Laser Linewidth Calculator

How Lasers Achieve Coherence

Laser devices generate directional, coherent light by stimulating an excited medium to emit photons at a specific wavelength. Two mirrors form an optical cavity that amplifies photons travelling along the resonator axis while suppressing off-axis radiation. This resonant amplification forces the emitted light toward a dominant frequency.

In an ideal scenario with no external disturbances, all photons would possess identical energy and wavelength. However, quantum mechanical uncertainty and thermal fluctuations in the cavity inevitably introduce frequency drift. Spontaneous emission adds random phase noise, while vibrations and temperature variations shift the cavity's resonant frequency. The cumulative effect is a finite spectral width rather than emission at a single frequency.

The narrower the linewidth, the greater the coherence and temporal stability of the output. Applications requiring high coherence—such as optical communications, laser cooling, or gravitational wave detection—demand careful thermal stabilisation and mechanical isolation to minimise these sources of broadening.

Calculating Spectral Linewidth

The spectral linewidth of a laser follows from the energy-time uncertainty principle in quantum mechanics. The linewidth depends on three critical parameters: the laser's operating frequency, the power of the resonant mode, and the cavity's intrinsic spectral response.

• Δν — Spectral linewidth (Hz) — the full width at half maximum (FWHM) of the power spectrum
• h — Planck constant (6.626 × 10⁻³⁴ J·s)
• ν — Fundamental laser frequency (Hz)
• Γ — Cavity linewidth (Hz) — the spectral width of the cavity resonance
• P — Power of the laser mode (W)

Linewidth Versus Bandwidth: Distinction and Conversion

Linewidth and bandwidth are related but distinct concepts. The linewidth (FWHM) represents the actual frequency range occupied by the laser's output power. The bandwidth describes the range of frequencies the cavity can support, determined by mirror reflectivity and resonator geometry.

Bandwidth is often specified in the laser's datasheet in wavelength units (nanometres) rather than frequency. Converting between wavelength bandwidth and frequency bandwidth requires the Lorentz transformation:

Δν_band = (c × Δλ) ÷ (λ₀² − Δλ²/4)

where c is the speed of light, Δλ is the wavelength bandwidth, and λ₀ is the centre wavelength. For small bandwidths where Δλ ≪ λ₀, this simplifies to approximately (c × Δλ) ÷ λ₀². This nonlinear relationship explains why narrow linewidths in the infrared correspond to wider wavelength spreads than comparable linewidths at ultraviolet frequencies.

Practical Considerations for Linewidth Measurements

Achieving and verifying narrow linewidth operation requires attention to several factors.

1. Temperature Stabilisation Dominates — Cavity length changes with thermal fluctuations, shifting the resonant frequency and broadening the spectrum. Lasers with <1 MHz linewidths typically require temperature control better than ±0.01 °C. Passive thermal isolation often proves insufficient; active feedback systems stabilising cavity temperature to a reference thermistor are standard in precision applications.
2. Mode-Hopping Introduces Discontinuities — At marginal operating points, a laser can jump between adjacent longitudinal modes, each with different frequencies. This discontinuous frequency shifting appears as sudden linewidth broadening and coherence loss. Maintaining stable pump power and avoiding mechanical vibration near resonance frequencies prevents mode-hopping.
3. Output Power Scales Linewidth Inversely — According to the linewidth formula, doubling the laser power halves the spectral linewidth. However, higher power operation increases thermal load and can paradoxically degrade linewidth if cooling is inadequate. Trade-offs between optical power, thermal management, and electrical efficiency determine achievable performance.
4. Measurement Bandwidth Affects Observed Values — Spectrometers and optical heterodyne systems have finite resolution. A laser with 1 MHz true linewidth may appear to have 10 MHz width if measured with a 5 MHz resolution instrument. Cross-checking with multiple measurement techniques and understanding instrument specifications is essential for accurate characterisation.

Example: Red Laser Pointer Linewidth

A typical red laser pointer operating at 635 nm (fundamental frequency ≈ 472 THz) with 5 mW output power and a cavity linewidth of 10 GHz exhibits:

Δν = (π × 6.626×10⁻³⁴ × 472×10¹² × (10×10⁹)²) ÷ 0.005
Δν ≈ 19.7 kHz

This ~20 kHz linewidth corresponds to a relative spectral purity of about 1 part in 10⁷—excellent by everyday standards but modest for precision spectroscopy. Military-grade solid-state lasers can achieve sub-kilohertz linewidths through advanced frequency stabilisation, while semiconductor lasers without external feedback typically exhibit megahertz-scale linewidths.