L30 · Follow a phase until a line appears

Laser Linewidth & Phase Noise

How does phase noise become a laser linewidth? The phase drifts randomly, so the longer you wait, the less the field keeps its earlier phase. That loss of coherence spreads the spectrum into a line of finite width. What you measure depends on the kind of noise, on the reference laser you beat against and on how long you observe; reading the width off the highest point of one spectrum is not enough.

Let the laser phase take a reproducible random walk, then look at the beat spectrum, frequency-noise density and coherence that come out of it. Change the photon lifetime, output power, technical noise and observation time, and compare.

Elapsed acquisition time $\text{—}$

Requested observation duration $\text{—}$

Chosen ideal quantum linewidth $\text{—}$

Mean cavity photon number $\text{—}$

Cold-cavity power linewidth $\text{—}$

Laser white-diffusion linewidth $\text{—}$

Separate white-only beat FWHM $\text{—}$

Separate white frequency-noise floor $\text{—}$

Separate white-noise coherence time $\text{—}$

Captured complex samples $\text{—}$

Optical spectrum bin spacing $\text{—}$

Four-window PSD bin spacing $\text{—}$

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

Captured mean beat frequency $\text{—}$

Captured frequency-increment RMS $\text{—}$

Captured coherence at longest lag $\text{—}$

Dominant selected mechanism $\text{—}$

L30

How does phase noise become a laser linewidth?

Laser output power

P_{\rm out}

0.5\,\mathrm{mW}

Total cavity photon lifetime

\tau_p

0.2\,\mathrm{ns}

Additional white-diffusion linewidth

\Delta\nu_w

0\,\mathrm{kHz}

Correlated frequency-noise RMS

\sigma_f

0\,\mathrm{kHz}

Frequency-noise correlation time

\tau_c

0.2\,\mathrm{ms}

Vibration frequency excursion

A_v

0\,\mathrm{kHz}

Vibration tone frequency

f_v

10\,\mathrm{kHz}

Independent reference linewidth

\Delta\nu_r

0\,\mathrm{kHz}

Record length as power of two

\log_2 N

13

Reproducible noise realization

s

23

Detail: test laser / receiver / reference

j

0

Compare a short record with a long one, then add reference linewidth. The beat carries the noise of both lasers, and one FFT peak from a finite record cannot pin down the linewidth of either laser.

This teaching model follows only the phase of a single-mode laser in a good cavity, with fixed amplitude and an assumed wavelength of one thousand sixty-four nanometers. The gain medium is ideal and fully inverted: spontaneous emission into the mode equals the photon loss rate, and output loss is half of the total loss. Output power and photon lifetime set the mean photon number in the cavity. The Lorentzian FWHM follows a Schawlow–Townes-type convention: the loss rate divided by four pi times that photon number. This ideal scaling is not a general formula for semiconductor, bad-cavity or multimode lasers. The model leaves out Henry linewidth enhancement, incomplete inversion, nonorthogonal modes, amplitude noise and thermal laser dynamics. The tested laser and the reference each get independent white phase diffusion, so their Lorentzian widths add in the beat. Extra technical frequency noise is a stationary Ornstein–Uhlenbeck process with a set RMS and correlation time. At each step, the noise value and its time integral are drawn together from their exact Gaussian covariance. Sinusoidal vibration is integrated analytically. Seeded pseudorandom Gaussian steps make each run reproducible; the result is a simulation, not measured data, and it does not follow individual spontaneous photons. An ideal coherent optical hybrid records both quadratures at one megasample per second, with a nominal positive beat of fifty kilohertz. Phase unwrapping is perfect in the model; real receiver phase slips, shot noise and anti-aliasing electronics are not simulated. The optical spectrum applies a Hann window to the captured complex samples. The one-sided frequency-noise PSD averages four nonoverlapping, mean-removed Hann windows of the recorded phase increments, so its bins are four times wider. Every diagnostic uses the same frozen record. The symmetric half-power span is measured on this finite record; it is not a Lorentzian FWHM fit. Field coherence is a lag average over the finite record and is statistically uncertain at long lags. The green coherence curve is a separate reference for white noise only. The continuous frequency-noise reference includes the stationary correlated noise but not the vibration line or finite sampling, so compare it only well below the Nyquist frequency. Changing the observation length changes the resolution and the statistical scatter, and different realizations differ even with identical source settings. Spatial layout and beam colors are schematic.

Start with the essentials

Focus question: How does phase noise become a laser linewidth?
One-sentence intuition: The shape of the line depends on the kind of noise, on the reference laser and on how long you observe; the tallest FFT bin alone does not tell you.

Core mathematical model

Ideal quantum-limited linewidth

\begin{aligned}\bar n&=\frac{P_{\rm out}\tau_p}{\eta_{\rm out}h\nu}\\\Delta\nu_q&=\frac{1}{4\pi\bar n\tau_p}\\\eta_{\rm out}&=\frac12\end{aligned}

In this ideal gain medium, spontaneous emission adds one photon to the lasing mode per photon lifetime. The numerical factor rests on these assumptions and does not hold for every laser.

Phase diffusion and the beat linewidth

\begin{aligned}d\phi&=2\pi f_b\,dt+\sqrt{2D}\,dW\\D&=\pi\Delta\nu_w\\\Delta\nu_b&=\Delta\nu_l+\Delta\nu_r\end{aligned}

When the laser and the reference each have independent white phase noise, their linewidths add in the beat note between them.

Technical noise with a correlation time

\begin{aligned}dX&=-\frac{X}{\tau_c}\,dt+\sqrt{\frac{2\sigma_f^2}{\tau_c}}\,dW\\\langle X(t)X(t+\tau)\rangle&=\sigma_f^2e^{-|\tau|/\tau_c}\end{aligned}

This frequency noise is stationary and stays correlated for about one correlation time. The model advances the noise and the phase it accumulates together, so every spectrum comes from the same record.

White and colored frequency-noise density

\begin{aligned}S_{\nu,w}(f)&=\Delta\nu_b/\pi\\S_{\nu,c}(f)&=\frac{4\sigma_f^2\tau_c}{1+(2\pi f\tau_c)^2}\\S_\nu&=S_{\nu,w}+S_{\nu,c}\end{aligned}

The green curve is a one-sided continuous reference; it leaves out vibration lines and sampling effects. The measured PSD is computed from the recorded phase increments.

Coherence from phase diffusion

\begin{aligned}|g^{(1)}(\tau)|&=e^{-\pi\Delta\nu_b|\tau|}\\\tau_{\rm coh}&=1/(\pi\Delta\nu_b)\end{aligned}

The exponential is the ensemble average for white noise alone. A coherence averaged over lags in one finite record can deviate from it.

Spectra of the recorded signal

\begin{aligned}z_n&=e^{i\phi_n}\\S_z(f)&\propto|\operatorname{DFT}(w_nz_n)|^2\\\delta f_{\rm opt}&=1/(N\Delta t)\\\delta f_{\rm noise}&=4\delta f_{\rm opt}\end{aligned}

The optical spectrum uses one Hann window over the whole record. The frequency-noise PSD averages four shorter windows. The half-power span is measured on the spectrum itself; it is not a fitted FWHM.

Common difficulties

One linewidth describes all noise

Typical misconceptionEvery broad line is a Lorentzian, and the width around its tallest bin is the laser's intrinsic linewidth.

Better mental modelColored noise and vibration change the shape of the line. Before you quote a width, state how long you observed and which measure of width you used.

A beat measures only the tested laser

Typical misconceptionYou can ignore the reference laser's noise when you read a beat note.

Better mental modelThe white-noise linewidths of two independent lasers add in the beat. Unless you know the reference is quiet, the beat cannot give you the linewidth of either laser on its own.

Run the experiment

01
Change power and photon lifetime
With the same seed, capture the Ideal quantum scaling, Brighter laser and Shorter photon lifetime presets.
What to observe: Higher power slows the ideal phase diffusion, and a shorter photon lifetime speeds it up.
02
Add technical noise
Capture the Correlated drift, Vibration sidebands and Noisy reference presets.
What to observe: Each source leaves a different mark: drift makes the phase wander slowly, vibration adds discrete sidebands, and a noisy reference broadens the beat.
03
Change the record length
Compare a short record with a long one, then press New noise realization.
What to observe: The apparent spectral width changes with frequency resolution and from one random realization to the next.