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Binomial theorem question

  • #1
I am trying to do a question from Eugene Hecht's Optics book, which goes something like this:

Given the following equations:

Cauchy's Equation:

[tex]n = C_1 + \frac{C_2}{\lambda^2} + \frac{C_3}{\lambda^4} + ... [/tex]

Sellmeier's Equation:

[tex]n^2 = 1 + \sum_{j} \frac{A_j\lambda^2}{\lambda^2-\lambda_0_j^2}
[/tex]

where the [tex]A_j[/tex] terms are constants and each [tex]\lambda_0_j[/tex] is the vacuum wavelength associated with a natural frequency [tex]v_0_j[/tex], such that [tex]\lambda_0_jv_0_j = c [/tex].

Show that where [tex]\lambda >> \lambda_0_j [/tex], Cauchy's Equation is an approximation of Sellmeier's Equation.

Now it also gives a hint which is as follows:

Write the above expression with only the first term in the sum; expand it by the binomial theorem; take the square root of [tex]n^2[/tex] and expand again.

From the hint, I gather that it means to rewrite Sellmeier's Equation as:

[tex]n^2 = 1 + \frac{A\lambda^2}{\lambda^2 - \lambda_0^2}[/tex]

From there though, I have no idea how to apply the binomial theorem to expand it. I just don't see how anything in that equation has the form [tex](x+y)^n[/tex], except for where n = 1.

If anyone can explain to me how to apply the binomial theorem to the equation, or if I've misunderstood what the hint means, it would be much appreciated.
 

Answers and Replies

  • #2
Galileo
Science Advisor
Homework Helper
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You can use the binomial theorem to expand [itex](1+x)^{1/2}[/itex] when x<<1.
 
  • #3
So you mean first take the square root of both sides, then expand it using the binomial theorem , letting [tex]x = \frac{A\lambda^2}{\lambda^2 - \lambda_0^2}[/tex], rather than first applying the binomial theorem, then taking the square root of both sides and then expanding again like the hint suggests?
 
  • #4
Dr Transport
Science Advisor
Gold Member
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Rewrite [tex] \frac{A_j\lambda^2}{\lambda^2-\lambda_0_j^2} [/tex] as

[tex] \frac{A_j}{\lambda^2}\frac{1}{1-\frac{\lambda_0_j^2}{\lambda^2}} [/tex] and expand the second part as

[tex] \frac{1}{1-x^2} \approx 1 - x^2 + x^4 - x^6 \ldots [/tex] where [tex] x = \frac{\lambda}{\lambda_0_j} [/tex]
 
Last edited:
  • #5
Aah, I didn't think to do that. Thanks, that was a great help.
 

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