[1211] | 1 | % GEANT4 Physics Reference Manual - Optical Photons |
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| 2 | % in LaTex 2e - adopted from GEANT3 manual by P. Gumplinger |
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| 3 | |
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| 4 | \section{Interactions of optical photons} |
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| 5 | |
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| 6 | Optical photons are produced when a charged particle traverses: |
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| 7 | |
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| 8 | \begin{enumerate} |
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| 9 | \item a dielectric material with velocity above the \v{C}erenkov |
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| 10 | threshold; |
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| 11 | \item a scintillating material. |
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| 12 | \end{enumerate} |
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| 13 | |
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| 14 | \subsection{Physics processes for optical photons} |
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| 15 | |
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| 16 | A photon is called optical when its wavelength is much greater than the |
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| 17 | typical atomic spacing, for instance when $\lambda \geq 10nm$ |
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| 18 | which corresponds to an energy $E \leq 100eV$\@. Production of an |
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| 19 | optical photon in a HEP detector is primarily due to: |
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| 20 | |
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| 21 | \begin{enumerate} |
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| 22 | \item \v{C}erenkov effect; |
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| 23 | \item Scintillation. |
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| 24 | \end{enumerate} |
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| 25 | |
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| 26 | Optical photons undergo three kinds of interactions: |
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| 27 | |
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| 28 | \begin{enumerate} |
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| 29 | \item Elastic (Rayleigh) scattering; |
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| 30 | \item Absorption; |
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| 31 | \item Medium boundary interactions. |
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| 32 | \end{enumerate} |
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| 33 | |
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| 34 | \subsubsection{Rayleigh scattering} |
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| 35 | |
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| 36 | For optical photons Rayleigh scattering is usually unimportant. For |
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| 37 | $\lambda=.2\mu m$ we have $\sigma_{Rayleigh} \approx .2b$ for $N_{2}$ or |
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| 38 | $O_{2}$ which gives a mean free path of $\approx1.7km$ in air and |
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| 39 | $\approx1m$ in quartz. Two important exceptions are aerogel, which is |
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| 40 | used as a \v{C}erenkov radiator for some special applications and large |
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| 41 | water \v{C}erenkov detectors for neutrino detection. |
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| 42 | |
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| 43 | The differential cross section in Rayleigh scattering, |
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| 44 | $d\sigma/d\Omega$, is proportional to $\cos^{2}\theta$, where $\theta$ |
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| 45 | is the polar angle of the new polarization with respect to the old |
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| 46 | polarization. |
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| 47 | |
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| 48 | \subsubsection{Absorption} |
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| 49 | |
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| 50 | Absorption is important for optical photons because it determines the |
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| 51 | lower $\lambda$ limit in the window of transparency of the radiator. |
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| 52 | Absorption competes with photo-ionization in producing the signal in the |
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| 53 | detector, so it must be treated properly in the tracking of optical |
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| 54 | photons. |
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| 55 | |
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| 56 | \subsubsection {Medium boundary effects} |
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| 57 | |
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| 58 | When a photon arrives at the boundary of a dielectric medium, its |
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| 59 | behaviour depends on the nature of the two materials which join at that |
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| 60 | boundary: |
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| 61 | |
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| 62 | \begin{itemize} |
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| 63 | \item Case dielectric $\rightarrow$ dielectric.\\ |
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| 64 | The photon can be transmitted (refracted ray) or reflected (reflected |
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| 65 | ray). In case where the photon can only be reflected, total internal |
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| 66 | reflection takes place. |
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| 67 | \item Case dielectric $\rightarrow$ metal.\\ |
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| 68 | The photon can be absorbed by the metal or reflected back into the |
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| 69 | dielectric. If the photon is absorbed it can be detected according to |
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| 70 | the photoelectron efficiency of the metal. |
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| 71 | \item Case dielectric $\rightarrow$ black material.\\ |
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| 72 | A black material is a tracking medium for which the user has not |
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| 73 | defined any optical property. In this case the photon is immediately |
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| 74 | absorbed undetected. |
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| 75 | \end{itemize} |
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| 76 | |
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| 77 | \subsection {Photon polarization} |
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| 78 | |
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| 79 | The photon polarization is defined as a two component vector normal to |
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| 80 | the direction of the photon: |
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| 81 | |
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| 82 | \begin{displaymath} |
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| 83 | {a_{1}e^{i\Phi_{1}} \choose a_{2}e^{i\Phi_{2}}} = |
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| 84 | e^{\Phi_{o}} |
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| 85 | {a_{1}e^{i\Phi_{c}} \choose a_{2}e^{-i\Phi_{c}}} |
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| 86 | \end{displaymath} |
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| 87 | |
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| 88 | where $\Phi_{c}= (\Phi_{1}-\Phi_{2})/2$ is called circularity and |
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| 89 | $\Phi_{o}=(\Phi_{1}+\Phi_{2})/2$ is called overall phase. Circularity |
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| 90 | gives the left- or right-polarization characteristic of the photon. RICH |
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| 91 | materials usually do not distinguish between the two polarizations and |
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| 92 | photons produced by the \v{C}erenkov effect and scintillation are |
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| 93 | linearly polarized, that is $\Phi_{c}=0$\@. |
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| 94 | |
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| 95 | The overall phase is important in determining interference effects between |
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| 96 | coherent waves. These are important only in layers of thickness comparable |
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| 97 | with the wavelength, such as interference filters on mirrors. The effects of |
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| 98 | such coatings can be accounted for by the empirical reflectivity factor for |
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| 99 | the surface, and do not require a microscopic simulation. GEANT4 does not |
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| 100 | keep track of the overall phase. |
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| 101 | |
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| 102 | Vector polarization is described by the polarization angle $\tan \Psi = |
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| 103 | a_{2}/a_{1}$\@. Reflection/transmission probabilities are sensitive to |
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| 104 | the state of linear polarization, so this has to be taken into account. |
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| 105 | One parameter is sufficient to describe vector polarization, but to |
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| 106 | avoid too many trigonometrical transformations, a unit vector |
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| 107 | perpendicular to the direction of the photon is used in GEANT4. The |
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| 108 | polarization vector is a data member of \texttt{G4DynamicParticle}. |
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| 109 | |
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| 110 | \subsection{Tracking of the photons} |
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| 111 | |
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| 112 | Optical photons are subject to in flight absorption, Rayleigh scattering |
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| 113 | and boundary action. As explained above, the status of the photon is |
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| 114 | defined by two vectors, the photon momentum ($\vec{p}=\hbar \vec{k}$) and |
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| 115 | photon polarization ($\vec{e}$). By convention the direction of the |
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| 116 | polarization vector is that of the electric field. Let also $\vec{u}$ be |
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| 117 | the normal to the material boundary at the point of intersection, |
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| 118 | pointing out of the material which the photon is leaving and toward the |
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| 119 | one which the photon is entering. The behaviour of a photon at the |
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| 120 | surface boundary is determined by three quantities: |
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| 121 | |
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| 122 | \begin{enumerate} |
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| 123 | \item refraction or reflection angle, this represents the kinematics of |
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| 124 | the effect; |
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| 125 | \item amplitude of the reflected and refracted waves, this is the |
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| 126 | dynamics of the effect; |
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| 127 | \item probability of the photon to be refracted or reflected, this is |
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| 128 | the quantum mechanical effect which we have to take into account if we |
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| 129 | want to describe the photon as a particle and not as a wave. |
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| 130 | \end{enumerate} |
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| 131 | |
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| 132 | As said above, we distinguish three kinds of boundary action, dielectric |
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| 133 | $\rightarrow$ black material, dielectric $\rightarrow$ metal, dielectric |
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| 134 | $\rightarrow$ dielectric. The first case is trivial, in the sense that |
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| 135 | the photon is immediately absorbed and it goes undetected. |
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| 136 | |
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| 137 | To determine the behaviour of the photon at the boundary, we will at |
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| 138 | first treat it as an homogeneous monochromatic plane wave: |
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| 139 | |
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| 140 | \begin{displaymath} |
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| 141 | \vec{E} = \vec{E}_{0}e^{i\vec{k} \cdot \vec{x}-i\omega t} |
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| 142 | \end{displaymath} |
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| 143 | \begin{displaymath} |
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| 144 | \vec{B} = \sqrt{\mu \epsilon} \frac{\vec{k} \times \vec{E}}{k} |
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| 145 | \end{displaymath} |
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| 146 | |
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| 147 | \subsubsection{Case dielectric $\rightarrow$ dielectric} |
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| 148 | |
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| 149 | In the classical description the incoming wave splits into a reflected |
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| 150 | wave (quantities with a double prime) and a refracted wave (quantities |
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| 151 | with a single prime). Our problem is solved if we find the following |
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| 152 | quantities: |
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| 153 | |
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| 154 | \begin{displaymath} |
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| 155 | \vec{E}' = \vec{E}_{0}' e^{i\vec{k}'\cdot \vec{x}-i\omega t} |
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| 156 | \end{displaymath} |
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| 157 | \begin{displaymath} |
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| 158 | \vec{E}'' = \vec{E}_{0}'' e^{i\vec{k}''\cdot \vec{x}-i\omega t} |
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| 159 | \end{displaymath} |
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| 160 | |
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| 161 | For the wave numbers the following relations hold: |
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| 162 | |
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| 163 | \begin{displaymath} |
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| 164 | |\vec{k}| = |\vec{k}''| = k = \frac{\omega}{c}\sqrt{\mu \epsilon} |
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| 165 | \end{displaymath} |
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| 166 | \begin{displaymath} |
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| 167 | |\vec{k}'| = k' = \frac{\omega}{c}\sqrt{\mu ' \epsilon '} |
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| 168 | \end{displaymath} |
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| 169 | |
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| 170 | Where the speed of the wave in the medium is $v=c/\sqrt{\mu \epsilon}$ |
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| 171 | and the quantity $n=c/v=\sqrt{\mu \epsilon}$ is called refractive index |
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| 172 | of the medium. The condition that the three waves, refracted, reflected |
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| 173 | and incident have the same phase at the surface of the medium, gives us |
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| 174 | the well known Fresnel law: |
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| 175 | |
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| 176 | \begin{displaymath} |
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| 177 | (\vec{k} \cdot \vec{x})_{surf} = (\vec{k}' \cdot \vec{x})_{surf} = |
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| 178 | (\vec{k}'' \cdot \vec{x})_{surf} |
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| 179 | \end{displaymath} |
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| 180 | |
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| 181 | \begin{displaymath} |
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| 182 | k \sin i = k' \sin r = k'' \sin r' |
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| 183 | \end{displaymath} |
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| 184 | |
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| 185 | where $i, r, r'$ are, respectively, the angle of the incident, refracted |
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| 186 | and reflected ray with the normal to the surface. From this formula the |
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| 187 | well known condition emerges: |
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| 188 | |
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| 189 | \begin{displaymath} |
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| 190 | i = r' |
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| 191 | \end{displaymath} |
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| 192 | \begin{displaymath} |
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| 193 | \frac{\sin i}{\sin r} = \sqrt{\frac{\mu ' \epsilon '}{\mu \epsilon}} = |
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| 194 | \frac{n'}{n} |
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| 195 | \end{displaymath} |
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| 196 | |
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| 197 | The dynamic properties of the wave at the boundary are derived from |
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| 198 | Maxwell's equations which impose the continuity of the normal components |
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| 199 | of $\vec{D}$ and $\vec{B}$ and of the tangential components of $\vec{E}$ |
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| 200 | and $\vec{H}$ at the surface boundary. The resulting ratios between the |
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| 201 | amplitudes of the the generated waves with respect to the incoming one |
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| 202 | are expressed in the two following cases: |
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| 203 | |
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| 204 | \begin{enumerate} |
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| 205 | |
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| 206 | \item a plane wave with the electric field (polarization vector) |
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| 207 | perpendicular to the plane defined by the photon direction and the |
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| 208 | normal to the boundary: |
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| 209 | |
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| 210 | \begin{displaymath} |
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| 211 | \frac{E_{0}'}{E_{0}} = \frac{2n\cos i}{n \cos i = \frac{\mu}{\mu '} |
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| 212 | n' \cos r} = \frac{2n \cos i}{n \cos i + n' \cos r} |
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| 213 | \end{displaymath} |
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| 214 | \begin{displaymath} |
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| 215 | \frac{E_{0}''}{E_{0}} = \frac{n \cos i - \frac{\mu}{\mu '} |
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| 216 | n' \cos r}{n \cos i + \frac{\mu}{\mu '}n' \cos r} = |
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| 217 | \frac{n \cos i - n' \cos r}{n \cos i + n' \cos r} |
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| 218 | \end{displaymath} |
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| 219 | |
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| 220 | where we suppose, as it is legitimate for visible or near-visible |
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| 221 | light, that $\mu/\mu ' \approx 1$; |
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| 222 | |
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| 223 | \item a plane wave with the electric field parallel to the above |
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| 224 | surface: |
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| 225 | |
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| 226 | \begin{displaymath} |
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| 227 | \frac{E_{0}'}{E_{0}} = \frac{2n \cos i}{\frac{\mu}{\mu '}n' |
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| 228 | \cos i + n \cos r} = \frac{2n \cos i}{n' \cos i + n \cos r} |
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| 229 | \end{displaymath} |
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| 230 | \begin{displaymath} |
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| 231 | \frac{E_{0}''}{E_{0}} = \frac{\frac{\mu}{\mu '}n' \cos i - n \cos r} |
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| 232 | {\frac{\mu}{\mu '}n' \cos i + n \cos r} = |
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| 233 | \frac{n' \cos i - n \cos r}{n' \cos i + n \cos r} |
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| 234 | \end{displaymath} |
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| 235 | |
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| 236 | with the same approximation as above. |
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| 237 | |
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| 238 | \end{enumerate} |
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| 239 | |
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| 240 | We note that in case of photon perpendicular to the surface, the |
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| 241 | following relations hold: |
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| 242 | |
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| 243 | \begin{displaymath} |
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| 244 | \frac{E_{0}'}{E_{0}} = \frac{2n}{n'+n} \qquad |
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| 245 | \frac{E_{0}''}{E_{0}} = \frac{n'-n}{n'+n} |
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| 246 | \end{displaymath} |
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| 247 | |
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| 248 | where the sign convention for the parallel field has been adopted. This |
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| 249 | means that if $n'>n$ there is a phase inversion for the reflected wave. |
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| 250 | |
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| 251 | Any incoming wave can be separated into one piece polarized parallel to |
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| 252 | the plane and one polarized perpendicular, and the two components |
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| 253 | treated accordingly. |
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| 254 | |
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| 255 | To maintain the particle description of the photon, the probability to |
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| 256 | have a refracted or reflected photon must be calculated. The constraint |
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| 257 | is that the number of photons be conserved, and this can be imposed via |
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| 258 | the conservation of the energy flux at the boundary, as the number of |
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| 259 | photons is proportional to the energy. The energy current is given by |
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| 260 | the expression: |
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| 261 | |
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| 262 | \begin{displaymath} |
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| 263 | \vec{S} = \frac{1}{2}\frac{c}{4\pi}\sqrt{\mu \epsilon} |
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| 264 | \vec{E} \times \vec{H} = \frac{c}{8\pi}\sqrt{\frac{\epsilon}{\mu}} |
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| 265 | E_{0}^{2}\hat{k} |
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| 266 | \end{displaymath} |
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| 267 | |
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| 268 | and the energy balance on a unit area of the boundary requires that: |
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| 269 | |
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| 270 | \begin{displaymath} |
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| 271 | \vec{S} \cdot \vec{u} = \vec{S}' \cdot \vec{u} - \vec{S}'' \cdot \vec{u} |
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| 272 | \end{displaymath} |
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| 273 | \begin{displaymath} |
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| 274 | S \cos i = S' cos r + S'' cos i |
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| 275 | \end{displaymath} |
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| 276 | \begin{displaymath} |
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| 277 | \frac{c}{8\pi}\frac{1}{\mu}nE_{0}^{2}\cos i = |
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| 278 | \frac{c}{8\pi}\frac{1}{\mu '}n'E_{0}'^{2}\cos r + |
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| 279 | \frac{c}{8\pi}\frac{1}{\mu}nE_{0}''^{2}\cos i |
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| 280 | \end{displaymath} |
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| 281 | |
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| 282 | If we set again $\mu /\mu ' \approx 1$, then the transmission |
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| 283 | probability for the photon will be: |
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| 284 | |
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| 285 | \begin{displaymath} |
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| 286 | T = (\frac{E_{0}'}{E_{0}})^{2} \frac{n' \cos r}{n \cos i} |
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| 287 | \end{displaymath} |
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| 288 | |
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| 289 | and the corresponding probability to be reflected will be $R=1-T$. |
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| 290 | |
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| 291 | In case of reflection, the relation between the incoming photon |
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| 292 | ($\vec{k},\vec{e}$), the refracted one ($\vec{k}', \vec{e}'$) and the |
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| 293 | reflected one ($\vec{k}'', \vec{e}''$) is given by the following |
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| 294 | relations: |
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| 295 | |
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| 296 | \begin{displaymath} |
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| 297 | \vec{q} = \vec{k} \times \vec{u} |
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| 298 | \end{displaymath} |
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| 299 | \begin{displaymath} |
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| 300 | \vec{e}_{\perp} = (\frac{\vec{e} \cdot |
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| 301 | \vec{q}}{|\vec{q}|}) \frac{\vec{q}}{|\vec{q}|} |
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| 302 | \end{displaymath} |
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| 303 | \begin{displaymath} |
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| 304 | \vec{e}_{\parallel} = \vec{e} - \vec{e}_{\perp} |
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| 305 | \end{displaymath} |
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| 306 | \begin{displaymath} |
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| 307 | e_{\parallel}' = e_{\parallel} \frac{2n \cos i}{n'\cos i + n \cos r} |
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| 308 | \end{displaymath} |
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| 309 | \begin{displaymath} |
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| 310 | e_{\perp|}' = e_{\perp} \frac{2n \cos i}{n \cos i + n' \cos r} |
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| 311 | \end{displaymath} |
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| 312 | \begin{displaymath} |
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| 313 | e_{\parallel}'' = \frac{n'}{n}e_{\parallel}' - e_{\parallel} |
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| 314 | \end{displaymath} |
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| 315 | \begin{displaymath} |
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| 316 | e_{\perp}'' = e_{\perp}' - e_{\perp} |
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| 317 | \end{displaymath} |
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| 318 | |
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| 319 | After transmission or reflection of the photon, the polarization vector |
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| 320 | is re-normalized to 1. In the case where $\sin r = n \sin i/n' > 1$ then |
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| 321 | there cannot be a refracted wave, and in this case we have a total |
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| 322 | internal reflection according to the following formulas: |
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| 323 | \begin{displaymath} |
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| 324 | \vec{k}'' = \vec{k} - 2(\vec{k} \cdot \vec{u})\vec{u} |
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| 325 | \end{displaymath} |
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| 326 | \begin{displaymath} |
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| 327 | \vec{e}'' = -\vec{e} + 2(\vec{e} \cdot \vec{u})\vec{u} |
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| 328 | \end{displaymath} |
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| 329 | |
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| 330 | \subsubsection{Case dielectric $\rightarrow$ metal} |
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| 331 | |
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| 332 | In this case the photon cannot be transmitted. So the probability for |
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| 333 | the photon to be absorbed by the metal is estimated according to the |
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| 334 | table provided by the user. If the photon is not absorbed, it is |
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| 335 | reflected. |
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| 336 | |
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| 337 | \begin{latexonly} |
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| 338 | |
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| 339 | \begin{thebibliography}{99} |
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| 340 | \bibitem{one} J.D.~Jackson, \emph{Classical Electrodynamics}, |
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| 341 | J.~Wiley \& Sons Inc., New York, 1975. |
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| 342 | \end{thebibliography} |
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| 343 | |
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| 344 | \end{latexonly} |
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| 345 | |
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| 346 | \begin{htmlonly} |
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| 347 | |
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| 348 | \subsection{Bibliography} |
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| 349 | |
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| 350 | \begin{enumerate} |
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| 351 | \item J.D.~Jackson, \emph{Classical Electrodynamics}, |
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| 352 | J.~Wiley \& Sons Inc., New York, 1975. |
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| 353 | \end{enumerate} |
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| 354 | |
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| 355 | \end{htmlonly} |
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| 356 | |
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