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Dispersion and Ultrashort Pulses

Sellmeier data for real glasses and crystals, a Lorentz oscillator that ties resonance, dispersion and absorption together, and exact FFT propagation of femtosecond pulses with the complex k(ω). Compare phase delay, group delay, GVD and third-order dispersion one term at a time.

Type exact values in the number boxes • Click a material plot (or focus it and use ←/→) to set the carrier wavelength • Share the setup with “Copy link”
λ₀ 800 nm Carrier
ng — Group index
β₂ — GVD (fs²/mm)
φ₂ — Total GDD (fs²)
Δt — Output FWHM
T — Energy transmitted

Pulse

Field and envelope in time peak-normalised to input

Solid: output envelope ±|A(z, τ)|; thin line: real field Re{A e−iω₀τ}; dashed: input envelope at z = 0. Amplitudes are divided by the input peak, so loss and broadening both lower the curve.

Power spectrum |Ã|², input peak = 1

Dashed: input; solid: output. Hatched bands lie outside the material data's validity interval.

Spectral phase constant + linear removed

φ(ω) − φ(ω₀) − φ′(ω₀)(ω − ω₀), shown where |Ã|² > 10⁻³. A parabola is GDD, an S-shaped cubic is TOD. Dashed: the input chirp alone.

Group delay across the spectrum τg = dφ/dω, retarded frame

Arrival time of each spectral component in the chosen frame. A tilted line is linear chirp (GDD); curvature is TOD. The spread of τg over the bandwidth sets the output duration once it exceeds the input duration.

Phase delay n z/c (β₀ z/ω₀)
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Group delay ng z/c (β₁ z)
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Carrier–envelope slip ω₀z(n − ng)/c
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GDD β₂ z / total with input chirp
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TOD β₃ z
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Dispersion length LD = τ₀²/|β₂|
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Input FWHM (transform limit / chirped)
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Output FWHM (numerical)
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Gaussian prediction (GDD only)
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Peak intensity out/in
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Energy transmitted
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Taylor residual beyond β₃ (rad)
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Spectrum outside valid data
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FFT grid
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Inverse-propagation error
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Material

Refractive and group index Sellmeier

Solid: n(λ); dashed: ng = n − λ dn/dλ. Vertical line: carrier λ₀. Click to set λ₀.

Group-velocity dispersion β₂ in fs²/mm

β₂ = d²k/dω² = λ³/(2πc²) d²n/dλ². β₂ > 0: normal (red leads); β₂ < 0: anomalous. Markers: zero-dispersion wavelengths.

Absorption κ = Im ñ

Extinction coefficient κ(λ); intensity absorption α = 4πκ/λ.

Prism at minimum deviation angles exaggerated

The incident beam is set to minimum deviation for λ₀. Output rays for the blue and red edges of the pulse spectrum (|Ã|² = 1 %) are drawn with their angular separation magnified by the stated factor.

n(λ₀) / κ(λ₀)
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Group index ng
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vp/c and vg/c
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β₁ (group delay per length)
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β₂ (GVD)
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β₃ (TOD)
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Absorption α(λ₀)
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Zero-GVD wavelength(s)
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Cursor
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Prism: min. deviation δmin
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Prism: dδ/dλ at λ₀
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Prism-pair GDD at L
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L that cancels this slab's GDD
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All built-in datasets at the current carrier λ₀ (Sellmeier form n² − 1 = Σ Biλ²/(λ² − Ci), λ in µm, Ci in µm²)
Material n ng β₂ (fs²/mm) β₃ (fs³/mm) Valid λ (µm) Source
💡 How to Use

Learn with this tool

Learning objectives

  • Tell phase velocity, group velocity and group-velocity dispersion apart, and compute ng, β₂ and β₃ from n(λ).
  • Predict how much a Gaussian pulse broadens in a given glass, using LD = τ₀²/|β₂|, and design a pre-chirp or prism pair that compensates it.
  • Use a Lorentz oscillator to connect an absorption line with normal and anomalous dispersion (Kramers–Kronig), and say when “group velocity” stops meaning “speed of the pulse”.

Prerequisites

How a pulse propagates

A linearly polarized plane-wave pulse travels through a homogeneous, isotropic, linear medium. Each frequency component propagates independently. The tool takes an FFT, applies the transfer function and inverse-transforms; sampling is the only approximation in the full-index model.

Field and spectral propagation

E(z,t)=Re⁡ ⁣{A(z,t)e−iω0t}E(z,t)=\operatorname{Re}\!\left\{A(z,t)e^{-i\omega_0t}\right\}
E~(z,ω)=E~(0,ω)eik(ω)z\tilde E(z,\omega)=\tilde E(0,\omega)e^{ik(\omega)z}
k(ω)=ωn~(ω)c,n~=n+iκk(\omega)=\frac{\omega\tilde n(\omega)}{c},\qquad\tilde n=n+i\kappa

Taylor approximation

The Taylor mode expands the propagation constant around the carrier. The coefficients come from the same material data as the full-index model.

Expansion around the carrier

k(ω)≈β0+β1Ω+β2Ω22+β3Ω36k(\omega)\approx\beta_0+\beta_1\Omega+\frac{\beta_2\Omega^2}{2}+\frac{\beta_3\Omega^3}{6}
Ω=ω−ω0,βm=dmkdωm∣ω0\Omega=\omega-\omega_0,\qquad\beta_m=\left.\frac{d^m k}{d\omega^m}\right|_{\omega_0}

Material models

The Sellmeier fit is lossless and applies only in the transparent region. Use wavelength in micrometres and CiC_i in square micrometres.

Sellmeier dispersion

n2−1=∑iBiλ2λ2−Ci,κ=0n^2-1=\sum_i\frac{B_i\lambda^2}{\lambda^2-C_i},\qquad\kappa=0

The Lorentz oscillator includes both absorption and the dispersion around the resonance. Choose the square-root branch with nonnegative absorption.

Lorentz oscillator

ε(ω)=ε∞+SωR2ωR2−ω2−iγω\varepsilon(\omega)=\varepsilon_\infty+\frac{S\omega_R^2}{\omega_R^2-\omega^2-i\gamma\omega}
n~=ε,κ≥0\tilde n=\sqrt\varepsilon,\qquad\kappa\ge0

Pulse width and sign conventions

The input is a Gaussian pulse. Its τ0\tau_0 is the 1/e intensity half-width; the full width at half maximum is larger by a factor of about 1.665.

Gaussian pulse and broadening

A(0,t)=e−t2/(2τ02),∣A∣2=e−t2/τ02A(0,t)=e^{-t^2/(2\tau_0^2)},\qquad |A|^2=e^{-t^2/\tau_0^2}
FWHM=2ln⁡2 τ0\mathrm{FWHM}=2\sqrt{\ln2}\,\tau_0
τ(z)=τ01+(zLD)2,LD=τ02∣β2∣\tau(z)=\tau_0\sqrt{1+\left(\frac z{L_D}\right)^2},\qquad L_D=\frac{\tau_0^2}{|\beta_2|}

The time dependence is e−iωte^{-i\omega t}; a positive spectral phase multiplies the field by eiφ(ω)e^{i\varphi(\omega)}.

Group delay and group-delay dispersion

τg=dφdω,GDD=d2φdω2=β2z\tau_g=\frac{d\varphi}{d\omega},\qquad\mathrm{GDD}=\frac{d^2\varphi}{d\omega^2}=\beta_2z
λ₀, ω₀
carrier vacuum wavelength and angular frequency
n, κ
real and imaginary parts of the complex index ñ
ng
group index c/vg = n − λ dn/dλ = n + ω dn/dω
βm
dmk/dωm at ω₀: β₁ = 1/vg (fs/mm), β₂ GVD (fs²/mm), β₃ TOD (fs³/mm)
φ₂, φ₃
GDD (fs²) and TOD (fs³) of the whole path, = β₂z and β₃z
τ₀, LD
1/e intensity half-width of the transform-limited pulse; dispersion length
α
intensity absorption coefficient 2 Im k = 4πκ/λ₀
A, δ
prism apex angle; deviation angle
Derivation: group delay, GDD and the broadening law

Expand k(ω) about ω₀. The output envelope is A(z, τ) = ∫ Ã(0, Ω) exp[i(β₀ + β₁Ω + β₂Ω²/2 + …)z − iΩt] dΩ/2π. The constant β₀z multiplies the whole field by a phase: it moves the carrier crests (phase velocity vp = ω₀/β₀ = c/n). The linear term shifts the envelope in time by β₁z. That is the group delay, vg = 1/β₁. In the retarded time τ = t − β₁z only the curvature terms remain.

For a Gaussian, Ã(0, Ω) ∝ exp(−Ω²τ₀²/2). Multiplying by exp(iβ₂zΩ²/2) gives exp[−Ω²(τ₀² − iβ₂z)/2]. Transforming back gives A ∝ exp[−τ²/2(τ₀² − iβ₂z)], whose intensity is exp[−τ²τ₀²/(τ₀⁴ + β₂²z²)]. So τ(z)² = τ₀²[1 + (β₂z/τ₀²)²] = τ₀²[1 + (z/LD)²]. The imaginary part of the exponent is a quadratic temporal phase, a linear chirp. For β₂ > 0 the instantaneous frequency rises from the front to the back of the pulse.

Group index: k = nω/c gives β₁ = (n + ω dn/dω)/c. With ω = 2πc/λ this becomes (n − λ dn/dλ)/c. Differentiating again gives β₂ = λ³/(2πc²) d²n/dλ².

Kramers–Kronig: causality makes ñ(ω) − n∞ analytic in the upper half-plane, so n(ω) − n∞ = (2/π) P∫₀^∞ ω′κ(ω′)/(ω′² − ω²) dω′. The dashed curve in the index plot for the Lorentz medium is this integral evaluated numerically from κ alone.

Prism at minimum deviation: the ray crosses symmetrically, so the internal angle is A/2 and n = sin[(δmin + A)/2]/sin(A/2). Differentiating at fixed incidence gives dδ/dλ = [2 sin(A/2)/cos((δmin + A)/2)] dn/dλ. A pair of prisms turns this angular dispersion into a path length that grows with wavelength. For a beam grazing the apexes this gives GDD ≈ −(λ³/2πc²)·8L(dn/dλ)² (Fork, Martinez & Gordon 1984), plus the positive material GDD of the glass the beam passes through.

Exercise 1 — How long is a 10 fs pulse after 1 cm of glass?

  1. Use β₂(800 nm) = 44.65 fs²/mm for N-BK7 to predict the output FWHM of a transform-limited 10 fs pulse after z = 10 mm.
  2. Run 10 fs → 1 cm BK7, then switch to Taylor mode and untick β₃.
  3. Read the output FWHM in exact mode and in β₂-only mode, and compare them with the Gaussian prediction.
  4. Why is the pulse more than ten times longer even though the spectrum did not change at all?
Show answer

τ₀ = 10/1.665 = 6.01 fs and LD = τ₀²/β₂ = 36.1/44.65 mm = 0.81 mm, so z/LD = 12.4 and FWHM ≈ 10 × √(1 + 12.4²) ≈ 124 fs. In β₂-only mode the tool gives 124.2 fs. The exact k(ω) gives ≈ 123.9 fs with a slightly asymmetric shape from TOD (β₃z ≈ 321 fs³). A lossless slab only multiplies each frequency by a phase, so |Ã|² is unchanged. The phases now differ between colours: red arrives ≈ 124 fs before blue, so the pulse is stretched and linearly up-chirped.

Exercise 2 — Phase delay, group delay, and the slipping carrier

  1. In fused silica at 800 nm, n = 1.4533 and ng = 1.4671. What thickness makes the carrier slip by half a cycle (π) under the envelope?
  2. Run Phase vs group. Then switch the frame between group and phase velocity.
  3. Read the carrier–envelope slip. In the phase frame, measure how far the envelope moves.
  4. Which Taylor term controls the carrier position, and which controls the envelope? Check by unticking β₀ or β₁ in Taylor mode.
Show answer

The slip is ω₀z(n − ng)/c, so |Δφ| = π needs z = λ₀/[2(ng − n)] = 0.8 µm/(2 × 0.01383) ≈ 28.9 µm. At z = 29 µm the tool shows ≈ −π: the carrier crest under the peak has become a trough. In the phase frame the envelope instead moves by (ng − n)z/c ≈ 1.34 fs, half a carrier period. β₀ sets the carrier phase and β₁ sets the envelope delay. Unticking β₁ in the group frame makes the envelope appear early by β₁z.

Exercise 3 — Limiting cases: no dispersion, and zero GVD

  1. (a) With a constant index n = 1.5, what happens to a 10 fs pulse after 30 µm (vacuum frame)? (b) At fused silica's zero-GVD wavelength, does a 10 fs pulse stay 10 fs long after 1 cm?
  2. Run Nondispersive, then Zero-GVD 1.27 µm.
  3. (a) Read the delay and the FWHM. (b) Read β₂z, β₃z and the output FWHM, and look at the group-delay plot.
  4. Why is “zero dispersion” not the same as “no broadening”?
Show answer

(a) With n constant, k = nω/c is exactly linear. The pulse is delayed by (n − 1)z/c = 50.0 fs and the FWHM stays 10.0 fs (to 10⁻⁹). Phase and group velocity are equal, so there is no carrier slip. (b) At λZD ≈ 1.273 µm, β₂ ≈ 0 but β₃z ≈ 740 fs³. The group delay is a parabola in ω: both spectral edges arrive late. The pulse grows to ≈ 13.9 fs with oscillating trailing sub-pulses. For short pulses the next Taylor term takes over.

Exercise 4 — Absorption, anomalous dispersion and “superluminal” group velocity

  1. For a Lorentz line at λR = 600 nm, where do you expect dn/dω < 0, and what does that do to ng?
  2. Run Near a resonance. Click the index plot close to 600 nm and watch ng.
  3. Find a wavelength where ng < 1 or ng < 0. Compare the transmitted energy and the pulse shape with the simple group delay β₁z.
  4. Does information travel faster than c there?
Show answer

Inside the absorption line (about ωR ± γ/2) n falls with frequency: anomalous dispersion. There ng can be below 1 or negative (≈ −22 at 600 nm here). The absorption length is then far shorter than z, and different parts of the spectrum are attenuated by different amounts, so the pulse that survives is reshaped. Its peak is not delayed by β₁z. The front of any signal still travels at c (Sommerfeld–Brillouin), so there is no causality violation. The Taylor picture fails because k varies by more than its first few derivatives across the bandwidth.

Worked example — pre-compensating a microscope objective

A Ti:sapphire oscillator delivers 20 fs (FWHM, transform-limited) pulses at 800 nm. The focusing objective contains the equivalent of 25 mm of N-BK7. What GDD must a prism pair supply, and what is the pulse at the focus without it?

  1. Material GDD: φ₂ = β₂z = 44.65 fs²/mm × 25 mm = 1116 fs². TOD: β₃z = 32.1 fs³/mm × 25 mm ≈ 803 fs³.
  2. Uncompensated: τ₀ = 20/1.665 = 12.0 fs, φ₂/τ₀² = 1116/144.3 = 7.74, so FWHM = 20 × √(1 + 7.74²) ≈ 156 fs. The peak intensity drops by the same factor, about 7.8.
  3. Compensation: the input chirp must be φ₂,in = −1116 fs². With N-BK7 prisms the angular term is ≈ −2856 fs² per metre of apex separation (see the prism readout), so L ≈ 1116/2856 m ≈ 39 cm, less whatever GDD the prism glass itself adds.
  4. Residual: TOD is not cancelled. With φ₃ = 803 fs³ and a 20 fs pulse, φ₃/τ₀³ ≈ 0.46, which leaves a few-percent broadening (the tool gives ≈ 20.7 fs) and a weak trailing satellite. Check it here: BK7, λ₀ = 800 nm, 20 fs, z = 25 mm, input chirp −1116 fs².

When the model fails

  • Outside the fitted interval: a Sellmeier formula is an interpolation. Beyond its range it can give n² < 0 near a pole or miss absorption bands entirely (for example water above ≈ 1.1 µm). The tool warns you when more than 10⁻⁴ of the pulse energy lies outside the interval, and drops components where n is undefined.
  • Near strong absorption: when αz ≳ 1 across the bandwidth, vg = 1/β₁ no longer predicts when the peak arrives, and ng < 1 or < 0 does not mean a signal outruns c. The energy velocity and the signal front are the meaningful speeds. The exact FFT result is still correct for the linear model.
  • Taylor truncation: β₂ and β₃ describe k(ω) only while Ω is small. For few-cycle pulses or long paths, compare Taylor and exact modes and watch the residual readout.
  • Nonlinear and spatial effects: self-phase modulation (the intensity-dependent index n₂I), diffraction, spatial chirp and the pulse-front tilt of a prism are not modelled. The field is one plane wave. A 1 µJ, 10 fs pulse focused in glass is strongly nonlinear.
  • Surfaces: Fresnel reflection losses at the slab and prism faces are ignored, and so are temperature dependence and birefringence (sapphire is the o-ray only).
  • Prism-pair formula: the angular term is the apex-grazing limit. A real compressor also includes the insertion-dependent glass path and higher orders.

References

  • I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965).
  • SCHOTT AG, Optical glass data sheets, N-BK7 (517642.251).
  • I. H. Malitson and M. J. Dodge, “Refractive index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. 62, 1405 (1972).
  • I. H. Malitson, “A redetermination of some optical properties of calcium fluoride,” Appl. Opt. 2, 1103–1107 (1963).
  • M. Daimon and A. Masumura, “Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region,” Appl. Opt. 46, 3811–3820 (2007).
  • R. L. Fork, O. E. Martinez and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984).
  • J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed., ch. 1–2.
  • G. P. Agrawal, Nonlinear Fiber Optics, 6th ed., ch. 3 (GVD, LD, TOD).
  • J. D. Jackson, Classical Electrodynamics, 3rd ed., §7.5 and §7.10 (Lorentz model, Kramers–Kronig); §7.11 (signal velocity).
  • Numerical values of the fits can be cross-checked at refractiveindex.info.