Coherence Time Calculator

Calculator

Choose a method, enter values, then submit. Results appear above this form.

Calculation method

Pick whichever matches your known parameters.

Coherence convention

Different texts use different constants.

Output unit

Pick a convenient engineering unit.

Linewidth Δν

Enter the spectral linewidth in frequency.

Δν unit

Central wavelength λ₀

Required for converting Δλ to Δν.

λ₀ unit

Wavelength linewidth Δλ

Small-bandwidth approximation is assumed.

Δλ unit

Quality factor Q

Uses Δν = ν₀ / Q.

Coherence length Lc

Lc corresponds to useful interference length.

Lc unit

Refractive index n

Approximates group velocity as c/n.

Example data table

Method	Inputs	Convention	Output τc
Frequency linewidth	Δν = 1 MHz	τ = 1/(πΔν)	≈ 318 ns
Wavelength linewidth	λ₀ = 1550 nm, Δλ = 0.01 nm	τ = 1/(πΔν)	≈ 255 ps
Quality factor	λ₀ = 1064 nm, Q = 1×10⁸	τ = 1/(2πΔν)	≈ 5.65 µs
Coherence length	Lc = 10 m, n = 1.47	τ = 1/Δν	≈ 49.0 ns

Values are illustrative and depend on convention and approximations.

Formula used

From frequency linewidth: τ_c = 1 / (k · Δν), where k ∈ {1, π, 2π}.
From wavelength linewidth: Δν ≈ (c / λ₀²) · Δλ, then τ_c = 1 / (k · Δν).
From quality factor: ν₀ = c / λ₀, Δν = ν₀ / Q, then τ_c = 1 / (k · Δν).
From coherence length: τ_c ≈ (n · L_c) / c, using v ≈ c/n.

Here c is the speed of light in vacuum. The small-bandwidth approximation is assumed when converting between Δλ and Δν.

How to use this calculator

Select a calculation method that matches your available specifications.
Choose the convention factor that your reference uses.
Enter values with correct units, then press Calculate.
Review the coherence time shown above the form.
Use Download CSV or Download PDF for reports.

Notes for advanced work

Linewidth may refer to FWHM, RMS, or equivalent noise bandwidth.
The factor k changes with spectral shape and definition.
For dispersive media, group index is more accurate than n.
For short pulses, coherence time differs from pulse duration.

Why coherence time matters in optical systems

Coherence time (τc) estimates how long a wave keeps a predictable phase relationship. It directly influences interference visibility, fringe contrast, coherent detection, and the ability to resolve fine optical path differences in sensors, metrology, and imaging. In communication links, it also affects phase noise tolerance and carrier recovery margins.

Link between spectral linewidth and coherence time

A narrow spectral linewidth produces a long τc. For many stationary sources, τc scales inversely with frequency linewidth Δν. A common estimate is τc ≈ 1/(πΔν), but the exact factor depends on how linewidth is defined and the assumed spectral shape.

Spectral-shape conventions and definition factors

Real spectra are rarely ideal. Lorentzian spectra often appear in phase-noise-dominated lasers, while Gaussian spectra are common when many independent broadening mechanisms add statistically. This calculator lets you choose a convention factor k so results match references that use 1/Δν, 1/(2πΔν), or 1/(πΔν).

From wavelength bandwidth to frequency bandwidth

If you know wavelength FWHM Δλ near a center wavelength λ0, the small-bandwidth approximation converts it to frequency bandwidth as Δν ≈ (c/λ0²)·Δλ. This works best when Δλ is much smaller than λ0 and dispersion is modest, such as many narrowband lasers.

Coherence length and propagation in media

Coherence length relates to coherence time by Lc ≈ vg·τc, where vg is group velocity. A simple estimate uses vg ≈ c/n. In fiber with n≈1.468, a τc of 1 ns corresponds to Lc of about 0.204 m, which helps set allowable path mismatch in interferometers.

Typical numbers to sanity-check results

Broadband emitters have very short τc: an LED with tens of nanometers bandwidth can yield τc on the order of femtoseconds. A laser with 1 MHz linewidth gives τc ≈ 0.318 µs using 1/(πΔν). An ultra-narrow 1 kHz laser can reach τc ≈ 0.318 ms.

Measurement and reporting considerations

Linewidth may be reported as FWHM, RMS, or “effective” noise bandwidth. Some instruments measure a short-term linewidth, while others capture long-term drift that broadens the spectrum. For meaningful comparisons, keep the same definition and averaging time when converting to τc.

Design use-cases for engineers and researchers

Use τc and Lc to estimate interference fringe visibility versus delay, set coherence gating windows, and assess coherent receiver performance. For interferometric sensing, keep path difference below Lc for strong fringes. In coherent radar and lidar, τc helps estimate range ambiguity from coherence gating. For pulsed sources, treat τc as spectral coherence, distinct from pulse width.

Frequently asked questions

1) What does coherence time represent?

It is the characteristic time over which the field phase remains correlated. Longer coherence time means the wave can interfere with delayed versions of itself over longer delays.

2) Which input method should I use?

Use frequency linewidth if you have Δν directly. Use wavelength bandwidth if you have Δλ and a center wavelength. Use coherence length if your system spec is given as an allowable path mismatch.

3) Why does changing the convention factor change τc?

Different textbooks and instruments define linewidth and coherence using different constants (π or 2π) and spectral shapes. The factor aligns the calculation with the specific definition used by your source.

4) Can I use the wavelength conversion for large bandwidths?

It is most accurate when Δλ is small compared with λ0. For very broad spectra, the linear conversion can introduce error; consider using a full spectrum model or measured Δν when available.

5) Does refractive index change coherence time?

Coherence time is a property of the source spectrum. The refractive index mainly affects coherence length because it changes propagation speed, so the same τc corresponds to different Lc in different media.

6) How can I estimate τc from time-domain data?

You can approximate τc from the decay of the field autocorrelation or fringe visibility versus delay. Fit the envelope (often exponential or Gaussian) and relate the fitted decay constant to τc.

7) Is coherence time the same as pulse duration?

No. Pulse duration describes temporal envelope length. Coherence time describes phase correlation set by spectral width. A short pulse can have high coherence if its spectrum is narrow, and vice versa.