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Sandbox Physics

L34 · Driven nonlinear ring

Microresonator Kerr Comb

Evolve a periodic complex Kerr field. Capture its pulse and discrete spectrum, scan detuning and test thermal drift, dispersion and pulse survival.

Interactive modelMicroresonator Kerr Comb
Slow time / field decay time—\text{—}
Current cold-cavity detuning—\text{—}
Effective detuning—\text{—}
Thermal resonance shift—\text{—}
Mean normalized intracavity power—\text{—}
Non-pump mode power fraction—\text{—}
Periodic field cells—\text{—}
Captured spectral modes—\text{—}
Illustrative mode spacing—\text{—}
Captured peak intensity—\text{—}
Captured localized peak count—\text{—}
Relative instantaneous PDE residual—\text{—}
Outer-third spectral energy fraction—\text{—}
Captured field description—\text{—}
Captured slow time—\text{—}

Physics tutorial

A comb emerges from one circulating field

BackgroundA continuous pump can support localized pulses in a passive Kerr ring without population inversion.

Why it mattersWhen does a driven ring keep a localized pulse?

Start with the essentials

Focus question
When does a driven ring keep a localized pulse?
One-sentence intuition
Pulse survival, formation and thermal access are distinct experiments.

Core mathematical model

Periodic mean field

∂τψ=−(1+iαeff)ψ+F+iγ∣ψ∣2ψ−iβ2∂θ2ψ\begin{aligned}\partial_\tau\psi={}&-(1+i\alpha_{\rm eff})\psi+F\\&+i\gamma|\psi|^2\psi-i\tfrac{\beta}{2}\partial_\theta^2\psi\end{aligned}

Anomalous dispersion is negative; the periodic field resolves interacting longitudinal modes.

Chosen thermal pole

H˙=(χ⟨∣ψ∣2⟩−H)/10αeff=αcold−H\begin{aligned}\dot H&=(\chi\langle|\psi|^2\rangle-H)/10\\\alpha_{\rm eff}&=\alpha_{\rm cold}-H\end{aligned}

Heating lowers the resonance and reduces the positive red detuning in this sign convention.

CW response

F2=P[1+(αeff−γP)2]F^2=P\left[1+(\alpha_{\rm eff}-\gamma P)^2\right]

The multivalued response supplies backgrounds; it does not replace the spatial evolution.

Modulation growth

Ωm=−αeff+2γP+12βm2λm=−1+(γP)2−Ωm2\begin{aligned}\Omega_m&=-\alpha_{\rm eff}+2\gamma P+\tfrac12\beta m^2\\\lambda_m&=-1+\sqrt{(\gamma P)^2-\Omega_m^2}\end{aligned}

A positive real part permits sideband growth from the specified perturbation.

A prepared approximate pulse

ψ≃ψcw+Asech⁡(bθ)A=2α eiφb=2α/∣β∣\begin{aligned}\psi&\simeq\psi_{\rm cw}+A\operatorname{sech}(b\theta)\\A&=\sqrt{2\alpha}\,e^{i\varphi}\\b&=\sqrt{2\alpha/|\beta|}\end{aligned}

This seeds the survival experiment only; the solver must sustain or destroy it.

One field, two domains

am=12π∫−ππψe−imθ dθ∑m∣am∣2=⟨∣ψ∣2⟩\begin{aligned}a_m&=\frac1{2\pi}\int_{-\pi}^{\pi}\psi e^{-im\theta}\,d\theta\\\sum_m|a_m|^2&=\langle|\psi|^2\rangle\end{aligned}

Parseval links the recorded pulse to the discrete spectrum; the comb is not an independent ideal drawing.

Common difficulties

A comb proves a soliton

Typical misconceptionAny regularly spaced sidebands certify a quiet solitary pulse.

Better mental modelPatterns also create combs; inspect the time envelope, residual and long evolution.

A seed is spontaneous formation

Typical misconceptionThe prepared pulse emerged spontaneously during the detuning scan.

Better mental modelPrepared pulses test survival. Scans begin from a perturbed CW field and insert no pulse.

Run the experiment

  1. 01

    Test survival

    Evolve the prepared single and double pulse presets for sixty decay times.

    What to observe: Compare localized peak count, a small residual and the measured Fourier spectrum.
  2. 02

    Remove support

    Repeat with normal dispersion or zero Kerr strength.

    What to observe: The same initial pulse can decay into a continuous field.
  3. 03

    Scan for formation

    Choose the scan start, scan to the target, then vary target, speed and heating.

    What to observe: The computed power history may show multiple pulses, steps or complete comb loss.