Sellmeier Equation Fit Calculator

Formula Used

The Sellmeier dispersion model relates refractive index to wavelength:

n²(λ) = A₀ + Σ_i=1..m [ (B_i λ²) / (λ² − C_i) ]

• λ is wavelength in micrometers (µm) internally.
• C_i uses units of µm².
• m is the number of terms you choose (1–3).
• The calculator fits coefficients by minimizing the sum of squared index errors.

How to Use This Calculator

1. Select your wavelength unit, then paste wavelength–index pairs.
2. Choose one to three terms based on your material behavior.
3. Keep C values in µm², even when the input unit is nm.
4. Adjust initial guesses if your fit stops early.
5. Click Fit Sellmeier Model to generate coefficients and plots.
6. Use the CSV and PDF buttons to export tables and summaries.

Example Data Table

Sample wavelength–index pairs for quick testing.

Dispersion fitting for measured refractive index

Optical materials rarely keep a constant refractive index across wavelength. Accurate dispersion models are vital for lens design, waveguide simulations, interferometry, and thin‑film stack optimization. This calculator fits your measured n(λ) data to a Sellmeier model and returns coefficients plus a prediction table across a chosen wavelength span.

What the Sellmeier model represents

Sellmeier terms approximate resonant contributions to the dielectric response. Each term includes a strength coefficient B and a pole location C (in µm²). When λ approaches √C, the model becomes steep; real materials show absorption features nearby, so your fitting range should avoid known resonances and strong absorption bands.

Selecting one, two, or three terms

A one‑term fit can describe mild dispersion over a narrow band. Two terms often balance flexibility and stability for common glasses. Three terms can capture broader wavelength spans, but may introduce parameter coupling, especially when data are sparse or noisy. If the fit stalls, reduce the term count or refine initial guesses.

Unit handling and why µm² matters

Wavelengths can be entered in nm or µm, but the internal equation uses λ in µm and C in µm². Keeping C in µm² maintains consistent scale and avoids extremely small numbers when fitting visible‑range data. If you switch the wavelength unit, you do not need to rescale your C guesses.

Data quality checks that improve fits

Use evenly spaced wavelengths when possible and include at least 8–12 points for multi‑term fits. Remove outliers caused by mis‑indexing, temperature drift, or measurement mode changes. If you combine datasets, ensure all indices correspond to the same polarization, temperature, and sample batch, since small offsets can bias coefficients.

Initial guesses and convergence behavior

The optimizer searches parameter space without requiring derivatives. Good starting values help avoid flat regions and near‑poles. Reasonable C guesses are often far from the fitting band (for example, very small µm² for UV poles, large µm² for IR poles), while B values typically remain positive. If convergence is slow, adjust C upward or downward and re‑run.

Interpreting RMSE and residual trends

RMSE summarizes average index error in absolute n units. For high‑precision metrology, errors on the order of 10⁻⁵ to 10⁻⁴ may matter, while many design tasks tolerate larger values. Inspect the point‑by‑point Δn table: systematic curvature in residuals often indicates insufficient term count or a mismatched wavelength range.

Using exports for reports and workflows

The CSV export is convenient for spreadsheets, scripts, and versioned lab notebooks. The PDF export captures coefficients, RMSE, and a curve preview for quick sharing. For downstream modeling, copy coefficients into your simulation tool and keep a note of the wavelength unit and the data range used during fitting.

FAQs

1) How many data points do I need?

For stable results, use at least 6 points for simple fits and 10 or more for two or three terms. More points reduce parameter coupling and improve the reliability of the residual pattern.

2) Why are C values entered in µm²?

The Sellmeier equation uses λ in micrometers, so pole locations naturally appear as µm². Keeping C in µm² prevents extreme scaling and makes initial guesses easier to choose across common wavelength bands.

3) What does “Converged” mean here?

It indicates the optimizer found a stable simplex with very small changes in the objective function. A “Stopped” status usually means the iteration limit was reached, often improved by better initial guesses or fewer terms.

4) My fitted curve spikes sharply. What should I do?

A spike often occurs when λ approaches √C for a term. Adjust the C guesses away from the measured band, reduce the term count, or restrict the fitting range to avoid resonant regions and absorption features.

5) Should I fit A0 or keep it fixed?

Keeping A0 fixed to 1.0 is common and can stabilize the fit. Fitting A0 may help when data include offsets or broader ranges, but it also adds freedom that can increase parameter correlation.

6) Can I use this for temperature‑dependent data?

Yes, but fit each temperature separately and label results clearly. Mixing temperatures can inflate errors and distort coefficients, especially for materials with strong thermo‑optic response.

7) How do I validate the fit?

Check RMSE, review Δn residuals for systematic trends, and compare predictions at wavelengths not used during fitting. If residuals show curvature, adjust term count or refit with improved range selection.