Optics 019 · Branched flow in a soap film

Branched Light in Random Media

Why does light crossing gently uneven material gather into branches instead of spreading evenly? Smooth, weak bumps in the refractive index act like many gentle lenses. Neighboring rays feel nearly the same bends, so they are focused into caustics long before their directions get scrambled, and the light gathers into branches. The first branches appear farther away when the bumps are wider and weaker.

Send green laser light through a soap film of uneven thickness. The flat beam gathers into bright branches at a distance set by how wide and strong the ripples are.

Disorder strength of the film $\text{—}$

Branching distance from the scaling law $\text{—}$

Where the light flickers most (simulated) $\text{—}$

Brightest point on the analysis line $\text{—}$

Largest scintillation index along the film $\text{—}$

Flicker peak over the scaling distance $\text{—}$

Points near the line brighter than five times the mean (fully random wave: 0.67 %) $\text{—}$

Root-mean-square ray angle at the line $\text{—}$

Mean effective index of the guided light $\text{—}$

Thinnest to thickest film $\text{—}$

Light power at 6 mm over the input $\text{—}$

Steps of 20 µm computed $\text{—}$

Model of the soap-film experiment of Patsyk, Sivan, Segev and Bandres (2020): 532 nm light guided inside a horizontal film of water (index 1.33) in air, 6 mm long and 2 mm wide, with a mean thickness of 250 nm. The thickness varies as a smooth Gaussian random field whose correlation falls off as a Gaussian over the chosen correlation length; it is held above 20 nm. Each thickness sets the effective index of the lowest guided mode, found from the symmetric slab-waveguide condition. The disorder strength is the root-mean-square of the resulting relative change in the squared index, the potential strength of the 2020 paper. A flat wave enters at one edge and is followed through the film by the split-step method for the forward Helmholtz equation, on 8192 points across the film (0.24 µm apart) in 20 µm steps, with the film repeating across its width. Light scattered backwards, absorption, drainage, evaporation and the slow drift of a real film are not included. Light running at steep angles, more than about 0.4 rad from the film axis on average, is bent too little by this forward method. The rays in the data row follow the ray limit of the same equation. The film is drawn to scale; the arrival of the light is slowed for viewing.

Physics tutorial

Watch a flat beam split into branches

BackgroundA soap film is a sheet of water a few hundred nanometres thick. Light sent into its edge stays trapped inside, the way light stays inside an optical fibre, and travels along the film.

Why it mattersWhere the film is thicker, more of the light runs inside the water and it travels more slowly, as if the film had a higher refractive index there. A real film is never even, so the light crosses a landscape of gentle, wide bumps. Patsyk and colleagues sent a flat green beam into such a film in 2020 and filmed it from above with a microscope.

Start with the essentials

Focus question: The bumps are gentle and placed at random. Why does the beam still gather into bright branches, and how far does it travel before they appear?
One-sentence intuition: Every bump is a weak lens, and neighbouring parts of the beam pass through nearly the same lenses. Their small bends add up until rays cross and pile light onto bright lines, the caustics. The distance this takes scales as $l_{\mathrm{c}}\,v_0^{-2/3}$ : proportional to the width of the bumps, and only weakly dependent on their strength.

Core mathematical model

Film thickness sets the index

\begin{gathered}\kappa\,\frac{t}{2}=\arctan\frac{\gamma}{\kappa}\\\kappa=k_0\sqrt{n_1^2-n_{\mathrm{eff}}^2},\quad\gamma=k_0\sqrt{n_{\mathrm{eff}}^2-1}\end{gathered}

The lowest guided mode of a water film of thickness $t$ and index $n_1=1.33$ in air. Its effective index $n_{\mathrm{eff}}$ runs from 1 for a vanishing film towards 1.33 for a thick one; at 532 nm it is 1.20 at 250 nm. $k_0=2\pi/\lambda$ .

The strength of the disorder

\begin{gathered}\eta=\frac{n_{\mathrm{eff}}^2-\langle n_{\mathrm{eff}}^2\rangle}{2\langle n_{\mathrm{eff}}^2\rangle}\\v_0=\sqrt{\langle\eta^2\rangle}\end{gathered}

The relative change of the squared index plays the part of a potential, and $v_0$ is its root-mean-square size, the potential strength of Patsyk et al. (2020). The default film, with a 60 nm thickness ripple, has $v_0=0.029$ . Its correlation length $l_{\mathrm{c}}$ is the width of the bumps.

Rays bend towards a higher index

\frac{dx}{dz}=\theta,\qquad\frac{d\theta}{dz}=\frac{\partial\eta}{\partial x}

For small angles $\theta$ to the film axis $z$ , a ray drifts across the film and turns towards larger $\eta$ . A single Gaussian bump of height $v_0$ and width $l_{\mathrm{c}}$ focuses like a lens of focal length $l_{\mathrm{c}}/(2\sqrt{\pi}\,v_0)$ , 1.2 mm for the default film. That grows faster than $l_{\mathrm{c}}\,v_0^{-2/3}$ as the bumps weaken, so in weak disorder the branches come from many bumps adding up.

Where the first branches form

d_0\sim l_{\mathrm{c}}\,v_0^{-2/3}

Kaplan (2002) showed that random lenses first bring neighbouring rays together after this distance. It fixes how the distance scales, not its prefactor. Averaged over four films, the flicker of the light here peaks at 0.6 to 0.8 times $l_{\mathrm{c}}\,v_0^{-2/3}$ for every strength and width tried, where a quarter of the traced rays have crossed a neighbour.

How strongly the light flickers

\begin{gathered}S(z)=\frac{\langle I^2\rangle}{\langle I\rangle^2}-1\\P(I>x\langle I\rangle)=e^{-x}\ \text{(fully random wave)}\end{gathered}

The scintillation index $S$ is averaged across the film at each distance $z$ . It is zero for an even sheet of light and peaks where the branches form (Patsyk et al. 2020). A fully random wave has $S=1$ and the exponential intensity distribution of the second line.

Common difficulties

Branches are not valleys in the film

Typical misconceptionEach branch follows a channel of thicker film that guides the light.

Better mental modelThe branches do not follow the valleys of the potential (Kaplan 2002). They are caustics, where many slightly bent rays cross; compare the film colours with the light in the scene.

Weak bumps still make branches

Typical misconceptionIf the ripples are weak enough, the beam will simply stay even.

Better mental modelWeaker ripples only push the first branches farther away, and slowly: three times weaker bumps move them about twice as far. Only a perfectly flat film leaves the light even.

Branching is not localization

Typical misconceptionThe light is trapped in the branches the way waves are trapped by strong disorder.

Better mental modelThe branches are paths the light keeps travelling along: the rays focus, cross and spread again, and each branch splits further on. Anderson localization instead stops a wave from spreading at all, through the interference of waves scattered many times.

Run the experiment

01
Watch the beam break up
Keep the default preset, A flat beam breaks into branches, and watch the light front cross the film.
What to observe: The even green sheet gathers into bright lines within the first millimetre, and the lines split again farther on. The flicker peaks 0.86 mm in, where the scaling distance is 1.28 mm.
02
Read the analysis line
Look at the two plots under the scene.
What to observe: At 2.5 mm the brightest point is 24 times the mean. 2.5 % of the points near the line are brighter than five times the mean; a fully random wave would give 0.67 %.
03
Weaken the ripples
Load Gentler ripples: branches come later.
What to observe: Halving the thickness ripple halves the strength to 0.014. The scaling distance grows only by the factor 1.6, to 2.1 mm, and the flicker peak moves to 2.0 mm.
04
Change the width of the ripples
Load Narrower ripples, then Wider ripples.
What to observe: With 80 µm bumps the flicker peaks 0.70 mm in; with 250 µm bumps, at 1.5 mm. The distance follows the width of the bumps.
05
Try another film
Load Another film, same statistics.
What to observe: The branches run in different places, but the flicker still peaks at 0.88 mm, next to the 0.86 mm of the first film.
06
Flatten the film
Load Flat film: no branches.
What to observe: The light stays an even sheet all the way through: the flicker is zero and the brightest point equals the mean.

Branched Light in Random Media

Watch a flat beam split into branches

Core mathematical model

Film thickness sets the index

The strength of the disorder

Rays bend towards a higher index

Where the first branches form

How strongly the light flickers

Common difficulties

Branches are not valleys in the film

Weak bumps still make branches

Branching is not localization

Run the experiment

Watch the beam break up

Read the analysis line

Weaken the ripples

Change the width of the ripples

Try another film

Flatten the film

Related experiments and articles

Magnetism, disorder and topological response