Pulse Duration & Bandwidth Calculator

The trade-off in one line

A pulse of light is a sum of waves of different frequencies. The shorter the pulse, the more frequencies it needs: a pulse that lasts τ must contain a spread of frequencies of at least about 1/τ, because waves closer together in frequency than that would stay in step for longer than τ and could not switch the light off that quickly. This is a property of the Fourier transform, not of lasers, and it is exact. For a pulse with power envelope of full width at half maximum τ and a spectrum of FWHM Δν,

where K depends only on the pulse shape, and the equality holds when every frequency component peaks at the same moment, which means the spectral phase is flat and the pulse is transform-limited (also called bandwidth-limited or Fourier-limited). The products for the common shapes are listed below. The Gaussian, sech², and Lorentzian values are those of Diels and Rudolph’s Table 1.1; the rectangular and one-sided exponential values are computed here from the envelopes (the table lists the square pulse by its half-width, which gives 0.443, half the FWHM value, and does not list the one-sided exponential). Every value is also recomputed from the envelope:

The numbers are closed forms. For a Gaussian field \(e^{-(t/\tau_G)^2}\) the intensity FWHM is \(\tau = \tau_G\sqrt{2\ln 2}\) and the spectral intensity is a Gaussian of FWHM \(\Delta\nu = \sqrt{2\ln 2}/(\pi\tau_G)\), so \(\Delta\nu\,\tau = 2\ln 2/\pi\). The spectrum of \(\operatorname{sech}(t/\tau_s)\) is \(\operatorname{sech}(\pi^2\tau_s\nu)\), giving \(4\,\text{arccosh}^2\sqrt{2}/\pi^2\). The one-sided exponential has a Lorentzian spectrum of FWHM \(1/(\pi\tau_e)\) and an intensity FWHM of \(\tau_e\ln 2/2\). The rectangular pulse has a sinc² spectrum whose half-maximum points are at \(x = 1.3916\), giving \(2x/\pi\). The calculator’s value of K comes from these expressions, and a numerical Fourier transform of each envelope reproduces them to better than 1 %.

Bandwidth is one quantity in several units. With \(\lambda_0\) the center wavelength, \(c\) the speed of light, and \(h\) Planck’s constant,

The first relation is the derivative of \(\lambda = c/\nu\) and is accurate while \(\Delta\lambda \ll \lambda_0\); the calculator flags spectra wider than 20 % of the center wavelength, where it is only indicative. The other two are exact, which is why a width in cm⁻¹ or GHz means the same thing at every wavelength and a width in nm does not: 1 nm is 40 cm⁻¹ at 500 nm but 8.8 cm⁻¹ at 1064 nm (see energy units in spectroscopy). The number of optical cycles is \(\tau/(\lambda_0/c)\), with \(\lambda_0/c = 2.67\) fs at 800 nm, and the coherence length is \(c\tau\).

Why K is a floor, not a formula

The inequality says what a pulse of duration τ can have at best. Most pulses have more bandwidth than that, for one of three reasons.

Chirp. If the frequency sweeps across the pulse, the different colors arrive at different times and the pulse is longer than its spectrum requires. For a Gaussian pulse with a linear chirp, the product grows to \(\Delta\nu\,\tau = K\sqrt{1 + C^2}\), where C is the chirp parameter (Diels and Rudolph; Siegman, ch. 9). Chirp is also what dispersion does: a transform-limited Gaussian of duration \(\tau_{TL}\) that acquires a group delay dispersion \(D\) comes out with

with the spectrum unchanged. When you enter both a measured duration and a measured spectrum, the calculator reports \(\Delta\nu\,\tau/K\), the duration the spectrum could support, and the C and D that would explain the excess. Widths alone cannot give the sign of D, and a product above K can also come from satellite pulses, a pedestal, or a spectral phase that is not quadratic, so treat C as an estimate of how much phase there is to remove, not as a diagnosis of its form. Measuring the phase takes an interferometric method such as FROG or SPIDER (Diels and Rudolph, ch. 9).

Several longitudinal modes. A laser cavity oscillates on discrete frequencies spaced by \(c/2L\), 150 MHz for a 1 m cavity. A free-running Q-switched Nd:YAG laser runs on every mode that fits under its 0.6 nm gain curve (Paschotta, “YAG lasers”), and its bandwidth is set by how many modes oscillate, not by the pulse length. Spectra-Physics quotes under 1 cm⁻¹ (30 GHz) for the unseeded Quanta-Ray Pro, against a transform limit of 44 MHz for its 10 ns pulse: 700 times the limit. The same laser with an injection seeder, which forces oscillation on one mode, is specified below 0.003 cm⁻¹ (90 MHz), within a factor of two of the limit. The beating of many modes also puts fast spikes on the pulse, so a seeded pulse is smoother as well as narrower (Paschotta, “Injection seeding”).

Incoherent light. A flashlamp, an LED, or amplified spontaneous emission can have a nanometers-wide spectrum and last microseconds. Those are many independent wave trains, each of them short, with random phases; the envelope you measure is not a single pulse and the inequality does not relate its length to its spectrum. The time–bandwidth product is a statement about coherent pulses.

So, when a datasheet or a paper gives a duration, three questions decide what the bandwidth is: how much bandwidth does the gain medium or any filter in the system let through (an upper bound on the spectrum, and a lower bound on the duration of a transform-limited pulse); is the pulse transform-limited (only a measurement of both widths, or of the spectral phase, can tell); and how much dispersion has it seen since (every millimetre of glass and, for the shortest pulses, every metre of air adds chirp). The methods below are different answers to those questions.

How the pulse lengths are made, and what each method does not do

1. Q-switching gives nanoseconds; the cavity, not the pulse, sets the spectrum. A Q-switch blocks the cavity while the gain medium stores energy, then opens it so the stored energy leaves in a single pulse of a few to a few hundred nanoseconds, a duration of the order of several cavity round trips (Paschotta, “Q-switching”; Siegman, ch. 26). Nothing in that process selects frequencies: the pulse builds up on every longitudinal mode with gain, so its spectrum is the gain curve clipped by the mode comb, hundreds or thousands of times the transform limit. To approach the limit, a nanosecond laser needs an extra frequency-selective element. Injection seeding injects a weak single-frequency continuous-wave beam into the cavity as the pulse builds up, so the pulse grows from that light rather than from noise and comes out on one mode (Paschotta, “Injection seeding”); intracavity etalons and gratings do the same more lossily. Single-longitudinal-mode operation brings an 8–12 ns Nd:YAG pulse to below 0.003 cm⁻¹ (Spectra-Physics), that is 90 MHz against a Gaussian limit of 37–55 MHz, a factor of 1.6–2.4. What seeding does not do is shorten the pulse: the duration is still set by the gain, the cavity, and the Q-switch.

2. Mode locking gives picoseconds to femtoseconds, and the gain bandwidth sets the floor. A mode-locked laser oscillates on many longitudinal modes with their phases locked together, so that they add up to one short pulse circulating in the cavity; the more modes that lock, the shorter the pulse (Paschotta, “Mode locking”; Siegman, ch. 27). The number of modes is limited by the gain bandwidth of the medium, so the shortest pulse a mode-locked laser can produce is roughly \(K/\Delta\nu_\text{gain}\). That is why the gain medium matters so much more for a mode-locked laser than for a Q-switched one:

¹ Paschotta, “YAG lasers.” ² Paschotta, “Laser glasses” (bandwidths “of the order of tens of nanometers”; the fs figure is K/Δν for 10–25 nm, not a value from the source). ³ Paschotta, “Tungstate lasers.” ⁴ Light Conversion PHAROS datasheet. ⁵ Paschotta, “Titanium–sapphire lasers” (tuning range 650–1100 nm). ⁶ Asaki et al. (1993). ⁷ Ell et al. (2001). ⁸ Sung et al. (2017). ⁹ Paschotta, “Dye lasers.” ¹⁰ Sirah Cobra-Stretch and Quantel TDL90 datasheets. ¹¹ Paschotta, “Excimer lasers”; Gigaphoton G41K datasheet (0.35 pm). ¹² Coherent COMPex datasheet. ¹³ Laser Components and AOG Crystal Yb:YAG crystal data sheets (emission linewidth 9 nm). The implied limit is K/Δν_gain with K = 0.315; real oscillators usually stay a factor of a few above it, because the full gain curve is not used and because gain narrowing, dispersion, and the mode-locking mechanism all favor longer pulses (Paschotta, “Gain narrowing”).

Two cautions. First, the full gain bandwidth is rarely used, so a medium’s implied limit is a bound, not a prediction: the 9–15 nm quoted for Yb:YAG implies 75–125 fs, but Yb:YAG lasers mostly run at several hundred femtoseconds to a picosecond. Second, a mode-locked oscillator can emit chirped pulses, especially when it is not soliton mode-locked, so even an oscillator pulse should be checked against its spectrum before it is called transform-limited (Paschotta, “Time–bandwidth product”).

3. Chirped-pulse amplification (CPA) does not shorten anything. Amplifying a femtosecond pulse to millijoules or joules directly would destroy the amplifier, because the peak power rises as the pulse shortens (see the pulse energy and peak power calculator and the damage threshold calculator). Strickland and Mourou’s solution (1985) is to stretch the pulse with a dispersive element before amplification, so that its peak power is low while it is amplified, and to recompress it afterwards with the opposite dispersion, usually a grating pair; modern systems stretch femtosecond pulses by a factor of 10³–10⁴, to hundreds of picoseconds or a nanosecond. The bandwidth is carried through the amplifier; CPA simply makes it possible to amplify a short pulse without damage or nonlinear distortion (Paschotta, “Chirped-pulse amplification”). The compressed pulse is, at best, as short as the input pulse, and in practice longer, because an amplifier with finite gain bandwidth amplifies the center of the spectrum more than the wings: this gain narrowing removes bandwidth and lengthens the compressed pulse (Paschotta, “Gain narrowing”). Petawatt Ti:sapphire systems, which amplify by factors of 10¹⁰ or more, counter it with spectral shaping and still end up near 20–30 fs, where the oscillator that seeds them might produce 10 fs: the CoReLS 4.2 PW laser delivers 19.4 fs pulses from an 84 nm amplified spectrum, 1.7 times the Gaussian limit (Sung et al. 2017). Uncompensated higher-order dispersion between the stretcher and the compressor leaves a pedestal and lengthens the pulse further.

4. Optical parametric chirped-pulse amplification (OPCPA) keeps bandwidth because the gain is not stored in a medium. Dubietis, Jonušauskas, and Piskarskas (1992) replaced the laser amplifier in a CPA chain with a nonlinear crystal pumped by a strong, longer pulse, in which the stretched seed pulse is amplified by parametric down-conversion. The gain bandwidth is then set by phase matching rather than by an atomic transition, and with noncollinear phase matching it can span hundreds of nanometers; the crystal stores no energy and absorbs almost nothing, so there is little heat and little gain narrowing (Paschotta, “Optical parametric chirped-pulse amplification”). OPCPA systems reach 4–5 fs compressed pulses, which a Ti:sapphire amplifier cannot. What OPCPA does not do is relax the pump requirements: the pump pulse must overlap the stretched seed in time and carry the energy the amplified pulse will have.

5. Post-compression is the only way below the gain-bandwidth limit, because it adds new bandwidth. Everything above conserves or loses bandwidth. Self-phase modulation creates it: an intense pulse in a nonlinear medium (a noble gas in a hollow-core fiber, a bulk plate in a multipass cell, or a solid-core fiber for low energies) modulates its own phase through the Kerr effect, and the time-dependent phase shift means new frequencies, red in the leading edge and blue in the trailing edge of the pulse (Paschotta, “Self-phase modulation”). The spectrally broadened pulse is longer and strongly chirped; chirped mirrors or a prism or grating pair then remove the chirp, and the pulse comes out shorter than it went in (Paschotta, “Pulse compression”). Nisoli, De Silvestri, and Svelto introduced the gas-filled hollow fiber for this in 1996, producing 10 fs pulses at high energy, and the same group compressed 20 fs Ti:sapphire pulses to below 5 fs the following year (Nisoli et al. 1997). Multipass cells do the same at high average power (Gröbmeyer et al. 2020). Post-compression does not change the pulse energy much (the hollow fiber transmits most of it) and cannot be applied before amplification, since it needs the high intensity; it is the last stage.

6. The other direction: spectroscopy trades time for linewidth on purpose. To resolve rotational lines, which are a few cm⁻¹ apart in light molecules and a small fraction of a cm⁻¹ in heavy ones, a laser needs a linewidth narrower than the spacing, and the Fourier relation then forces the pulse to last nanoseconds. Resonance-enhanced multiphoton ionization (REMPI) and laser-induced fluorescence (LIF) are therefore mostly done with nanosecond dye lasers or optical parametric oscillators pumped by seeded Nd:YAG lasers: a 5 ns pulse may be as narrow as 88 MHz (0.003 cm⁻¹), and commercial dye lasers reach 0.03–0.1 cm⁻¹ with one or two gratings, a line Hänsch’s 1972 design first pushed below 0.004 Å (about 0.01 cm⁻¹). Picosecond lasers sit in between: a 1–2 ps Gaussian pulse has a transform-limited width of 7–15 cm⁻¹, which still resolves vibrational bands, hundreds of cm⁻¹ apart, but not rotational lines, and it gives time resolution on the scale of vibrational energy flow. Femtosecond pulses give the time resolution to watch nuclear motion but span 100 cm⁻¹ or more, so they excite several vibrational levels at once, which is the basis of wavepacket spectroscopy. A pulse that is narrow in frequency and short in time at the same time does not exist; the result row “what this bandwidth means for spectroscopy” says which side of the trade-off you are on.

Dispersion: how glass lengthens a pulse

In a transparent material the refractive index \(n\) falls with wavelength, so the group velocity \(v_g\) of a pulse depends on its color. The group velocity dispersion is the rate of change of the inverse group velocity with angular frequency, which in terms of \(n(\lambda)\) is

The calculator takes \(n(\lambda)\) from each material’s Sellmeier equation, differentiates it analytically for the Sellmeier forms and with a five-point stencil for the other fitted forms, and reports GVD in fs²/mm. The coefficients come from the refractiveindex.info database (Polyanskiy 2024), which is in the public domain, and from the original papers it cites: fused silica from Malitson (1965), measured 0.21–3.71 µm (the database extends the fit to 6.7 µm); N-BK7 from the SCHOTT catalog, 0.30–2.5 µm; sapphire (ordinary ray) from Malitson (1962), 0.27–5.6 µm; CaF₂ from Li (1980), 0.15–12 µm; MgF₂ (ordinary ray) from Dodge (1984), 0.20–7.0 µm; YAG from Zelmon et al. (1998), 0.4–5.0 µm; KDP (ordinary ray) from Zernike (1964), 0.21–1.53 µm; BBO (ordinary ray) from Tamošauskas et al. (2018), 0.19–5.2 µm; and air at 15 °C and 1 atm from Ciddor (1996), 0.30–1.69 µm. Outside these ranges the result is an extrapolation and the calculator says so. As checks, the Malitson equation gives 36.16 fs²/mm for fused silica at 800 nm, where Edmund Optics quotes 36 fs²/mm and Paschotta about 35 fs²/mm, and 18.97 fs²/mm at 1030 nm, the value Gröbmeyer et al. (2020) use for fused silica at 1 µm; at 1550 nm it gives −28 fs²/mm, matching the same paper, with the zero crossing near 1.27 µm.

A positive GDD delays the blue components relative to the red and stretches a transform-limited Gaussian pulse to \(\tau_{TL}\sqrt{1 + (4\ln 2\,D/\tau_{TL}^2)^2}\) (Siegman, ch. 9; Weiner, ch. 4). The effect scales as \(1/\tau^2\): 10 mm of fused silica at 800 nm (362 fs²) lengthens a 100 fs pulse by 0.5 %, a 30 fs pulse to 45 fs, and a 10 fs pulse to about 100 fs; for a 10 fs pulse even 1 mm gives 14 fs, and the 21 fs²/m of air matters over a few metres. For nanosecond pulses dispersion is irrelevant. Compensation means adding the same GDD with the opposite sign, which prism pairs, grating pairs, and chirped mirrors provide; below about 20 fs the third-order dispersion, which these do not cancel automatically, has to be managed as well (Paschotta, “Dispersion compensation”). The stretched-duration result assumes the pulse is transform-limited going in. A pulse that already carries chirp of the same sign broadens more than this, and one with the opposite chirp is compressed, which is exactly how a compressor works.

Worked examples

Ti:sapphire amplifier. A 100 fs Gaussian pulse at 800 nm has a transform-limited bandwidth of 0.4413/100 fs = 4.41 THz, which is 147 cm⁻¹, 18.3 meV, and \(\Delta\lambda = (800\ \text{nm})^2 \times 4.41\ \text{THz}/c = 9.4\) nm. It contains 37 optical cycles and is 30 µm long. If the spectrometer shows 12 nm, the product is 1.27 times the limit; the spectrum could support 79 fs, and a linear chirp with C = 0.79, equivalent to a GDD of about ±1 750 fs² or 49 mm of fused silica, would explain the difference. A 20 fs pulse at the same wavelength needs 47 nm and has 7.5 cycles; a 5 fs pulse would need 190 nm at 800 nm, a quarter of the center wavelength, which is why few-cycle spectra are described in frequency rather than in nm.

Nanosecond spectroscopy. A 5 ns Gaussian pulse can be as narrow as 88 MHz, or 0.0029 cm⁻¹, which at 560 nm is 0.09 pm. A dye laser specified at 0.06 cm⁻¹ (1.8 GHz) is therefore 20 times above the limit of its 5 ns pulse, and a seeded Nd:YAG at 0.003 cm⁻¹ with a 10 ns pulse is twice its limit of 44 MHz. Both are narrower than the Doppler width of a room-temperature gas line, 1–3 GHz in the visible (see the Doppler broadening calculator), so the sample, not the laser, limits the resolution.

Picosecond middle ground. A 1.5 ps Gaussian pulse has a transform-limited width of 294 GHz = 9.8 cm⁻¹. It resolves the vibrational bands of a molecule but spans the rotational lines of N₂ (4.0 cm⁻¹ apart) and of everything heavier; only H₂ and HCl, with 122 and 21 cm⁻¹ between lines, would still show rotational structure.

Assumptions and limits

• Widths are FWHM of the intensity in time and of the spectral intensity in frequency. Datasheets sometimes quote 1/e, 1/e², or full widths at 95 % energy; convert these first. A sech² pulse with parameter τ_s has FWHM 1.763 τ_s; for a Gaussian, the intensity FWHM is 1.665 times the intensity 1/e half-width and 1.177 times the intensity 1/e² half-width (which equals the field 1/e half-width τ_G of the table above).
• The transform limit depends on the shape, and a real pulse rarely has exactly one of the five shapes. An autocorrelation cannot distinguish Gaussian from sech², and the two products differ by 40 %, so quote the shape with the limit.
• Chirp from widths alone. The chirp parameter and the equivalent GDD assume a Gaussian with a purely linear chirp. Higher-order phase, pre- or post-pulses, and a pedestal all raise the product and are not distinguished. Nor is the sign of the chirp.
• Wavelength conversions are first-order. Δλ = λ₀²Δν/c is exact only in the limit of a narrow spectrum; above Δλ/λ₀ ≈ 0.2 (few-cycle pulses) the calculator marks the nm value as approximate. Air wavelengths and vacuum wavelengths differ by 0.03 %, which is negligible here.
• Dispersion is second order only. The broadening formula uses GDD and a transform-limited Gaussian input. Third-order dispersion, which matters below about 20 fs and for thick materials, and the chirp of a non-transform-limited input are not included. Birefringent crystals are given for the ordinary ray only; the Sellmeier fits hold within the ranges listed above, and air is taken as dry standard air at 15 °C.
• Coherent pulses. The time–bandwidth relation applies to a single coherent pulse. Multimode nanosecond pulses, amplified spontaneous emission, and incoherent sources have products far above K for reasons that have nothing to do with chirp.

References

Primary sources

1. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed., Academic Press (2006), ch. 1, “Fundamentals,” doi:10.1016/B978-012215493-5/50002-1 (Table 1.1, the time–bandwidth products of the Gaussian, sech², and Lorentzian pulses, and of the square pulse in a half-width convention; chirped Gaussian pulses; dispersive broadening), and ch. 9, “Diagnostic techniques,” doi:10.1016/B978-012215493-5/50010-0 (autocorrelation, FROG, SPIDER).
2. C. Rullière (ed.), Femtosecond Laser Pulses: Principles and Experiments, 2nd ed., Springer (2005), doi:10.1007/b137908 (pulse shapes and their spectra; mode locking; dispersion).
3. A. E. Siegman, Lasers, University Science Books (1986), ch. 9, “Linear pulse propagation” (Gaussian pulse broadening by dispersion), ch. 26, “Laser Q-switching,” and ch. 27, “Active laser mode coupling.”
4. A. M. Weiner, Ultrafast Optics, Wiley (2009), doi:10.1002/9780470473467, ch. 4 (dispersion and pulse propagation).
5. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985), doi:10.1016/0030-4018(85)90120-8 (chirped-pulse amplification).
6. A. Dubietis, G. Jonušauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. 88, 437–440 (1992), doi:10.1016/0030-4018(92)90070-8 (OPCPA).
7. M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68, 2793–2795 (1996), doi:10.1063/1.116609 (the gas-filled hollow-fiber compressor).
8. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipőcs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997), doi:10.1364/OL.22.000522 (the few-cycle marker on the chart).
9. M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M. Murnane, “Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 18, 977–979 (1993), doi:10.1364/OL.18.000977 (10.95 fs, 62 nm bandwidth centered at 780 nm: the Ti:sapphire oscillator marker).
10. R. Ell, U. Morgner, F. X. Kärtner, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, M. J. Lederer, A. Boiko, and B. Luther-Davies, “Generation of 5-fs pulses and octave-spanning spectra directly from a Ti:sapphire laser,” Opt. Lett. 26, 373–375 (2001), doi:10.1364/OL.26.000373.
11. J. H. Sung, H. W. Lee, J. Y. Yoo, J. W. Yoon, C. W. Lee, J. M. Yang, Y. J. Son, Y. H. Jang, S. K. Lee, and C. H. Nam, “4.2 PW, 20 fs Ti:sapphire laser at 0.1 Hz,” Opt. Lett. 42, 2058–2061 (2017), doi:10.1364/OL.42.002058 (83 J in 19.4 fs from an 84 nm FWHM amplified spectrum: the petawatt marker on the chart).
12. T. W. Hänsch, “Repetitively pulsed tunable dye laser for high resolution spectroscopy,” Appl. Opt. 11, 895–898 (1972), doi:10.1364/AO.11.000895 (the grating-and-etalon narrowband pulsed dye laser, bandwidth below 0.004 Å).
13. S. Gröbmeyer, K. Fritsch, B. Schneider, M. Poetzlberger, V. Pervak, J. Brons, and O. Pronin, “Self-compression at 1 µm wavelength in all-bulk multi-pass geometry,” Appl. Phys. B 126, 159 (2020), doi:10.1007/s00340-020-07506-4 (multipass-cell compression; fused silica GVD of 18.97 fs²/mm at 1 µm and −28 fs²/mm at 1550 nm, the checks for the Sellmeier differentiation).
14. M. N. Polyanskiy, “Refractiveindex.info database of optical constants,” Sci. Data 11, 94 (2024), doi:10.1038/s41597-023-02898-2 (the CC0 database the Sellmeier coefficients come from; files main/SiO2/nk/Malitson.yml, specs/schott/optical/N-BK7.yml, main/Al2O3/nk/Malitson.yml, main/CaF2/nk/Li.yml, main/MgF2/nk/Dodge-o.yml, main/Y3Al5O12/nk/Zelmon.yml, main/KH2PO4/nk/Zernike-o.yml, main/BaB2O4/nk/Tamosauskas-o.yml, other/mixed gases/air/nk/Ciddor.yml, main-branch snapshot of 2026-11-06; each coefficient set was checked against the table published with the original paper, and the wavelength ranges given above are those of the original measurements).
15. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965), doi:10.1364/JOSA.55.001205; I. H. Malitson, “Refraction and dispersion of synthetic sapphire,” J. Opt. Soc. Am. 52, 1377–1379 (1962), doi:10.1364/JOSA.52.001377; H. H. Li, “Refractive index of alkaline earth halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 161–290 (1980), doi:10.1063/1.555616; M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl. Opt. 23, 1980–1985 (1984), doi:10.1364/AO.23.001980; D. E. Zelmon, D. L. Small, and R. Page, “Refractive-index measurements of undoped yttrium aluminum garnet from 0.4 to 5.0 µm,” Appl. Opt. 37, 4933–4935 (1998), doi:10.1364/AO.37.004933; F. Zernike, “Refractive indices of ammonium dihydrogen phosphate and potassium dihydrogen phosphate between 2000 Å and 1.5 µ,” J. Opt. Soc. Am. 54, 1215–1220 (1964), doi:10.1364/JOSA.54.001215 (Table IV(a), indices relative to air); G. Tamošauskas, G. Beresnevičius, D. Gadonas, and A. Dubietis, “Transmittance and phase matching of BBO crystal in the 3–5 μm range and its application for the characterization of mid-infrared laser pulses,” Opt. Mater. Express 8, 1410–1418 (2018), doi:10.1364/OME.8.001410; P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566–1573 (1996), doi:10.1364/AO.35.001566; SCHOTT AG, optical glass catalog (Zemax format, 2017-01-20b), N-BK7 (the Sellmeier equations behind the dispersion results).
16. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules, Van Nostrand Reinhold (1979), doi:10.1007/978-1-4757-0961-2 (the rotational constants of H₂, HCl, N₂, NO, and I₂ behind the spectroscopy reading).

Manufacturer specifications (the chart markers)

• Spectra-Physics, Quanta-Ray Pro Series pulsed Nd:YAG lasers, datasheet (1064 nm pulse width 8–12 ns; linewidth <1.0 cm⁻¹ standard, <0.003 cm⁻¹ injection seeded), copy hosted by MIT.
• Quantel, TDL90 tunable dye laser, datasheet (pulse duration 4–5 ns with a Brilliant pump or 8–10 ns with the YG980; linewidth at 560 nm 0.8 cm⁻¹ standard, 0.08, 0.06, and 0.05 cm⁻¹ with the narrow-bandwidth and dual-grating options), velocimetry.net; Sirah Lasertechnik, Cobra and Cobra-Stretch linewidth specifications (0.03–0.1 cm⁻¹ with gratings and 5 cm⁻¹ with prisms, by option; tuning range 370–920 nm with the prism option), sirah.com.
• Coherent, COMPex excimer laser series, datasheet (pulse duration 20 ns typical FWHM), coherent.com; Gigaphoton, G41K KrF excimer laser (248 nm, 0.35 pm FWHM line-narrowed), gigaphoton.com.
• Light Conversion, PHAROS femtosecond lasers (1030 ± 10 nm; pulse duration <290 fs to <100 fs by model, tunable to 10 ps), lightcon.com.
• Edmund Optics, UV fused silica ultrafast dispersing prisms, specification sheet (group velocity dispersion 36 fs²/mm at 800 nm), edmundoptics.co.uk.
• Laser Components, Yb:YAG laser crystals, data sheet (emission linewidth 9 nm), lasercomponents.com; AOG Crystal, Yb:YAG crystal (emission linewidth 9 nm), aogcrystal.com.

Free introductions

• R. Paschotta, RP Photonics Encyclopedia: “Time–bandwidth product,” doi:10.61835/bfb; “Bandwidth-limited pulses,” doi:10.61835/zft; “Chirp,” doi:10.61835/j6t; “Group velocity dispersion,” doi:10.61835/9gj (fused silica about +35 fs²/mm at 800 nm); “Group delay dispersion,” doi:10.61835/bhj; “Sellmeier formula,” doi:10.61835/do2; “Dispersion compensation,” doi:10.61835/qni.
• R. Paschotta, RP Photonics Encyclopedia: “Q-switching,” doi:10.61835/fdh; “Nanosecond lasers,” doi:10.61835/36m; “Injection seeding,” doi:10.61835/zj0; “Mode locking,” doi:10.61835/dm2; “Gain bandwidth,” doi:10.61835/r1g; “Gain narrowing,” doi:10.61835/uib; “Chirped-pulse amplification,” doi:10.61835/3ts; “Optical parametric chirped-pulse amplification,” doi:10.61835/19r; “Pulse compression,” doi:10.61835/cb7; “Self-phase modulation,” doi:10.61835/pqf; “Hollow-core fibers and capillaries,” doi:10.61835/x34.
• R. Paschotta, RP Photonics Encyclopedia, gain media: “YAG lasers,” doi:10.61835/7vp (Nd:YAG 0.6 nm and Yb:YAG 15 nm gain bandwidths); “Laser glasses,” doi:10.61835/6ic; “Tungstate lasers,” doi:10.61835/2vq; “Titanium–sapphire lasers,” doi:10.61835/2j5; “Dye lasers,” doi:10.61835/hbd; “Excimer lasers,” doi:10.61835/qp6.