# Interference in random media: Coherent backscattering and Anderson localization

Disordered photonic systems are media in which the dielectric constant varies randomly in space, as for example a suspension of scattering micro-particles in a liquid, as it is the case for ordinary milk, or a grind crystal powder. The phase of the scattered waves is random, and thus one can model light propagation with a multiple scattering process which leads to a random walk (see Fig. 1). A di usion model which considers the transport of photons like hard balls with no internal degree of freedom can be appropriate. But this simple picture is not complete as some interference efects can survive the multiple scattering.
When one shines coherent light on a system with disorder, the scattered light has a “grainy” random intensity pattern with very bright and dark spots. This is a signature of preserved phase coherence upon multiple scattering.

Speckle

At the bright spots many scattered waves interfere constructively while at the dark spots they cancel out. This effect is called speckle and can be observed with bulk systems and surfaces (see Fig. 2). If we average over many system configurations, as for example in a turbid liquid, we smooth out the profile and wash out the interference pattern.

In the figure below the speckle pattern as a function of the illuminating beam diameter is shown. The beam size determines the number of scattering channels the light couples to. From left to right: the more the channels the more the ‘diffusive modes’ and therefore the finer the interference patterns. In the last image the average of many disorder realization is shown. As a result of the averaging the individual speckle spots disappear, but a bigger peak in the center survives: it is the coherent backscattering cone.

Some coherent effects can survive the ensemble averaging. One of the most evident is coherent backscattering, an interference responsible for up to a doubling of the backscattered intensity and which thus decreases the optical conduction of the medium.

A more conventional way of looking at a coherent backscattering profile is making a 1D cut to the 2D image here above. If we do it for a cbs measured from paper we obtain:

A fit of the profile leads to the value of the transport mean free path, which is in our case 25 micron.

Coherent backscattering

One of the most robust interference phenomena that survives multiple scattering is coherent backscattering or weak localization of light. In weak localization, interference of the direct and reverse paths leads to a net reduction of light transport in the forward direction, similar to the weak localization phenomenon for electrons in disordered (semi)conductors and often seen as the precursor to Anderson (or strong) localization of light. Weak localization of light can be detected since it is manifest as an enhancement of light intensity in the backscattering direction. This substantial enhancement is called the cone of coherent backscattering.

Coherent backscattering enhancement.

Weak localization has been initially studied for electronic systems, in the 1970s, where it has been reported as quantum interference (coherent echo) between electronic waves multiple scattered by impurities in conductors. Weak localization manifests itself as an anomaly of the resistance of a conducting thin film. In electronic systems it has arisen much interest, as it is one of those unique cases where the superposition principle of quantum mechanics leads to observable consequences in the properties of macroscopic systems.

At the beginning of the 1980s, this concept based on interference has been successfully exported also to light waves, which instead of electrons show very weak photon to photon interaction, have a much longer coherence time and are extremely sensitive to interference effects.
On the other hand in optical experiment, it is hard to measure quantities like the total conductance, but it is possible, due to the long coherence length, to observe its counterpart, i.e. directly the interference-induced increase diffusion in backscattering: the coherent backscattering cone.

Since the first experimental observation of coherent backscattering from colloidal suspensions, the phenomenon has been successfully studied for electromagnetic waves in strongly scattering powders, cold atom gases, two-dimensional random systems of rods, randomized laser materials, disordered liquid crystals, chaotic cavities photonic crystals and even sea bottom. The phenomenon is typical of any wave which is multiply scattered, and it has indeed been observed also for mechanical waves: acoustic waves in macroscopic disordered systems and even seismic waves propagating in the earth crust.

Counter-propagating light paths that give rise to coherent backscattering.

Weak localization has its origin in the interference between direct and reverse paths in the backscattering direction. When a multiply scattering medium is illuminated by a laser beam, the scattered intensity results from the interference between the amplitudes associated with the various scattering paths; for a disordered medium, the interference terms are washed out when averaged over many sample configurations, except in a narrow angular range around exact backscattering where the average intensity is enhanced.
This phenomenon, is the result of many two-waves interference patterns $I(\theta,\phi) = I_0 \, (1 + \zeta\, \cos (\mathbf{d} \cdot \Delta k_b ))$ where I0 is the total intensity neglecting interferences, ζ the contrast of the interference, θ and φ the angles with respect to the backscattering direction, and we assume no additional phase difference along the reverse paths exists. The incident and scattered k-vector kbi and $k_{bf}$ are written as kbi = (0,0,k)( − ksinθcosφ, − ksinθsinφ, − kcosθ). This interference is generated by counter-propagating light paths with entering-exiting distance $\mathbf{d}$ and initial and final k-vectors kbi,kbf such that Δkb = Kbf + kbi. Maximum interference is obtained when the counter-propagating paths have the same amplitude (and thusζ = 1), and only at θ = 0.
Reciprocity is an important ingredient as it ensures the equality of the direct and reverse path amplitudes. For example, as predicted it was experimentally observed that the presence of an external magnetic field breaks the reciprocity and results in a decrease of the coherent backscattering enhancement as well as some rather complicated behavior of the cone shape. Here onwards we will assume ζ = 1.

The cone is the Fourier transform of the spatial distribution of the intensity of the scattered light on the sample surface, when the latter is illuminated by a point-like source.
The enhanced backscattering relies on the constructive interference between reverse paths. One can make an analogy with a Young’s interference experiment, where two diffracting slits would be positioned in place of the “input” and “output” scatterers. If the slits are backlit with a plane wave (of wavevector kbi), the interference produces a sinusoidal fringe pattern in the far field, with a maximum intensity at θ = 0 and a fringe spacing inversely proportional to the transverse spacing between the scatterers. The coherent backscattering cone comes from a superpositions of many of these fringes, which are all in phase only at θ = 0.

Coherent backscattering arises from many two-wave interference patterns which are all in phase at θ = 0.
In the figure, the full profile (full line) and the interference that results from only double scattering (dashed line), and a few lower scattering orders (dotted and mixed lines) are shown. The line profile is obtained with a Monte Carlo simulation for scalar waves. In the top panel, a few two-paths interference patterns are shown before averaging.

The triangular top of the coherent backscattered profile is sensitive to the long and diffusive paths which have very far entrance-exit points, and thus short spatial frequency contribution. This can be evident if we look at the phase difference between two paths, which is $\Delta \phi = \frac{2\pi}{\lambda}(\mathbf{d}\cdot \Delta k_b)$ and which can be simplified in the small θ limit (milliradians) into $\Delta \phi \approx \frac{2\pi}{\lambda} \theta \Delta r$ If we replace in Eq. Δr for the mean separation between the first and last scattered, which is of the order of the transport mean free path ($\ell_t$), the phase difference between the reciprocal paths becomes:
$\Delta \phi \approx \frac{2\pi}{\lambda}\ \theta \, \ell_t.$

Under diffusion approximation, one can find for the cone an opening angle (at full width half maximum) which is $W \simeq \frac{0.7}{k\ell_t}$.
The first scattering orders (small N), are very important as they get less dephased. The factor 0.7 of Eq. comes from the averaging of many scattering order contributions, and it has a value close to unity: double scattering dominates.

The wings are influenced by the lower spatial frequencies, and therefore by the short paths due to low order scattering. The top and full width at half maximum are determined by the transport mean free path ($\ell_t$), which is an averaged quantity, whereas the wings are strongly influenced by the details of the not-averaged single scattering differential cross section. In Fig. , the profiles of the interference that results from the different contribution are shown. The line profile is obtained with a Monte Carlo simulation for scalar waves. In the top panel, few two-paths interference patterns before averaging show the different spatial frequency contributions.

Monte Carlo simulation of coherent backscattering for linearly polarized light.

Anderson localization of light

If the disorder is so strong that the light is scattered before completing an optical cycle, then Anderson localization can occur, a phase transition which brings the system into a regime where diffusive transport is inhibited (see Fig. xx). P. Anderson won the Nobel prize in 1977 for his investigations into this very important issue [P.W. Anderson, Absence of di usion in certain random lattices, Phys. Rev. 109 1492 (1958); P.W. Anderson, Phil. Mag. B 52, 505 (1985).].