L31 · Feedback without mirrors

Random Laser

Can scattering hold light long enough to lase? It can. Light that scatters many times wanders through the pumped medium along a long, tangled path and picks up gain the whole way. With enough gain it grows past threshold without any mirrors. That diffuse growth is one effect; sharp random resonances selected by interference are a different one.

Let multiple scattering do the job of mirrors. Change the sample size, the gain and the pumped core, and record how the light grows, saturates and diffuses out of the sample.

Resolved optical threshold candidates $\text{—}$

Selected coherent wavelength $\text{—}$

Selected outgoing-mode threshold $\text{—}$

Selected mode participation fraction $\text{—}$

Outgoing boundary residual $\text{—}$

Candidates above unsaturated gain threshold $\text{—}$

Elapsed model time $\text{—}$

Optical thickness $\text{—}$

Diffusion coefficient $\text{—}$

Full-inversion gain threshold $\text{—}$

Initial principal growth rate $\text{—}$

Second diffusion growth rate $\text{—}$

Third diffusion growth rate $\text{—}$

Mean normalized energy $\text{—}$

Total normalized escape flux $\text{—}$

Current net energy growth rate $\text{—}$

Central excited fraction $\text{—}$

Captured time samples $\text{—}$

Initial gain regime $\text{—}$

Diffusion approximation $\text{—}$

Explore the mechanism

Can scattering hold light long enough to lase?

Apparatus: sphere / coherent layers

b

0

Layer refractive-index disorder

\delta n

0.65

Layer realization seed

s

4

Layer fully inverted gain

g_w

3000\,\mathrm{cm^{-1}}

Layer mode by ascending threshold

j_w

0

Sphere radius

R

1\,\mathrm{mm}

Sphere transport mean free path

\ell_t

0.05\,\mathrm{mm}

Sphere fully inverted gain

g_0

0.8\,\mathrm{mm^{-1}}

Pump relative to decay

p

2

Sphere core / layer pump fraction

a/R

0.85

Sphere background absorption

\alpha

0.02\,\mathrm{mm^{-1}}

Detail: sample / pump / collector

j

0

Compare a small pumped core with pumping the whole volume. A positive initial growth rate means the unsaturated state is unstable, but the output it reaches still has to saturate. Diffusion cannot predict the spectral peaks that interference selects.

The Lab compares incoherent and coherent feedback in two separately defined samples. The sphere solves radial intensity diffusion on finite volumes, with a prescribed saturable inversion and an absorbing outer surface. Its energy and its total escape over the whole sphere are normalized; they are not watts or the counts of a single detector. Its three diffusion eigenvectors can take negative values and serve only as a basis for the intensity. The coherent sample is a stack of 32 disordered dielectric layers, drawn with the same relative widths and refractive indices that the scalar transfer matrix uses. The outside medium and the mean layer index are both 1.8, and layer lengths are in micrometres. A finite wavelength scan without gain supplies starting points for damped Newton solves, which find a real frequency and a positive gain with light only leaving the stack. The resulting fields interfere, and their thresholds and participation lengths depend on the disorder. They are the linear threshold modes the solver resolved, not a complete spectrum or a prediction of saturated multimode lasing. The pump covers the central layers; that pumped fraction is defined on its own and does not carry the sphere’s geometric meaning. The calculation assumes one dimension, normal incidence, scalar gain without dispersion and fixed layers, so it leaves out three-dimensional disorder, vector polarization and nonlinear competition between modes. The optical calculation is not a reconstruction of the diffuse escape data. Moving markers are only illustrative.

Start with the essentials

Focus question: Can scattering hold light long enough to lase?
One-sentence intuition: Diffusion explains how trapped light grows overall, but sharp random resonances come from interference, which diffusion cannot predict.

Core mathematical model

Light energy diffuses and grows

\begin{aligned}\partial_t W&=D\nabla^2W+v(g_0n-\alpha)W+s n\\D&=v\ell_t/3\end{aligned}

More scattering lowers the diffusion coefficient, so light takes longer to escape.

The inversion saturates

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

This normalized rate law, chosen for the Lab, limits the gain. It is not a quantum model of any particular material.

Threshold for a uniformly pumped sphere

\begin{aligned}W(R)&=0\\\lambda_1&=v(g-\alpha)\\&\quad-\frac{D\pi^2}{R^2}\\g_{\rm th}&=\alpha+\frac{\pi^2\ell_t}{3R^2}\end{aligned}

The boundary here absorbs perfectly and ignores the extrapolation length. This formula uses the effective unsaturated gain. The full-inversion gain threshold readout also accounts for the initial excited fraction and the pumped core, using a numerical eigenvalue.

Spatial modes of the intensity

\begin{aligned}\mathcal L&=D\nabla^2+v(g-\alpha)\\\mathcal L\Phi_j&=\lambda_j\Phi_j\\\Phi_j(r)&\propto\sin(j\pi r/R)/r\end{aligned}

The radial sine form applies to uniform unsaturated gain; a pumped core uses numerical eigenvectors. Higher-order basis functions change sign, but the physical intensity remains nonnegative.

Energy balance and escape through the surface

\begin{aligned}\bar U&=\frac{3}{R^3}\int_0^R Wr^2\,dr\\\dot{\bar U}&=\bar G+\bar S-\bar A-\bar F\\\bar F&=-\frac{3D}{R}\left.\partial_rW\right|_R\end{aligned}

The page divides energy and total surface escape by the sphere volume. Fluxes between neighbouring cells cancel; the remaining outer flux per volume is the reported output.

When diffusion is valid

\begin{aligned}R/\ell_t&\gg1\\\max(g_0,\alpha)\ell_t&\ll1\end{aligned}

With weak scattering the diffusion picture breaks down, and the model does not add coherent resonances to make up for it.

A separate layered sample with interference

\begin{aligned}E_j(z)&=A_je^{ikn_jz}\\&\quad+B_je^{-ikn_jz}\\n_j&=n_{j,r}-ig_j/(2k)\\\binom{A_R}{B_R}&=M(k,g)\binom{A_L}{B_L}\end{aligned}

The refractive indices and widths of the seeded layers set the reflection at each interface and the phase of the light. This sample is not an optical reconstruction of the diffusing sphere.

Threshold modes with light only leaving

\begin{aligned}A_L&=B_R=0\\M_{22}(k,g_{\rm th})&=0\\\mathcal P&=\frac{(\int |E|^2dz)^2}{L\int |E|^4dz}\end{aligned}

The solver looks for a real frequency and a positive gain that satisfy both outgoing-boundary conditions. The participation ratio measures how concentrated a mode is in space. The modes found are candidates: the list is not complete, and saturation is not included.

Common difficulties

Random lasing must show speckle

Typical misconceptionThe smooth diffusion map should show narrow, random spectral spikes.

Better mental modelThe sphere shows how the ensemble-averaged intensity diffuses. To find threshold modes selected by interference, switch to the separate coherent layer sample; those modes are not derived from the sphere data.

A positive growth rate means endless growth

Typical misconceptionAbove the linear threshold, the output keeps growing without limit.

Better mental modelStimulated emission uses up the local inversion, so the gain drops and the output saturates.

Run the experiment

01
Cross the threshold
Compare Diffuse feedback and Below threshold. Each preset automatically records twenty nanoseconds; the replay illustrates surface escape.
What to observe: When the initial growth rate is positive, the light grows at first and then levels off at a finite, saturated output.
02
Change how long light stays
Keep the gain fixed and shrink the sample or weaken the scattering.
What to observe: Light escapes faster, and the threshold goes up.
03
Change the pump region
Compare the Small pump core and Whole-volume pump presets.
What to observe: When the outer shell is not pumped, the leading spatial mode needs more gain to keep going.
04
Look at coherent modes
Open Advanced controls and choose Show coherent layers. Pick a mode, then change the layer seed or the pumped fraction; the separate optical calculation updates automatically.
What to observe: Each mode has its own wavelength, threshold and participation. Compare these, and the outgoing-boundary residual, with the layer stack shown in the apparatus.