L28 · Let one laser set another’s rhythm

Injection Locking

Change detuning and injection strength, disturb the phase or close the shutter. Record an I/Q beat spectrum and find where lock turns into phase slips.

Elapsed model time $\text{—}$

Noise-free locking half-width $\text{—}$

Wrapped relative phase $\text{—}$

Accumulated relative phase turns $\text{—}$

Deterministic instantaneous frequency $\text{—}$

Separate long-time pulling reference $\text{—}$

Noise-free stable phase $\text{—}$

Linearized phase recovery time $\text{—}$

Normalized I-channel intensity $\text{—}$

Normalized Q signal $\text{—}$

Injected-to-slave power ratio $\text{—}$

Captured I/Q samples $\text{—}$

Finite-record frequency resolution $\text{—}$

Captured half-power spectral span $\text{—}$

Captured relative-phase coherence $\text{—}$

Captured mean relative frequency $\text{—}$

Model regime $\text{—}$

L28

How can a weak master control a stronger slave?

Free slave minus master frequency

\Delta f

2\,\mathrm{MHz}

Injected-to-slave power ratio

10\log_{10}\rho

-30\,\mathrm{dB}

Assumed field coupling rate

\kappa

100\,\mathrm{MHz}

Free slave phase-diffusion linewidth

\Delta\nu_s

0\,\mathrm{MHz}

Initial relative phase

\phi_0

-120\,\mathrm{{}^\circ}

Injection shutter: blocked / open

b

1

Reproducible noise realization

s

17

Detail: slave / receiver / attenuator

j

0

Compare free diffusion with the noisy locked preset using the same seed and linewidth. Near the boundary, noise can cause phase slips even though the noise-free reference admits lock.

A weak-injection, fixed-amplitude single-mode Adler phase model, not a full semiconductor carrier–photon simulation. Relative phase is slave minus ideal master; positive detuning means the free slave frequency is higher. The assumed field coupling rate times the square root of injected-to-slave power ratio gives the noise-free locking half-width. The ratio is limited to one percent or less; coupling efficiency is absorbed in this illustrative rate. Optical powers and spatial mode matching stay fixed. White slave phase diffusion is chosen so the isolated field has the specified Lorentzian linewidth; the ideal master has no phase noise. Seeded pseudorandom Gaussian increments make runs reproducible; the noise trajectory represents a model realization, not experimental data. The deterministic reference assumes asymptotic time and zero noise. Noise can produce phase slips within that reference interval, and a marginal boundary has no positive restoring rate. A shutter action keeps the reached phase but starts a new record and clock. An ideal coherent optical hybrid supplies both quadratures; the signed spectrum transforms recorded complex phase samples with a Hann window. It is a finite-record periodogram, not a generated ideal Lorentzian curve. Its half-power span is a window-dependent spectral diagnostic, not intrinsic linewidth or a hardware stability certificate. Up to 4096 final samples at 400 megasamples per second give a resolution of about 98 kilohertz. Zero mean relative frequency alone does not prove phase coherence. Amplitude noise, gain saturation, linewidth enhancement, carrier dynamics, multiple longitudinal modes, spatial injection, technical drift, detector noise and chaotic strong-injection regimes are omitted. No claims about their locking boundaries are made. Apparatus dimensions, beam colors and routing are schematic.

Start with the essentials

Focus question: How can a weak master control a stronger slave?
One-sentence intuition: Lock is a restoring force on relative phase; just outside the boundary it becomes periodic phase slipping.

Core mathematical model

The weak-injection phase equation

\begin{aligned}\dot\phi&=2\pi\Delta f-K\sin\phi\\K&=2\pi\kappa\sqrt{\rho}\end{aligned}

Relative phase and free detuning are slave minus master. Field injection scales as the square root of power.

A stable phase exists inside the boundary

\begin{aligned}|\Delta f|&<\kappa\sqrt{\rho}\\\phi_*&=\arcsin\frac{\Delta f}{\kappa\sqrt{\rho}}\end{aligned}

At equality the restoring rate vanishes; the boundary is marginal rather than robust lock.

Measure the recovery time

\begin{aligned}\tau^{-1}&=K\cos\phi_*\\\dot\epsilon&=-\epsilon/\tau\end{aligned}

Small disturbances decay slowly near the edge. Large disturbances may wrap into a neighboring phase well.

Outside lock, frequency is pulled

\begin{aligned}\bar f_{\rm rel}&=\operatorname{sgn}(\Delta f)\sqrt{u}\\u&=\Delta f^2-\kappa^2\rho\end{aligned}

This is the noise-free long-time average outside the locking region, not an instantaneous derivative.

Add free-running phase diffusion

\begin{aligned}d\phi&=\dot\phi\,dt+\sqrt{2D}\,dW\\D&=\pi\Delta\nu_s\end{aligned}

The chosen diffusion gives an isolated Lorentzian field linewidth. Locking confines relative phase, but noise can cause slips.

Acquire both quadratures

\begin{aligned}z_n&=\cos\phi_n+i\sin\phi_n\\I_n&=1+\cos\phi_n\\\delta f&=1/(N\delta t)\end{aligned}

The recorded signed spectrum transforms the complex samples with a Hann window; its width includes the finite observation time.

Common difficulties

Locked phases must be equal

Typical misconceptionLock requires zero relative phase.

Better mental modelA constant phase offset balances detuning against injection. The stable offset changes with detuning.

One narrow FFT peak proves intrinsic linewidth

Typical misconceptionThe periodogram width is a certified laser linewidth.

Better mental modelThe Hann window, finite record and ideal noiseless master affect the diagnostic. Use coherence and phase slips alongside it.

Run the experiment

01
Catch and disturb the slave
Step the locked preset, disturb its phase, then capture a run.
What to observe: The phase returns to a stable offset rather than necessarily becoming zero.
02
Open the boundary
Capture frequency pulling, then weaken injection. Block injection and keep the phase already reached.
What to observe: Beyond the boundary phase winds and the beat stays nonzero, although it is pulled toward the master.
03
Separate noise from resolution
Capture free diffusion and noisy lock with the same noise seed. Then inspect the noise-driven-slip preset.
What to observe: Confining relative phase concentrates the recorded spectrum. Near the edge, noise can create slips inside the deterministic range.