The Sellmeier equation provides accurate refractive index calculations across a wide wavelength range. Includes GVD, group index, and dispersion curve with preset glass material coefficients.
The Sellmeier equation is the industry standard for accurate refractive index modeling across broad wavelength ranges. It models each material as a sum of Lorentz oscillator resonances — terms B_i × λ² / (λ² − C_i²) — where each (B_i, C_i) pair represents a resonance with strength B and resonance wavelength √C. The classical Sellmeier form uses three terms (six coefficients) that fit virtually every transparent optical glass to within ~0.001% across the visible and near-IR. This precision is what enables modern lens design software to predict optical performance to fractions of a wavelength.
This calculator implements the standard three-term Sellmeier form with presets for BK7 (borosilicate crown glass, the workhorse of optical design), fused silica (the standard for UV transmission), and SF11 (dense flint, used in achromat doublets). For other materials, enter custom B₁, C₁, B₂, C₂, B₃, C₃ coefficients — you can find tabulated values in optical glass catalogs (Schott, Ohara, CDGM) and online databases like RefractiveIndex.INFO.
Sellmeier is essential for designing modern multi-element lenses, ultrafast laser optics, fiber-optic systems, and any application where wavelength-dependent index accuracy matters. Where Cauchy gives "good enough" approximations in the visible, Sellmeier gives publishable-precision values across UV, visible, and near-IR.
**Scenario:** Design an achromat doublet at 587.6 nm with BK7 crown + SF11 flint. What refractive indices and Abbe numbers to use? **Calculation:** BK7 at 587.6: n = 1.5168. SF11 at 587.6: n = 1.7847. BK7 Abbe V_d = (1.5168−1)/(1.5224−1.5143) = 0.5168/0.0081 = 63.8. SF11 V_d = (1.7847−1)/(1.7980−1.7783) = 0.7847/0.0197 = 39.8. Achromat condition: P_BK7/V_BK7 + P_SF11/V_SF11 = 0 for chromatic aberration cancellation. **Result:** A standard BK7/SF11 achromat with V values 63.8 and 39.8 can null chromatic aberration at two wavelengths (F and C lines). The ~24-point Abbe gap is the source of the cancellation; smaller gap → harder to achromatize. This pairing is the most common achromat configuration in commercial telescope and camera objectives.
**Scenario:** A laser at 248 nm (KrF excimer) needs to transmit through fused silica. What is n at 248 nm, and is the material transparent? **Calculation:** λ = 0.248 µm. Fused silica Sellmeier: n² − 1 = 0.6962×0.0615/(0.0615−0.00468) + 0.4079×0.0615/(0.0615−0.01351) + ... ≈ 0.756 + 0.522 + small ≈ 1.28 → n² = 2.28 → n = 1.510. **Result:** Fused silica n at 248 nm ≈ 1.51 (higher than visible 1.46 due to UV dispersion). Fused silica is transparent down to ~190 nm — uniquely UV-friendly among common glasses. Most other glasses (BK7, SF11) absorb below ~330 nm. This is why excimer laser optics, photolithography (193 nm), and UV spectroscopy use fused silica.
**Scenario:** A 100 fs laser pulse at 800 nm propagates through 1 cm of BK7. What's the group delay and pulse broadening from GVD? **Calculation:** BK7 at 800 nm: n ≈ 1.510, n_g ≈ 1.526 (about 1% higher than phase index). Group delay over 1 cm: 0.01 × n_g/c = 0.01 × 1.526 / (3×10⁸) = 51 ns. GVD ≈ +44 fs²/mm at 800 nm. Pulse spread: Δτ ≈ √(τ₀² + (GVD × L / τ₀)²) = √(100² + (44×10/100)²) = √(10000 + 19) ≈ 100.1 fs. **Result:** 100 fs pulse barely broadens through 1 cm BK7 — only ~0.1 fs added. But through 10 cm of glass: broadening becomes ~10 fs (10%). Through 1 m of glass: broadening ~3.4 ps (33×). For ultrafast experiments, even modest glass thickness in pulse paths matters. Solutions: use prism pair pre-compression, or use achromatic doublets with cancelling GVD.
**Use the Sellmeier equation for:**
- **Lens design**: precision lens systems (camera, telescope, microscope) where wavelength accuracy matters. - **Ultrafast laser system design**: pulse stretching/compression through dispersive elements. - **Optical fiber design**: zero-dispersion wavelength, GVD compensation. - **Photolithography**: 193 nm and shorter wavelengths where Sellmeier is essential. - **UV and IR spectroscopy**: material transmission across non-visible bands. - **Crystal optics**: birefringent calculations (different Sellmeier for each crystal axis). - **Material comparison**: precise comparison of optical properties across glasses.
**Optical glass families and use:**
**Crown glasses (high V):** - BK7: workhorse for visible-light optics, achromat element. - Fused silica: UV transmission, very low coefficient of thermal expansion. - BAK1, BAK4: rugged crown for prisms. - N-BAF10: better UV than BK7.
**Flint glasses (low V):** - SF11: standard achromat partner for BK7. - SF57: heavier flint, slightly larger dispersion correction. - N-LASF31A: high-index dense flint, used in compact zoom lenses. - Lanthanum flints: rare-earth-doped glasses for special optics.
**Specialty glasses:** - ED (extra-low dispersion) glasses: fluorite-equivalent, premium camera apochromats. - Phosphate glasses: athermal optical systems (mating with thermal coefficient). - Chalcogenide glasses: mid-IR (3–14 µm) transmission, used in thermal imaging.
**Why Sellmeier > Cauchy:**
- Models actual material physics (Lorentz oscillators) vs empirical curve fitting. - Works across UV, visible, and IR — Cauchy only in narrow visible band. - Standard form in optical glass catalogs. - Required for precision (camera lens design at λ/10 wavefront tolerances).
**Modified Sellmeier forms:**
- **Schott formula**: alternative six-coefficient form used in some catalogs. - **Cauchy + Sellmeier hybrid**: mixed for special wavelength ranges. - **Tabulated index**: when even Sellmeier is insufficient (some birefringent crystals). - **Modified Sellmeier with temperature**: T-dependent index for thermal lensing.
**Useful zero-GVD wavelengths:**
- BK7 zero-GVD: ~1.32 µm. - Fused silica zero-GVD: ~1.27 µm. - Standard single-mode fiber (silica): 1.31 µm — chosen so chromatic dispersion vanishes in the operating band. - Dispersion-shifted fiber: zero-GVD at 1.55 µm to coincide with EDFA gain peak.
Calculate refractive index vs wavelength using the Cauchy dispersion equation n(λ) = A + B/λ² + C/λ⁴.
Calculate prism deviation angle, minimum deviation, and angular dispersion for optical prisms.
Calculate reflectance of a single-layer thin-film optical coating at normal incidence with quarter-wave and half-wave analysis.
Calculate diffraction angles, angular dispersion, and resolving power using the grating equation mλ = d(sinα + sinβ).
Convert between Raman shift (cm⁻¹), scattered wavelength (nm), and excitation wavelength for Raman spectroscopy.
Convert between Numerical Aperture (NA) and f-number. NA = 1/(2×f/#) for small angles, or NA = n×sin(θ).
Refractive Index
1.518522
Group Index
1.546197
GVD
77.04 fs²/mm
| Parameter | Value |
|---|---|
| Refractive Index n(λ) | 1.518522 |
| Wavelength | 550 nm (0.5500 μm) |
| Group Index ng | 1.546197 |
| GVD (approx) | 77.04 fs²/mm |
| dn/dλ | -0.050318 /μm |
| Material | BK7 |
| B₁ | 1.03961212 |
| C₁ | 0.00600070 |
| B₂ | 0.23179234 |
| C₂ | 0.02001791 |
| B₃ | 1.01046945 |
| C₃ | 103.56065300 |