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LightRays/lois.tm

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Initial commit.
2020-03-04 22:14:44 +00:00
<TeXmacs|1.99.11>
<style|<tuple|base|french>>
<\body>
<doc-data|<doc-title|Notes techniques>>
<\subsection>
<name|2d> vs. <name|3d>
</subsection>
L'intensit<69> d'une source omnidirectionnelle <name|2d> diminue en
<math|<frac*|1|R>>, et non en <math|<frac*|1|R<rsup|2>>>
<math|\<rightarrow\>> ce n'est pas une source <name|3d>-isotrope, mais
plut<75>t une source focalis<69>e dans la direction <math|z> (comme un
phare-fente) ou un barre lumineuse infinie dans la direction <math|z>.
Conjecture : les r<>sultats de la simulation sont les m<>mes que en <name|3d>
avec des objets infinis suivant <math|z>.
<subsection|Angles modulo <math|<with|math-condensed|true|2*\<pi\>>>>
<\compact>
De fa<66>on g<>n<EFBFBD>rale, on ne ram<61>ne jamais syst<73>matiquement les angles \S
g<>om<6F>triques \T <20> <math|<around*|[|0,<with|math-condensed|true|2*\<pi\>>|[>>
ou <math|<around*|[|-\<pi\>,+\<pi\>|[>>, car cela casse les potentielles
relations d'ordres entre les angles (e.g.
<math|-0.1*\<pi\>\<leqslant\>\<theta\>\<leqslant\>+0.1*\<pi\>> n'est
<em|pas> <20>quivalent <20> <math|1.9*\<pi\>\<leqslant\>\<theta\>\<leqslant\>0.1*\<pi\>>,
qui n'est jamais vrai). Exceptions :
<\itemize-dot>
<item>angles d'incidences : <math|i\<in\><around*|[|-<frac|\<pi\>|2>,+<frac|\<pi\>|2>|]>>
syst<73>matiquement
<item>angle du rayon, ramen<65> <20> <math|<around*|[|0,<with|math-condensed|true|2*\<pi\>>|[>>
<20> chaque <20>mission, pour <20>viter d'accumumuler des tours
</itemize-dot>
</compact>
<subsection|Intersection segment / demi-droite>
<\padded-center>
<image|<tuple|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
</padded-center>
Segment d'extr<74>mit<69>s <math|A> et <math|B> : <math|A*s+<around*|(|1-s|)>*B>
pour <math|s\<in\><around*|[|0,1|]>>.
Demi-droite port<72>e par <math|<wide|u|\<vect\>><rsub|\<alpha\>>> d'origine
<math|O> : <math|O+t*<wide|u|\<vect\>><rsub|\<alpha\>>> pour
<math|t\<in\><around*|[|0,\<infty\>|[>>.
Intersection :
<\equation*>
\<exists\>s<rsub|<text|i>>,t<rsub|<text|i>><space|1em>:<space|1em>A*s<rsub|<text|i>>+<around*|(|1-s<rsub|<text|i>>|)>*B<separating-space|0.2em>=<separating-space|0.2em>O+t<rsub|<text|i>>*<wide|u|\<vect\>><rsub|\<theta\>><space|1em>\<Leftrightarrow\><space|1em><wide|B\<nospace\>A|\<vect\>>*s<rsub|<text|i>>+t<rsub|<text|i>>*<wide|u|\<vect\>><rsub|\<alpha\>>*<separating-space|0.2em>=<separating-space|0.2em><wide|B\<nospace\>O|\<vect\>>
</equation*>
<\equation*>
\<Longleftrightarrow\><space|1em>\<exists\>s<rsub|<text|i>>\<in\><around*|[|0,1|]>,t<rsub|<text|i>>\<in\><around*|[|0,\<infty\>|[><space|1em>:<space|1em><bmatrix|<tformat|<table|<row|<cell|x<rsub|a>-x<rsub|b>>|<cell|-x<rsub|\<alpha\>>>>|<row|<cell|y<rsub|a>-y<rsub|b>>|<cell|-y<rsub|\<alpha\>>>>>>>*<bmatrix|<tformat|<table|<row|<cell|s<rsub|<text|i>>>>|<row|<cell|t<rsub|<text|i>>>>>>>=<bmatrix|<tformat|<table|<row|<cell|x<rsub|o>-x<rsub|b>>>|<row|<cell|y<rsub|o>-y<rsub|b>>>>>>
</equation*>
En ignorant le cas o<> le segment et la demi-droite sont parall<6C>les, il
suffit de r<>soudre ce syst<73>me lin<69>aire. Si <math|s\<nin\><around*|[|0,1|]>>
ou si <math|t\<less\>0>, alors il n'y a pas intersection. Sinon, le point
d'intersection est <math|P=O+t<rsub|<text|i>>*<wide|u|\<vect\>><rsub|\<alpha\>>>,
et l'angle d'incidence est <math|i=\<alpha\>-\<theta\><rprime|'>>, o<>
<math|\<theta\>=\<angle\><wide|B\<nospace\>A|\<vect\>>> et
<math|\<theta\><rprime|'>=\<theta\>-<frac|\<pi\>|2>>. L'angle de la normale
est <math|\<theta\><rsub|<text|normale>>=\<theta\><rprime|'>+\<pi\>>.
<subsection|Construction d'un arc de cercle <20> partir de
<math|<around*|(|A,B,R|)>>>
<\wide-tabular>
<tformat|<cwith|1|-1|1|-1|cell-valign|t>|<cwith|1|1|1|1|cell-rsep|20pt>|<table|<row|<\cell>
Apr<70>s translation de <math|-<wide|a|\<vect\>>> et rotation d'un angle
<math|-\<theta\><rsub|0>>, on suppose que <math|A> est l'origine et que
<math|B> est <20> la verticale de <math|A> (<math|x<rprime|'><rsub|B>=0>).
Le centre <math|C> de l'arc de cercle voulu est l'intersection (de
droite ici) des ceux cercles centr<74>s en <math|A> et <math|B> de rayons
<math|R>. On voit imm<6D>diatement que
<math|y<rprime|'><rsub|C>=A\<nospace\>B/2>. Le centre <math|C> est
finalement d<>fini par la solution n<>gative de
<math|x<rprime|'><rsub|C><rsup|2>+y<rprime|'><rsub|C><rsup|2>=R<rsup|2>>,
c'est-<2D>-dire
<\equation*>
x<rprime|'><rsub|C>=-<sqrt|R<rsup|2>-y<rprime|'><rsub|C><rsup|2>>=-<sqrt|R<rsup|2>-<around*|(|A\<nospace\>B/2|)><rsup|2>><space|1em><text|d<>finie
si <math|A\<nospace\>B\<leqslant\>2*R>>
</equation*>
Pour obtenir <math|\<theta\><rprime|'><rsub|A>> et
<math|\<theta\><rprime|'><rsub|B>>, on a simplement
<\equation*>
tan<around*|(|\<mp\>\<theta\><rprime|'><rsub|A,B>|)>=<frac|y<rprime|'><rsub|C>|-x<rprime|'><rsub|C>>=1/<sqrt|<around*|(|2*R/A\<nospace\>B|)><rsup|2>-1>
</equation*>
</cell>|<\cell>
<image|<tuple|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
</cell>>>>
</wide-tabular>
On obtient finalement <math|\<theta\><rsub|A,B>=\<theta\><rprime|'><rsub|A,B>+\<theta\><rsub|0>>
et <math|C> apr<70>s rotation d'un angle <math|+\<theta\><rsub|0>> puis une
translation de <math|+<wide|a|\<vect\>>>.
<subsection|Intersection arc de cercle / demi-droite>
<\padded-center>
<image|<tuple|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
</padded-center>
On suppose que <math|\<theta\><rsub|A>\<less\>\<theta\><rsub|B>>. La
distance <math|d=O\<nospace\>C> est non sign<67>e ici, et les angles ne sont
pas absolus comme avec le segment, mais relatifs <20> l'axe
<math|<wide|O\<nospace\>C|\<vect\>>> (on peut regarder la figure dans
n'importe quel sens). L'arc est de rayon de courbure
<math|R=C\<nospace\>A=C\<nospace\>B>. Clairement, si
<math|<around*|\||\<alpha\>|\|>\<gtr\><frac|\<pi\>|2>> et <math|d\<gtr\>R>,
il n'y a pas intersection.
Pour le premier point d'intersection avec le disque (<math|P> ici),
<\equation*>
<wide*|d<phantom|<around*|\|||<right|.>>>tan
\<alpha\>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|1>>+<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|2>>>=<wide*|<around*|(|+R*sin
\<theta\><rsub|1>|)>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|1>>>+<wide*|tan
\<alpha\>\<cdot\><around*|(|-R*cos \<theta\><rsub|1>|)>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|2>>><space|1em>\<Longleftrightarrow\><space|1em>b\<cdot\>sin<around*|(|\<alpha\>|)>=sin<around*|(|\<theta\><rsub|1>-\<alpha\>|)><space|1em><text|o<>><space|1em>b=<frac|d|R>
</equation*>
La demi-droite intersecte l'arc <math|<wide|A\<nospace\>B|\<invbreve\>>> si
<math|\<theta\><rsub|1>> est d<>fini, donc si<\footnote>
On a alors <math|<around*|\||sin<around*|(|\<theta\><rsub|<text|t>>-\<alpha\><rsub|<text|t>>|)>|\|>=1>,
d'o<> <math|\<theta\><rsub|<text|t>>=\<alpha\><rsub|<text|t>>\<pm\><frac|\<pi\>|2>>,
d'o<>\
<\equation*>
cos \<theta\><rsub|<text|t>>=cos<around*|(|\<alpha\><rsub|<text|t>>\<pm\><tfrac|\<pi\>|2>|)>=\<mp\>sin<around*|(|\<alpha\><rsub|<text|t>>|)>=\<mp\><frac|R|d>
</equation*>
On peut retrouver cet angle <math|<math|\<theta\><rsub|<text|t>>>> en
<20>crivant la condition de tengentialit<69> rayon/cercle au point <math|P> :
<\equation*>
<bmatrix|<tformat|<table|<row|<cell|R*cos
\<theta\><rsub|<text|t>>>>|<row|<cell|\<pm\>R*sin*\<theta\><rsub|<text|t>>>>>>><separating-space|0.2em>=<separating-space|0.2em>O+t*<rigid|<wide|n|\<vect\>><rsub|\<theta\><rsub|<text|t>>>><rsup|\<bot\>><separating-space|0.2em>=<separating-space|0.2em><bmatrix|<tformat|<table|<row|<cell|-d>>|<row|<cell|0>>>>>+t*<bmatrix|<tformat|<table|<row|<cell|sin
\<theta\><rsub|<text|t>>>>|<row|<cell|\<mp\>cos
\<theta\><rsub|<text|t>>>>>>>
</equation*>
Pour <math|y>, on a <math|R*sin \<theta\><rsub|<text|t>>=-t*cos
\<theta\><rsub|<text|t>>> donc <math|t=-R*sin
\<theta\><rsub|<text|t>>/cos \<theta\><rsub|<text|t>>>. Alors, pour
<math|x>, on a
<\equation*>
R*cos \<theta\><rsub|<text|t>>=-d+<around*|<left|(|-1>|-R*<frac|sin
\<theta\><rsub|<text|t>>|cos \<theta\><rsub|<text|t>>>|<right|)|-1>>*sin*\<theta\><rsub|<text|t>><space|1em>\<Longleftrightarrow\><space|1em>R*cos<rsup|2>
\<theta\><rsub|<text|t>>=-d*cos \<theta\><rsub|<text|t>>-R*sin<rsup|2>
\<theta\><rsub|<text|t>><space|1em>\<Longleftrightarrow\><space|1em>cos
\<theta\><rsub|<text|t>>=-<frac|R|d>*<around*|(|cos<rsup|2>+sin<rsup|2>|)>=-<frac|R|d>
</equation*>
Et puisque <math|sin<around*|(|arccos<around*|(|x|)>|)>=<sqrt|1-x<rsup|2>>>
est ind<6E>pendant du signe de <math|x>, on peut tout aussi bien prendre
<math|cos \<theta\><rsub|<text|t>>=+R/d>.
</footnote>
<\wide-tabular>
<tformat|<cwith|1|-1|1|-1|cell-valign|t>|<table|<row|<\cell>
<\compact>
<\equation*>
b*<around*|\||\<space\>sin<around*|(|\<alpha\>|)>|\|>\<leqslant\>1<space|1em>\<Longleftrightarrow\><space|1em><around*|\||\<alpha\>|\|>\<less\>\<alpha\><rsub|<text|t>><separating-space|0.2em>=<separating-space|0.2em>arcsin<around*|(|R/d|)>
</equation*>
lorsque <math|d\<gtr\>R>. Pour <math|d\<less\>R> (<math|O> <20>
l'int<6E>rieur du cercle), il y a toujours une solution, <20>videmment.
</compact>
Pour le deuxi<78>me point d'intersection avec le disque, on a
<\equation*>
<wide*|d<phantom|<around*|\|||<right|.>>>tan
\<alpha\>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|1>>+<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|2>>>=<wide*|<around*|(|+R*sin
\<theta\><rsub|2>|)>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|1>+<decorated|2>+<decorated|3>>>-<wide*|tan
\<alpha\>\<cdot\><around*|(|+R*cos
\<theta\><rsub|2>|)>|\<wide-underbrace\>><rsub|<with|ornament-shape|rounded|padding-below|0.0fn|padding-above|0.0fn|ornament-vpadding|0.8spc|ornament-hpadding|0.8spc|<decorated|3>>>
</equation*>
<text-dots> ce qui revient au m<>me :
<math|b\<cdot\>sin<around*|(|\<alpha\>|)>=sin<around*|(|\<theta\><rsub|2>-\<alpha\>|)>>.
Cette <20>quation poss<73>de en effet, dans
<math|\<theta\>\<in\><around*|[|0,<with|math-condensed|true|2*\<pi\>>|[>>,
nos<space|3em><line-break>deux solutions (courbes
<math|\<theta\><around*|(|\<alpha\>|)>> bi-valu<6C>es).
Au final, les solution sont
<\equation*>
<tabular|<tformat|<table|<row|<cell|\<theta\><rsub|1>=-arcsin<around*|(|b\<cdot\>sin<around*|(|\<alpha\>|)>|)>+\<alpha\>+\<pi\>>>|<row|<cell|\<theta\><rsub|2>=+arcsin<around*|(|b\<cdot\>sin<around*|(|\<alpha\>|)>|)>+\<alpha\>>>>>>
</equation*>
Pour <math|d\<gtr\>R>, la solution correcte est
<math|\<theta\><rsub|1>> si <math|\<theta\><rsub|A>\<less\>\<theta\><rsub|1>\<less\>\<theta\><rsub|B>>,
et sinon, <20>ventuellement <math|\<theta\><rsub|2>> si
<math|\<theta\><rsub|A>\<less\>\<theta\><rsub|2>\<less\>\<theta\><rsub|B>>.
Pour <math|d\<less\>R>, la seule solution correcte est
<math|\<theta\><rsub|2>> (et si on veut rester dans
<math|<around*|[|0,<with|math-condensed|true|2*\<pi\>>|[>>, on ajoute
<math|<with|math-condensed|true|2*\<pi\>>> <20> <math|\<theta\><rsub|2>>
pour <math|\<alpha\>\<less\>0>).
Les angles d'incidence respectifs sont
</cell>|<\cell>
<image|<tuple|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
</cell>>>>
</wide-tabular>
<\equation*>
i<rsub|1>=<around*|(|\<pi\>-\<theta\><rsub|1>|)>+\<alpha\><space|1em><text|et><space|1em><tfrac|\<pi\>|2>=\<alpha\>+<around*|(|-i<rsub|2>|)>+<around*|(|<tfrac|\<pi\>|2>-\<theta\><rsub|2>|)><separating-space|0.2em>\<Leftrightarrow\><separating-space|0.2em>i<rsub|2>=\<alpha\>-\<theta\><rsub|2>
</equation*>
<subsection|R<>flexion et r<>fraction sur une interface
<math|n<rsub|1>\<rightarrow\>n<rsub|2>>>
R<>flexion :
<\equation*>
<text|toujours,<space|1em>avec un angle><space|1em>i<rsub|<text|r>>=i<space|1em>\<Rightarrow\><space|1em>\<alpha\><rsub|<text|r>>=\<theta\><rsub|<text|normale>>-i
</equation*>
R<>fraction (transmission) :
<\equation*>
n<rsub|1>*sin<around*|(|i|)>=n<rsub|2>*sin<around*|(|i<rsub|<text|t>>|)><space|2em><stack|<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|<text|toujours
lorsque <math|n<rsub|1>\<leqslant\>n<rsub|2>>>>>|<row|<cell|<text|pour
<math|i\<leqslant\>i<rsub|<text|cr>>=arcsin<around*|(|1/\<gamma\>|)>>
lorsque <math|n<rsub|1>\<gtr\>n<rsub|2>>>>>>>><space|1em><text|avec><space|1em>\<gamma\>\<assign\><frac|n<rsub|1>|n<rsub|2>>
</equation*>
<\equation*>
\<Rightarrow\><space|1em>\<alpha\><rsub|<text|t>><separating-space|0.2em>=<separating-space|0.2em>\<theta\><rsub|<text|anti-normale>>+i<rsub|<text|t>><separating-space|0.4em>=<separating-space|0.4em><around*|(|\<theta\><rsub|<text|normale>>+\<pi\>|)>+arcsin<around*|<left|(|-1>|\<gamma\>\<cdot\>sin<around*|(|i|)>|<right|)|-1>>
</equation*>
Coefficients de r<>flexion en puissance (c'est juste
<math|R=<around*|\||\<rho\>|\|><rsup|2>>) :
<\equation*>
R<rsub|<text|TE>><separating-space|0.2em>=<separating-space|0.2em><around*|\||<frac|n<rsub|1>*cos
i-n<rsub|2>*cos i<rsub|<text|t>>|n<rsub|1>*cos i+n<rsub|2>*cos
i<rsub|<text|t>>>|\|><rsup|2><space|1em><text|et><space|1em>R<rsub|<text|TM>><separating-space|0.2em>=<separating-space|0.2em><around*|\||<frac|n<rsub|2>*cos
i-n<rsub|1>*cos i<rsub|<text|t>>|n<rsub|2>*cos i+n<rsub|1>*cos
i<rsub|<text|t>>>|\|><rsup|2>
</equation*>
d<>velopp<70>s<\footnote>
<\equation*>
*i<rsub|<text|t>>=arcsin<around*|<left|(|-1>|<frac|n<rsub|1>|n<rsub|2>>*sin<around*|(|i|)>|<right|)|-1>><space|1em>\<Longrightarrow\><space|1em>cos<around*|(|*i<rsub|<text|t>>|)>=<choice|<tformat|<table|<row|<cell|<sqrt|1-\<gamma\><rsup|2>*sin<rsup|2>
i>>|<cell|<text|si <math|i\<leqslant\>i<rsub|<text|cr>>>>>>|<row|<cell|\<mathi\><sqrt|\<gamma\><rsup|2>*sin<rsup|2>
i-1>>|<cell|<text|si <math|i\<gtr\>i<rsub|<text|cr>>>>>>>>><space|1em><text|avec><space|1em>x=<frac|n<rsub|1>|n<rsub|2>>*sin<around*|(|i|)>
</equation*>
</footnote> :
<\equation*>
R<rsub|<text|TE>,<text|TM>>=<choice|<tformat|<table|<row|<cell|<around*|(|<frac|a-b|a+b>|)><rsup|2>>|<cell|<text|si
<math|i\<leqslant\>i<rsub|<text|cr>>>>>>|<row|<cell|1<space|1em><stack|<tformat|<table|<row|<cell|<very-small|<text|(refl.<nbsp>int.<nbsp>tot.)>>>>>>>>|<cell|<text|si
<math|i\<gtr\>i<rsub|<text|cr>>>>>>>>><space|1em><text|avec><space|1em><stack|<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|a<rsub|<text|TE>>=\<gamma\>*cos
i>>|<row|<cell|a<rsub|<text|TM>>=\<gamma\><rsup|-1>*cos
i>>>>><space|1em><text|et><space|1em>b=<sqrt|1-\<gamma\><rsup|2>*sin<rsup|2>
i>
</equation*>
R<>flexion nulle (<math|a=b>) si :
<\itemize-dot>
<item><math|R<rsub|<text|TE>>=0<space|1em>\<Longleftrightarrow\><space|1em>cos<rsup|2>
i+sin<rsup|2> i=1=\<gamma\><rsup|2><space|2em>\<Longleftrightarrow\><space|1em>n<rsub|1>=n<rsub|2>>
<item><math|R<rsub|<text|TM>>=0<space|1em>\<Longleftrightarrow\><space|1em>cos<rsup|2>
i+\<gamma\><rsup|4>*sin<rsup|2> i=\<gamma\><rsup|2><space|1em>\<Longleftrightarrow\><space|1em>n<rsub|1>=n<rsub|2><space|1em><text|ou><space|1em>i=i<rsub|<text|Brewster>>=arctan<around*|<left|(|-1>|<frac|n<rsub|2>|n<rsub|1>>|<right|)|-1>>>
<small|<\eqnarray*>
<tformat|<table|<row|<cell|<text|car><space|3em>R<rsub|<text|TM>>=0>|<cell|\<Longleftrightarrow\>>|<cell|cos<rsup|2>+\<gamma\><rsup|4>*sin<rsup|2><separating-space|0.2em>=<separating-space|0.2em>\<gamma\><rsup|2><separating-space|0.2em>=<separating-space|0.2em>\<gamma\><rsup|2>*cos<rsup|2>+\<gamma\><rsup|2>*sin<rsup|2>>>|<row|<cell|>|<cell|\<Longleftrightarrow\>>|<cell|<around*|(|1-\<gamma\><rsup|2>|)>*cos<rsup|2>=\<gamma\><rsup|2>*<around*|(|1-\<gamma\><rsup|2>|)>*sin<rsup|2>>>|<row|<cell|>|<cell|\<Longleftrightarrow\>>|<cell|\<gamma\>=1<space|1em><text|ou><space|1em>tan<rsup|2>=\<gamma\><rsup|-2>>>>>
</eqnarray*>>
</itemize-dot>
Coefficients de transmission en puissance (ce n'est <em|pas><\footnote>
Coefficients de r<>flexion en amplitude :
<\equation*>
\<rho\><rsub|<text|TE>><separating-space|0.2em>=<separating-space|0.2em><frac|n<rsub|1>*cos
i-n<rsub|2>*cos i<rsub|<text|t>>|n<rsub|1>*cos i+n<rsub|2>*cos
i<rsub|<text|t>>><space|1em><text|et><space|1em>\<rho\><rsub|<text|TM>><separating-space|0.2em>=<separating-space|0.2em><frac|n<rsub|2>*cos
i-n<rsub|1>*cos i<rsub|<text|t>>|n<rsub|2>*cos i+n<rsub|1>*cos
i<rsub|<text|t>>>
</equation*>
d<>velopp<70>s :
<\equation*>
\<rho\><rsub|<text|TE>,<text|TM>>=<choice|<tformat|<table|<row|<cell|<frac|a-b|a+b>>|<cell|<text|si
<math|i\<leqslant\>i<rsub|<text|cr>>>>>>|<row|<cell|<frac|a-\<mathi\>*b|a+\<mathi\>*b><space|1em><stack|<tformat|<cwith|1|-1|1|-1|cell-lsep|0pt>|<cwith|1|-1|1|-1|cell-rsep|0pt>|<cwith|1|-1|1|-1|cell-bsep|0pt>|<cwith|1|-1|1|-1|cell-tsep|0pt>|<table|<row|<cell|<stack|<tformat|<table|<row|<cell|<very-small|<text|(de
norme 1)>>>>>>>>>>>>>|<cell|<text|si
<math|i\<gtr\>i<rsub|<text|cr>>>>>>>>><space|1em><text|avec><space|1em><stack|<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|a<rsub|<text|TE>>=\<gamma\>*cos
i>>|<row|<cell|a<rsub|<text|TM>>=\<gamma\><rsup|-1>*cos
i>>>>><space|1em><text|et><space|1em>b=<sqrt|<around*|\||1-\<gamma\><rsup|2>*sin<rsup|2>
i|\|>>
</equation*>
Coefficients de transmission en amplitude :
<\equation*>
\<tau\><rsub|<text|TE>><separating-space|0.2em>=<separating-space|0.2em><frac|2*n<rsub|1>*cos
i|n<rsub|1>*cos i+n<rsub|2>*cos i<rsub|<text|t>>><space|1em><text|et><space|1em>\<tau\><rsub|<text|TM>><separating-space|0.2em>=<separating-space|0.2em><frac|2*n<rsub|1>*cos
i|n<rsub|2>*cos i+n<rsub|1>*cos i<rsub|<text|t>>>
</equation*>
d<>velopp<70>s :
<\equation*>
\<tau\><rsub|<text|TE>,<text|TM>>=<choice|<tformat|<table|<row|<cell|<frac|c|a+b>>|<cell|<text|si
<math|i\<leqslant\>i<rsub|<text|cr>>>>>>|<row|<cell|<frac|c|a+\<mathi\>*b><space|1em><very-small|<text|(ampl.
plasmon)>>>|<cell|<text|si <math|i\<gtr\>i<rsub|<text|cr>>>>>>>>><space|1em><text|avec><space|1em><stack|<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<table|<row|<cell|c<rsub|<text|TE>>=2*\<gamma\>*cos
i>>|<row|<cell|c<rsub|<text|TM>>=2*cos
i>>>>><space|2em>\<longrightarrow\><space|1em>T=<frac|n<rsub|2>*cos
i<rsub|<text|t>>|n<rsub|1>*cos i>*<around*|\||\<tau\>|\|><rsup|2>
</equation*>
</footnote> juste <math|T=<around*|\||\<tau\>|\|><rsup|2>>) :
<\equation*>
T<separating-space|0.2em>=<separating-space|0.2em>1-R
</equation*>
par conservation de l'<27>nergie. Ici, l'histoire d'angle de vue de
l'interface ne rentre pas en compte, puisqu'on a seulement des rayons
<math|\<rightarrow\>> l'int<6E>gration spatiale est d<>j<EFBFBD> faite, et on manipule
des <em|puissances>, pas des intensit<69>s. Cela est coh<6F>rent avec le fait
d'oublier la formule de l'<27>clairement <math|\<cal-E\>=I*cos \<theta\>>.
<subsection|Objets diffusants : BRDF>
<slink|https://en.wikipedia.org/wiki/Bidirectional_reflectance_distribution_function>
Propri<72>t<EFBFBD> d'une <name|brdf> \S <name|2d> \T
<math|f<around*|(|i,i<rsub|<text|r>>|)>> :
<\itemize-dot>
<item>r<>ciprocit<69> : <math|f<around*|(|i,i<rsub|<text|r>>|)>=f<around*|(|i<rsub|<text|r>>,i|)>><space|1em>(crucial
pour avoir l'<27>quivalence <name|2d> <math|\<leftrightarrow\>> <name|3d>
<math|z>-invar. : les rayons \S perdus \T hors du plan sont compens<6E>s par
tous rayons \S hors plan \T (venant des points <math|z\<neq\>0> des
sources) r<>fl<66>chis dans le plan)
<item>positivit<69> : <math|f<around*|(|i,i<rsub|<text|r>>|)>\<gtr\>0>
<item>conservation de l'<27>nergie : <math|<smash|<with|math-display|true|\<forall\>i,<separating-space|0.2em><big|int><rsub|-<frac|\<pi\>|2>><rsup|<frac|\<pi\>|2>>\<mathd\>i<rsub|<text|r>>
f<around*|(|i,i<rsub|<text|r>>|)>=2\<pi\>>>> (les absorptions sont r<>gl<67>s
avec la variable d'alb<6C>do d<>di<64>e)
</itemize-dot>
\;
</body>
<\initial>
<\collection>
<associate|page-medium|paper>
</collection>
</initial>
<\references>
<\collection>
<associate|auto-1|<tuple|1|1>>
<associate|auto-2|<tuple|2|1>>
<associate|auto-3|<tuple|3|1>>
<associate|auto-4|<tuple|4|2>>
<associate|auto-5|<tuple|5|3>>
<associate|auto-6|<tuple|6|3>>
<associate|auto-7|<tuple|7|4>>
<associate|footnote-1|<tuple|1|2>>
<associate|footnote-2|<tuple|2|4>>
<associate|footnote-3|<tuple|3|4>>
<associate|footnr-1|<tuple|1|2>>
<associate|footnr-2|<tuple|2|4>>
<associate|footnr-3|<tuple|3|4>>
</collection>
</references>
<\auxiliary>
<\collection>
<\associate|toc>
<with|par-left|<quote|1tab>|1.<space|2spc><with|font-shape|<quote|small-caps>|2d>
vs. <with|font-shape|<quote|small-caps>|3d>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1>>
<with|par-left|<quote|1tab>|2.<space|2spc>Angles modulo
<with|font-family|<quote|rm>|<with|mode|<quote|math>|<with|math-condensed|<quote|true>|2*\<pi\>>>>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2>>
<with|par-left|<quote|1tab>|3.<space|2spc>Intersection segment /
demi-droite <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3>>
<with|par-left|<quote|1tab>|4.<space|2spc>Intersection arc de cercle /
demi-droite <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-4>>
<with|par-left|<quote|1tab>|5.<space|2spc>Construction d'un arc de
cercle <20> partir de <with|font-family|<quote|rm>|<with|mode|<quote|math>|<around*|(|A,B,R|)>>>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-5>>
<with|par-left|<quote|1tab>|6.<space|2spc>R<>flexion et r<>fraction sur
une interface <with|font-family|<quote|rm>|<with|mode|<quote|math>|n<rsub|1>\<rightarrow\>n<rsub|2>>>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-6>>
<with|par-left|<quote|1tab>|7.<space|2spc>Objets diffusants : BRDF
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-7>>
</associate>
</collection>
</auxiliary>
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