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[tex] \frac{n_1}{s_o} + \frac{n_2}{s_i} = \frac{n_2-n_1}{R} [/tex]

for Gaussian optics from the following equation

[tex] \frac{n_1}{l_o} + \frac{n_2}{l_i} = \frac{1}{R} \left( \frac{n_2s_i}{l_i} - \frac{n_1 s_o}{l_o} \right)[/tex]

by approximating

[tex] l_o = \sqrt{R^2 + \left( s_o + R \right) ^2 - 2R\left( s_o + R \right) \cos \phi} [/tex]

and

[tex] l_i = \sqrt{R^2 + \left( s_i - R \right) ^2 - 2R\left( s_i - R \right) \cos \phi} [/tex]

with the aid of

[tex] \cos \phi \approx 1 [/tex]

[tex] \hline [/tex]

Here is what I have:

[tex] l_o \approx \sqrt{R^2 + \left( s_o + R \right) ^2 - 2R\left( s_o + R \right) } = \sqrt{R^2 + s_o ^2 + R^2 + 2s_o R - 2s_o R - 2R^2 } = s_o [/tex]

and

[tex] l_i \approx \sqrt{R^2 + \left( s_i - R \right) ^2 - 2R\left( s_i - R \right) } = \sqrt{R^2 + s_i ^2 + R^2 -2s_i R - 2Rs_i + 2R^2} = 2R-s_i [/tex]

But, I expected to find [tex] l_i \approx s_i [/tex] .... what I have doesn't seem to work out. I can't find where I made a mistake, though.

Thanks

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# Homework Help: Gaussian Optics / Paraxial Approximation

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