Le coefficient de réflexion sur une surface dépend de l'orientation du champ électrique.
k
→
∧
E
→
=
c
B
→
{\displaystyle {\overrightarrow {k}}\wedge {\overrightarrow {E}}=c{\overrightarrow {B}}}
E
=
c
B
=
c
0
n
B
=
1
ε
μ
B
=
Z
H
=
μ
ε
H
{\displaystyle E=c\,B={\frac {c_{0}}{n}}B={\frac {1}{\sqrt {\varepsilon \,\mu }}}B=Z\,H={\sqrt {\frac {\mu }{\varepsilon }}}H}
La composante tangentielle du champ électrique
E
→
{\displaystyle {\overrightarrow {E}}}
(1) et la composante tangentielle du champ magnétique
H
→
=
B
→
μ
{\displaystyle {\overrightarrow {H}}={\frac {\overrightarrow {B}}{\mu }}}
(2) doit être continue de part et d'autre de la frontière formée par le dioptre[ 1] . La composante normale de
ε
E
→
{\displaystyle \varepsilon {\overrightarrow {E}}}
et la composante normale de
B
→
=
μ
H
→
{\displaystyle {\overrightarrow {B}}=\mu {\overrightarrow {H}}}
est continue de part et d'autre de la frontière[ 1] .
Cas 1 : champ électrique perpendiculaire au plan d'incidence.
modifier
En projetant sur le plan tangent à la surface,
(1) donne
E
i
⊥
+
E
r
⊥
=
E
t
⊥
{\displaystyle {E_{i_{\perp }}}+{E_{r_{\perp }}}={E_{t_{\perp }}}}
(3)
et (2) donne
−
B
i
μ
i
cos
θ
i
+
B
r
μ
r
cos
θ
r
=
−
B
t
μ
t
cos
θ
t
⇔
−
E
i
⊥
c
i
μ
i
cos
θ
i
+
E
r
⊥
c
r
μ
r
cos
θ
r
=
−
E
t
⊥
c
t
μ
t
cos
θ
t
{\displaystyle -{\frac {B_{i}}{\mu _{i}}}\cos \theta _{i}+{\frac {B_{r}}{\mu _{r}}}\cos \theta _{r}=-{\frac {B_{t}}{\mu _{t}}}\cos \theta _{t}\Leftrightarrow -{\frac {E_{i_{\perp }}}{c_{i}\,\mu _{i}}}\cos \theta _{i}+{\frac {E_{r_{\perp }}}{c_{r}\,\mu _{r}}}\cos \theta _{r}=-{\frac {E_{t_{\perp }}}{c_{t}\,\mu _{t}}}\cos \theta _{t}}
,
ce qui permet d'obtenir :
(
E
i
⊥
−
E
r
⊥
)
n
i
μ
i
cos
θ
i
=
E
t
⊥
n
t
μ
t
cos
θ
t
{\displaystyle \left(E_{i_{\perp }}-E_{r_{\perp }}\right){\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}=E_{t_{\perp }}{\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}
(4)
Pour trouver le coefficient de réflexion :
(
E
i
⊥
−
E
r
⊥
)
n
i
μ
i
cos
θ
i
=
(
E
i
⊥
+
E
r
⊥
)
n
t
μ
t
cos
θ
t
{\displaystyle \left(E_{i_{\perp }}-E_{r_{\perp }}\right){\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}=\left(E_{i_{\perp }}+E_{r_{\perp }}\right){\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}
r
⊥
=
E
r
⊥
E
i
⊥
=
n
i
μ
i
cos
θ
i
−
n
t
μ
t
cos
θ
t
n
i
μ
i
cos
θ
i
+
n
t
μ
t
cos
θ
t
{\displaystyle r_{\perp }={\frac {E_{r_{\perp }}}{E_{i_{\perp }}}}={\frac {{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}-{\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}{{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}+{\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}}}
.
r
⊥
=
E
r
⊥
E
i
⊥
=
n
i
μ
i
cos
θ
i
−
n
t
μ
t
cos
θ
t
n
i
μ
i
cos
θ
i
+
n
t
μ
t
cos
θ
t
{\displaystyle r_{\perp }={\frac {E_{r_{\perp }}}{E_{i_{\perp }}}}={\frac {{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}-{\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}{{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}+{\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}}}
.
Pour trouver le coefficient de transmission :
(
2
E
i
⊥
−
E
t
⊥
)
n
i
μ
i
cos
θ
i
=
E
t
⊥
n
t
μ
t
cos
θ
t
{\displaystyle \left(2E_{i_{\perp }}-E_{t_{\perp }}\right){\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}=E_{t_{\perp }}{\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}
t
⊥
=
E
t
⊥
E
i
⊥
=
2
n
i
μ
i
cos
θ
i
n
i
μ
i
cos
θ
i
+
n
t
μ
t
cos
θ
t
{\displaystyle t_{\perp }={\frac {E_{t_{\perp }}}{E_{i_{\perp }}}}={\frac {2{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}}{{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}+{\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}}}
t
⊥
=
E
t
⊥
E
i
⊥
=
2
n
i
μ
i
cos
θ
i
n
i
μ
i
cos
θ
i
+
n
t
μ
t
cos
θ
t
{\displaystyle t_{\perp }={\frac {E_{t_{\perp }}}{E_{i_{\perp }}}}={\frac {2{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}}{{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}+{\frac {n_{t}}{\mu _{t}}}\cos \theta _{t}}}}
.
Cas 2 : champ électrique parallèle au plan d'incidence.
modifier
En projetant sur le plan tangent à la surface,
(1) donne
E
i
∥
cos
θ
i
−
E
r
∥
cos
θ
r
=
E
t
∥
cos
θ
t
{\displaystyle {E_{i_{\parallel }}}\cos \theta _{i}-{E_{r_{\parallel }}}\cos \theta _{r}={E_{t_{\parallel }}}\cos \theta _{t}}
(5)
et (2) donne
B
i
μ
i
+
B
r
μ
r
=
B
t
μ
t
{\displaystyle {\frac {B_{i}}{\mu _{i}}}+{\frac {B_{r}}{\mu _{r}}}={\frac {B_{t}}{\mu _{t}}}}
ce qui permet d'obtenir :
(
E
i
∥
+
E
r
∥
)
n
i
μ
i
=
E
t
∥
n
t
μ
t
{\displaystyle \left(E_{i_{\parallel }}+E_{r_{\parallel }}\right){\frac {n_{i}}{\mu _{i}}}=E_{t_{\parallel }}{\frac {n_{t}}{\mu _{t}}}}
(6).
Pour trouver le coefficient de réflexion :
E
t
∥
=
(
E
i
∥
−
E
r
∥
)
cos
θ
i
cos
θ
t
{\displaystyle {E_{t_{\parallel }}}=\left({E_{i_{\parallel }}}-{E_{r_{\parallel }}}\right){\frac {\cos \theta _{i}}{\cos \theta _{t}}}}
(
E
i
∥
+
E
r
∥
)
n
i
μ
i
=
(
E
i
∥
−
E
r
∥
)
cos
θ
i
cos
θ
t
n
t
μ
t
{\displaystyle \left(E_{i_{\parallel }}+E_{r_{\parallel }}\right){\frac {n_{i}}{\mu _{i}}}=\left({E_{i_{\parallel }}}-{E_{r_{\parallel }}}\right){\frac {\cos \theta _{i}}{\cos \theta _{t}}}{\frac {n_{t}}{\mu _{t}}}}
(
E
i
∥
+
E
r
∥
)
n
i
μ
i
cos
θ
t
=
(
E
i
∥
−
E
r
∥
)
n
t
μ
t
cos
θ
i
{\displaystyle \left(E_{i_{\parallel }}+E_{r_{\parallel }}\right){\frac {n_{i}}{\mu _{i}}}{\cos \theta _{t}}=\left({E_{i_{\parallel }}}-{E_{r_{\parallel }}}\right){\frac {n_{t}}{\mu _{t}}}{\cos \theta _{i}}}
E
i
∥
(
n
i
μ
i
cos
θ
t
−
n
t
μ
t
cos
θ
i
)
=
E
r
∥
(
n
i
μ
i
cos
θ
t
+
n
t
μ
t
cos
θ
i
)
{\displaystyle E_{i_{\parallel }}\left({\frac {n_{i}}{\mu _{i}}}{\cos \theta _{t}}-{\frac {n_{t}}{\mu _{t}}}{\cos \theta _{i}}\right)=E_{r_{\parallel }}\left({\frac {n_{i}}{\mu _{i}}}{\cos \theta _{t}}+{\frac {n_{t}}{\mu _{t}}}{\cos \theta _{i}}\right)}
r
∥
=
E
r
∥
E
i
∥
=
n
t
μ
t
cos
θ
i
−
n
i
μ
i
cos
θ
t
n
t
μ
t
cos
θ
i
+
n
i
μ
i
cos
θ
t
{\displaystyle r_{\parallel }={\frac {E_{r_{\parallel }}}{E_{i_{\parallel }}}}={\frac {{\frac {n_{t}}{\mu _{t}}}\cos \theta _{i}-{\frac {n_{i}}{\mu _{i}}}\cos \theta _{t}}{{\frac {n_{t}}{\mu _{t}}}\cos \theta _{i}+{\frac {n_{i}}{\mu _{i}}}\cos \theta _{t}}}}
.
r
∥
=
E
r
∥
E
i
∥
=
n
t
μ
t
cos
θ
i
−
n
i
μ
i
cos
θ
t
n
t
μ
t
cos
θ
i
+
n
i
μ
i
cos
θ
t
{\displaystyle r_{\parallel }={\frac {E_{r_{\parallel }}}{E_{i_{\parallel }}}}={\frac {{\frac {n_{t}}{\mu _{t}}}\cos \theta _{i}-{\frac {n_{i}}{\mu _{i}}}\cos \theta _{t}}{{\frac {n_{t}}{\mu _{t}}}\cos \theta _{i}+{\frac {n_{i}}{\mu _{i}}}\cos \theta _{t}}}}
.
Pour trouver le coefficient de transmission :
(5) permet d'écrire
E
r
∥
=
E
i
∥
−
E
t
∥
cos
θ
t
cos
θ
i
{\displaystyle {E_{r_{\parallel }}}={E_{i_{\parallel }}}-{E_{t_{\parallel }}}{\frac {\cos \theta _{t}}{\cos \theta _{i}}}}
,
et (6) peut alors donner
(
2
E
i
∥
−
E
t
∥
cos
θ
t
cos
θ
i
)
n
i
μ
i
=
E
t
∥
n
t
μ
t
{\displaystyle \left(2E_{i_{\parallel }}-{E_{t_{\parallel }}}{\frac {\cos \theta _{t}}{\cos \theta _{i}}}\right){\frac {n_{i}}{\mu _{i}}}=E_{t_{\parallel }}{\frac {n_{t}}{\mu _{t}}}}
,
ou encore
2
E
i
∥
n
i
μ
i
cos
θ
i
−
E
t
∥
n
i
μ
i
cos
θ
t
=
E
t
∥
n
t
μ
t
cos
θ
i
{\displaystyle 2E_{i_{\parallel }}{\frac {n_{i}}{\mu _{i}}}{\cos \theta _{i}}-{E_{t_{\parallel }}}{\frac {n_{i}}{\mu _{i}}}{\cos \theta _{t}}=E_{t_{\parallel }}{\frac {n_{t}}{\mu _{t}}}{\cos \theta _{i}}}
t
∥
=
E
r
∥
E
i
∥
=
2
n
i
μ
i
cos
θ
i
n
t
μ
t
cos
θ
i
+
n
i
μ
i
cos
θ
t
{\displaystyle t_{\parallel }={\frac {E_{r_{\parallel }}}{E_{i_{\parallel }}}}={\frac {2{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}}{{\frac {n_{t}}{\mu _{t}}}\cos \theta _{i}+{\frac {n_{i}}{\mu _{i}}}\cos \theta _{t}}}}
t
∥
=
E
r
∥
E
i
∥
=
2
n
i
μ
i
cos
θ
i
n
t
μ
t
cos
θ
i
+
n
i
μ
i
cos
θ
t
{\displaystyle t_{\parallel }={\frac {E_{r_{\parallel }}}{E_{i_{\parallel }}}}={\frac {2{\frac {n_{i}}{\mu _{i}}}\cos \theta _{i}}{{\frac {n_{t}}{\mu _{t}}}\cos \theta _{i}+{\frac {n_{i}}{\mu _{i}}}\cos \theta _{t}}}}
.
Perméabilités magnétiques égales à celle du vide
modifier
Dans la plupart des cas en optique les matériaux ont une perméabilité magnétique égale à celle du vide :
μ
i
≈
μ
t
≈
μ
0
{\displaystyle \mu _{i}\approx \mu _{t}\approx \mu _{0}}
. Alors les coeffcients de réflexion et de transmission, selon la polarisation, deviennent :
r
⊥
=
n
i
cos
θ
i
−
n
t
cos
θ
t
n
i
cos
θ
i
+
n
t
cos
θ
t
{\displaystyle r_{\perp }={\frac {n_{i}\cos \theta _{i}-n_{t}\cos \theta _{t}}{n_{i}\cos \theta _{i}+n_{t}\cos \theta _{t}}}}
et
r
∥
=
n
t
cos
θ
i
−
n
i
cos
θ
t
n
t
cos
θ
i
+
n
i
cos
θ
t
{\displaystyle r_{\parallel }={\frac {n_{t}\cos \theta _{i}-n_{i}\cos \theta _{t}}{n_{t}\cos \theta _{i}+n_{i}\cos \theta _{t}}}}
,
t
⊥
=
2
n
i
cos
θ
i
n
i
cos
θ
i
+
n
t
cos
θ
t
{\displaystyle t_{\perp }={\frac {2\,n_{i}\cos \theta _{i}}{n_{i}\cos \theta _{i}+n_{t}\cos \theta _{t}}}}
et
t
∥
=
2
n
i
cos
θ
i
n
t
cos
θ
i
+
n
i
cos
θ
t
{\displaystyle t_{\parallel }={\frac {2\,n_{i}\cos \theta _{i}}{n_{t}\cos \theta _{i}+n_{i}\cos \theta _{t}}}}
.
Polarisation circulaire
modifier
Si on considère que la polarisation de la lumière incidente est parfaitement circulaire,
E
i
→
=
E
i
⊥
u
⊥
→
+
E
i
∥
u
∥
→
=
1
2
E
i
u
⊥
→
+
1
2
E
i
u
∥
→
{\displaystyle {\overrightarrow {E_{i}}}=E_{i_{\perp }}{\overrightarrow {u_{\perp }}}+E_{i_{\parallel }}{\overrightarrow {u_{\parallel }}}={\frac {1}{\sqrt {2}}}E_{i}{\overrightarrow {u_{\perp }}}+{\frac {1}{\sqrt {2}}}E_{i}{\overrightarrow {u_{\parallel }}}}
,
alors
E
r
2
=
r
⊥
2
E
i
⊥
2
+
r
∥
2
E
i
∥
2
=
1
2
E
i
2
(
r
⊥
2
+
r
∥
2
)
{\displaystyle E_{r}^{2}=r_{\perp }^{2}E_{i_{\perp }}^{2}+r_{\parallel }^{2}E_{i_{\parallel }}^{2}={\frac {1}{2}}E_{i}^{2}\left(r_{\perp }^{2}+r_{\parallel }^{2}\right)}
.Les facteurs de réflexion et de transmission peuvent être exprimés :
R
=
1
2
(
r
⊥
2
+
r
∥
2
)
=
1
2
(
n
i
cos
θ
i
−
n
t
cos
θ
t
n
i
cos
θ
i
+
n
t
cos
θ
t
)
2
+
1
2
(
n
t
cos
θ
i
−
n
i
cos
θ
t
n
t
cos
θ
i
+
n
i
cos
θ
t
)
2
{\displaystyle R={\frac {1}{2}}\left(r_{\perp }^{2}+r_{\parallel }^{2}\right)={\frac {1}{2}}\left({\frac {n_{i}\cos \theta _{i}-n_{t}\cos \theta _{t}}{n_{i}\cos \theta _{i}+n_{t}\cos \theta _{t}}}\right)^{2}+{\frac {1}{2}}\left({\frac {n_{t}\cos \theta _{i}-n_{i}\cos \theta _{t}}{n_{t}\cos \theta _{i}+n_{i}\cos \theta _{t}}}\right)^{2}}
,
T
=
1
2
(
t
⊥
2
+
t
∥
2
)
=
1
2
(
2
n
i
cos
θ
i
n
i
cos
θ
i
+
n
t
cos
θ
t
)
2
+
1
2
(
2
n
i
cos
θ
i
n
t
cos
θ
i
+
n
i
cos
θ
t
)
2
{\displaystyle T={\frac {1}{2}}\left(t_{\perp }^{2}+t_{\parallel }^{2}\right)={\frac {1}{2}}\left({\frac {2\,n_{i}\cos \theta _{i}}{n_{i}\cos \theta _{i}+n_{t}\cos \theta _{t}}}\right)^{2}+{\frac {1}{2}}\left({\frac {2\,n_{i}\cos \theta _{i}}{n_{t}\cos \theta _{i}+n_{i}\cos \theta _{t}}}\right)^{2}}
.
Incidence normale
modifier
Dans le cas particulier d'une incidence normale, le plan d'incidence n'est plus définit ce qui n'a pas d'importance car les angles sont tous nuls
θ
i
=
θ
r
=
θ
t
=
0
{\displaystyle \theta _{i}=\theta _{r}=\theta _{t}=0}
; ceci permet d'obtenir une expression simple des coefficients de réflexion et de transmission
r
⊥
=
−
r
∥
=
n
i
−
n
t
n
i
+
n
t
{\displaystyle r_{\perp }=-r_{\parallel }={\frac {n_{i}-n_{t}}{n_{i}+n_{t}}}}
et
t
⊥
=
t
∥
=
2
n
i
n
i
+
n
t
{\displaystyle t_{\perp }=t_{\parallel }={\frac {2\,n_{i}}{n_{i}+n_{t}}}}
,
et des facteurs de réflexion et de transmission quelle que soit la polarisation
R
=
(
n
i
−
n
t
n
i
+
n
t
)
2
{\displaystyle R=\left({\frac {n_{i}-n_{t}}{n_{i}+n_{t}}}\right)^{2}}
et
T
=
(
2
n
i
n
i
+
n
t
)
2
{\displaystyle T=\left({\frac {2\,n_{i}}{n_{i}+n_{t}}}\right)^{2}}
.
Incidence normale, couche unique
modifier
Au niveau du premier dioptre, sachant que :
(1) donne
E
0
+
E
0
′
=
E
1
+
E
1
′
{\displaystyle {E_{0}}+{E_{0}^{'}}={E_{1}}+{E_{1}^{'}}}
,
et (2) donne
(
E
0
−
E
0
′
)
n
0
=
(
E
1
−
E
1
′
)
n
1
{\displaystyle \left({E_{0}}-{E_{0}^{'}}\right){n_{0}}=\left({E_{1}}-{E_{1}^{'}}\right){n_{1}}}
,
On obtient :
(
E
0
(
0
)
E
0
′
(
0
)
)
=
1
2
(
1
+
n
1
/
n
0
1
−
n
1
/
n
0
1
−
n
1
/
n
0
1
+
n
1
/
n
0
)
(
E
1
(
0
)
E
1
′
(
0
)
)
=
M
01
(
E
1
(
0
)
E
1
′
(
0
)
)
{\displaystyle {\binom {E_{0}(0)}{E_{0}^{'}(0)}}={\frac {1}{2}}{\begin{pmatrix}1+n_{1}/n_{0}&1-n_{1}/n_{0}\\1-n_{1}/n_{0}&1+n_{1}/n_{0}\end{pmatrix}}{\binom {E_{1}(0)}{E_{1}^{'}(0)}}=\mathbf {M} _{01}{\binom {E_{1}(0)}{E_{1}^{'}(0)}}}
.
Le retard à l'intérieur de la couche 1 introduit :
(
E
1
(
0
)
E
1
′
(
0
)
)
=
(
e
−
i
k
1
d
1
0
0
e
+
i
k
1
d
1
)
(
E
1
(
d
1
)
E
1
′
(
d
1
)
)
=
M
1
(
E
1
(
d
1
)
E
1
′
(
d
1
)
)
{\displaystyle {\binom {E_{1}(0)}{E_{1}^{'}(0)}}={\begin{pmatrix}\mathrm {e} ^{-\mathrm {i} k_{1}d_{1}}&0\\0&\mathrm {e} ^{+\mathrm {i} k_{1}d_{1}}\end{pmatrix}}{\binom {E_{1}(d_{1})}{E_{1}^{'}(d_{1})}}=\mathbf {M} _{1}{\binom {E_{1}(d_{1})}{E_{1}^{'}(d_{1})}}}
.
Ce qui donne pour l'ensemble dioptre et couche :
(
E
0
(
0
)
E
0
′
(
0
)
)
=
M
01
M
1
(
E
1
(
d
1
)
E
1
′
(
d
1
)
)
{\displaystyle {\binom {E_{0}(0)}{E_{0}^{'}(0)}}=\mathbf {M} _{01}\mathbf {M} _{1}{\binom {E_{1}(d_{1})}{E_{1}^{'}(d_{1})}}}
.
Au niveau du dernier dioptre :
(
E
1
(
d
1
)
E
1
′
(
d
1
)
)
=
1
2
(
1
+
n
s
/
n
1
1
−
n
s
/
n
1
1
−
n
s
/
n
1
1
+
n
s
/
n
1
)
(
E
s
(
d
1
)
E
s
′
(
d
1
)
)
=
M
1
s
(
E
s
(
d
1
)
E
s
′
(
d
1
)
)
{\displaystyle {\binom {E_{1}(d_{1})}{E_{1}^{'}(d_{1})}}={\frac {1}{2}}{\begin{pmatrix}1+n_{s}/n_{1}&1-n_{s}/n_{1}\\1-n_{s}/n_{1}&1+n_{s}/n_{1}\end{pmatrix}}{\binom {E_{s}(d_{1})}{E_{s}^{'}(d_{1})}}=\mathbf {M} _{1s}{\binom {E_{s}(d_{1})}{E_{s}^{'}(d_{1})}}}
,
avec
E
s
′
=
0
{\displaystyle E_{s}^{'}=0}
car il n'y a pas de retour de la lumière à ce niveau,
(
E
0
(
0
)
E
0
′
(
0
)
)
=
M
01
M
1
M
1
s
(
E
s
(
d
1
)
0
)
{\displaystyle {\binom {E_{0}(0)}{E_{0}^{'}(0)}}=\mathbf {M} _{01}\mathbf {M} _{1}\mathbf {M} _{1s}{\binom {E_{s}(d_{1})}{0}}}
Pour la longueur d'onde
λ
0
{\displaystyle \lambda _{0}}
ciblée
modifier
d
1
=
λ
0
4
n
1
{\displaystyle d_{1}={\frac {\lambda _{0}}{4\,n_{1}}}}
de sorte que
k
1
d
1
=
π
/
2
{\displaystyle k_{1}d_{1}=\pi /2}
et
e
i
k
1
d
1
=
i
{\displaystyle \mathrm {e} ^{\mathrm {i} k_{1}d_{1}}=\mathrm {i} }
(
E
0
(
0
)
E
0
′
(
0
)
)
=
i
2
(
−
(
1
+
n
1
/
n
0
)
(
1
−
n
1
/
n
0
)
−
(
1
−
n
1
/
n
0
)
(
1
+
n
1
/
n
0
)
)
(
E
1
(
d
1
)
E
1
′
(
d
1
)
)
{\displaystyle {\binom {E_{0}(0)}{E_{0}^{'}(0)}}={\frac {\mathrm {i} }{2}}{\begin{pmatrix}-(1+n_{1}/n_{0})&(1-n_{1}/n_{0})\\-(1-n_{1}/n_{0})&(1+n_{1}/n_{0})\end{pmatrix}}{\binom {E_{1}(d_{1})}{E_{1}^{'}(d_{1})}}}
(
E
1
(
d
1
)
E
1
′
(
d
1
)
)
=
M
1
s
(
E
s
(
d
1
)
0
)
=
E
s
(
d
1
)
2
(
1
+
n
s
/
n
1
1
−
n
s
/
n
1
)
{\displaystyle {\binom {E_{1}(d_{1})}{E_{1}^{'}(d_{1})}}=\mathbf {M} _{1s}{\binom {E_{s}(d_{1})}{0}}={\frac {E_{s}(d_{1})}{2}}{\binom {1+n_{s}/n_{1}}{1-n_{s}/n_{1}}}}
(
E
0
(
0
)
E
0
′
(
0
)
)
=
−
i
E
s
(
d
1
)
2
(
n
s
/
n
1
+
n
1
/
n
0
n
s
/
n
1
−
n
1
/
n
0
)
{\displaystyle {\binom {E_{0}(0)}{E_{0}^{'}(0)}}=-{\frac {\mathrm {i} \,E_{s}(d_{1})}{2}}{\binom {n_{s}/n_{1}+n_{1}/n_{0}}{n_{s}/n_{1}-n_{1}/n_{0}}}}
r
=
E
0
′
E
0
=
n
s
/
n
1
−
n
1
/
n
0
n
s
/
n
1
+
n
1
/
n
0
{\displaystyle r={\frac {E_{0}^{'}}{E_{0}}}={\frac {n_{s}/n_{1}-n_{1}/n_{0}}{n_{s}/n_{1}+n_{1}/n_{0}}}}
R
=
(
E
0
′
E
0
)
2
=
(
n
s
/
n
1
−
n
1
/
n
0
n
s
/
n
1
+
n
1
/
n
0
)
2
{\displaystyle R=\left({\frac {E_{0}^{'}}{E_{0}}}\right)^{2}=\left({\frac {n_{s}/n_{1}-n_{1}/n_{0}}{n_{s}/n_{1}+n_{1}/n_{0}}}\right)^{2}}
Le traitement est optimisé si
n
s
/
n
1
=
n
1
/
n
0
⇔
n
1
=
n
0
n
s
{\displaystyle n_{s}/n_{1}=n_{1}/n_{0}\Leftrightarrow n_{1}={\sqrt {n_{0}\,n_{s}}}}
.
Si
n
0
=
1
{\displaystyle n_{0}=1}
pour l'air et
n
s
=
1
,
52
{\displaystyle n_{s}=1{,}52}
pour le verre :
n
1
=
1
,
23
{\displaystyle n_{1}=1{,}23}
. MgF2 : 1,38.
Incidence quelconque, polarisation circulaire
modifier
Au niveau du premier dioptre, sachant que :
(1) donne
E
0
+
E
0
′
=
E
1
+
E
1
′
{\displaystyle {E_{0}}+{E_{0}^{'}}={E_{1}}+{E_{1}^{'}}}
,
et (2) donne
(
E
0
−
E
0
′
)
n
0
cos
θ
0
=
(
E
1
−
E
1
′
)
n
1
cos
θ
1
{\displaystyle \left({E_{0}}-{E_{0}^{'}}\right){n_{0}}\cos \theta _{0}=\left({E_{1}}-{E_{1}^{'}}\right){n_{1}}\cos \theta _{1}}
,