Update textbook.fr.md

12.temporary_ins/96.electromagnetism-in-media/20.reflexion-refraction-at-interfaces/10.boundary-conditions/10.main/textbook.fr.md

@@ -231,8 +231,8 @@ conductor. For the perfect conductor, as *σ* , its penetration depth
 transmission can occur, the wave is totally reflected. The incident
 and reflected waves are:
 
-![]()   
-![]()   
+![](electromag-in-media-reflexion-transmission-fig-33a.jpg)   
+![](electromag-in-media-reflexion-transmission-fig-33b.jpg)     
 $`\overrightarrow{E}`$*~r~* $`\overrightarrow{B}`$*~r~*
 medium 2 perfect conductor
 
@@ -286,8 +286,8 @@ time-averaged values is indeed *\<* $`\overrightarrow{S}`$*~tot~ \>~t~*= 0.
 Due to the discontinuity of the magnetic field, a surface current
 density must
 
-![]()   
-![]()   
+![](electromag-in-media-reflexion-transmission-fig-34a.jpg)   
+![](electromag-in-media-reflexion-transmission-fig-34b.jpg)     
 _Figure 3.4: Left: The incident, reflected and resulting wave at a_
 _particular time. Right: The superposition of several resulting waves_
 _at different times. See the video of the simulation on the_
@@ -305,8 +305,8 @@ $`\overrightarrow{j}`$*~s~* = 0). The incident ware is given by
 
 @@@@@@@@ $`\quad (equ. 3.28)`$
 
-![]() 
-![]()   
+![](electromag-in-media-reflexion-transmission-fig-35a.jpg)   
+![](electromag-in-media-reflexion-transmission-fig-35b.jpg)     
 _Figure 3.5: Left: Normal incidence at the boundary between twp perfect_
 _dielectrics. Right the total fields in medium 1 at a few different_
 _times showing a partial standing wave. The minimum amplitude of_
@@ -345,16 +345,20 @@ and
 
 @@@@@@@@@
 
-#### chap2 Reflection and transmission at oblique incidence
+<br>
+
+------------------
+
+#### 3.3 - Reflection and transmission at oblique incidence
 
 We now turn to the more general case of an oblique incidence at an
 arbitrary angle *θ~i~*. Before that we will need a few definitions and
 considerations.
 
-![](media/image182.png)    
+![](electromag-in-media-reflexion-transmission-fig-36.jpg)   
 _Figure 3.6: Plane of incidence, *s* and *p* polarisations._
 
-##### chap5 Plane of incidence
+##### Plane of incidence
 
 We first define the **plane of incidence** as the plane the contains
 the incidence wavevector and the normal to the interface separating
@@ -376,7 +380,7 @@ Two special cases arise:
 Any other polarisation state can be decomposed into the sum of a TM
 and TE wave.
 
-##### chap5 Laws of reflection and refraction
+##### Laws of reflection and refraction
 
 We will derive here the laws of reflection and refraction by making
 use of the boundary conditions which are independent on the
@@ -389,7 +393,7 @@ demonstrate it here. Let's consider the the general situation as of an
 inci- dent, reflected and refracted wave as depicted in figure
 [3.7.]. The three plane
 
-![]()   
+![](electromag-in-media-reflexion-transmission-fig-37.jpg)   
 _Figure 3.7: General case of reflection and refraction_
 
 monochromatic wave are:
@@ -431,6 +435,9 @@ As **r***~s~* belongs to the interface, (**k***~i~* **k***~r~*) is
 normal to it. This means that the vectors **k***~i~*, **k***~r~* and
 the normal belong to the same plane, i.e. the plane of incidence.
 
+!! *First law of reflexion*   
+!! ....
+
 iii. Equation [3.38](#_bookmark65) can be recast as
 
 @@@@@@@@ $`\quad (equ. 3.40)`$
@@ -447,11 +454,20 @@ c.  @@@@@@@
 From the first two relations we get *θ~i~* = *θ~r~* as \|*k~i~*\| =
 \|*k~r~*\|.
 
+!! *Second law of reflexion*   
+!! ....
 
 From the first and the third we get \|*k~i~*\| sin *θ~i~* = \|*k~t~*\|
 sin *θ~t~* or
 
-#### chap2 Fresnel's laws & Brewster's angle
+!! *Snell law* 
+!! ....
+
+<br>
+
+--------------
+
+#### Fresnel's laws & Brewster's angle
 
 From the previous section we can write the incident, reflected and
 transmitted waves as:
@@ -466,10 +482,11 @@ Likewise for a TM incident wave.
 
 ##### chap3 TE Wave TM Wave
 
-![](media/image184.png) ![](media/image187.png)    
+![](electromag-in-media-reflexion-transmission-fig-38a.jpg)   
+![](electromag-in-media-reflexion-transmission-fig-38b.jpg)   
 _Figure 3.8: Configuration for a TE and TM incidence._
 
-__**chap5 TE Wave**__
+__TE Wave__
 
 $`\overrightarrow{E}`$ is tangential and $`\overrightarrow{B}`$ is contained in the plane of incidence.
 From figure [3.8](#_bookmark67) we can write using the two tangential
@@ -487,7 +504,10 @@ which by solving the previous equations can be evaluated to:
 
 @@@@@@@@ $`\quad (equ. 3.43)`$
 
-__**chap5 TM Wave**__
+! *Remarks*   
+! ....
+
+__TM Wave__
 
 $`\overrightarrow{E}`$ is s contained in the plane of incidence and $`\overrightarrow{B}`$ is
 transverse. From figure [3.8](#_bookmark67) we can write using the two
@@ -501,7 +521,8 @@ coefficients for a TM (/I or *p*) wave as:
 
 @@@@@@@@ $`\quad (equ. 3.44)`$
 
-chap4 Brewster's angle
+
+##### Brewster's angle
 
 Let's make a few considerations on the consequences of the Fresnel
 relations by plotting the reflection coefficients for *s* and *p*
@@ -514,7 +535,8 @@ a.  If @@@@@@@
 
 b.  If @@@@@@@@
 
-chap5 Case a)
+
+##### Case a)
 
 In this case, we consider the range 0 *θ*~1~ *π/*2 for the incident
 angle *θ*~1~. Correspondingly, the refraction angle *θ*~2~ will vary
@@ -522,12 +544,16 @@ in the range 0 *θ*~2~ *θ*2*max*. Using Snell we have:
 
 @@@@@@@@@@@
 
-![]()   
+![](electromag-in-media-reflexion-transmission-fig-39a.jpg)    
+![](electromag-in-media-reflexion-transmission-fig-39b.jpg)    
 _Figure 3.9: The snell law for *n*~1~ *\< n*~2~ and *n*~1~ *\n*~2~._
 
-Angle of incidence (deg) Angle of incidence (deg)
 
 
+![](electromag-in-media-reflexion-transmission-fig-310a.jpg)    
+![](electromag-in-media-reflexion-transmission-fig-310b.jpg)    
+![](electromag-in-media-reflexion-transmission-fig-310a.jpg)    
+![](electromag-in-media-reflexion-transmission-fig-310b.jpg)    
 _Figure 3.10: Plot of the reflection_
 _coefficients (*r*~⊥~ and *r*I/) and the correspond- ing reflectivities_
 _(*R*~⊥~ and *R*I/) for case a (top) and case b (bottom)._
@@ -555,14 +581,15 @@ suggests different applications:
     wave thought an interface using a proper polarisation and
     incidence angle.
 
-![](media/image191.png)    
+![]()    
 _Figure 3.11: An illustration of the_
 _polarisation of light that is incident on an interface at Brewster's_
 _angle._
 _[https://en.wikipedia.org/wiki/Brewster'](https://en.wikipedia.org/wiki/Brewster%27s_angle)_
 _[s_angle](https://en.wikipedia.org/wiki/Brewster%27s_angle)_
 
-chap5 Case b)
+
+##### Case b)
 
 In this case, total reflections will occur for angles *θ*~1~
 *θ*1*lim*. The considera- tions on the Brewster's angle stated above
@@ -595,7 +622,7 @@ the transmitted and reflected beams are perpendicular, see figure
 to show that *θ~B~ \< θ*1*lim*, i.e. the Brewster's angle is
 always smaller than the limit angle for total reflection.
 
-chap4 Total internal reflection
+##### Total internal reflection
 
 Let's consider the oblique incidence shown in figure
 [3.8.](#_bookmark67) The transmitted field is given by
@@ -641,7 +668,11 @@ but also
 
 i.e. the reflected wave is totally reflected with a phase shift.
 
-chap2 Oblique incidence onto a perfect conductor
+<br>
+
+----------------
+
+#### Oblique incidence onto a perfect conductor
 
 We consider now a linearly polarised plane wave travelling in a
 perfect dielec- tric making an oblique incidence onto a perfect
@@ -654,7 +685,9 @@ separately. As before, being the second medium a perfect conductor, no
 electromagnetic fields can exists in it. We will deal only with the
 incident and reflected fields.
 
-chap5 TE Wave Incident fields
+##### TE Wave 
+
+##### Incident fields
 
 Considering the situation depicted in figure [3.13,](#_bookmark74) the
 incident wavevector is given by: