Sellmeier equations & group delay dispersion

[Voxynn]

Hi,

I'm trying to take this Sellmeier equation,

n^2 (λ)=1+ (B_1 λ^2)/(λ^2-C_1 )+(B_2 λ^2)/(λ^2-C_2 )+(B_3 λ^2)/(λ^2-C_3 )

for which i have several sets of constants for prospective glasses, and convert it into a group delay dispersion graph, with axes of fs^2 vs wavelength.

How do i rearrange the equation above (or convert the resultant graph) into a form i can use to find the GDD for the glasses?

If it's easier, i could use the GVD instead of the GDD.

Thanks!

Voxynn

 

[elect_eng]

Dispersion is generally some type of a derivative of refractive index. Sometimes people are interested in first order, or second order dispersion, and it's possible to have different definitions for each type (i.e. first or second derivatives with respect to either wavelength or frequency). Just take your definition of dispersion (in equation form) and apply it using derivatives and the chain rule if necessary.

 

[charng]

,

The Sellmeier equation is a widely used empirical formula for calculating the refractive index of a material as a function of wavelength. The equation takes into account the material's dispersion properties, which is the variation of the refractive index with wavelength. The Sellmeier equation is typically used for glasses and other transparent materials.

To convert the Sellmeier equation into a group delay dispersion (GDD) graph, you will need to use the second derivative of the refractive index (n) with respect to wavelength (λ). This can be calculated using the following formula:

d²n/dλ² = (2*B_1*λ)/(λ^2-C_1 )^2 + (2*B_2*λ)/(λ^2-C_2 )^2 + (2*B_3*λ)/(λ^2-C_3 )^2

Once you have the second derivative, you can plot it on the y-axis (fs^2) and the wavelength on the x-axis to get a GDD graph. This graph will show you the variation of the group delay (the time it takes for a light pulse to travel through the material) with respect to wavelength. The GDD graph can also be converted to a group velocity dispersion (GVD) graph by taking the inverse of the second derivative.

Alternatively, you could also use the Sellmeier equation to directly calculate the GDD or GVD by taking the second or first derivative of the refractive index with respect to wavelength, respectively. This will give you a numerical value for the GDD or GVD at a specific wavelength.

I hope this helps you in your research on prospective glasses. Good luck!