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L31 · Feedback without mirrors

Random Laser

Replace mirrors with multiple scattering. Change sample size, gain and pumped core, then capture growth, saturation and diffuse escape.

Interactive modelRandom Laser
Elapsed model time—\text{—}
Optical thickness—\text{—}
Diffusion coefficient—\text{—}
Initial gain threshold—\text{—}
First diffusion growth rate—\text{—}
Second diffusion growth rate—\text{—}
Third diffusion growth rate—\text{—}
Mean normalized energy—\text{—}
Total normalized escape flux—\text{—}
Current energy growth ledger—\text{—}
Central excited fraction—\text{—}
Captured time samples—\text{—}
Initial gain regime—\text{—}
Diffusion approximation—\text{—}

Physics tutorial

Scattering creates a long residence time

BackgroundA disordered gain medium can obtain feedback without a conventional mirror pair.

Why it mattersReplace mirrors with multiple scattering. Change sample size, gain and pumped core, then capture growth, saturation and diffuse escape.

Start with the essentials

Focus question
Can scattering hold light long enough to lase?
One-sentence intuition
Separate diffuse growth from interference-selected random resonances.

Core mathematical model

Diffuse light

∂tW=D∇2W+v(g0n−α)W+snD=vℓt/3\begin{aligned}\partial_t W&=D\nabla^2W+v(g_0n-\alpha)W+s n\\D&=v\ell_t/3\end{aligned}

Scattering lowers the diffusion coefficient and slows escape.

Saturate the inversion

n˙=p(r)(1−n)−n(1+W)τnp(r)=p0(r<a)\begin{aligned}\dot n&=\frac{p(r)(1-n)-n(1+W)}{\tau_n}\\p(r)&=p_0\quad(r<a)\end{aligned}

This chosen normalized rate law limits gain; it is not a material-specific quantum model.

Uniform-pump analytic limit

W(R)=0λ1=vg−Dπ2R2gth=α+π2ℓt3R2\begin{aligned}W(R)&=0\\\lambda_1&=vg-\frac{D\pi^2}{R^2}\\g_{\rm th}&=\alpha+\frac{\pi^2\ell_t}{3R^2}\end{aligned}

This ideal absorbing boundary omits extrapolation length. The page also solves the nonuniform pumped core numerically.

Spatial intensity modes

(D∇2+vg)Φj=λjΦjΦj(r)∝sin⁡(jπr/R)/r\begin{aligned}(D\nabla^2+vg)\Phi_j&=\lambda_j\Phi_j\\\Phi_j(r)&\propto\sin(j\pi r/R)/r\end{aligned}

The displayed eigenvectors form a diffusion basis. Their signed higher orders are not negative physical intensity.

Account for the escaped energy

U˙=G+S−A−FF=−4πR2D∂rW∣R\begin{aligned}\dot U&=G+S-A-F\\F&=-4\pi R^2D\left.\partial_rW\right|_R\end{aligned}

The finite-volume interfaces cancel internally, leaving the same boundary flux used by the output record.

Check the approximation

R/ℓt≫1,gℓt≪1R/\ell_t\gg1,\qquad g\ell_t\ll1

Weak scattering leaves the diffusion regime; the model does not invent coherent resonances to fill this gap.

Common difficulties

Random means optical speckle

Typical misconceptionThe smooth diffusion map should contain narrow random spectral spikes.

Better mental modelThose require a wave interference model; this Lab displays ensemble intensity diffusion.

A positive rate means infinite light

Typical misconceptionThe linear threshold predicts unlimited final output.

Better mental modelStimulated depletion lowers local inversion and saturates the nonlinear state.

Run the experiment

  1. 01

    Cross threshold

    Compare diffuse feedback with the below-threshold preset, then capture.

    What to observe: Positive initial growth develops into a finite saturated output.
  2. 02

    Change residence time

    Shrink the sample or weaken scattering at unchanged gain.

    What to observe: Escape becomes faster and threshold rises.
  3. 03

    Change the pump region

    Compare a small central core with whole-volume pumping.

    What to observe: Unpumped shells increase the gain required to sustain the leading spatial mode.