}\right]}^2,$$

$$D_{\infty }(z)=D_{\infty }(0)\exp (-2z/\xi )$$

$$D(z)={\left[D_{\infty }(z{)}^2+D_{\infty }(L-z{)}^2\right]}^{1/2}$$

$$D_{\infty }(z)=D_{\infty }(0)\frac{1}{1+z/\eta }$$

$$⟨I(z){⟩}_{\mathrm{SC}}$$

$$\hat{H}$$

$$G_{e_{jm}{e}_{n{m}^{\prime }}}$$

$$\mathcal{R}(\omega )={\left[(\omega -{\omega }_0)1+({\Gamma }_0/2)G\right]}^{-1},$$

$$P(\mathbf{r})$$

$$⟨P(z)⟩=\frac{1}{\pi R_1^2}\underset{r_{\perp }<R_1<R}{\int }⟨P(\mathbf{r}=\left\{{\mathbf{r}}_{\perp },z\right\})⟩d^2{\mathbf{r}}_{\perp }.$$

$${\Lambda }_{\pm }=i\pm \frac{\exp (i{k}_0\Delta r)}{k_0\Delta r},$$

$${\mathbf{\psi }}_{\pm }=\frac{1}{\sqrt{2}}{\left\{1,\pm 1\right\}}^T.$$

Full text

Studies of wave propagation in disordered media mainly focus on the scattering problem in which one is interested in determining a relation between incident and scattered waves outside the disordered sample and often even in the far field of it sheng06 ; akkermans07 . Transmission and reflection coefficients of disordered media have been extensively studied in this context, including their statistics and correlations akkermans07 . Scattered waves outside the medium are not only easier to measure, they are also relevant for understanding practically important quantities, such as the electrical conductance of metals dugdale95 or the whiteness of paints palmer89 , as well as for developing applications for complex material scheffold03 or biological tissue durduran10 sensing, imaging through opaque, turbid media katz14 , or cryptography goorden14 . In contrast, the spatial distribution of wave intensity inside a disordered medium has attracted much less attention even though it is important for such prospective applications of disordered materials as light harvesting in solar cells vynck12 , random lasing wiersma08 , optical frequency conversion fischer06 or photoacoustic tomography wang12 . For three-dimensional (3D) media we know that the average intensity exhibits diffusive behavior for weak disorder and hence, in the absence of absorption, decays linearly with the depth inside a disordered layer (slab) illuminated by a plane wave sheng06 ; akkermans07 . However, nothing is known at the moment about the way in which this linear behavior is modified when the disorder becomes strong enough for reaching a critical point of the Anderson localization transition (a mobility edge) and crossing it to enter the Anderson localization regime anderson58 ; lagendijk09 .

The spatial distribution of the average wave intensity $\langle I({\mathbf{r}})\rangle$ inside a strongly disordered medium of length $L$ illuminated by a monochromatic wave has been studied theoretically for a one-dimensional (1D) medium gazaryan69 ; lang73 ; abram79 ; mello16 and for a quasi-one dimensional (quasi-1D) waveguide zhao13 ; tiggelen17 ; cheng17 . In both cases, the behavior of $\langle I({\mathbf{r}})\rangle=\langle I(z)\rangle$ differs from a simple exponential decay with the distance $z$ from the sample boundary. This suggests that the exponential decay of eigenmodes in space does not directly map to the exponential decay of the average intensity. Instead, $\langle I(z)\rangle$ exhibits a step-like shape, first remaining virtually constant with $z$, then dropping sharply in a narrow region around the middle of the disordered sample $z=L/2$, and finally remaining low for $L/2<z<L$. A tendency towards such a behavior has been experimentally observed by Yamilov et al. in two-dimensional (2D) quasi-1D waveguides yamilov14 .

In this Letter we use ab initio numerical simulations of wave scattering in large 3D ensembles of point scatterers and the local diffusion theories of Anderson localization to discover two important results. First, we show that the behavior that was previously found for $\langle I(z)\rangle$ in 1D and quasi-1D samples, generalizes to 3D slabs, provided that the disorder is strong enough for reaching Anderson localization. Two competing local diffusion theories—the self-consistent (SC) theory of Anderson localization and the supersymmetric (SUSY) field theory—yield analytic expressions for $\langle I(z)\rangle$ as a function of $z/L$ that are parameterized by a single parameter $L/\xi$, where $L$ is the slab thickness and $\xi$ is the localization length. Second, we compute $\langle I(z)\rangle$ at a mobility edge, i.e. in the critical regime that does not exist in low-dimensional systems. Analytic expressions for $\langle I(z)\rangle$ following from SC and SUSY theories become scale-independent for $L$ much exceeding the mean free path $\ell$. By repeating calculations for light scattering by atoms in a strong magnetic field we demonstrate that our results are universal and hold beyond the scalar wave model. This completes the palette of behaviors expected for $\langle I(z)\rangle$ for any disorder strength, any dimensionality of space, and for both scalar and vector waves. Comparison of SC theory with numerical simulations and SUSY theory confirms its validity at the mobility edge but reveals its deficiency in the Anderson localization regime. Understanding limitations of SC theory is important in view of its applications for interpretation of 3D acoustic cobus16 ; cobus18 and cold-atom jendr12 experiments as well as of large-scale numerical simulations of light localization haberko18 .

We consider a monochromatic plane wave $\psi_{0}({\mathbf{r}})=\exp(ikz)$ incident at $z=0$ on a disordered sample (slab) confined between the planes $z=0$ and $z=L$ and having a shape of a cylinder of length (thickness) $L$, radius $R\gg L$ and volume $V=\pi R^{2}L$. We denote the frequency of the wave by $\omega$ and its wave number by $k=\omega/c$, where $c$ is the speed of the wave in the homogeneous medium by which the sample is surrounded. Our point-scatterer model assumes that the sample is simply an ensemble of $N\gg 1$ identical resonant point scatterers with a polarizability $\alpha(\omega)=-(\Gamma_{0}/2)/(\omega-\omega_{0}+i\Gamma_{0}/2)$ located at random positions $\{{\mathbf{r}}_{m}\}$, $m=1,\ldots,N$, inside the slab. The resonance width $\Gamma_{0}$ is assumed to be much smaller than the resonance frequency $\omega_{0}$ of an individual scatterer. A vector $\bm{\psi}=[\psi({\mathbf{r}}_{1}),\ldots,\psi({\mathbf{r}}_{N})]^{T}$ of wave amplitudes at scatterer positions obeys foldy45 ; lax51

[TABLE]

where $\bm{\psi}_{0}=[\psi_{0}({\mathbf{r}}_{1}),\ldots,\psi_{0}({\mathbf{r}}_{N})]^{T}$ and

[TABLE]

The solution of Eq. (1) reads

[TABLE]

We compute the average intensity $\langle I({\mathbf{r}})\rangle$ inside the sample by averaging $|\psi({\mathbf{r}}_{m})|^{2}$ over all ${\mathbf{r}}_{m}$ inside a small volume around ${\mathbf{r}}$ and over many (up to $5\times 10^{5}$) random and statistically independent scatterer configurations $\{{\mathbf{r}}_{m}\}$. In addition, $\langle I({\mathbf{r}})\rangle$ is averaged over a sufficiently large circular area of radius $R_{1}$ around the sample axis ($1/k\ll R_{1}<R$) in order to obtain $\langle I(z)\rangle$ which is independent of ${\mathbf{r}}_{\perp}=\{x,y\}$ and mimics the average intensity in a disordered slab of infinite transverse extent $R\to\infty$.

We have extensively studied Anderson localization in the model defined by Eqs. (1–3) in our previous works skip16prb ; skip18ir . In particular, we have found that spatially localized modes appear in a narrow frequency band between two density-dependent mobility edges $\omega_{c}^{\mathrm{I}}=\omega_{c}^{\mathrm{I}}(\rho/k_{0}^{3})$ and $\omega_{c}^{\mathrm{II}}=\omega_{c}^{\mathrm{II}}(\rho/k_{0}^{3})$ for scatterer number densities $\rho=N/V$ exceeding a critical value $\rho_{c}\simeq k_{0}^{3}/4\pi$, where $k_{0}=\omega_{0}/c$. We will use these previous results to study $\langle I(z)\rangle$ in the localized regime by choosing the frequency $\omega\in(\omega_{c}^{\mathrm{I}},\omega_{c}^{\mathrm{II}})$ and in the critical regime for $\omega=\omega_{c}^{\mathrm{I}}$ or $\omega=\omega_{c}^{\mathrm{II}}$.

The results of the point-scatterer model (1–3) will be compared to two competing local diffusion theories of Anderson localization tiggelen00 ; cherroret08 ; tian08 ; tian10 ; tian13 . In these theories, the average intensity of a wave $\langle I({\mathbf{r}})\rangle$ obeys a diffusion equation with a position-dependent diffusivity $D({\mathbf{r}})$:

[TABLE]

where $S({\mathbf{r}})$ describes the distribution of wave sources in the medium. In 3D, the position dependence of $D({\mathbf{r}})$ in Eq. (4) arises only for strong disorder and can be found in two different ways. First, SC theory of localization vollhardt80 ; vollhardt92 ; tiggelen00 yields $D({\mathbf{r}})$ determined self-consistently via the return probability $P({\mathbf{r}},{\mathbf{r}}^{\prime}={\mathbf{r}})$ found as a solution of Eq. (4) with $S({\mathbf{r}})=\delta({\mathbf{r}}-{\mathbf{r}}^{\prime})$ and an appropriate cut-off procedure to regularize the unphysical divergence of the solution for ${\mathbf{r}}^{\prime}={\mathbf{r}}$ tiggelen00 ; cherroret08 :

[TABLE]

where $D_{B}$ is the bare value of $D$ in the absence of localization effects and $K$ is the effective wave number in the disordered medium. In a slab, Eqs. (4) and (S2) should be solved with appropriate boundary conditions for $P({\mathbf{r}},{\mathbf{r}}^{\prime})$ sm . A second approach is based on field-theoretic, SUSY methods and has been mainly developed for 1D and quasi-1D media tian08 ; tian10 ; tian13 . It does not provide a simple microscopic expression or an equation for $D({\mathbf{r}})$ that would hold for any sample geometry, but it yields a scaling relation between $D_{\infty}(z)$ in the semi-infinite medium and $D(z)$ in a slab of finite thickness $L$ tian10 .

Interestingly enough, both SC and SUSY theories yield $D_{\infty}(z)=D(0)\exp(-2z/\xi)$ in the Anderson localization regime ($K\ell<1$) note1 , but solutions for the slab geometry differ. SC theory yields a result that for $L\gg\xi$ is well described by an interpolation formula $D_{\mathrm{SC}}(z)=[D_{\infty}(z)^{1/2}+D_{\infty}(L-z)^{1/2}]^{2}$ sm . The SUSY approach yields a different result: $D_{\mathrm{SUSY}}(z)=D_{\infty}[z(L-z)/L]$ tian10 . In both cases, flux conservation implies that the diffusive flux $J_{\mathrm{dif}}$ given by the Fick’s law $J_{\mathrm{dif}}=-D(z)\partial\langle I(z)\rangle/\partial z$ is independent of $z$ for $z\gg\ell$. Integrating the Fick’s law yields

[TABLE]

where the precise value of $I_{0}$ depends on the details of conversion of the incident plane wave into diffuse radiation near the sample surface $z=0$. Supplemented with a boundary condition $\langle I(L)\rangle=0$ akkermans07 , Eq. (6) yields

[TABLE]

where $\mathrm{erf}(x)$ is the error function.

The point-scatterer model and SC and SUSY theories yield consistent results for the distribution of the average intensity $\langle I(z)\rangle$ inside the disordered slab in the localized regime. As we see from Fig. 1, $\langle I(z)\rangle$ does not decay exponentially with $z$ as one could expect from naive considerations, but instead exhibits a rapid drop near the middle of the slab, while varying much slower near its boundaries. Such a behavior is similar to that found previously in 1D gazaryan69 ; lang73 ; abram79 ; mello16 ; zhao13 and quasi-1D tiggelen17 ; cheng17 media.

Even though both SC and SUSY theories provide good and, in fact, hardly distinguishable fits to the numerical data, only SUSY model consistently yields the same (within error bars) best-fit values of $\xi$ for different $L$ as we show in the inset of Fig. 1. The underlying problem of SC model is best demonstrated by computing the width $\delta$ of the spatial region in which the average intensity changes rapidly near the middle of the sample:

[TABLE]

We find $\delta_{\mathrm{SC}}=\xi$ and $\delta_{\mathrm{SUSY}}=\sqrt{(\pi/8)L\xi}$ for $L\gg\xi$, which predict different scalings of $\delta$ with $L$. The need for different values of $\xi$ to fit the numerical data corresponding to different $L$ with SC theory signals that the scaling that it predicts for $\delta$ is wrong. In contrast, SUSY theory yields the correct scaling for $\delta$ and describes the data in Fig. 1 with a single value of $\xi$ for all $L$.

An interesting regime that is not accessible in low-dimensional systems is the critical one. In order to study it in the framework of the point-scatterer model (1), we choose the frequency of the wave $\omega$ exactly at one of the the mobility edges $\omega_{c}^{\mathrm{I,II}}$ determined in Ref. skip18ir . The resulting spatial distributions of $\langle I(z)\rangle$ are shown in Fig. 2 by symbols. To study the critical regime using the local diffusion theories, we note that for a semi-infinite medium ($L\to\infty$) one finds $D(z)=D_{\infty}(z)\simeq D(0)/(1+z/\eta)$ tiggelen00 with a decay length $\eta\sim\ell$. For a slab of finite thickness $L$, the results of SC theory may be nicely interpolated by $D(z)=[D_{\infty}(z)^{2}+D_{\infty}(L-z)^{2}]^{1/2}$ sm , whereas another option is to extrapolate the relation $D_{\mathrm{SUSY}}(z)=D_{\infty}[z(L-z)/L]$ tian10 ; tian13 to the mobility edge. Proceeding in the same way as for deriving Eqs. (7) and (8), we obtain expressions for $\langle I(z)\rangle$ that depend on $z/L$ and $L/\eta$. The full expression following from SC theory is quite cumbersome and we reproduce it elsewhere sm whereas the SUSY result is simpler:

[TABLE]

Comparison of these results with numerical simulations of the model (1) is shown in Fig. 2. The agreement is less striking than in the localized regime but it improves when $L$ increases. In addition, variations of the best-fit $\eta$ with $L$ are similar for SC and SUSY theories but differ from SC theory expectations sm . Universal, parameter-free intensity distributions follow in the limit of $L\gg\eta$:

[TABLE]

These two expressions are very close when plotted as functions of $z/L$. Their lack of any characteristic length scale can be seen as a consequence of the fractal character of critical eigenmodes evers08 . By analogy with other models of Anderson localization evers08 ; rodriguez09 , we expect the critical eigenmodes of our model defined by Eqs. (1–3) to be multifractal, but evidencing this would require analysis of higher-order statistical moments and spatial correlations of intensity in addition to the study of its average value.

Comparison of results corresponding to the two mobility edges $\omega=\omega_{c}^{\mathrm{I}}$ and $\omega=\omega_{c}^{\mathrm{II}}$ suggests that the behaviors of our point-scatterer model at these frequencies are quite different. First, the mean free path $\ell$ can be estimated as a position of the maximum of $\langle I(z)\rangle$ in Figs. 2(a) or (b) and turns out to be considerably larger at the second, high-frequency mobility edge. As a consequence, the results presented in Fig. 2(b) correspond to shorter optical thicknesses $L/\ell$ than the data in Fig. 2(a). Second, the sample-to-sample fluctuations of intensity at the second mobility edge are much stronger than at the first one. This is illustrated in Fig. 3(a) where we show the relative intensity fluctuation $\sigma=\sqrt{\langle\delta I({\mathbf{r}})^{2}\rangle}/\langle I(z)\rangle$, where $\delta I({\mathbf{r}})=I({\mathbf{r}})-\langle I(z)\rangle$, as a function of $z$ for $k_{0}L=12$. We attribute this difference to subradiant states localized on pairs of closely located scatterers and surviving multiple scattering only for small interatomic distances and hence large frequency shifts $(\omega-\omega_{0})/\Gamma_{0}$ skip16prb . Figure 3(b) shows the inverse participation ratio $\mathrm{IPR}_{n}=\sum_{m}|\psi_{n}({\mathbf{r}}_{m})|^{4}$ of eigenvectors $\bm{\psi}_{n}$ of the matrix ${\hat{G}}(\omega_{0})$ (quasimodes) as a function of their frequencies $\omega$ and decay rates $\Gamma$. Note that whereas the low-frequency mobility edge I defines a sharp transition between extended states for $\omega<\omega_{c}^{\mathrm{I}}$ (low IPR, light grey points) and localized states for $\omega_{c}^{\mathrm{I}}<\omega<\omega_{c}^{\mathrm{II}}$ (high IPR, dark grey and black points), there are localized states on both sides from the mobility edge II. However, the physical origin of quasimode localization is different for $\omega_{c}^{\mathrm{I}}<\omega<\omega_{c}^{\mathrm{II}}$ (localization due to strong scattering appearing only for $N\gg 1$ and $\rho>\rho_{c}$) and $\omega>\omega_{c}^{\mathrm{II}}$ (localization that exist for any $N\geq 2$ and any $\rho$). This difference is manifest in the scaling properties of quasimode properties with sample size skip16prb . Its link with existence of two-atom subradiant states is further confirmed by a more detailed analysis sm .

In conclusion, we have found analytic formulas for the spatial distribution of average wave intensity $\langle I(z)\rangle$ inside a thick 3D slab of strongly disordered medium illuminated by a monochromatic plane wave. In the Anderson localization regime, $\langle I(z)\rangle$ exhibits a step-like shape and drops sharply within a region of width $\delta\sim\sqrt{L\xi}$ in the middle of the sample. At a mobility edge, $\langle I(z)\rangle$ takes a universal, parameter-free shape as a function of $z/L$. Comparison of ab initio numerical simulations with local diffusion theories allowed us to reveal a deficiency of SC theory for description of Anderson localization in 3D. A realistic physical system in which Anderson localization of light can be observed is a large ensemble of cold atoms in a strong magnetic field skip15 ; skip18 . Repeating all the calculations presented above for this system yields very similar results sm . In a cloud of two-level cold atoms, intensity of light is proportional to the population of the excited state and therefore its spatial distribution can be imaged by the so-called diffraction-contrast imaging turner05 allowing for state-selective imaging of atoms sheludko08 , by monitoring a slow spontaneous decay of the excited state to a third, auxiliary level, or by probing the excited level by a weak probe beam resonant with a transition to a higher-energy state. In a dielectric disordered system, spatial distribution of optical intensity can be imaged by optoacoustic methods karabutov99 .

Numerical calculations of the spatial distributions of average intensity and of the transmission coefficients were carried out with the financial support of the Russian Science Foundation (Project No. 17-12-01085). IMS acknowledges the hospitality of the LPMMC where a part of this work has been performed with a financial support of the Centre de Physique Théorique de Grenoble-Alpes (CPTGA).