L35 · The field remembers

External Feedback & Laser Chaos

When does a delayed reflection destabilize a laser? It depends on delay and return phase as well as strength. The old field adds to the present field and changes carrier depletion; that delayed response can turn steady output into oscillations or irregular dynamics. Feedback strength alone does not locate the stability boundary.

Move the delay mirror and compare laser fluctuations with their field spectrum. Test how return phase and ideal coupling change the acquired history.

Finite-time largest history growth rate $\text{—}$

First-half history growth rate $\text{—}$

Second-half history growth rate $\text{—}$

Post-transient diagnostic duration $\text{—}$

Weak-feedback noise linewidth reference $\text{—}$

Recorded long-lag diffusion linewidth $\text{—}$

Nonoverlapping phase blocks $\text{—}$

Integrated time / photon lifetime $\text{—}$

End-of-record master intensity $\text{—}$

End-of-record 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 $\text{—}$

The laser follows the single-mode Lang–Kobayashi equations for the field and the carriers, and the complex field history over the delay is kept. Time is measured in photon lifetimes. Each local stage of the fourth-order Runge–Kutta (RK4) integration uses interpolated values of the past field. A separate, exact linearized system runs along the same trajectory. Every ten lifetimes it rescales the current carrier perturbation together with the whole delayed-field history, and the initial transient is left out. The cumulative and half-record growth rates are finite-time estimates. Checks of time step and record length support the selected chaotic examples; they do not cover every parameter setting and are not a mathematical proof of an attractor. The raw separation without rescaling and the spectral spread over a finite window are reported separately. A different model, with a noisy phase and an adiabatic delay, runs only when weak feedback has a single steady solution. It assumes a solitary linewidth of one ten-thousandth of the inverse photon lifetime. The variance of phase changes over long lags gives the noise linewidth, which is compared with the low-frequency analytic limit. This model is not extended to strong-feedback chaos, and it does not replace the field spectrum with a noise formula. The response laser is linked to the master by an ideal, instantaneous field-error coupling. Multimode competition, a calibrated spontaneous-emission strength, gain compression, polarization and real detector noise are left out. The beam paths and moving markers are schematic. Each selection integrates six thousand photon lifetimes and captures 1024 samples; replay leaves those data fixed. Mirror distance follows delay on a schematic length scale. Blue and green intensities follow the recorded master and response, and the gold return samples the retained field at the delayed time. Its brightness uses a fixed logarithmic map of the normalized forcing strength. The feedback coefficient is a field coupling rate, not a mirror power reflectance. The response delay loop is collapsed inside its module; the cable denotes ideal error coupling. No calibrated photon-flight time or detector bandwidth is inferred from the animation.

Start with the essentials

Focus question: When does a delayed reflection destabilize a laser?
One-sentence intuition: The returned field carries both a memory of the past and a phase, and both affect stability; stronger feedback does not necessarily make the laser quieter.

Core mathematical model

Field equation with delayed feedback

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

The return retains the stored complex amplitude and its phase.

Carrier rate equation

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

The same field drives both the carrier depletion and the refractive-index response.

The laser without feedback

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

With nothing returned, the steady intensity above threshold equals the excess pump.

Steady external-cavity solutions

\begin{aligned}\Phi&=\phi+\Omega\tau\\N_s&=-\kappa\cos\Phi\\\Omega&=-\kappa(\alpha\cos\Phi+\sin\Phi)\end{aligned}

Even when the intensity is steady, the optical frequency can be shifted.

Spectrum of a windowed record

\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}

Summed over all frequency bins, the spectrum gives back the window-weighted power. This is Parseval's theorem, and the Lab checks it separately.

An ideally coupled response laser

\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}

When both lasers obey identical equations, an exactly synchronized solution exists. A mismatch can destroy it.

Rescaling the whole delay history

\begin{aligned}H(t)&=\frac1\tau\int_{t-\tau}^t|\delta E(s)|^2ds\\\|\delta X\|^2&=\delta N^2+H(t)\\\lambda_T&=\frac1T\sum_j\log r_j\end{aligned}

The exact linearized equations run along the same trajectory as the laser. Each rescaling covers the entire stored history; the initial transient is left out, and the two halves of the record and a finer time step are compared. The result is still a finite-time estimate.

Noise linewidth with weak feedback, computed separately

\begin{aligned}d\varphi&=-a(\varphi-\varphi_\tau)dt\\&\quad+\sqrt{2\pi\Delta\nu_0}\,dW\\\Delta\nu_{\rm weak}&=\frac{\Delta\nu_0}{(1+a\tau)^2}\end{aligned}

This adiabatic model with a noisy phase runs only when weak feedback leaves a single steady solution. The variance of the recorded phase over long lags measures phase diffusion; it is not the width of a chaotic spectrum.

Common difficulties

A broad spectrum means chaos

Typical misconceptionIf the spectrum is broad, the laser must be on a chaotic attractor.

Better mental modelA finite record, a transient or periodic sidebands can also broaden a spectrum. Use the growth rate of the rescaled full history, and check it with a finer step and in both halves of the record. Even then, one finite record does not prove a long-time attractor.

More feedback is always better

Typical misconceptionA stronger returned beam always stabilizes the laser.

Better mental modelThe delay and the return phase also change stability, so compare what the recorded dynamics actually do.

Run the experiment

01
Compare automatically acquired records
Choose No returned field, then Weak feedback. Each selection integrates six thousand photon lifetimes and acquires its final 1024 samples.
What to observe: Both can retain a nearly steady intensity. The narrow finite-window field spectrum can shift even when there is no resolved intensity modulation.
02
Change the delay and the phase
Drag the delay mirror or compare Delayed dynamics with Changed return phase. Longer delay remains in Advanced controls.
What to observe: The intensity, the field coherence and the spectra all come from the same simulated trajectory.
03
Test synchronization
Turn up Ideal field-error coupling, then add Response pump mismatch.
What to observe: Compare the recorded field error as well as the intensity traces. Replay reads a fixed trajectory; explicit further computation is in Advanced controls.
04
Tell deterministic growth from noise
Open Advanced controls. With Changed return phase, choose Extend chaos diagnostic. Record weak-feedback noise separately for No returned field and Weak feedback.
What to observe: Check that small perturbations grow in both halves of the record, and that weak feedback slows the phase diffusion.