PowerWiki: OOEET_ZakonyGeometrickeOptiky

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Dopadá-li na těleso zářivý tok [{MathInline fontsize='12' latex='\\Phi_e'}], potom se jeho část [{MathInline fontsize='12' latex='\\Phi_a'}] pohltí, část [{MathInline fontsize='12' latex='\\Phi_r'}] odrazí a část [{MathInline fontsize='12' latex='\\Phi_{tr}'}] tělesem pronikne. Podle zákona o zachování energie musí platit:

Dopadá-li na těleso zářivý tok [{LTMath fontsize='12' latex='\\Phi_e'}], potom se jeho část [{LTMath fontsize='12' latex='\\Phi_a'}] pohltí, část [{LTMath fontsize='12' latex='\\Phi_r'}] odrazí a část [{LTMath fontsize='12' latex='\\Phi_{tr}'}] tělesem pronikne. Podle zákona o zachování energie musí platit:

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Dělíme-li všechny členy [{MathInline fontsize='12' latex='\\Phi_e'}], dostaneme:

Dělíme-li všechny členy [{LTMath fontsize='12' latex='\\Phi_e'}], dostaneme:

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[{MathInline fontsize='12' latex='\\frac{\\Phi_a}{\\Phi_e}=\\alpha_r\\quad\\ldots'}] je součinitel pohltivosti (absorptance)\\

[{MathInline fontsize='12' latex='\\frac{\\Phi_r}{\\Phi_e}=\\rho_r\\quad\\ldots'}] je součinitel odrazivosti (reflektance)\\

[{MathInline fontsize='12' latex='\\frac{\\Phi_{tr}}{\\Phi_e}=\\tau_r\\quad\\ldots'}] je součinitel propustnosti (transmitance).

[{LTMath fontsize='12' latex='\\frac{\\Phi_a}{\\Phi_e}=\\alpha_r\\quad\\ldots'}] je součinitel pohltivosti (absorptance)\\

[{LTMath fontsize='12' latex='\\frac{\\Phi_r}{\\Phi_e}=\\rho_r\\quad\\ldots'}] je součinitel odrazivosti (reflektance)\\

[{LTMath fontsize='12' latex='\\frac{\\Phi_{tr}}{\\Phi_e}=\\tau_r\\quad\\ldots'}] je součinitel propustnosti (transmitance).

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* [{MathInline fontsize='12' latex='\\theta_t'}] je úhel lomu

* [{LTMath fontsize='12' latex='\\theta_t'}] je úhel lomu

* [{LTMath 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%%.

Pro odrazivost vlny paralelně polarizované na rovinu dopadu platí

[{LTMath fontsize='12' maxheight='450'

\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}

}]

[{LTMath fontsize='12'

R_\perp = r_\perp^2 = \left(\frac{\sin(\theta_i-\theta_t)}{\sin(\theta_i+\theta_t)}\right)^2 }]

Pro propustnost platí

[{LTMath fontsize='12'

\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}

}]

Pro odrazivost vlny kolmo polarizované na rovinu dopadu platí

[{LTMath fontsize='12' maxheight='450'

\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}

}]

[{LTMath fontsize='12'

R_\parallel = r_\parallel^2 = \frac{\tan^2(\theta_i-\theta_t)}{\tan^2(\theta_i+\theta_t)} }]

[{LTMath fontsize='12'

t_\parallel=\frac{2\sin(\theta_t)\cos(\theta_i)}{\cos(\theta_i - \theta_t)\sin(\theta_i + \theta_t)} }]

Součinitel odrazivosti je možno uvažovat aritmetický průměr součinitel odrazivosti pro paralelně a kolmo polarizované paprsky

[{LTMath fontsize='12'

\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] }]