\Section{optics}
    \desc{Optics}{}{}
    \desc[german]{Optik}{}{}

    \Subsection{insulator}
        \desc{Dielectrics and Insulators}{}{}
        \desc[german]{Dielektrika und Isolatoren}{}{}

        \begin{formula}{eom}
            \desc{Equation of motion}{Nuclei remain quasi static, electrons respond to field}{$u$ \GT{dislocation}, $\gamma = \frac{1}{\tau}$ \GT{dampening}, \QtyRef{momentum_relaxation_time}, \QtyRef{electric_field}, \ConstRef{charge}, $\omega_0$ \GT{resonance_frequency}, \ConstRef{electron_mass}}
            \desc[german]{Bewegungsgleichung}{Kerne bleiben quasi-statisch, Elektronen beeinflusst durch äußeres Feld}{}
            \eq{m_\txe \odv{u}{t^2} = -e\E - m_\txe \gamma \pdv{u}{t} - m_\txe\omega_0^2 u}
        \end{formula}

        \begin{formula}{lorentz}
            \desc{Drude-Lorentz model}{Dipoles treated as classical harmonic oscillators}{$N$ number of oscillators (atoms), $\omega_0$ resonance frequency }
            \desc[german]{Drude-Lorentz-Model}{Dipole werden als klassische harmonische Oszillatoren behandelt}{}
            \eq{\epsilon_\txr(\omega) = 1+\chi + \frac{Ne^2}{\epsilon_0 m_\txe} \left(\frac{1}{\omega^2-\omega^2-i\gamma\omega}\right)}
            \eq{
                \complex{\epsilon}_\txr(0) &\to 1+\chi + \frac{Ne^2}{\epsilon_0 m_\txe \omega_0^2} \\
                \complex{\epsilon}_\txr(\infty) &= \epsilon_\infty = 1+\chi
            }
            \fig{img/cm_optics_absorption_dielectric.pdf}
        \end{formula}

        \begin{formula}{clausius-mosotti}
            \desc{Clausius-Mosotti relation}{for dense optical media: local field from external contribution + field from other dipoles}{$\chi_\txA$ \qtyRef[susecptibility]{susecptibility} of one atom, \QtyRef{relative_permittivity}, $N$ number of dipoles (atoms)}
            \desc[german]{Clausius-Mosotti Beziehung}{}{}
            \eq{\frac{(\epsilon_\txr - 1)}{\epsilon_\txr + 2} = \frac{N\chi_\txA}{3}}
        \end{formula}


    \Subsection{metal}
        \desc{Metals and doped semiconductors}{}{}
        \desc[german]{Metalle und gedopte Halbleiter}{}{}

        \begin{formula}{plasma_frequency}
            \desc{Plasma frequency}{For metals and doped semiconductors.}{$\epsilon_\infty$ high frequency \qtyRef[permittivity]{permittivity}, \ConstRef{vacuum_permittivity}, \QtyRef{effetive_mass}, \ConstRef{charge}, $n$ \qtyRef{charge_carrier_density}}
            \desc[german]{Plasmafrequenz}{In Metallen dotierten Halbleitern}{$\epsilon_\infty$ Hochfrequenz-\qtyRef{permittivity}, \ConstRef{vacuum_permittivity}, \QtyRef{effetive_mass}, \ConstRef{charge}, $n$ \qtyRef{charge_carrier_density}}
            \eq{\omega_\txp = \left(\frac{en^2}{\epsilon_0 \epsilon_\infty \meff}\right)^{1/2}}
            \ttxt{\eng{
                Characteristic frequency for collective motion in external field.\\
                For free charge carriers: perfect screening (reflection) of the external field for $\omega < \omega_\txp$.
            }\ger{
               Charakteristische Frequenz der kollektiven Bewegung im externen Feld.\\
               Für freie Ladungsträger: perfekte Abschirmung (Reflektion) des äußeren Feldes bei $\omega<\omega_\txp$
            }}
        \end{formula}

        \begin{formula}{dielectric_function}
            \desc{\qtyRef{complex_dielectric_function}}{for a free electon plasma}{$\omega_\txp$ \fRef{::plasma_frequency}, $\omega_0$ \GT{resonance_frequency}, $\gamma = \frac{1}{\tau}$ \GT{dampening}, \QtyRef{momentum_relaxation_time}}
            \desc[german]{}{für ein Plasma aus freien Elektronen}{}
            \eq{
                \complex{\epsilon}_\txr(\omega) = 1 - \frac{\omega_\txp}{\omega_0^2 + i\gamma\omega} \\
            }
        \end{formula}


        \begin{formula}{absorption}
            \desc{Free charge carrier absorption}{Exemplary values}{$\omega_\txp$ \fRef{::plasma_frequency}, \QtyRef{refraction_index_real}, \QtyRef{refraction_index_complex}, \QtyRef{absorption_coefficient}, $R$ \fRef{ed:optics:reflectivity}}
            \desc[german]{Freie Ladungsträger}{Beispielwerte}{}
            \fig{img/cm_optics_absorption_free_electrons.pdf}
            \TODO{Include equations? aus adv sc ex 10/1c}
        \end{formula}

        \TODO{relation to AC and DC conductivity}

        \Subsubsection{sc_interband}
            \desc{Interband transitions in semiconductors}{}{}
            % \desc[german]
            \begin{formula}{selection}
                \desc{Selection rule}{}{}
                \desc[german]{Auswahlregel}{}{}
                \ttxt{\eng{
                    Parity of wave functions must be opposite for the transition to occur
                }\ger{
                    Die Wellenfunktionen müssen unterschiedliche Parität haben, damit der Übergang möglich ist
                }}
            \end{formula}

            \begin{formulagroup}{transition_rate}
                \desc{Transition rate}{}{}
                \desc[german]{Übergangsrate}{}{}
                \begin{formula}{absorption}
                    \desc{Absorption}{}{}
                    \desc[german]{Absorption}{}{}
                    \eq{W_{1\to2} = \frac{2\pi}{\hbar} \abs{M_{12}}^2 \delta(E_1-E_2+\hbar\omega)}
                \end{formula}
                \begin{formula}{emission}
                    \desc{Emission}{}{}
                    \desc[german]{Emission}{}{}
                    \eq{W_{2\to1} = \frac{2\pi}{\hbar} \abs{M_{12}}^2 \delta(E_1-E_2-\hbar\omega)}
                \end{formula}
                \TODO{stimulated vs spontaneous}
                \TODO{matrix element stuff, kane energy}
                \begin{formula}{possible_matrix_elements}
                    \desc{Matrix elements}{}{}
                    \desc[german]{}{}{}
                    \eq{\Braket{p_x|\hat{p}_x|s} = \Braket{p_y|\hat{p}_y|s} = \Braket{p_z|\hat{p}_z|s} \neq 0}
                    \TODO{heavy holes, light holes}
                \end{formula}

            \end{formulagroup}

            \begin{formula}{jdos}
                \desc{Joint density of states}{}{}
                % \desc[german]{}{}{}
                \ttxt{\eng{
                    Desribes the density of states of an optical transition by combining the electron states in the valence band and hole states in the conduction band.
                }\ger{
                    Beschreibt die Zustandsdichte eines optischen Übergangs durch kombinieren der Elektronenzustände im Valenzband und der Lochzustände im Leitungsband.
                }}
                \eq{}
            \end{formula}

            \begin{formula}{absorption_coefficient_direct}
                \desc{\qtyRef{absorption_coefficient}}{For a direct semiconductor}{\QtyRef{angular_frequency}, \QtyRef{refraction_index_real}, \QtyRef{permittivity_complex}}
                \desc[german]{}{Für direkte Halbleiter}{}
                \eq{
                    \alpha &= \frac{\omega}{\nreal c} \epsreal \\
                    \left(\hbar\omega\alpha\right)^2 \propto \hbar\omega-\Egap
                }
            \end{formula}

            \begin{formula}{absorption_coefficient_indirect}
                \desc{\qtyRef{absorption_coefficient}}{For an indirect semiconductor}{\QtyRef{angular_frequency}, $E_\txp$ phonon energy, $E_\text{ig}$ indirect gap}
                \desc[german]{}{Für indirekte Halbleiter}{\QtyRef{angular_frequency}, $E_\txp$ Phononenergie, $E_\text{ig}$ indirekte Bandlücke}
                \eq{
                    \sqrt{\hbar\omega\alpha} \propto \hbar\omega \mp E_\txp - E_\text{ig}
                }
            \end{formula}


            \begin{formulagroup}{quantum_well}
                \desc{Interband absorption in quantum wells}{}{}
                \desc[german]{Interbandabsorption in Quantum Wells}{}{}
                \TODO{TODO}
            \end{formulagroup}

            \begin{formulagroup}{exciton}
                \desc{\fRef{cm:sc:exciton} absorption}{}{\QtyRef{band_gap}, $E_\text{binding}$ \fRef{cm:sc:exciton:binding_energy}}
                \desc[german]{\fRef{cm:sc:exciton} Absorption}{}{}
                \begin{formula}{absorption}
                    \desc{Absorption}{}{}
                    \desc[german]{Absorption}{}{}
                    \ttxt{\eng{
                            Due to binding energy, exciton absorption can happen below the \absRef[band gap energy]{band_gap} \Rightarrow Sharp absorption peak below $\Egap$.
                            At high (room) temperatures, excitons are ionized by collisions with phonons.
                    }\ger{
                        Aufgrund der Bindungsenergie kann die Exzitonenabsorption unterhalb der \absRef[Bandlückenenergie]{band_gap} auftreten \Rightarrow scharfer Absorptionspeak unterhalb von $\Egap$.
                        Bei hohen (Raum) Temperaturen werden Exzitons durch Kollisionen mit Phononen ionisiert.
                    }}
                    \eq{\hbar\omega = \Egap - E_\text{binding}}
                \end{formula}
            \end{formulagroup}



            \TODO{dipole approximation}