Optics 019 · Ray worlds, boundaries, and natural optics

Branched Light in Random Media

A volumetric random-medium observatory integrates a dense plane-wave manifold through an analytic correlated index gradient. A colored field surface, computed fold-caustic sparks, output-density trace, and twelve-seed first-caustic ensemble explain why weak smooth disorder produces branching before diffusion.

First detected caustic distance

L_{\mathrm{first}}

$0.50$

Characteristic or ensemble distance

L_b

$50\%$

Peak ray-density gain

G_{\rho}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Branched Light in Random Media

BackgroundFor a weak, slowly varying index field, a nearly horizontal ray obeys the paraxial eikonal equation $\frac{\mathrm d^2y}{\mathrm dx^2}=\frac{1}{n}\frac{\partial n}{\partial y}$ . Neighboring rays feel correlated gradients, so they focus collectively into folds instead of executing independent random walks.

Why it mattersWhy does weak smooth disorder focus parallel rays into branches instead of simple diffusion?

Start with the essentials

Focus question: Why does weak smooth disorder focus parallel rays into branches instead of simple diffusion?
One-sentence intuition: Across an ensemble the first-branch distance scales as $L_b\propto\ell_c\sigma_n^{-2/3}$ . The exponent is robust, while the order-unity prefactor depends on how correlation length and fluctuation strength are defined.

Core mathematical model

Paraxial random-medium ray equation

\frac{\mathrm d^2y}{\mathrm dx^2}=\frac{1}{n(x,y)}\frac{\partial n(x,y)}{\partial y}

The solver evaluates the analytic gradient of the same seeded field displayed under the rays at every integration substep.

Fold-caustic condition

J=\frac{\partial y(x;y_0)}{\partial y_0}=0

When the launch-coordinate map loses local invertibility, neighboring rays compress and reverse order, producing a geometric caustic.

Characteristic branching scale

L_b\propto\ell_c\left(\frac{n_0}{\sigma_n}\right)^{2/3}

Stronger smooth disorder produces folds sooner, while a longer correlation scale moves them farther downstream.

Common difficulties

Treating branches as material channels

Typical misconceptionThe bright paths must be high-index fibers embedded in the random medium.

Better mental modelBranches are caustic folds of a ray manifold. They move between realizations and can cross the same smooth field regions; no discrete channel boundary is present.

Run the experiment

01
Scene 1: Weak random field
Start with weak disorder and compare the seeded index landscape with the gradual bending of the sparse ray manifold.
What to observe: Weak disorder bends nearby rays coherently because they sample nearly the same gradient over one correlation length.
02
Scene 2: Random caustic network
Increase fluctuation strength and locate gold points where neighboring ray order compresses or reverses.
What to observe: Detector peaks coincide with manifold crowding, while gold fold points appear upstream where the coordinate map first becomes singular.
03
Scene 3: Branch-distance statistics
Open the ensemble scene and compare twelve first-caustic samples with the theoretical characteristic-distance scaling.
What to observe: Individual first-caustic distances fluctuate substantially; the scaling law belongs to their characteristic ensemble behavior, not one seed.