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Sandbox Physics

L35 · The field remembers

External Feedback & Laser Chaos

Return a stored complex field to a semiconductor laser. Record intensity, optical and RF spectra, nearby-history separation and an ideally coupled response.

Interactive modelExternal Feedback & Laser Chaos
Time / photon lifetime—\text{—}
Current master intensity—\text{—}
Current excess carriers—\text{—}
Stored return delay—\text{—}
Captured samples—\text{—}
Recorded mean intensity—\text{—}
Relative intensity deviation—\text{—}
Field coherence at return delay—\text{—}
Windowed spectral standard deviation—\text{—}
Frequency-bin spacing—\text{—}
Response field error / master power—\text{—}
Log separation relative to initial—\text{—}
Spectral / window-weighted power—\text{—}
Captured time / photon lifetime—\text{—}

Physics tutorial

A delayed field adds both memory and phase; stronger feedback need not mean a quieter laser.

BackgroundReturn a stored complex field to a semiconductor laser. Record intensity, optical and RF spectra, nearby-history separation and an ideally coupled response.

Why it mattersWhen does a delayed reflection destabilize a laser?

Start with the essentials

Focus question
When does a delayed reflection destabilize a laser?
One-sentence intuition
A delayed field adds both memory and phase; stronger feedback need not mean a quieter laser.

Core mathematical model

Complex delayed field

E˙=(1+iα)NE+κe−iϕE(t−τ)\begin{aligned}\dot E&=(1+i\alpha)NE\\&\quad+\kappa e^{-i\phi}E(t-\tau)\end{aligned}

The return is the stored complex amplitude, not an intensity proxy.

Carrier balance

TN˙=P−N−(1+2N)∣E∣2I=∣E∣2\begin{aligned}T\dot N&=P-N-(1+2N)|E|^2\\I&=|E|^2\end{aligned}

Carrier depletion and refractive response share the same field.

Solitary limit

κ=0Nss=0,Iss=P\begin{aligned}\kappa&=0\\N_{\rm ss}&=0,\quad I_{\rm ss}=P\end{aligned}

With no return, the above-threshold steady intensity equals excess pump.

External-cavity solution

Φ=ϕ+ΩτNs=−κcos⁡ΦΩ=−κ(αcos⁡Φ+sin⁡Φ)\begin{aligned}\Phi&=\phi+\Omega\tau\\N_s&=-\kappa\cos\Phi\\\Omega&=-\kappa(\alpha\cos\Phi+\sin\Phi)\end{aligned}

Even a stationary intensity can have a shifted optical frequency.

Windowed measurement

Sk=∣F[wE]k∣2M∑nwn2∑kSk=∑nwn2∣En∣2∑nwn2\begin{aligned}S_k&=\frac{|\mathcal F[wE]_k|^2}{M\sum_n w_n^2}\\\sum_k S_k&=\frac{\sum_n w_n^2|E_n|^2}{\sum_n w_n^2}\end{aligned}

The all-bin spectrum obeys a separately checked Parseval ledger.

Ideal response link

E˙r=F(Er,Nr,Er,τ)+K(E−Er)ε=⟨∣Er−E∣2⟩⟨∣E∣2⟩\begin{aligned}\dot E_r&=F(E_r,N_r,E_{r,\tau})\\&\quad+K(E-E_r)\\\varepsilon&=\frac{\langle|E_r-E|^2\rangle}{\langle|E|^2\rangle}\end{aligned}

Identical dynamics admit an exact synchronization manifold; mismatch may destroy it.

Common difficulties

Every broad spectrum is chaos

Typical misconceptionA broad spectrum proves a chaotic attractor.

Better mental modelFinite records, transients and periodic sidebands also broaden spectra. Nearby-history growth is supporting evidence, not an asymptotic exponent.

More feedback is always better

Typical misconceptionA stronger returned beam always stabilizes the laser.

Better mental modelDelay and phase change stability; compare actual recorded dynamics.

Run the experiment

  1. 01

    Record before judging

    Compare isolated and weak feedback after evolving.

    What to observe: Check stationary intensity and coherent narrow finite-window spectra.
  2. 02

    Change memory and phase

    Compare delayed, changed-phase and long-delay presets.

    What to observe: Intensity, field coherence and spectra all come from the same trajectory.
  3. 03

    Test synchronization

    Enable ideal coupling, then add response pump mismatch.

    What to observe: Compare the measured field error rather than visual similarity.