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How to approximate this relation?

  1. Dec 2, 2008 #1


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    I got a relation as follow

    [tex]\lambda_k = \frac{2 n(\lambda_k) L}{k}[/tex]

    where [tex]\lambda_k[/tex] is a wavelength at mode k, k is integer, n is the index of reflection, L is a constant. I am trying to find the change of wavelength between two adjacent mode approximately, the answer will be

    [tex]\Delta\lambda_k \approx \frac{\lambda_k^2}{2n_gL}[/tex]

    [tex]n_g = n(\lambda_k) - \left.\lambda_k\frac{dn}{d\lambda}\right|_{\lambda_k}[/tex]

    I have no idea how to achieve this. Please give me some hint. Thanks
  2. jcsd
  3. Dec 2, 2008 #2


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    I don't see exactly how to do it right away, but [itex]n_g[/itex] looks an awful lot like a first-order Taylor expansion of n around [itex]\lambda_k[/itex]. So maybe if you rewrite [tex]n(\lambda_k) = \frac{k \lambda_k}{2 L}[/tex]
    and then do some expansion of the left hand side for
    [tex]n(\lambda_{k + 1}) = n(\lambda_k) + \Delta\lambda_k \frac{dn(\lambda_k)}{d\lambda_k} + \text{ higher order}[/tex].
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