Laser Linewidth and Bandwidth Calculator

Laser Linewidth and Bandwidth Calculator for the Schawlow–Townes limit, Γ from mirror reflectivity, exact Δλ to Δν conversion, coherence length, and the bandwidth of a short pulse.

Laser Linewidth and Bandwidth Calculator Modes

The Laser Linewidth and Bandwidth Calculator works in seven modes, picked from the Solve For list. Each mode takes the center wavelength (or the optical frequency in THz) and one or two other values, then reports the same set of linked results.

Under the main result, four cards turn the same Δν into numbers you can use at the bench. They give the wavenumber and wavelength width, the coherence length and time, the widths at −10 and −20 dB with the matching self-heterodyne beat, and the cavity Q, finesse or natural linewidth. A dB chart, a table of the same Δν at seven common laser lines and a step list sit in the panels below.

Semiconductor lasers need the Henry α factor, which sits in the Advanced panel. The mirror mode adds the group index and the internal loss there too. The calculator runs in any phone browser, so it works as a laser linewidth and bandwidth calculator app without an install, and inputs never leave the device.

The Laser Linewidth and Bandwidth Calculator covers the spectrum of the beam, not its shape in space. The Laser Beam Divergence Calculator handles how the beam spreads, and the Laser Beam Expander Calculator sizes the optics to cut that spread. The Laser Brightness Calculator turns power and beam quality into radiance.

Linewidth vs Bandwidth

Laser linewidth is the full width at half maximum (FWHM) of the optical spectrum of one oscillating mode. It is usually quoted in Hz, kHz or MHz, and it tells you how pure that single frequency is. The RP Photonics entry on linewidth defines it as the FWHM of the power spectral density of the emitted field.

Laser bandwidth is the wider term. It can mean the gain bandwidth of the medium, the span of all oscillating modes, or the spectrum of a pulse, and it is often quoted in nm. A He–Ne tube, for example, has a gain bandwidth near 1.5 GHz while one of its modes can be far narrower.

Frequency and bandwidth also differ in a simple way. Frequency is where the light sits, about 473.8 THz for a 632.8 nm line. Bandwidth is how far the light spreads around that point. “Bandwidth” and “band width” mean the same thing, and the joined spelling is the standard one.

How to Calculate Laser Linewidth

The quantum limit of a single-frequency laser comes from the Schawlow–Townes equation. Its modern form uses FWHM values for both the laser line and the cold-cavity bandwidth, and it already includes Lax’s factor of one half for a laser running above threshold.

$$\Delta\nu = \frac{\pi h \nu \Gamma^2}{P}$$

Here h is the Planck constant (6.62607015 × 10⁻³⁴ J·s, exact in the SI), ν is the laser frequency and P is the output power. Γ is the FWHM bandwidth of the passive cavity. It is a frequency width in Hz, not the cavity Q, even though the two are linked by Q = ν/Γ.

The 1958 paper by Schawlow and Townes wrote the result as 4πhν(Δν_c)²/P with half widths. Converting both widths to FWHM and applying the factor of two from Lax gives the πhνΓ²/P form above, as explained in the Schawlow–Townes linewidth article.

Take a 632.8 nm laser with Γ = 1 GHz and P = 1 W. The frequency is c ÷ λ = 473.755 THz, so Δν = π × 6.62607015×10⁻³⁴ × 4.73755×10¹⁴ × (10⁹)² ÷ 1 = 0.9862 Hz. That line has a cavity Q of 473,800 and a coherence length of 96,760 km.

A red laser pointer is a more everyday case. With 635 nm, Γ = 10 GHz and P = 5 mW, the quantum limit works out to 19.66 kHz. Plugging 1 GHz into that example by mistake gives 19.66 Hz, so the Γ unit is worth a second look before you trust the answer.

Henry Factor for Laser Diodes

Semiconductor lasers couple gain and phase, which widens the line beyond Schawlow–Townes. Henry’s 1982 theory multiplies the result by 1 + α², where α is the linewidth enhancement factor.

$$\Delta\nu_{diode} = \frac{\pi h \nu \Gamma^2}{P}\left(1 + \alpha^2\right)$$

An α of 4 widens the line 17 times. That is why monolithic diode lasers usually sit in the megahertz range, while a careful single-frequency solid-state laser measures a few kilohertz over 1 s. Both values are still far above their quantum limit, because vibration, temperature drift and pump noise add technical broadening the equation does not include.

Cavity Linewidth From Mirrors

When Γ is not known, the mirror mode of the Laser Linewidth and Bandwidth Calculator builds it from the cavity. The free spectral range is c ÷ (2nL), where n is the group index of the cavity and L its length. The exact Airy FWHM then follows from the mirror reflectivities.

$$\Gamma = FSR \cdot \frac{2}{\pi}\arcsin\left(\frac{1 – \sqrt{R_1 R_2}}{2\left(R_1 R_2\right)^{1/4}}\right)$$

Take a 300 µm Fabry–Pérot diode at 1550 nm with a group index of 3.6 and two cleaved facets. A facet in a material of index 3.6 reflects about 32%, from (n − 1)² ÷ (n + 1)², so use 30% on each side. The FSR is 138.8 GHz, Γ comes out at 61.25 GHz, and the finesse is only 2.27.

At 10 mW that cavity gives a Schawlow–Townes linewidth of 151 kHz. Adding α = 4 raises it to 2.57 MHz, with a coherence length near 37 m. The result lands in the megahertz range expected for a diode, and the Index of Refraction Calculator helps when you need n from another material.

Mirrors are not the only loss. Absorption and scattering inside the cavity take a share of the light on every pass, and the Internal Loss (αᵢ) field in cm⁻¹ folds that loss into the round trip.

$$\rho = \sqrt{R_1 R_2}\,e^{-\alpha_i L}$$

The same diode with αᵢ = 10 cm⁻¹ drops ρ from 0.300 to 0.222. Γ widens to 85.71 GHz and the finesse falls to 1.62, so the quantum limit nearly doubles to 296 kHz, or 5.03 MHz with α = 4. Cavities this lossy sit at the edge of the formula, since the Schawlow–Townes result assumes losses well below 1 per round trip.

A resonance needs ρ of at least 0.172, the point where the Airy peaks still reach half height. With mirrors alone that means R₁ × R₂ ≥ 2.94%, and with two 30% facets it caps αᵢ at about 18.6 cm⁻¹ for a 300 µm chip. Above that limit the calculator flags the mirror fields instead of giving a Γ.

Wavelength Bandwidth Formula

Spectrometers report a width in nm, while noise and RF work need Hz. The common shortcut in the wavelength bandwidth formula divides by λ², and it holds well for narrow lines such as a single laser mode.

$$\Delta\nu \approx \frac{c}{\lambda^2}\Delta\lambda$$

The exact version sets the band evenly in wavelength around λ and takes the frequency gap between its two edges. The convert mode of the Laser Linewidth and Bandwidth Calculator uses this form in both directions.

$$\Delta\nu = \frac{c}{\lambda – \Delta\lambda/2} – \frac{c}{\lambda + \Delta\lambda/2} = \frac{c\,\Delta\lambda}{\lambda^2 – \Delta\lambda^2/4}$$

A 411.6 nm helium–cadmium line with a 1 nm band gives 1.770 THz. For a broadband source the two forms part ways. A 400 nm band centered at 1000 nm gives 124.9 THz exactly but 119.9 THz from the shortcut, a 4% error that the calculator flags with a warning.

The table below works as a quick wavelength to frequency bandwidth calculator for common lines, using the same exact rule as the Laser Linewidth and Bandwidth Calculator. It lists the frequency width of a 1 nm band and the wavelength width of a 1 GHz line.

Wavenumber widths need no approximation at all, because wavenumber is proportional to frequency. One cm⁻¹ equals 29.98 GHz at any wavelength. For a plain wavelength to frequency calculator answer, ν = c ÷ λ is enough, and the calculator shows that step first in its solution panel.

Coherence Length From Linewidth

Coherence length is how far the light travels before its phase memory fades. For a Lorentzian line, the shape set by white frequency noise, the 1/e definition from the coherence length article gives this result.

$$L_c = \frac{c}{\pi\,\Delta\nu}$$

A Gaussian line, typical of Doppler-broadened gas lines and superluminescent diodes, reaches the same 1/e point later. Its coherence length is longer by a factor of 2√(ln 2), about 1.665.

$$L_c = \frac{2\sqrt{\ln 2}\,c}{\pi\,\Delta\nu}$$

Many sources quote Lc ≈ λ²/Δλ, which equals c/Δν and drops the π. That shortcut is an order-of-magnitude estimate and runs about three times longer than the FWHM-based Lorentzian value. Used as a coherence length calculator, the coherence mode works backward from the length you need to the widest Δν that still delivers it.

In a Michelson interferometer the light crosses the arm difference twice. The usable arm mismatch is therefore half the coherence length, and the calculator lists it on the same card. Holography follows the same rule, since the reference and object paths must stay within one coherence length of each other.

3 dB, 6 dB and 20 dB Widths

The 3 dB point is used because 10 log₁₀(0.5) = −3.01 dB, which is exactly half power. The −3 dB bandwidth and the FWHM of a power spectrum are the same number. The 6 dB width is the quarter-power width, and the −20 dB width sits at 1% of the peak.

How much wider a line gets at lower levels depends on its shape. A Lorentzian keeps strong wings, so its −20 dB width is almost ten times its FWHM. A Gaussian falls fast and only reaches 2.58 times.

The Line Shape in dB chart in the Laser Linewidth and Bandwidth Calculator draws both shapes at the same FWHM. It makes clear why a Lorentzian line measured at −20 dB can look ten times wider than its quoted linewidth, and the third result card gives that −20 dB width in Hz.

Laser Linewidth Measurement

A grating spectrum analyzer suits wide lines, roughly above 10 GHz. A 0.02 nm resolution at 1550 nm is already 2.5 GHz, so a narrow diode line only shows the instrument’s own width. The Diffraction Grating Calculator shows where that resolving limit comes from.

For single-frequency lasers the usual method is the delayed self-heterodyne test. Part of the beam passes a long fiber delay, part is shifted by an acousto-optic modulator, and the two are beat on a photodiode. The self-heterodyne method needs a delay longer than the coherence length so the two arms are uncorrelated.

The beat of two uncorrelated Lorentzian copies is twice as wide as the laser line. Its −20 dB width is 2√99, or 19.90, times the laser FWHM, so dividing the measured −20 dB beat width by 19.9 gives Δν.

A Gaussian line widens by √2 under the same test, so its −20 dB beat is 3.646 times the laser FWHM. A 100 kHz laser has a coherence length of 954 m in vacuum, and the fiber delay must beat that figure after dividing by the fiber’s group index.

The beat mode of the Laser Linewidth and Bandwidth Calculator runs this step in reverse. A 2 MHz beat read at −20 dB on a Lorentzian line gives a laser linewidth of 100.5 kHz. The third card then shows the 201 kHz beat FWHM you should see on the same trace.

Strong 1/f frequency noise breaks the simple rule, and a too-short delay then reads a linewidth that is too small. Beating against a second, quieter laser avoids the delay line altogether and resolves sub-kilohertz lines.

Pulse Bandwidth and Transform Limit

A short pulse cannot have a narrow spectrum. The time–bandwidth product sets the minimum, with both widths taken as FWHM. It is 2 ln 2/π ≈ 0.441 for a Gaussian pulse and about 0.315 for a sech² pulse.

$$\Delta\nu = \frac{TBP}{\tau}$$

A 100 fs Gaussian pulse at 800 nm needs at least 4.413 THz, which is 9.42 nm of spectrum. The same pulse with a sech² shape needs 3.148 THz, or 6.72 nm. The pulse spans 37.5 optical cycles and is 30 µm long in space.

Chirp only adds spectrum, so a measured Δν·τ above these values means the pulse is chirped. The pulse mode of the Laser Linewidth and Bandwidth Calculator reports this limit along with the bandwidth for the other pulse shape.

Ultrafast sources also sit at the far end of laser bandwidth, with terahertz-wide spectra where a single-mode diode has megahertz.

Typical Laser Linewidths

The three families most people mean by laser types are gas, solid-state and semiconductor lasers. Their linewidths span many orders of magnitude, as the published figures below show.

The solid-state, diode and stabilized figures come from the RP Photonics linewidth article, and the He–Ne value from its He–Ne laser entry. A He–Ne tube’s Brewster windows fix its polarization, a topic covered by the Brewster’s Angle Calculator.

For VCSELs, a Chalmers study of 1060 nm single-mode VCSELs measured lines as narrow as 6 MHz. A 660 nm single-mode VCSEL datasheet caps spectral width at 100 MHz. A 940 nm VCSEL array datasheet lists 0.5 nm typical, a width of about 170 GHz.

A VCSEL is a vertical cavity surface-emitting laser, a diode that emits from the chip surface between two Bragg mirrors. Those mirror stacks work on the same interference rule as the Bragg’s Law Calculator, and the short cavity favors single-frequency operation.

Natural and Doppler Linewidth

An isolated atom at rest emits a Lorentzian line whose width is set by the lifetime τ of its excited state. This natural linewidth is the floor for any line from that transition.

$$\Delta\nu_{nat} = \frac{1}{2\pi\tau}$$

The rubidium-87 D2 line at 780.24 nm has τ = 26.23 ns, which gives 6.07 MHz. That matches the 2π × 6.0666 MHz in Steck’s rubidium D line data. Natural broadening is the lifetime mechanism, and the natural linewidth is the FWHM it produces.

Thermal motion in a gas spreads the line much further. The result is Gaussian, and its FWHM follows the formula in the Doppler broadening article, with k_B the Boltzmann constant, T the gas temperature and m the atomic mass.

$$\Delta\nu_D = \frac{2\nu_0}{c}\sqrt{\frac{2\ln 2\,k_B T}{m}}$$

Rubidium-87 vapor at 300 K, with a mass of 86.909 u, gives a Doppler width of 511.3 MHz. That is about 84 times its natural width, and the mean atom speed is 270 m/s. Neon at its natural atomic mass of 20.18 u and a 400 K discharge gives 1.511 GHz at 632.8 nm, in line with the 1.5 GHz He–Ne gain bandwidth.

When both widths matter, the line is a Voigt profile, the Gaussian and Lorentzian combined. The Doppler mode reports its FWHM from the Olivero–Longbothum fit, accurate to about 0.02%.

$$\Delta\nu_V \approx 0.5346\,\Delta\nu_{nat} + \sqrt{0.2166\,\Delta\nu_{nat}^2 + \Delta\nu_D^2}$$

For the rubidium case the Voigt width is 514.5 MHz, so the natural part adds only 3 MHz. Enter the temperature in K or °C, and the steps panel shows the conversion to kelvin before the Doppler formula.

Laser Bandwidth Questions

These answers cover the questions people search most often next to a laser linewidth result. Each one uses numbers you can reproduce in the Laser Linewidth and Bandwidth Calculator with the same inputs.

What is the bandwidth of a laser?

It depends on what is being counted. A single-mode laser has a linewidth from below 1 Hz in stabilized systems to megahertz in diodes. A multimode laser’s bandwidth covers all its running modes, and a pulsed laser’s bandwidth is set by its pulse length through the time–bandwidth product.

Datasheets often give the wider figure in nm. Enter that value in the convert mode of the Laser Linewidth and Bandwidth Calculator to see it in GHz, along with the coherence length it allows.

What does 20 MHz bandwidth mean for a laser?

A 20 MHz linewidth at 1550 nm is only 0.160 pm wide, far below what a grating analyzer can show. As a Lorentzian line it keeps phase coherence over 4.77 m, with a coherence time of 15.9 ns.

That is enough for a 2.4 m arm mismatch in a Michelson interferometer. It would not suit coherent sensing over hundreds of meters, which needs a line in the kilohertz range.

How do I convert frequency to bandwidth in nm?

Divide the frequency width by c and multiply by λ², or use the exact inverse when the band is wide. At 1550 nm, 1 GHz equals 8.014 pm and 1 MHz equals 8.014 fm.

Pick a Δν unit in the convert mode and the calculator returns Δλ directly. It also shows how far the linear shortcut strays from the exact value, which only matters once the band is a few percent of λ.

Are lasers AC or DC?

Most continuous lasers run from DC. A He–Ne tube runs a DC glow discharge near 1 kV at about 10 mA, and a laser diode is driven by a steady DC current.

The light itself is an oscillating field, about 473.8 THz for a 632.8 nm line. Linewidth measures how much that optical frequency wanders, which is why current noise in a diode driver shows up directly as extra linewidth.

Why is a laser linewidth so far above the Schawlow–Townes value?

The equation covers quantum noise only. Mirror vibration, thermal drift of the cavity length, pump fluctuations and, in diodes, the α factor all add more. A diode with a 151 kHz quantum limit can show 2.6 MHz from α = 4 alone, before any technical noise.

Raising power and narrowing the cavity bandwidth lower the quantum limit. Real narrow-linewidth lasers also need stable mechanics and, at the very top, locking to a reference cavity.