L30 · Follow a phase until a line appears

Laser Linewidth & Phase Noise

Record a reproducible phase walk and compute its beat spectrum, frequency-noise density and coherence. Change photon lifetime, power, technical noise and observation time.

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 short and long records, then add reference linewidth. A beat measures both lasers, and one finite FFT peak cannot certify the linewidth of either one.

A fixed-amplitude, single-mode, good-cavity phase-diffusion teaching model at an assumed wavelength of one thousand sixty-four nanometers. The chosen fully inverted ideal gain has spontaneous emission into the mode equal to its photon loss rate. Output loss is half of total loss. Mean cavity photon number follows output power and photon lifetime; the specified Schawlow–Townes-type diffusion convention divides the loss rate by four pi times that photon number to define Lorentzian FWHM. This is a declared ideal scaling, not a universal formula for semiconductor, bad-cavity or multimode lasers. Henry enhancement, incomplete inversion, nonorthogonal modes, amplitude noise and thermal laser dynamics are omitted. Independent white phase diffusion is generated for the tested laser and reference, so their Lorentzian widths add in the relative beat. Additional technical frequency noise is a stationary Ornstein–Uhlenbeck process with specified RMS and correlation time. Its endpoint and time integral are sampled jointly with exact Gaussian covariance; sinusoidal vibration is integrated analytically. Seeded pseudorandom Gaussian increments give a reproducible simulated realization, not measured data or an individual spontaneous photon-event simulation. An ideal coherent optical hybrid records both quadratures at one megasample per second with a nominal positive beat of fifty kilohertz. Phase unwrapping is available perfectly in the model; real receiver slips, shot noise and alias prevention electronics are not simulated. The optical periodogram uses a Hann window on the captured complex samples. The one-sided frequency-noise PSD averages four nonoverlapping, mean-removed Hann windows of recorded phase increments and therefore has four times coarser bin spacing. All diagnostics use the same frozen record. The symmetric half-power span is a finite-record diagnostic, not a Lorentzian FWHM fit. Field coherence is a finite-record lag average and remains statistically uncertain at long lag. A separate green coherence reference includes white noise only. The continuous frequency-noise reference includes stationary correlated noise but excludes the vibration line and finite sampling effects; compare it well below Nyquist. Changing observation length changes resolution and statistical variability; noise realizations differ even with identical source parameters. Spatial geometry and optical colors are schematic.

Start with the essentials

Focus question: How does phase noise become a laser linewidth?
One-sentence intuition: Line shape depends on the noise process, reference laser and observation window, not just the tallest FFT bin.

Core mathematical model

Declare an ideal quantum scaling

\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}

The chosen ideal gain supplies one spontaneous mode photon per loss time. The factor and assumptions are declared, not universal.

Diffuse phase

\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}

Independent white phase diffusion in laser and reference adds in the relative beat linewidth.

Give technical noise a memory

\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 stationary correlated frequency noise is integrated jointly with its endpoint, rather than drawing unrelated spectra.

Separate white and colored 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 one-sided continuous reference omits vibration lines and sampling effects. The acquired PSD uses recorded phase increments.

Turn phase diffusion into coherence

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

This exponential is the white-noise-only ensemble reference. A finite lag average need not lie exactly on it.

Transform the actual record

\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; frequency PSD averages four shorter windows. Half-power span is not fitted FWHM.

Common difficulties

One linewidth describes all noise

Typical misconceptionEvery broad line is Lorentzian and its tallest-bin width is intrinsic.

Better mental modelColored noise and vibration change line shape; define the measurement window and diagnostic before assigning a width.

A beat measures only the tested laser

Typical misconceptionReference noise can be ignored when reading a beat.

Better mental modelIndependent white widths add. Without a known quiet reference, the beat cannot identify either laser alone.

Run the experiment

01
Dilute phase kicks
Capture quantum, bright and shorter-lifetime presets with the same seed.
What to observe: Higher power reduces ideal diffusion; shorter photon lifetime increases it.
02
Add mechanisms
Capture correlated drift, vibration and a noisy reference.
What to observe: Slow phase wandering, discrete sidebands and a broadened reference beat have different causes.
03
Change the observation
Compare short and long records, then draw a new realization.
What to observe: Resolution and statistical variation affect the apparent spectral span.