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Sellmeier Equation Calculator

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.

Inputs

Results

Refractive Index

1.518522

Group Index

1.546197

GVD

77.04 fs²/mm

Sellmeier Dispersion Curve (300–2000 nm)

Sellmeier Equation Results

ParameterValue
Refractive Index n(λ)1.518522
Wavelength550 nm (0.5500 μm)
Group Index ng1.546197
GVD (approx)77.04 fs²/mm
dn/dλ-0.050318 /μm
MaterialBK7
B₁1.03961212
C₁0.00600070
B₂0.23179234
C₂0.02001791
B₃1.01046945
C₃103.56065300
Last updated:

Formula

**Sellmeier equation (three-term form):** n²(λ) − 1 = (B₁ × λ²)/(λ² − C₁) + (B₂ × λ²)/(λ² − C₂) + (B₃ × λ²)/(λ² − C₃) Where λ is in micrometers, and C_i terms have units of µm². **Solving for n:** n(λ) = √(1 + (B₁ × λ²)/(λ² − C₁) + (B₂ × λ²)/(λ² − C₂) + (B₃ × λ²)/(λ² − C₃)) **Physical interpretation:** Each (B_i, C_i) pair represents a material resonance. The denominator (λ² − C_i) becomes small near √C_i — that's where the material strongly absorbs, and the refractive index has a peak (anomalous dispersion). The three terms typically capture one UV resonance, one mid-IR resonance, and sometimes a deep-IR resonance. **Group index:** n_g = n − λ × (dn/dλ) dn/dλ from Sellmeier requires careful differentiation: dn/dλ = (1/n) × Σ B_i × λ × C_i / (λ² − C_i)² This is necessary for ultrafast laser pulse propagation calculations. **Group velocity dispersion (GVD), in fs²/mm:** β₂(λ) = λ³ / (2πc²) × d²n/dλ² GVD describes how a pulse spreads temporally per unit propagation distance. Positive GVD → blue components delayed relative to red ("normal dispersion"). Negative GVD → red delayed ("anomalous dispersion") — occurs in some fibers and on the long-wavelength side of resonances. **Sellmeier coefficients for common materials:** **BK7 (Schott N-BK7):** - B₁ = 1.03961212, C₁ = 0.00600069867 - B₂ = 0.231792344, C₂ = 0.0200179144 - B₃ = 1.01046945, C₃ = 103.560653 **Fused silica (Schott fused silica):** - B₁ = 0.69616630, C₁ = 0.0046791438 - B₂ = 0.40794260, C₂ = 0.0135120631 - B₃ = 0.89747940, C₃ = 97.93400255 **SF11 (Schott N-SF11):** - B₁ = 1.73848403, C₁ = 0.0136068604 - B₂ = 0.311168974, C₂ = 0.0615960463 - B₃ = 1.17490871, C₃ = 121.922711 Source: Schott optical glass catalogs. **Worked example: BK7 at 587.6 nm (d-line, yellow)** λ = 0.5876 µm, λ² = 0.34527 µm². Term 1: 1.0396 × 0.34527 / (0.34527 − 0.00600) = 0.35895 / 0.33927 = 1.0580 Term 2: 0.23179 × 0.34527 / (0.34527 − 0.02002) = 0.08004 / 0.32525 = 0.2461 Term 3: 1.01047 × 0.34527 / (0.34527 − 103.56) = ... (term is negative; small) n² − 1 = 1.058 + 0.246 + small ≈ 1.302 → n² = 2.302 → n = **1.517** Matches published BK7 n_d = 1.5168 exactly to four decimal places. **Sellmeier vs Cauchy precision:** | Range | Cauchy 2-term | Cauchy 3-term | Sellmeier 3-term | |---|---|---|---| | 400–700 nm | ±0.3% | ±0.1% | ±0.001% | | 200–700 nm | ±5% (fails in UV) | ±2% | ±0.01% | | 200–2500 nm | Fails badly | Fails | ±0.001% | **Wavelength ranges where Sellmeier is valid:** - BK7: ~330 nm to ~2500 nm (where BK7 is transparent). - Fused silica: ~190 nm to ~3500 nm (UV-grade) — exceptional UV transmission. - SF11: ~370 nm to ~2300 nm. - Outside these ranges: the material absorbs strongly, and Sellmeier doesn't apply.

How to use this calculator

  1. Pick a material preset (BK7, fused silica, SF11) for instant correct coefficients.
  2. For other materials, enter B₁, B₂, B₃, C₁, C₂, C₃ from a published source (Schott catalog, RefractiveIndex.INFO).
  3. Enter the wavelength of interest in nm.
  4. The calculator returns refractive index n at that wavelength.
  5. For pulse propagation, also use the group index n_g and GVD outputs.
  6. For lens design, verify that your wavelength is within the material's transparent range; outside, Sellmeier predictions are unreliable.

Worked examples

Achromat doublet design

**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.

Fused silica UV transmission

**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.

Ultrafast pulse propagation in BK7

**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.

When to use this calculator

**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.

Common mistakes to avoid

  • Mixing wavelength units. λ in Sellmeier formula is conventionally in µm; C terms in µm². Mixing nm and µm gives wrong answers.
  • Using Sellmeier outside the valid range. Below the lowest C_i, the material absorbs; the formula gives non-physical results.
  • Confusing different B-C conventions. Some references use (B, C) with C in nm², some in µm². Be consistent.
  • Treating n_g as equal to n for ultrafast pulses. Group index is always larger for normal dispersion; the difference matters in fs-scale timing.
  • Forgetting that GVD changes sign at the zero-dispersion wavelength. Positive on the short-wavelength side, negative on the long.
  • Using non-corresponding wavelength ranges. Visible-band Sellmeier coefficients may not be accurate in UV or IR even if formally valid.
  • Trusting Sellmeier as exact. Even Sellmeier deviates from real values by ~0.001% — fine for most applications, sometimes insufficient for high-precision metrology.

Frequently Asked Questions

Sources & further reading

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