L28 · Let one laser set another’s rhythm

Injection Locking

How can a weak master laser make a stronger slave laser follow its frequency? The injected field restores the relative phase toward a steady offset. With weak injection and little noise, a slave inside the locking range follows the master frequency. Stronger injection widens that range. Outside it the phase keeps slipping; near the edge noise can also cause slips inside the noise-free range.

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

Elapsed model time $\text{—}$

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

Final 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 laser make a stronger slave laser follow its frequency?

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 phase diffusion with a noisy lock, using the same seed and linewidth. Near the edge, noise can cause phase slips even where the noise-free model would lock.

This is the Adler phase model for weak injection into a single-mode laser with fixed amplitude, not a full simulation of carriers and photons in a semiconductor laser. Relative phase is slave minus an ideal master, and a positive detuning means the free-running slave has the higher frequency. The noise-free locking half-width is an assumed field coupling rate times the square root of the injected-to-slave power ratio. That ratio is at most one percent, and the coupling efficiency is folded into the illustrative rate. Optical powers and spatial mode matching stay fixed. The slave has white phase diffusion, chosen so that on its own it has the set Lorentzian linewidth; the ideal master has no phase noise. Seeded pseudorandom Gaussian steps make each run reproducible, and a noisy trajectory is one realization of the model, not experimental data. The deterministic reference assumes long times and no noise. Noise can still cause phase slips inside that reference range, and exactly at the boundary the restoring rate falls to zero. Blocking or opening the injection keeps the current phase but starts a new record and clock. An ideal coherent optical hybrid gives both quadratures, and the signed spectrum is the Hann-windowed transform of the recorded complex phase samples. It is a periodogram of a finite record, not an ideal Lorentzian curve drawn in. Its half-power width depends on the window; it is a spectral diagnostic, not the intrinsic linewidth or proof that real hardware is stable. Up to 4096 final samples at 400 megasamples per second give a resolution of about 98 kilohertz. A mean relative frequency of zero does not by itself prove phase coherence. The model leaves out amplitude noise, gain saturation, linewidth enhancement, carrier dynamics, multiple longitudinal modes, spatial injection effects, technical drift, detector noise and chaotic behavior under strong injection, and it makes no claim about locking boundaries in those cases. Apparatus dimensions, beam colors and routing are schematic.

Start with the essentials

Focus question: How can a weak master laser make a stronger slave laser follow its frequency?
One-sentence intuition: Locking acts like a restoring force on the relative phase; just outside the locking range, the phase slips periodically instead.

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}

Both the relative phase and the free-running detuning are taken as slave minus master. The injected field grows with the square root of the injected power.

Inside the locking range there is a stable phase

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

At the edge, where the two sides are equal, the restoring rate drops to zero, so the lock there is marginal, not robust.

How fast the phase recovers

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

Near the edge, small disturbances die away slowly. A large disturbance can push the phase over into the neighboring well.

Outside the locking range the 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 long-time average relative frequency without noise, not the instantaneous rate of phase change.

Phase noise of the free-running slave

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

The diffusion strength is chosen so that the slave on its own has a Lorentzian linewidth. Locking holds the relative phase in place, but noise can still cause slips.

Recording 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 signed spectrum is the Fourier transform of the complex samples with a Hann window. Its width includes the effect of the finite observation time.

Common difficulties

Locked phases must be equal

Typical misconceptionLocking means the relative phase is zero.

Better mental modelA constant phase offset lets the injection balance the detuning. The stable offset changes with the detuning.

A narrow FFT peak gives the intrinsic linewidth

Typical misconceptionThe width of the periodogram is the true laser linewidth.

Better mental modelThe Hann window, the finite record and the ideal noiseless master all affect this measurement. Check the coherence and the phase slips as well.

Run the experiment

01
Lock the slave, then disturb it
Choose Locked, then click Disturb phase. The new response is recorded automatically; compare the first microsecond with the stable phase readout.
What to observe: The phase settles back to a stable offset, which is not necessarily zero.
02
Go past the locking edge
Choose Frequency pulling, then weaken the injection. Click Block injection; a new record starts from the reached phase.
What to observe: Past the edge, the phase keeps winding and the beat does not vanish, although the slave frequency is pulled toward the master.
03
Separate noise from resolution
Compare Free phase diffusion with Suppress relative noise, then choose Noise-driven slips. Noise seed and other secondary settings are in Advanced controls.
What to observe: Holding the relative phase in place concentrates the recorded spectrum. Near the edge, noise can cause slips even inside the range where the noise-free model locks.

Injection Locking

A weak field can stop a phase from running away