# Sellmeier Equation Calculator

Created by Dominik Czernia, PhD
Last updated: Jul 01, 2022

Use the Sellmeier equation calculator to estimate the relationship between refractive index and wavelength based on Sellmeier's coefficients. The tool uses the empirical expression, meaning that you should obtain all constants from an experiment or check the Sellmeier coefficients database.

In the following article, we explain the Sellmeier dispersion formula and give some examples of Sellmeiers equations:

1. Sellmeier equation for quartz
2. Sellmeier equation for silicon
3. Sellmeier equation for BBO (Beta-Barium Borate)

Be sure to check the advanced mode of the Sellmeier equation calculator. We allowed you to change the addictive constant $A$ if you would like to use the two-term form of the refractive index Sellmeier equation.

You can find more information about the refractive index in our index of refraction calculator, which provides all the essential knowledge you will probably need.

## Sellmeier equation formula

The Sellmeier equation derivation results from a simple physical model of a damped oscillator. We have a tool dedicated to the damping ratio formula for those interested in knowing more about it. Here, we show two forms of the Sellmeier dispersion formula:

Three-term form

This is one of the most commonly used variants:

$n^2 = 1 + \sum_{i=1}^3 \frac{B_i \lambda^2}{\lambda^2 - C_i}$

where:

• $n$ - Refractive index;
• $\lambda$ - Wavelength of light;
• $B_i$ - Sellmeier "B" dimensionless coefficients that are the oscillator strengths of transition; and
• $C_i$ - Sellmeier "C" coefficients that are the squares of the respective transition energies (as photon wavelengths), usually expressed in $\text{ μm²}$. One of our tools explains how to calculate wavelength from energy.

🙋 By knowing the experimental or calculated absorption spectrum of a substance, it is possible to predict the refractive index.

Two-term form

In this form, the additional $A$ coefficient is an approximation of the short-wavelength absorption contributions to the refractive index at longer wavelengths.

$n^2 = A + \sum_{i=1}^2 \frac{B_i \lambda^2}{\lambda^2 - C_i}$

The dispersion relation in physics usually describes how energy depends on momentum (or frequency on wavenumber). The is an excellent place to start if you want to understand what this physical quantity means.

## How to use the Sellmeier equation calculator

The Sellmeier equation uses the Kramers-Kronig relationship between optical absorption and refractive index to calculate the wavelength-dependent refractive index based on constants relating to key absorbance. Let's see how to apply the Sellmeier equation for the quartz crystal (SiO2) in our tool:

1. Check the to find needed coefficients.

2. For quartz, the coefficients are equal to:

• $B_1 = 0.6961663$
• $B_2 = 0.4079426$
• $B_3 = 0.8974794$
• $C_1 = 0.00467914826 \text{ μm²}$
• $C_2 = 0.0135120631 \text{ μm²}$
• $C_3 = 97.9340025 \text{ μm²}$
3. Fill in the wavelength of interest, along with Sellmeier coefficients in the corresponding boxes.

4. The Sellmeier equation calculator will show you the refractive index $n$ and its square $n^2$.

5. To use the two-term form Sellmeier equation for refractive index, use the advanced mode available in our calculator.

## Sellmeier equation for silicon

The Sellmeier equation for silicon has the following three-term form:

$n^2 = 1 + \sum_{i=1}^3 \frac{B_i \lambda^2}{\lambda^2 - C_i}$

where the Sellmeier coefficients are equal to:

• $B_1 = 10.6684293$
• $B_2 = 0.0030434748$
• $B_3 = 1.54133408$
• $C_1 = 0.0909121907 \text{ μm²}$
• $C_2 = 1.28766018 \text{ μm²}$
• $C_3 = 1,218,816 \text{ μm²}$

The above values are from the .

## Sellmeier equation for BBO

The Sellmeier equation for BBO is slightly different from the standard Sellmeier equation derivation we talk about above. It has the following two-term form:

$n^2 = A + \frac{B_1 \lambda^2}{\lambda^2 - C_1} - B_2 \lambda^2$

where the Sellmeier coefficients are equal to:

• $A = 2.7405$
• $B_1 = 0.0184$
• $B_2 = 0.0155 \text{ 1/μm²}$
• $C_1 = 0.0179 \text{ μm²}$

The above values are from the .

Dominik Czernia, PhD
Wavelength
μm
B1
B2
B3
C1
μm²
C2
μm²
C3
μm²
Refractive index (n²)
Refractive index (n)
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