The laser linewidth calculator above converts a laser's spectral width between the two units it is quoted in — frequency bandwidth in hertz and wavelength bandwidth in nanometres or picometres — and derives the coherence length, coherence time and optical Q factor that follow. It handles three different line-shape conventions, because coherence length is not a single unambiguous number and published figures for the same laser can differ by a factor of three depending on which definition was used.
Arb Digital publishes free engineering calculators, and this one carries the same warning as the rest of the laser set. Laser safety is governed by IEC 60825-1 internationally and by ANSI Z136.1 in the United States. The class printed on the equipment label determines the engineering and administrative controls that apply. No beam should ever be viewed directly and no beam should ever be aimed at a person, and invisible wavelengths — which include almost every narrow-linewidth telecom source — are more dangerous than visible ones because there is no blink reflex to protect the eye.
What This Laser Linewidth Calculator Does
A laser does not emit a single frequency. Its output occupies a narrow band, and the width of that band is the linewidth. It matters because it sets how far apart two points on a beam path can be while the light at them still interferes predictably, which is the entire basis of interferometry, coherent optical communication, laser ranging, holography and optical coherence tomography.
The tool takes the centre wavelength and the linewidth in whichever unit you have, converts it to the other, and computes the coherence time and coherence length under your chosen line-shape convention. It also reports the optical Q factor, which is the centre frequency divided by the linewidth — the same figure of merit used for a resonator, and a convenient dimensionless way to compare sources at different wavelengths.
How to Use It
- Enter the centre wavelength in nanometres. Use the vacuum wavelength; the refractive index field handles the medium separately.
- Say whether you are entering a frequency bandwidth or a wavelength bandwidth, then pick a matching unit from the second dropdown.
- Enter the linewidth value. Narrow-linewidth sources are usually quoted in kilohertz or megahertz; diode and broadband sources in nanometres.
- Choose a line-shape convention. Lorentzian with the pi factor is the standard for a laser above threshold; the simple form is common in datasheets.
- Set the refractive index if the light is travelling in a medium rather than vacuum, and read the coherence length, coherence time and Q factor.
The Formula: How Linewidth and Coherence Length Are Calculated
The centre frequency follows from the wavelength through ν = c / λ, where c is the speed of light in vacuum, exactly 299,792,458 metres per second because the metre is defined from it — a definition maintained through the constants published by the National Institute of Standards and Technology, whose applied physics work includes the optical frequency metrology that makes hertz-level linewidth measurement possible at all. Differentiating that relation gives the conversion between the two bandwidth units:
Δλ = λ² Δν / c and Δν = c Δλ / λ²
The factor of λ² is the reason the same frequency linewidth corresponds to a much larger wavelength spread in the infrared than in the visible. It also means you can never convert between the two without knowing the centre wavelength.
Coherence time is the reciprocal of the linewidth, but with a prefactor that depends on the line shape:
Lorentzian: τc = 1 / (π Δν) · Gaussian: τc = √(2 ln 2 / π) / Δν ≈ 0.664 / Δν · Simple: τc = 1 / Δν
Coherence length is then Lc = c τc / n, where n is the refractive index of the medium the light is travelling in. The Lorentzian form is the physically correct one for a single-mode laser operating above threshold, where the line shape is set by spontaneous-emission phase diffusion. The Gaussian form suits a line broadened by many independent contributions, such as Doppler broadening in a gas. The simple form, which is equivalent to Lc = λ²/Δλ, is the one most often printed without comment on a datasheet.
The optical Q factor is Q = ν / Δν, exactly analogous to the Q of an electrical resonator. It is a useful cross-check because it is dimensionless and independent of wavelength, so it lets you compare a visible and an infrared source directly.
Work the defaults through by hand. A 1550 nm source with a 1 MHz linewidth, Lorentzian, in vacuum. The centre frequency is 299,792,458 / 1.55 × 10⁻⁶ = 1.934 × 10¹⁴ Hz, about 193.4 THz. The wavelength bandwidth is (1.55 × 10⁻⁶)² × 10⁶ / 2.998 × 10⁸ = 8.014 × 10⁻¹⁵ m, which is 8.014 femtometres or 0.008 picometres. The coherence time is 1 / (π × 10⁶) = 318.3 nanoseconds, so the coherence length is 2.998 × 10⁸ × 3.183 × 10⁻⁷ = 95.44 metres. The Q factor is 1.934 × 10¹⁴ / 10⁶ = 1.934 × 10⁸.
Laser Safety Is Governed by a Standard, Not by This Page
This page computes spectral quantities rather than irradiance, but it belongs to a set of tools describing real laser sources, and the same rules apply. Laser radiation damages eyes and skin, and retinal injury from a visible or near-infrared beam can be instantaneous and permanent. The governing documents are IEC 60825-1, "Safety of laser products — Part 1: Equipment classification and requirements", and ANSI Z136.1, "American National Standard for Safe Use of Lasers". They are cited here by number and title rather than reproduced, because the limits inside them only mean anything within their full methodology.
The class marked on the equipment is the practical entry point. Classification is done by the manufacturer under the standard, and the class dictates the controls: enclosures, interlocks, key switches, beam stops, controlled areas, wavelength-specific eyewear and documented training. Nothing on this page substitutes for that. If a figure from this tool or its companions ever needs comparing against a maximum permissible exposure, that comparison is a qualified laser safety officer's job.
Linewidth work carries a specific hazard worth naming. Narrow-linewidth sources cluster in the telecom bands around 1310 and 1550 nm, and in the 1064 nm region, all of which are completely invisible. There is no visible spot, no blink reflex and no aversion response. Fibre-coupled equipment adds the risk of an open connector or a broken fibre emitting into a room with nothing to indicate it is live. Treat every fibre end as energised, never look into a connector or a fibre end, and never aim a beam at a person. University environment, health and safety programmes such as Stanford Environmental Health & Safety set out how these controls are organised in practice.
Why Coherence Length Has No Single Correct Value
This is the most useful thing on the page. Ask three sources for the coherence length of a laser with a 1 MHz linewidth and you can get 95 m, 199 m or 300 m, all of them defensible. The spread comes entirely from the line-shape prefactor: the Lorentzian convention divides by pi, the Gaussian convention multiplies by about 0.664, and the simple convention uses one.
None of these is wrong. They answer slightly different questions about where fringe visibility has fallen to a particular level for a particular spectral profile. What is wrong is quoting a coherence length without saying which convention produced it, or comparing two sources whose figures were computed under different conventions.
There is a second ambiguity underneath. Linewidth itself is a measurement result, and it depends on the observation time over which it was measured. Technical noise — temperature drift, acoustic pickup, current-source noise — broadens the apparent line as the averaging time grows, so the same laser can measure 1 kHz over a microsecond and 100 kHz over a second. A serious specification states the measurement time along with the number.
What Linewidth Actually Limits in Practice
In interferometry, coherence length sets the maximum optical path difference over which fringes remain visible. A displacement interferometer with an unequal path length of a few metres needs a source whose coherence length comfortably exceeds it, which is why stabilised helium-neon and narrow-linewidth fibre sources dominate metrology.
In coherent optical communication, linewidth sets the phase-noise floor that the receiver's carrier recovery has to track. Higher-order modulation formats pack more bits into each symbol and are correspondingly less tolerant of phase noise, so the linewidth requirement tightens sharply as the constellation grows.
In optical coherence tomography the requirement is inverted. Axial resolution improves as coherence length shortens, so those instruments deliberately use broadband sources with linewidths measured in tens of nanometres. A narrow line would ruin the depth resolution. The same is true of white-light interferometry, where a short coherence length is what localises the fringe packet.
And in laser ranging and lidar, the coherent detection range is bounded by coherence length, because the returning light has to still interfere with the local oscillator after a round trip that may be kilometres long.
Where This Sits Next to the Other Optics Tools
This page covers the spectral side of beam quality. The spatial side is handled elsewhere: the laser spot size calculator gives beam radius, Rayleigh range and focused spot, and the laser brightness calculator gives radiance and beam parameter product. The laser beam expander calculator covers the telescope that trades diameter against divergence. Linewidth and M² are independent quantities: a beam can be spatially perfect and spectrally broad, or the reverse.
For the wave and photon relations underneath, the wavelength calculator converts between wavelength, frequency and period, the frequency converter handles unit changes across the enormous range involved, and the photon energy calculator gives energy per photon. The optical density calculator covers attenuation through a filter.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Quoting a coherence length without naming the convention — Lorentzian, Gaussian and simple definitions differ by up to a factor of three for the same laser.
- Converting bandwidth units without the centre wavelength — the conversion carries a factor of wavelength squared, so it is different at every wavelength.
- Ignoring the measurement time behind a linewidth figure — technical noise broadens the apparent line as averaging time grows, and a number without a time is incomplete.
- Forgetting the refractive index — coherence length inside silica fibre is shorter than in vacuum by about a factor of 1.47.
- Assuming a narrow line means a good beam — spectral and spatial quality are independent, and linewidth says nothing at all about M².
Related Free Tools From Arb Digital
Pair this with the laser spot size calculator for beam radius and Rayleigh range, the laser brightness calculator for radiance and beam parameter product, and the laser beam expander calculator for the divergence trade. The wavelength calculator, frequency converter and photon energy calculator handle the underlying wave and photon relations, and the optical density calculator covers filter attenuation. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the width of the narrow band of frequencies a laser actually emits, rather than the single frequency an idealised laser would emit. It is quoted either as a frequency bandwidth in hertz or as a wavelength bandwidth in nanometres or picometres, and the two are linked by the centre wavelength through a factor of wavelength squared.
Because coherence length depends on the assumed line shape. The Lorentzian convention divides by pi, the Gaussian convention multiplies by about 0.664, and the simple convention uses the reciprocal of the linewidth directly. All three are defensible answers to slightly different questions, and they can differ by a factor of three, so the convention must be stated alongside the number.
Multiply the frequency bandwidth by the square of the centre wavelength and divide by the speed of light. The wavelength squared term is essential, which is why the conversion cannot be done without knowing the centre wavelength and why the same frequency linewidth is a much wider wavelength band in the infrared than in the visible.
No. Linewidth describes spectral purity and the beam quality factor describes spatial quality, and they are independent. A source can have a superb single-frequency line and a poor multimode spatial profile, or a perfect fundamental-mode beam with a broad spectrum. Which matters depends entirely on the application.
Because some techniques rely on interference only over a very short path difference. Optical coherence tomography and white-light interferometry get their depth resolution from a short coherence length, so they deliberately use broadband sources with linewidths measured in tens of nanometres. A narrow-linewidth laser would destroy their axial resolution.
It is the centre frequency divided by the linewidth, exactly as for an electrical resonator. Because it is dimensionless it lets you compare sources at completely different wavelengths on the same scale. A 1 MHz linewidth at 1550 nm corresponds to a Q of nearly two hundred million.
Yes. Coherence time is a property of the source and does not change, but coherence length is coherence time multiplied by the speed of light in the medium. Inside silica fibre with a refractive index of about 1.47, the coherence length is shorter than the vacuum figure by that factor.
No. It computes spectral quantities only and issues no safety verdict of any kind. Laser safety is governed by IEC 60825-1 and ANSI Z136.1, the class marked on the equipment determines the required controls, and any exposure assessment is the work of a qualified laser safety officer. Never view a beam directly and never look into a fibre connector.
This tool is provided for educational and preliminary engineering use only. It converts spectral quantities and does not assess laser safety, does not publish maximum permissible exposure values and issues no safety verdict. Laser safety is governed by IEC 60825-1 and ANSI Z136.1, and the class marked on the equipment determines the required controls. Never view a laser beam directly, never look into a fibre connector or fibre end, and never aim a beam at a person. Any exposure assessment must be made by a qualified laser safety officer working to the governing standard.
