Coherence Length Calculator
Interference fringes vanish when the two arms of an interferometer differ by more than the light stays in step with itself. That distance is the coherence length, set almost entirely by how narrow the source’s spectrum is: the narrower the linewidth, the longer the coherence.
Enter the linewidth in hertz and this calculator returns the coherence length in metres from Lc = c / (π x Δν), the Lorentzian convention of RP Photonics and Saleh and Teich, at the SI speed of light. If your datasheet quotes a wavelength bandwidth instead, switch the tool to wavelength mode: the centre wavelength and Δλ are converted, and the same π convention applied. One warning, because most pages skip it: coherence length has several competing definitions differing by small numerical factors, and this page states its convention openly and shows what the others give.
How it was calculated
Read the result as the Lorentzian convention: Lc = c / (π x Δν), the length where coherence has fallen to 1/e (RP Photonics). The 2π form returns exactly half; the rough estimate c / Δν returns about 3.14 times more. The table below gives all three; wavelength mode applies the same convention as Lc = λ² / (π x Δλ).
Quick Answer
Key takeaways
- Coherence length is coherence time times the speed of light, shrinking in exact inverse proportion to linewidth: halve the linewidth, double the length.
- Published formulas differ by factors of π and 2π depending on the spectral shape assumed and where the cutoff is drawn. A calculator that hides its factor hides up to 6x in the answer.
- In wavelength mode the centre wavelength and its bandwidth Δλ drive the result through Lc = λ² / (π x Δλ): the same π convention, reached by a different door.
- Use the FWHM linewidth from the datasheet, in the mode that matches it. Mixing Δλ into a frequency formula is the most common way to get a wildly wrong answer.
- Quick Answer
- How to use the coherence length calculator
- The formula this calculator uses
- Worked example: a 10 MHz laser linewidth
- Why textbooks quote different coherence lengths for the same source
- Where the number decides the design
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
- References
How to use the coherence length calculator
The "Linewidth given as" switch picks the input mode; only that mode's fields are shown. In frequency mode, enter the linewidth in hertz in the Δν field; that number drives the calculation. A 10 MHz linewidth is entered as 10000000, a 100 kHz source as 100000. In wavelength mode, enter the centre wavelength λ and the wavelength bandwidth Δλ, both in metres (550 nm is 0.00000055; 1 nm of bandwidth is 0.000000001); the tool converts the bandwidth to a frequency linewidth, applies the same π formula, and shows the conversion in the steps. The speed of light comes pre-filled at the exact SI value, 299,792,458 m/s, from the NIST CODATA fundamental constants listing; leave it unless you are deliberately modelling a medium. Press Calculate for the coherence length in metres (exponential notation), with the substitution underneath.
The formula this calculator uses
Coherence length is the distance light travels during its coherence time, Lc = c x τc. Everything after that is a choice about how coherence time relates to linewidth.
This is the Lorentzian form: it assumes a Lorentzian spectrum, the line shape of a source whose optical phase undergoes a random walk, and marks the length where the coherence function has fallen to 1/e. RP Photonics publishes this definition, and Saleh and Teich use the same relation for the path difference where fringe visibility falls to one half. The 2π version, built on the angular-frequency form Δω = 2πΔν, returns exactly half; it comes from one-radian phase-criterion derivations, not the Lorentzian linewidth definition, so it appears below as a comparison, not as the tool's own form.
Worked example: a 10 MHz laser linewidth
The calculator is pre-filled with this case; the first row below is exactly what it returns.
| Convention | Formula | Result at Δν = 10 MHz |
|---|---|---|
| This calculator (Lorentzian, RP Photonics) | c / (π x Δν) | 9.542690 m |
| Angular-frequency form | c / (2π x Δν) | 4.771345 m |
| Order-of-magnitude estimate | c / Δν | 29.979246 m |
| Wavelength form of this calculator | λ² / (π x Δλ) | Same as c / (π x Δν), by the Δν = cΔλ / λ² conversion |
Lengths scale inversely with linewidth under every convention: at 1 MHz the three formulas give 95.43 m, 47.71 m, and 299.79 m; at 1 GHz, a broad diode source, 9.54 cm, 4.77 cm, and 29.98 cm. The ratios never change; only the strictness does.
Wavelength mode has its own worked example, also pre-filled: 550 nm centre wavelength, 1 nm bandwidth. The tool converts first: Δν = c x Δλ / λ² = 299,792,458 x 0.000000001 / (0.00000055)², about 991 GHz. The π formula then gives Lc = λ² / (π x Δλ) = 9.628874e-5 m, about 96.3 µm: a micrometre-scale coherence device, fit for OCT rather than holography.
Why textbooks quote different coherence lengths for the same source
Coherence does not switch off at a wall; it decays, and each definition picks a different point on the decay. The RP Photonics coherence length reference defines it for a Lorentzian spectrum as the length where the coherence function has fallen to 1/e, producing the c / (π x Δν) form, and notes the literature often drops the π for order-of-magnitude estimates. A Gaussian spectrum gives another factor. Saleh and Teich's Fundamentals of Photonics, in the spectral width and fringe visibility problem, uses the same relation, tied to the path difference where fringe visibility falls to one half.
The practical rule: compare like with like. A coherence length quoted without a convention can be off by a factor of 2 to 6 from the one in your lab notebook, with no measurement error involved. Find a datasheet's definition before using its number.
Where the number decides the design
Interferometry. Fringes stay visible only while the arm path difference stays well inside the coherence length. A Michelson setup working over metre-scale differences needs a source whose coherence length clears them on the strictest convention you trust.
Holography. Reference and object beams travel different paths to the plate, so source coherence length caps the scene depth you can record. This is why holography waited for lasers: a filtered lamp's coherence length is micrometres to millimetres.
Optical coherence tomography. OCT inverts the logic: its axial resolution is a deliberately short coherence length, and broadband sources a few micrometres long slice tissue layer by layer.
Fibre and free-space links. Source coherence interacts with dispersion and interferometric noise in sensing systems, and sits alongside the signalling rate (the territory of the baud rate calculator) when a coherent link is budgeted. On the bench, the neighbouring tools are the ball lens focal length calculator for focusing elements and the aperture area calculator for beam openings.
What competing calculators miss
The tools reviewed here are richer in features, and quieter about what matters most.
Miss 1: the hidden factor. Rival tools variously build in λ² / Δλ, λ² / (n x Δλ), λ² / (2 x Δλ), or c / Δν, each presented as the formula, with no mention that other definitions differ by factors of π or 2π. Compare two tools and you get answers up to 6x apart, unexplained. This page prints its Lorentzian π convention in the calculator header, in the result, and in a side-by-side table.
Miss 2: one domain only. Tools in this niche split into camps: wavelength-only tools that leave frequency-specified laser users to convert by hand, and frequency-only tools that do the reverse. This calculator takes either input: a frequency linewidth in hertz, or a centre wavelength with its bandwidth, with the conversion shown in the steps.
Miss 3: tables that fail their own formula. One reviewed page prints a table listing a 1 GHz bandwidth in vacuum as 300 metres of coherence length, where its own c / Δν formula gives 0.30 m: a thousandfold error in a reference table. Every number on this page is produced by the calculator's own code and checked against it.
One honest limit: a single linewidth number cannot capture a source with a complicated spectrum, such as a frequency comb or a multimode laser with uneven mode spacing. For those, the coherence function itself must be measured or modelled.
Frequently Asked Questions
It is the distance over which a light wave stays phase-coherent with itself: beyond it, two copies of the beam delayed by that much no longer produce stable interference fringes. It equals the coherence time multiplied by the speed of light, Lc = c x coherence time.
Lc = c / (π x Δν), the Lorentzian convention published by RP Photonics and used in Saleh and Teich's Fundamentals of Photonics: the length where the coherence function has fallen to 1/e, for a Lorentzian spectrum with FWHM linewidth Δν. The stricter 2π form returns exactly half; the rough estimate c / Δν returns about 3.14 times as much. Wavelength mode runs the identical convention as Lc = λ² / (π x Δλ).
Switch "Linewidth given as" to wavelength bandwidth, then enter the centre wavelength λ and the wavelength bandwidth Δλ, both in metres. The tool converts the bandwidth to a frequency linewidth with Δν = c x Δλ / λ² and applies the same π formula: the same calculation as Lc = λ² / (π x Δλ). In frequency mode the wavelength fields are hidden and play no part, because a linewidth given in hertz needs no conversion.
Use Δν = c x Δλ / λ², all in SI units: c in metres per second, λ and Δλ in metres, giving Δν in hertz. Example: at λ = 632.8 nm with Δλ = 0.001 nm, Δν is about 749 MHz, and the coherence length is about 12.7 cm under the π convention. Wavelength mode performs this conversion for you and shows it in the steps.
Enter the full width at half maximum (FWHM) linewidth from the source datasheet, which is what the standard formulas assume. If a datasheet quotes a coherence length instead, work backwards from its own convention if it states one; treat unstated conventions as approximate.
It spans an enormous range because linewidths do. Narrow-linewidth lasers reach tens to hundreds of metres or more; a 10 MHz source gives 9.54 m, as the worked example shows. Broadband sources such as LEDs sit at the other end, typically micrometres to millimetres, and white light is shorter still, around a micrometre.
Yes. The formulas use the vacuum speed of light. In a medium with refractive index n, light travels at c / n, so the coherence length there is the vacuum value divided by n. Model it by entering the reduced speed in the speed of light field: about 2.05e8 m/s for optical fibre with n near 1.46.
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References
- RP Photonics, "Coherence Length": defines coherence length as coherence time times the speed of light, gives the Lorentzian formula c / (π x Δν) with the 1/e criterion, and notes the π is often dropped for order-of-magnitude estimates. RP Photonics: Coherence Length. Accessed October 2026.
- National Institute of Standards and Technology, "Fundamental Physical Constants," 2022 CODATA adjustment: the speed of light in vacuum, 299,792,458 m/s, exact by definition, which pre-fills this calculator. NIST CODATA: Fundamental Physical Constants. Accessed October 2026.
- Saleh and Teich, Fundamentals of Photonics, Chapter 11 problem on the effect of spectral width on fringe visibility, presented by Vaia: for a Lorentzian line shape the coherence length is c / (π x Δν), tied to the path difference where visibility falls to one half. Vaia: Effect of Spectral Width on Fringe Visibility. Accessed October 2026.