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Hello PF,
I'm reading a paper for a project. In the paper they derive an equation for the effective refractive index ##n=\sqrt{\epsilon^{e} \mu^{e}}## of two stacked layers ##(n_1^2 = \epsilon_1 \mu_1, a)## and ##(n_2^2 = \epsilon_2 \mu_2, b)## where ##a,b## are the lengths and in my case ##\mu_1=\mu_2=1##. One material is silica glass the other is air.
So using Maxwell's equations they derive this equation for ##n##: $$\frac{\alpha_2}{\mu_2} \tan\left(\frac{b\alpha_2}{2}\right)=\frac{\alpha_1}{\mu_1} \tan\left(\frac{a\alpha_1}{2}\right)$$
where ##\alpha_i=k\sqrt{n_i^2n^2}=k\sqrt{\epsilon_i\epsilon^{e}}##.
(I have attached a picture showing the setting used for the derivation.)
But since it can not be solved analytically they do the expansion ##tan(x) \approx x + \frac{1}{3}x^3## and get this: $$ \epsilon^e = \frac{a\epsilon_1+b\epsilon_2}{a+b} + \frac{k^2b^2a^2}{12(a+b)^2}(\epsilon_1\epsilon_2)^2$$
I'm trying to retrace the steps to get the equation for ##\epsilon^{e}## to then be able to expand it to third order. ##(\tan(x) \approx x+\frac{1}{3} x^3 + \frac{2}{15}x^5)##
##\epsilon^{e}_{1,2} = \frac{a_1\pm\sqrt D}{2a_0}, D=a_1^24a_0a_2##
When I do the second order expansion, as in the paper, I get the coefficients for the equation ##a_0x^2+a_1x+a_2=0## to be:
##a_0 = \frac{k^2 (b^3+a^3)}{12}##
##a_1 = (b+a)  \frac{1}{12} 2k^2 (\epsilon_2 b^3+\epsilon_1 a^3)##
## a_2 = \epsilon_1 a + \epsilon_2 b + \frac{1}{12} k^2 (\epsilon_1^2 a^3 + \epsilon_2^2 b^3)##
But when I put in the values from the experiment:
## a = 92\cdot10^{6} m , b = 58\cdot10^{6} m, 1400 GHz, \epsilon_1 = 1 F/m, \epsilon_2 = 3.75 F/m## and solve the equation for ##\epsilon^{e}_{1,2}##.
I am not getting the same result as if I would use the equation derived in the paper. And I can't seem to figure out how to reduce the expression, I get when substituting the coefficients into ##\epsilon^{e}_{1,2} = \frac{a_1\pm\sqrt D}{2a_0}, D=a_1^24a_0a_2##, to the equation given in the paper.
I have attached the paper as well in case there is something missing in my post.
To summarize: How is the equation for ## \epsilon^e ## derived? Because I seem to be getting wrong results
I'm reading a paper for a project. In the paper they derive an equation for the effective refractive index ##n=\sqrt{\epsilon^{e} \mu^{e}}## of two stacked layers ##(n_1^2 = \epsilon_1 \mu_1, a)## and ##(n_2^2 = \epsilon_2 \mu_2, b)## where ##a,b## are the lengths and in my case ##\mu_1=\mu_2=1##. One material is silica glass the other is air.
So using Maxwell's equations they derive this equation for ##n##: $$\frac{\alpha_2}{\mu_2} \tan\left(\frac{b\alpha_2}{2}\right)=\frac{\alpha_1}{\mu_1} \tan\left(\frac{a\alpha_1}{2}\right)$$
where ##\alpha_i=k\sqrt{n_i^2n^2}=k\sqrt{\epsilon_i\epsilon^{e}}##.
(I have attached a picture showing the setting used for the derivation.)
But since it can not be solved analytically they do the expansion ##tan(x) \approx x + \frac{1}{3}x^3## and get this: $$ \epsilon^e = \frac{a\epsilon_1+b\epsilon_2}{a+b} + \frac{k^2b^2a^2}{12(a+b)^2}(\epsilon_1\epsilon_2)^2$$
Homework Statement
I'm trying to retrace the steps to get the equation for ##\epsilon^{e}## to then be able to expand it to third order. ##(\tan(x) \approx x+\frac{1}{3} x^3 + \frac{2}{15}x^5)##
Homework Equations
##\epsilon^{e}_{1,2} = \frac{a_1\pm\sqrt D}{2a_0}, D=a_1^24a_0a_2##
The Attempt at a Solution
When I do the second order expansion, as in the paper, I get the coefficients for the equation ##a_0x^2+a_1x+a_2=0## to be:
##a_0 = \frac{k^2 (b^3+a^3)}{12}##
##a_1 = (b+a)  \frac{1}{12} 2k^2 (\epsilon_2 b^3+\epsilon_1 a^3)##
## a_2 = \epsilon_1 a + \epsilon_2 b + \frac{1}{12} k^2 (\epsilon_1^2 a^3 + \epsilon_2^2 b^3)##
But when I put in the values from the experiment:
## a = 92\cdot10^{6} m , b = 58\cdot10^{6} m, 1400 GHz, \epsilon_1 = 1 F/m, \epsilon_2 = 3.75 F/m## and solve the equation for ##\epsilon^{e}_{1,2}##.
I am not getting the same result as if I would use the equation derived in the paper. And I can't seem to figure out how to reduce the expression, I get when substituting the coefficients into ##\epsilon^{e}_{1,2} = \frac{a_1\pm\sqrt D}{2a_0}, D=a_1^24a_0a_2##, to the equation given in the paper.
I have attached the paper as well in case there is something missing in my post.
To summarize: How is the equation for ## \epsilon^e ## derived? Because I seem to be getting wrong results
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