# Anderson localization of vector waves

S.E. Skipetrov

Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France

https://lpmmc.cnrs.fr/skipetrov

## Motivation

Experiments on 3D Anderson localization of classical waves are performed with vector waves:

* **Microwaves & Light**
  Wiersma, Lagendijk et al. (1997)
  Chabanov, Genack et al. (2000)
  Störzer, Sperling,
  Maret et al. (2006, 2013, 2016)

<figure><img src="image_1.png" /></figure>

* **Elastic waves (sound in solids)**
  Hu, Hildebrand, Cobus,
  Page et al. (2008-)

<figure><img src="image_2.png" /></figure>

## Point scatterer model

$N \gg 1$ scatterers
at density $\rho$

Wave field
on the scatterers

<figure><img src="image_3.png" /></figure>

Resonant point scatterers
= "atoms"

<figure><img src="image_4.png" /></figure>

"Polarizability":

$$ \alpha(\omega) = -(\Gamma_0/2)/(\omega - \omega_0 + i\Gamma_0/2) $$

## Scalar Green's matrix

$$ G_{jn}(\omega) = i\delta_{jn} + (1 - \delta_{jn}) \frac{\exp(ik|r_n - r_j|)}{k|r_n - r_j|} $$

$$ \hat{G}\psi_n = \Lambda_n \psi_n $$

<figure><img src="image_5.png" /></figure>

## Thouless conductance

<figure><img src="image_6.png" /></figure>

Thouless parameter
dimensionless conductance
$\delta\omega \sim \text{Im}\Lambda - \text{"mode width"}$

Scaling theory:

**Extended modes:**
$g$ grows with $N$

**Localized modes:**
$g$ decreases with $N$

## Finite-size scaling

$$ \text{Percentile } g_q: \quad q = \int_0^{g_q} p(g)dg $$

$$ \ln g_q = F_q(R/\xi) $$

$$ \xi \propto (\text{Re}\Lambda - \text{Re}\Lambda_c)^{-\nu} $$

<figure><img src="image_7.png" /></figure>

## No Anderson localization for light

<figure><img src="image_8.png" /></figure>

Green's matrix $G$ describes
propagation of light
between pairs of scatterers
$r_{jn} = r_n - r_j$

$$ G_{jn}^{\mu\nu} = i\delta_{jn}\delta_{\mu\nu} + (1 - \delta_{jn}) \frac{3 e^{ik_0 r_{jn}}}{2 k_0 r_{jn}} \left[ P(ik_0 r_{jn})\delta_{\mu\nu} + Q(ik_0 r_{jn}) \frac{r_{jn}^{\mu} r_{jn}^{\nu}}{r_{jn}^2} \right] $$

$\mu, \nu = x, y, z$

$$ P(x) = 1 - 1/x + 1/x^2, \quad Q(x) = -1 + 3/x - 3/x^2 $$

<figure><img src="image_9.png" /></figure>

$g_q$ always increases with $N$ at a constant density $\rho$

<figure><img src="image_10.png" /></figure>

## Light in a magnetic field

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Phase diagram</figcaption></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

The value of critical exponent following from the fits is close to $\nu \simeq 1.57$ expected for the 3D orthogonal symmetry class.

We conclude that the observed transition is likely to belong to the same symmetry class as the **Anderson transition in a system of spinless electrons**.

<figure><img src="image_17.png" /><figcaption>Phase diagram</figcaption></figure>

## Green's function in the near field

Scalar wave:
$$ G_{\text{scalar}} = \frac{e^{ik_0 r}}{k_0 r} $$

<figure><img src="image_18.png" /></figure>

Light:

$$ \hat{G}_{\text{vector}} = \frac{3 e^{ik_0 r}}{2 k_0 r} \left[ \underbrace{\left(1 - \frac{\mathbf{r} \otimes \mathbf{r}}{r^2}\right)}_{\text{Far field}} + \underbrace{\left(\frac{i}{k_0 r} - \frac{1}{(k_0 r)^2}\right) \left(1 - 3 \frac{\mathbf{r} \otimes \mathbf{r}}{r^2}\right)}_{\text{Near field}} \right] $$

Light in a strong magnetic field, $\Delta m = \pm 1$:

$$ G_{B \to \infty}^{\Delta m = \pm 1} = \frac{3 e^{ik_0 r}}{2 k_0 r} \left[ \frac{1}{2} (1 + \cos^2 \theta) - \frac{1}{2} \left( \frac{i}{k_0 r} - \frac{1}{(k_0 r)^2} \right) (1 - 3 \cos^2 \theta) \right] $$

Elastic wave:

$$ \hat{G}_{\text{elastic}} = \frac{1}{8\pi\rho_0 c_s^2 r} \left\{ \left[ 1 + \left( \frac{c_s}{c_p} \right)^2 \right] \mathbf{1} + \left[ 1 - \left( \frac{c_s}{c_p} \right)^2 \right] \frac{\mathbf{r} \otimes \mathbf{r}}{r^2} \right\} $$

## Conclusions

<table><thead><tr><th></th><th>Scalar waves<sup>[2]</sup></th><th>Light B = 0<sup>[1]</sup></th><th>Light large B<sup>[4]</sup></th><th>Elastic waves<sup>[3]</sup></th></tr></thead><tbody><tr><th>Critical exponent</th><td>ν ≈ 1.6</td><td>No localization</td><td>ν ≈ 1.6</td><td>ν ≈ 1.6</td></tr><tr><th>Universality class</th><td>orthogonal</td><td></td><td>orthogonal despite broken TR invariance</td><td>orthogonal</td></tr></tbody></table>

## References

1. Phys. Rev. Lett. **112**, 023905 (2014)

2. Phys. Rev. B **94**, 064202 (2016)

3. Phys. Rev. B **98**, 064206 (2018)

4. Phys. Rev. Lett. **121**, 093601 (2018)

5. Phys. Rev. Lett. **114**, 053902 (2015)

6. Phys. Rev. B **98**, 064207 (2018)

This work was funded by the
Agence Nationale de la Recherche
(project ANR-14-CE26-0032 LOVE):
https://lpmmc.cnrs.fr/spip.php?article646

<figure><img src="image_19.png" /></figure>

UNIVERSITÉ
Grenoble
Alpes

LPMC
laboratoire
de physique et
de modélisation
des milieux condensés

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