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* [{MathInline fontsize='12' latex='\\theta_i'}] je úhel dopadu |
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Abychom mohli vyhodnotit jak velká část paprsku byla odražena a jaká |
lomena, použijeme Fresnelovy rovnice, které počítají energetické |
poměry odražených a lámaných paprsků na rovinném rozhraní dvou |
neabsorbujících, homogenních, izotropních, lineárních, nemagnetických |
dielektrik s indexy lomu n%%sub 1%% a n%%sub 2%%. |
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Pro odrazivost vlny paralelně polarizované na rovinu dopadu platí |
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[{Math fontsize='12' maxheight='450' |
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\begin{split} |
r_\perp |
&= \frac{\frac{n_1}{\mu_1}\cos(\theta_i)-\frac{n_2}{\mu_2}\cos(\theta_t)} |
{\frac{n_1}{\mu_1}\cos(\theta_i)+\frac{n_2}{\mu_2}\cos(\theta_t)} \qquad\qquad\qquad \mu_1\approx\mu_2\\ |
&= \frac{{n_1}\cos(\theta_i)-{n_2}\cos(\theta_t)} |
{{n_1}\cos(\theta_i)+{n_2}\cos(\theta_t)}\\ |
&= \frac{\cos(\theta_i)-\frac{n_2}{n_1}\cos(\theta_t)} |
{\cos(\theta_i)+\frac{n_2}{n_1}\cos(\theta_t)} |
\qquad\qquad{\small \frac{n_2}{n_1}=\frac{\sin(\theta_i)}{\sin(\theta_t)}}\\ |
&= \frac{\cos(\theta_i)-\frac{\sin(\theta_i)}{\sin(\theta_t)}\cos(\theta_t)} |
{\cos(\theta_i)+\frac{\sin(\theta_i)}{\sin(\theta_t)}\cos(\theta_t)} |
=\frac{\frac{\cos(\theta_i)}{\cos(\theta_i)}-\frac{\sin(\theta_i)}{\cos(\theta_i)}\frac{\cos(\theta_t)}{\sin(\theta_t)}} |
{\frac{\cos(\theta_i)}{\cos(\theta_i)}+\frac{\sin(\theta_i)}{\cos(\theta_i)}\frac{\cos(\theta_t)}{\sin(\theta_t)}} |
=\frac{1-\frac{\tan(\theta_i)}{\tan(\theta_t)}}{1+\frac{\tan(\theta_i)}{\tan(\theta_t)}} |
=\frac{\tan(\theta_t)-\tan(\theta_i)}{\tan(\theta_t)+\tan(\theta_i)}\\ |
&= -\frac{\sin(\theta_i-\theta_t)}{\sin(\theta_i+\theta_t)} |
\end{split} |
}] |
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[{Math fontsize='12' |
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R_\perp = r_\perp^2 = \left(\frac{\sin(\theta_i-\theta_t)}{\sin(\theta_i+\theta_t)}\right)^2 }] |
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Pro propustnost platí |
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[{Math fontsize='12' |
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\begin{split} |
t_\perp |
&= \frac{2\frac{n_1}{\mu_1}\cos(\theta_i)} |
{\frac{n_1}{\mu_1}\cos(\theta_i)+\frac{n_2}{\mu_2}\cos(\theta_t)} \qquad\qquad\qquad \mu_1\approx\mu_2\\ |
&= \frac{2{n_1}\cos(\theta_i)} |
{{n_1}\cos(\theta_i)+{n_2}\cos(\theta_t)} |
= -\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_i+\theta_t)} |
\end{split} |
}] |
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Pro odrazivost vlny kolmo polarizované na rovinu dopadu platí |
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[{Math fontsize='12' maxheight='450' |
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\begin{split} |
r_\parallel |
&= \frac{\frac{n_2}{\mu_2}\cos(\theta_i)-\frac{n_1}{\mu_1}\cos(\theta_t)} |
{\frac{n_2}{\mu_2}\cos(\theta_i)+\frac{n_1}{\mu_1}\cos(\theta_t)} \qquad\qquad\qquad \mu_1\approx\mu_2\\ |
&=\frac{n_2\cos(\theta_i)-n_1\cos(\theta_t)}{n_2\cos(\theta_i)+n_1\cos(\theta_t)} |
=\frac{\frac{\sin(\theta_i)}{\sin(\theta_t)}\cos(\theta_i)-\cos(\theta_t)} |
{\frac{\sin(\theta_i)}{\sin(\theta_t)}\cos(\theta_i)+\cos(\theta_t)}\\ |
&=\frac{\sin(\theta_i)\cos(\theta_i)-\cos(\theta_t)\sin(\theta_t)} |
{\sin(\theta_i)\cos(\theta_i)+\cos(\theta_t)\sin(\theta_t)}\\ |
&= \frac{\tan (\theta_i - \theta_t)}{\tan (\theta_i + \theta_t)} |
\end{split} |
}] |
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[{Math fontsize='12' |
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R_\parallel = r_\parallel^2 = \frac{\tan^2(\theta_i-\theta_t)}{\tan^2(\theta_i+\theta_t)} }] |
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[{Math fontsize='12' |
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t_\parallel=\frac{2\sin(\theta_t)\cos(\theta_i)}{\cos(\theta_i - \theta_t)\sin(\theta_i + \theta_t)} }] |
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Součinitel odrazivosti je možno uvažovat aritmetický průměr součinitel odrazivosti pro paralelně a kolmo polarizované paprsky |
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[{Math fontsize='12' |
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\rho_r=\frac{1}{2}(R_\perp+R_\parallel) = \frac{1}{2}\left[\frac{\tan^2(\theta_i-\theta_t)}{\tan^2(\theta_i+\theta_t)}+ |
\frac{\sin^2(\theta_i-\theta_t)}{\sin^2(\theta_i+\theta_t)}\right] }] |
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