Formula and Method for Laser Linewidth and Bandwidth
A laser's output is never a perfectly pure single frequency — spontaneous emission causes the phase of the light field to diffuse randomly over time, spreading the optical power over a narrow but finite range of wavelengths or frequencies. That spread, measured at half the peak power (full width at half maximum, FWHM), is the linewidth. It can be quoted either as a wavelength spread Δλ (picometers or nanometers) or a frequency spread Δν (hertz) — the two describe the same physical bandwidth. This calculator converts between them and derives the coherence time, coherence length, and quality factor that follow from the linewidth.
How the calculation works
Frequency and wavelength are related by ν = c/λ. Differentiating gives the small-signal conversion Δν = c·Δλ / λ², which turns a wavelength-domain linewidth into a frequency-domain linewidth (or the reverse). For a laser whose linewidth comes from phase diffusion driven by spontaneous emission — the standard model behind the Schawlow-Townes linewidth — the optical spectrum has a Lorentzian shape, and the field's phase memory decays exponentially with time constant (coherence time) τc = 1 / (π·Δν). Multiplying by the speed of light gives the coherence length Lc = c·τc = c / (π·Δν), the maximum path-length difference over which the light can still interfere with a delayed copy of itself. Finally, the quality factor Q = ν / Δν = λ / Δλ expresses the linewidth as a fraction of the optical carrier frequency — a convenient, unit-independent measure of spectral purity.
Common mistakes
- Mixing up Δλ and Δν: a linewidth of "1 nm" and a linewidth of "1 GHz" are wildly different fractions of the spectrum — always convert to the same domain before comparing two lasers.
- Forgetting the lineshape assumption: τc = 1/(πΔν) and Lc = c/(πΔν) assume a Lorentzian lineshape (phase-diffusion broadening). A Gaussian-broadened source (e.g., Doppler-broadened gas lasers, or lasers dominated by 1/f technical noise) has a different numeric prefactor, roughly 0.66×c/Δν instead of about 0.32×c/Δν.
- Using Δλ ≥ λ: the Δν = cΔλ/λ² conversion is a linear (small-signal) approximation and breaks down once the linewidth becomes comparable to the wavelength itself, which never happens for a real single-mode laser but can happen from a typo.
Real-world applications
- Fiber-optic communications (DWDM) rely on Δν to set channel spacing so adjacent wavelength channels do not overlap.
- Spectroscopy and metrology need laser linewidth narrower than the spectral features being resolved, or the measurement itself washes out the signal.
- Interferometry, LIDAR, and optical coherence tomography (OCT) need round-trip path differences to stay within the coherence length Lc to see interference fringes.
- Laser stabilization and frequency-comb work quote the quality factor Q as a quick, comparable measure of how "pure" a given laser source is.