Optics 092 · Waveguides, structured light, and modern optics

Light in Disorder

How can interference in disordered material send extra light straight back, or stop light from spreading sideways? In a weakly scattering slab, light takes random walks, and each multiply scattered path interferes with the same path run in reverse, adding a narrow bright cone straight back: coherent backscattering. In a row of coupled waveguides with fixed random detunings, interference keeps light launched into one guide from spreading sideways. That is transverse Anderson localization.

Reciprocal scattering paths build a backscattering cone, and photon-conserving random flights separate the collision length from the transport length. A disordered waveguide array evolves unitarily; read its complex output, the spreading averaged over many disorder samples and the computed eigenmodes.

Back peak or transverse spread

E(0),\sigma

$0.50$

Transport length or array power

\ell^*,P

$50\%$

Energy balance or eigenmode participation

R+T+U,\mathcal P

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Light in Disorder

Array detuning half-width / coupling

\Delta/C

3

Slab depth; not array length

L

4\,\mathrm{mm}

Mean collision free path in slab

\ell_s

0.80\,\mathrm{mm}

Mean scattering cosine in slab

g

0

Frozen disorder / path ensemble seed

s

7

Array output distance

z

18\,\mathrm{mm}

Reciprocal-pair visibility; not absorption

v

1

Array input guide; drag source in viewport

n_0

0

Disorder can send light back toward the source, and it can also hold a wavepacket near where it entered. Here these are two separate experiments. First, trace sampled scattering paths in reverse to see the backscattering cone appear. Then launch light into one guide of an array, watch its complex field evolve, and compare how strongly frozen disorder confines it with a separate classical diffusion reference.

The backscattering part uses a scalar weak-scattering slab approximation; the array is a separate, finite one-dimensional tight-binding model. No exponential envelope is imposed on the localized light. The Lab does not model a bulk mobility edge, is not calibrated to a material and does not reproduce a specific experiment.

Physics tutorial

Audit the physics of Light in Disorder

BackgroundA peak in reflected intensity and a wavepacket that stops spreading are two different measurements. In a disordered waveguide array, neighboring guides exchange energy coherently: $i\frac{da_n}{dz}=\delta\beta_n a_n+C(a_{n-1}+a_{n+1})$ . Random detunings, frozen in place, can stop the light from spreading sideways through interference. A separate weakly scattering slab shows the backscattering produced by reciprocal path pairs. The slab does not solve localization in a bulk medium.

Why it mattersHow can interference in disordered material send extra light straight back, or stop light from spreading sideways?

Start with the essentials

Focus question: How can interference in disordered material send extra light straight back, or stop light from spreading sideways?
One-sentence intuition: When spreading stops, that does not mean power was absorbed: $\sum_n|a_n(z)|^2=1$ . Follow the actual complex amplitudes, look at more than one disorder realization, and tell an output pattern apart from an eigenmode. The calculation does not assume an exponential localization length.

Core mathematical model

The collision length and the transport length

p(s)=\ell_s^{-1}e^{-s/\ell_s},\quad g=\langle\cos\theta\rangle,\quad\ell^*=\frac{\ell_s}{1-g},\quad B=e^{-L/\ell_s}

The slab model uses random flights in three dimensions, lengths in millimeters, a plane wave at normal incidence and a normalized Henyey–Greenstein phase function. Its boundaries are index-matched and it extends infinitely sideways. The model leaves out absorption, Fresnel reflection at the surfaces, vector polarization, repeated near-field coupling and the Mie response of any particular particle. Because it assumes weak scattering, it cannot support a claim about a bulk mobility edge.

Reversed path pairs make the backscattering cone

E(\alpha)\simeq1+v\frac{\sum_{p\ge2}w_p\cos[k\alpha(x_{p,\rm last}-x_{p,\rm first})]}{W_1+W_{\ge2}},\quad k=\frac{2\pi}{\lambda_0}

After each collision, a next-event estimator weights the chance of escaping exactly backward. Near the backward direction, the phase difference picked up outside the slab depends on the horizontal distance between the first and last scattering points. The finite slab, single scattering and the chosen pair visibility all affect the enhancement you see. The narrow-cone approximation leaves out angle-dependent ladder weights and the second-order angular phase, and it adds no interference correction that would keep total flux conserved. The half-width on the positive-angle side is measured from the numbers, not set by a formula. The green light has a vacuum wavelength of 532 nanometers.

Frozen disorder and unitary propagation

\delta\beta_n\sim\mathcal U[-\Delta,\Delta],\quad\mathbf a(z)=e^{-iHz}\mathbf e_{n_0},\quad\sigma^2=\sum_n(n-n_0)^2\langle|a_n|^2\rangle

The array has 81 guides with open ends, a pitch of sixteen micrometers and real, uniform nearest-neighbor coupling of 0.4 per millimeter. Each guide has its own independent, static detuning on the matrix diagonal. The program diagonalizes this real symmetric matrix, and its eigenvalues and eigenvectors produce every field along the array and every detector value. The common propagation constant is removed. Our phase sign is the opposite of the one in Lahini’s equation; for a real input in a single guide, the intensities are the same. The displayed ensemble has twelve realizations, so it is not a noise-free infinite average. Power reaching the edge guides warns that the finite size is affecting the result.

Classical hopping is a comparison, not a required stage

\sigma^2_{\rm ordered}=2(Cz)^2,\quad\partial_z p_n=\Gamma(p_{n-1}+p_{n+1}-2p_n),\quad\sigma^2_{\rm hop}\simeq2\Gamma z

In the ordered chain the spreading is ballistic. The separate classical process conserves probability with reflecting ends, and its hopping rate is set equal to the optical coupling. On an infinite chain its variance grows linearly, not quadratically. It is not a dephasing model calibrated against experiment. In a static one-dimensional Anderson lattice, ballistic and localized parts can appear together without passing through a diffusive stage. This Lab does not infer three-dimensional localization from low transmission.

The participation number counts occupied guides

\mathcal P(\mathbf v)=\frac{(\sum_n|v_n|^2)^2}{\sum_n|v_n|^4},\quad H\mathbf v_j=\beta_j\mathbf v_j

The mode that overlaps most with the launch guide is shown separately from the propagated packet. A low participation number means the power sits in few guides. Saturation caused by the finite size alone does not prove Anderson localization, and the participation number is not a fitted localization length. Matrix residuals, disorder ensembles, ordered analytic limits and probability conservation are each checked separately. Display gain, the compressed axial scale and the phase colors do not enter any physical readout.

Common difficulties

Does a dark output mean the waves localized?

Typical misconceptionIf the intensity decays exponentially or the transmission is low, that proves Anderson localization.

Better mental modelNo. Absorption, ordinary scattering, finite boundaries and the detector geometry can all reduce a signal. Here the confined output is compared with coherent propagation in the ordered array, with an ensemble, with eigenmodes and with conserved power. The dilute slab makes no claim about bulk localization.

Run the experiment

01
Scene 1: Reciprocal paths in a slab
Begin with the slab. Lower the reciprocal-pair visibility to zero: the angular peak disappears, but the reflection and transmission counts do not change. Change the mean collision path or the mean scattering cosine; the transport length and the cone width both respond, for different reasons.
What to observe: In the weak-scattering approximation, a backscattering cone is evidence for reciprocal paths. It does not prove localization in the bulk. The cone’s peak and the integrated reflected power are different quantities.
02
Scene 2: An ordered array spreads ballistically
Select the ordered array and increase the propagation distance: two ballistic wings spread outward. Then select frozen disorder and compare the actual output, the twelve-realization average, the spreading history and an eigenmode computed from the same matrix.
What to observe: The seed and the launch position change the result. Compare actual outputs with the ensemble; a preselected exponential envelope would hide the fluctuations.
03
Scene 3: Frozen disorder confines the wave
Change the seed, or drag the array source along the input face. The local result changes, but total power does not disappear. Freeze one acquisition, then change the controls: the saved angular estimate or complex output should stay the same.
What to observe: Unlike the geometric branched flow in Lab 019, the array evolves complex amplitudes and depends on interference. The slab and the array are two separate apparatus models.