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Sellmeier's equation & Least-squares Fitting

  1. Oct 24, 2013 #1
    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. jcsd
  3. Oct 24, 2013 #2

    DrDu

    User Avatar
    Science Advisor

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