# Sellmeier's equation & Least-squares Fitting

1. Oct 24, 2013

### yo56

1. The problem statement, all variables and given/known data
Determine the values of Sellmeier's coefficients (S and λ0) using the least-squares method and Sellmeier's dispersion equation:
n2=1+Sλ2/(λ202)

2. Relevant equations
n2=1+Sλ2/(λ202)

3. The attempt at a solution
I understand how to use the least-squares method with a simple equation like y=mx+b (see below), but when trying to do the same thing with Sellmeier's equation, I get confused with where to put the summation symbols. Also, I am not sure if I use the squared version of Sellmeier's equation or to take the square root of it.

With y=mx+b, I know to take a y' point on the line corresponding to one of the points. A general equation would be achieved, i.e. Di=yi-y'i --> Di=yi-mxi+b. You would then square the equation, take the summation of it, and then the derivatives with respect to m and b to minimize, setting both equations equal to 0:

(1) Ʃ(yixi)-mƩ(xi2)-bƩ(xi)=0
(2) Ʃ(yi)-mƩ(xi)-bn=0, n is the number of data points

The solved linear equations yielded:
m=[Ʃ(xi)][Ʃ(yi)]-nƩ(yixi)/[[Ʃ(xi)]2-nƩ(xi2)]
b=[Ʃ(yixi)][Ʃ(xi)]-[Ʃ(xi2)][Ʃ(yi)]/[[Ʃ(xi)]2-nƩ(xi2)]

With Sellmeier's however, I am achieving some complex starting equations. Is there some mathematical "trick" I don't know about? Thanks for any help!

2. Oct 24, 2013

### DrDu

You could try to massage your equation somehow so that $1/\lambda^2=x$ and $y=1/(n^2-1)$.