How to calculate chromatic dispersion with the Sellmeier equation lets you precisely determine the refractive index n of an optical material as a function of wavelength λ. The formula was derived in 1872 by Wilhelm Sellmeier (University of Göttingen) as an empirical improvement over Cauchy (1836) and Lorentz, based on the classical theory of resonant oscillators. The standard three-term form is n²(λ) = 1 + B₁λ²/(λ²-C₁) + B₂λ²/(λ²-C₂) + B₃λ²/(λ²-C₃), with λ in micrometers (µm). For Schott BK7 glass, B₁=1.03961212, B₂=0.231792344, B₃=1.01046945, C₁=0.00600069867, C₂=0.0200179144, C₃=103.560653. It underpins the design of camera lenses, fiber optics, EUV lithography systems, advanced microscopy, and lasers — used by Carl Zeiss SMT, Schott Mainz, Corning, OFS, ASML, IBM, Intel, and academic labs at MIT, Stanford, Caltech, Bell Labs.

1. The Sellmeier equation and Abbe number

The Sellmeier equation describes anomalous and normal dispersion of a dielectric material via sums of resonant terms: each Cᵢ is the squared resonance wavelength (in µm²) and Bᵢ the oscillator strength. The Abbe number ν_d = (n_d - 1)/(n_F - n_C) — where n_d is at 587.6 nm (Fraunhofer D), n_F at 486.1 nm (F) and n_C at 656.3 nm (C) — quantifies dispersion: ν_d > 50 indicates "crown" glasses (BK7 ν_d=64.17, low dispersion), ν_d < 50 indicates "flint" glasses (SF11 ν_d=25.76, high dispersion). Group velocity dispersion (GVD) ∝ d²n/dλ² is critical for ultrafast laser pulse compression (CPA, Mourou-Strickland Nobel 2018) and for fiber-optic communications.

2. Sellmeier coefficients for typical materials

• Fused silica SiO₂ (Malitson 1965) — B₁=0.6961663, B₂=0.4079426, B₃=0.8974794, C₁=0.0684043, C₂=0.1162414, C₃=9.896161 µm². Range 200-3500 nm. Basis of Corning telecom fiber.
• BK7 borosilicate (Schott) — Carl Zeiss SMT standard glass, range 350 nm-2000 nm.
• SF11 heavy flint (Schott) — high dispersion for achromats, ν_d=25.76.
• Sapphire Al₂O₃ (Tropf 1997) — high hardness, range 300 nm-5000 nm.
• Silicon Si (Tatian 1984) — B₁=10.6684293, B₂=0.003043475, B₃=1.54133408. Range 1100-7000 nm, basis of integrated photonics.
• Germanium Ge — IR transparency 2-15 µm, FLIR thermal imaging.
• ZnSe — CO₂ laser windows (10.6 µm), range 600 nm-21 µm.
• CVD diamond — extreme IR optics, range 230 nm-100 µm, n=2.42 at 590 nm.

3. Validity range and corrections

The Sellmeier equation is valid for isotropic transparent materials in the spectral range of applicability (typically 200 nm - 5 µm for silica, 400 nm - 2 µm for BK7). Near absorption resonances (UV for most glasses, IR for molecular bonds), Sellmeier diverges and the full Lorentz model with absorption is needed. For birefringent crystals (calcite, quartz, BBO, KTP), separate Sellmeier equations are needed for ordinary n_o and extraordinary n_e rays. For doped glasses or laser crystals (Nd:YAG, Yb:KGW, Ti:sapphire), Sellmeier requires correction for active-ion absorptions. Group dispersion and group velocity dispersion (GVD) require dn/dλ and d²n/dλ², critical for ultrashort laser pulses at LCLS Stanford, European XFEL Hamburg, ELI Beamlines. For temperature variations, dn/dT (~10⁻⁶ K⁻¹ typical) is needed. Codes Zemax, Code V, FRED, Lumerical FDTD implement all corrections.

4. Worked numerical examples

Example 1 — Fused silica at λ=1550 nm (telecom). λ=1.55 µm; λ²=2.4025. n²=1 + 0.6961663·2.4025/(2.4025-0.0684043²) + 0.4079426·2.4025/(2.4025-0.1162414²) + 0.8974794·2.4025/(2.4025-9.896161). Result: n_silica(1550) ≈ 1.4440 — matches Corning SMF-28 single-mode fiber tables.

Example 2 — BK7 at λ=587.56 nm (Na D line). Standard Schott coefficients yield n_d ≈ 1.5168 — IUPAC reference for standard glass. Abbe number ν_d = (1.5168-1)/(1.5224-1.5143) ≈ 63.8 (close to tabulated 64.17).

Example 3 — Silicon at λ=1310 nm (telecom O-band). Tatian 1984 coefficients yield n_Si(1310) ≈ 3.5037. For silicon photonics waveguides at IBM, Intel, IMEC Leuven, CEA-Leti this is the baseline for designing couplers, MZI modulators, and Ge/Si photodetectors.

5. Practical applications of Sellmeier

• Optical lens design: Carl Zeiss SMT, Schott Mainz, Hoya, Nikon, Canon, Pentax use Sellmeier for every element of achromatic, apochromatic, microscope, and EUV lenses
• Telecom fiber optics: AT&T, Verizon, Comcast, Lumen, Vodafone backbone Internet at 1310/1550 nm with chromatic dispersion calculated via Sellmeier on doped silica
• Integrated photonics: silicon waveguides at 1310/1550 nm designed via Sellmeier at CEA-Leti, IMEC Leuven, IBM Almaden, Intel Hillsboro
• DUV/EUV lithography: ASML EUV scanners use Sellmeier for projection optics design (Mo/Si multilayer mirrors)
• Ultrafast lasers: LCLS Stanford, European XFEL, BELLA Berkeley, ELI Beamlines use Sellmeier for dispersion management with gratings, prisms, dispersive mirrors in fs and attosecond lasers
• Confocal microscopy: Leica, Olympus, Nikon, Zeiss immersion lenses use Sellmeier for chromatic aberration correction

6. Comparison with other dispersion formulas

The Cauchy equation (1836) n(λ) = A + B/λ² + C/λ⁴ is a simplified approximation valid only in the visible, replaced by Sellmeier for extended range. The Conrady equation n(λ) = n₀ + A/λ + B/λ^3.5 is occasionally used for optical glasses. The Schott equation n²(λ) = a₀ + a₁λ² + a₂λ⁻² + a₃λ⁻⁴ + a₄λ⁻⁶ + a₅λ⁻⁸ is internally used in Schott Glass Catalog. The Herzberger formula for IR glasses is a 6-parameter variant. For laser crystals, NIST and Brookhaven Standard use semi-empirical formulas. Group dispersion requires d/dλ of Sellmeier. All algorithms are implemented in Zemax OpticStudio, Code V, FRED, Lumerical FDTD for industrial optical design.

7. Special cases and advanced materials

For birefringent crystals (calcite CaCO₃, crystalline quartz, BBO, KTP, LiNbO₃), separate Sellmeier equations are used for ordinary n_o and extraordinary n_e rays — fundamental for polarizers, λ/4 plates, and second-harmonic generation (SHG) at LCLS, European XFEL, ELI Beamlines. For nonlinear phase-matching crystals, the propagation angle is computed from Sellmeier to optimize conversion efficiency. For photonic crystal fibers (PCF) and hollow-core fibers, dispersive properties combine material Sellmeier with geometric structure, developed at OFS Specialty Photonics, Corning, Heriot-Watt, IPHT Jena. For doped materials (Yb:KGW, Nd:YAG, Ti:sapphire), Sellmeier requires correction for laser-active ions. Spatial dispersion in metamaterials and topological photonics requires k-dependent formalism beyond classical Sellmeier.

Updated July 2026