formelsammlung/src/ed/optics.tex

\Section{optics}
    \desc{Optics}{Properties of light and its interactions with matter}{}
    \desc[german]{Optik}{Ausbreitung von Licht und die Interaktion mit Materie}{}

    \TODO{adv. sc slide 427 classification}


    \begin{formulagroup}{refraction_index}
        \desc{Refraction index}{Macroscopic}{\QtyRef{relative_permittivity}, \QtyRef{relative_permeability}, \ConstRef{vacuum_speed_of_light}, \QtyRef{phase_velocity}}
        \desc[german]{Brechungsindex}{Macroscopisch}{}
        \begin{formula}{definition}
            \desc{Definition}{}{}
            \desc[german]{Definition}{}{}
            \quantity{\complex{n}}{}{s}
            \eq{
                \complex{n} = \nreal + i\ncomplex
            }
        \end{formula}

        \begin{formula}{real}
            \desc{Real part}{}{}
            \desc[german]{Reller Teil}{}{}
            \quantity[refraction_index_real]{\nreal}{}{s}
            \eq{
                n = \sqrt{\epsilon_\txr \mu_\txr}
            }
            \eq{
                n = \frac{c_0}{c_\txM}
            }
        \end{formula}
        \begin{formula}{complex}
            \desc{Extinction coefficient}{Complex part of the refraction index. Describes absorption in a medium}{\GT{sometimes} $\kappa$}
            \desc[german]{Auslöschungskoeffizient}{Komplexer Teil des Brechungsindex. Beschreibt Absorption im Medium}{}
            \quantity[refraction_index_complex]{\ncomplex}{}{s}
        \end{formula}
    \end{formulagroup}

    \begin{formula}{reflectivity}
        \desc{Reflectivity}{}{\QtyRef{refraction_index}}
        \desc[german]{Reflektion}{}{}
        \eq{
            R = \abs{\frac{\complex{n}-1}{\complex{n}+1}}
        }
    \end{formula}

    \begin{formula}{snell}
        \desc{Snell's law}{}{$\nreal_i$ \qtyRef{refraction_index_real}, $\theta_i$ incidence angle (normal to the surface)}
        \desc[german]{Snelliussches Brechungsgesetz}{}{$n_i$ \qtyRef{refraction_index}, $\theta_i$ Einfallswinkel (normal zur Fläche)}
        \eq{\nreal_1 \sin\theta_1 = \nreal_2\sin\theta_2}
    \end{formula}

    \begin{formula}{group_velocity}
        \desc{Group velocity}{Velocity with which the envelope of a wave propagates through space}{\QtyRef{angular_frequency}, \QtyRef{angular_wavenumber}}
        \desc[german]{Gruppengeschwindigkeit}{Geschwindigkeit, mit sich die Einhülende einer Welle ausbreitet}{}
        \eq{
            v_\txg \equiv \pdv{\omega}{k}
        }
    \end{formula}
    \begin{formula}{phase_velocity}
        \desc{Phase velocity}{Velocity with which a wave propagates through a medium}{\QtyRef{angular_frequency}, \QtyRef{angular_wavenumber}, \QtyRef{wavelength}, \QtyRef{time_period}}
        \desc[german]{Phasengeschwindigkeit}{Geschwindigkeit, mit der sich eine Welle im Medium ausbreitet}{}
        \hiddenQuantity{v_\txp}{\m\per\s}{}
        \eq{
            v_\txp = \frac{\omega}{k} = \frac{\lambda}{T}
        }
    \end{formula}

    \begin{formula}{absorption_coefficient}
        \desc{Absorption coefficient}{Intensity reduction while traversing a medium, not necessarily by energy transfer to the medium}{\QtyRef{refraction_index_complex}, \ConstRef{vacuum_speed_of_light}, \QtyRef{angular_frequency}}
        \desc[german]{Absoprtionskoeffizient}{Intensitätsverringerung beim Druchgang eines Mediums, nicht zwingend durch Energieabgabe an Medium}{}
        \quantity{\alpha}{\per\cm}{s}
        \eq{
            \alpha &= 2\ncomplex \frac{\omega}{c}
        }
        \TODO{Is this equation really true in general?}
    \end{formula}


    \begin{formula}{intensity}
        \desc{Electromagnetic radiation intensity}{Surface power density}{$S$ \fRef{ed:em:poynting}}
        \desc[german]{Elektromagnetische Strahlungsintensität}{Flächenleistungsdichte}{}
        \quantity{I}{\watt\per\m^2=\k\per\s^3}{s}
        \eq{I = \abs{\braket{S}_t}}
    \end{formula}

    % \begin{formula}{lambert_beer_law}
    %     \desc{Beer-Lambert law}{Intensity in an absorbing medium}{$E_\lambda$ extinction, \QtyRef{absorption_coefficient}, \QtyRef{concentration}, $d$ Thickness of the medium}
    %     \desc[german]{Lambert-beersches Gesetz}{Intensität in einem absorbierenden Medium}{$E_\lambda$ Extinktion, \QtyRef{refraction_index_complex}, \QtyRef{concentration}, $d$ Dicke des Mediums}
    %     \eq{
    %         E_\lambda = \log_{10} \frac{I_0}{I} = \kappa c d \\
    %     }
    % \end{formula}
    \begin{formula}{lambert_beer_law}
        \desc{Beer-Lambert law}{Intensity in an absorbing medium}{\QtyRef{intensity}, \QtyRef{absorption_coefficient}, $z$ penetration depth}
        \desc[german]{Lambert-beersches Gesetz}{Intensität in einem absorbierenden Medium}{\QtyRef{intensity}, \QtyRef{absorption_coefficient}, $z$ Eindringtiefe}
         \eq{
            \d I = -I_0 \alpha(\omega) \d z\\
            I(z) = I_0 \e^{-\alpha(\omega) z}
        }
    \end{formula}

    \begin{formulagroup}{permittivity_complex}
        \desc{Complex relative \qtyRef[permittivity]{permittivity}}{Complex dielectric function\\Microscopic, response of a single atom to an EM wave}{\QtyRef{refraction_index_real}, \QtyRef{refraction_index_complex}}
        \desc[german]{Komplexe relative \qtyRef{permittivity}}{Komplexe dielektrische Funktion\\Mikroskopisch, Verhalten eines Atoms gegen eine EM-Welle}{}
        \begin{formula}{definition}
            \desc{Definition}{}{}
            \desc[german]{Definition}{}{}
            \eq{\epsilon_\txr &= \epsreal + i\epscomplex}
        \end{formula}

        \begin{formula}{real}
            \desc{Real part}{}{}
            \desc[german]{Realteil}{}{}
            \eq{\epsreal &= {\nreal}^2 - {\ncomplex}^2}
            \hiddenQuantity[permittivity_real]{\epsreal}{}{}
        \end{formula}

        \begin{formula}{complex}
            \desc{Complex part}{}{}
            \desc[german]{Komplexer Teil}{}{}
            \eq{\epscomplex &= 2\nreal \ncomplex}
            \hiddenQuantity[permittivity_complex]{\epscomplex}{}{}
        \end{formula}
    \end{formulagroup}