Material refractive index varies with wavelength and therefore causes the group

velocity to vary; it is classified as material dispersion. The wavelength dependence of

refractive index can be expressed by Sellmeier’s equation. Waveguide dispersion is

the result of wavelength-dependence of the propagation constant of the optical

waveguide. It is important in single-mode waveguides. The larger the wavelength, the

more the fundamental mode will spread from the core into the cladding. This causes

the fundamental mode to propagate faster. The material and waveguide dispersion

effect is important when the grating filter has a broadband response, such as a codirectional waveguide coupler and a long period fiber grating. In these cases, the

waveguide mode constants are re-calculated for different wavelength by considering

material dispersion.

## Host materials

Optical telecommunication fibers are usually made from silica glasses. The high purity

glass is called the host material or substrate. Its bulk refractive index is usually the

fiber cladding refractive index. The fiber core is formed by adding dopant materials to

the host material.

## Dopant materials

To change the refractive index of optical fiber, pure silica is often doped with dopants.

For example, adding germanium can result in an increase in the refractive index,

while adding fluorine reduces it. The refractive index of doped material can be

determined by the linear relationship between the doped material’s mole percentage

and permittivity.

Assume that n0 is the refractive index of the host material and n1 is the refractive

index of m1 mole-percentage doped material. Then, the refractive index n of m molepercentage doped material can be interpolated as:

Note that this formula can also extrapolate (the case where m > m_{1} )

## Material dispersion calculation

In OptiGrating 4.2, when the refractive index at a central wavelength λ _{0}, n(λ_{0}) and

the host and dopant material dispersion curves, n_{host}(λ) and n_{dopant}(λ) are

defined, the dependence of the refractive index with wavelength, n(λ), is calculated

based on the following equation:

This equation is the same as (11), with the fraction *m/m*_{1} estimated by comparing,

at the centre wavelength λ_{0}, the refractive index of the given material with the index

of a doped material with known Sellmeier coefficients n_{dopant}. OptiGrating uses

Equation 12 to handle the case where the Sellmeier coefficients are known for

material of just one doping concentration. If the material in question has the same

dopant, but with an unknown concentration, the fraction in Equation 12 will estimate

the concentration by comparing the given index *n(λ _{0})* with the reference index

*n*. With the doping concentration estimated this way, the refractive index

_{dopant}(λ_{0})at other wavelengths is accurately estimated by Equation 12. On the other hand, in

the case where the Sellmeier coefficients are exacly known for the given material, the

user can enterthem himself in OptiGrating’s Sellmeier Coefficient Library. OptiGrating

still uses Equation 12, since in this case the fraction is calculated as unity, and the left

hand side is assigned to

*n*for all wavelengths directly.

_{dopant}(λ_{0})