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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