Update textbook.fr.md

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

@@ -116,7 +116,7 @@ _Rectangular waveguide with a TE mode._
 
 <br>
 
-#### $`\mathbf{TE}`$ modes
+#### 3 $`\mathbf{TE}`$ modes
 
 
 ##### Mode numbering
@@ -219,7 +219,7 @@ _Fig. 4.4: The geometric interpretation of the dispersion._
 
 Similarly we obtain for the group velocity:
 
-$`v_{\varphi}=\drac{\partial \omega}{\partial k_z}=c\,\sqrt{1-\dfrac{\omega_c^2}{\omega^2}}`$
+$`v_{\varphi}=\dfrac{\partial \omega}{\partial k_z}=c\,\sqrt{1-\dfrac{\omega_c^2}{\omega^2}}`$
 
 __Geometrical interpretation__
 
@@ -229,49 +229,48 @@ _Fig. 4.5: The geometric interpretation of the dispersion._
 We can understand the propagating behaviour of the $`TE`$ wave using a
 simple interpretation:
 
--   For *ω* = *ω~c~*, *k~z~* = *k* sin *θ* =0. This implies *θ* = 0. The
-    wave is doing a normal incidence on the plates (no *z*
-    progression) and *v~g~* = 0. In regions close to *ω~c~* the
+*  For $`\omega=\omega_c\,, k_z=k\,\sin\theta = 0`$.   
+   This implies $`\theta=0`$. The
+    wave is doing a normal incidence on the plates (no $`z`$
+    progression) and $`v_g=0`$. In regions close to $`\omega_c`$ the
     waveguides is highly dispersive.
 
--   For *ω* , *k~z~ ^[ω]{.underline}^* , *θ π/*2 and *v~g~ c*. The TE
-    waves tends to have very large incidence angles and the guide
-    behaves essentially as vacuum,
+* For $`\omega\longrightarrow\infty\,,k_z\longrightarrow\frac{\omega}{c}•,\theta\longrightarrow\pi/2`$
+  and $`v_g\longrightarrow c`$.   
+  The $`TE`$ waves tends to have very large incidence angles and the guide
+    behaves essentially as vacuum, i.e. dispersionless.
 
-i.e. dispersionless.
+* We can rewrite the dispersion relation for a $`TE_{0,n}`$ mode using
+  wavelengths. Eq [4.5] becomes:
+  <br>
+  $`\dfrac{1}{\lambda_z}=\sqrt{\big(\dfrac{1}{\lambda}\big)^2-\big(\dfrac{1}{\lambda_y}\big)^2}`$   
+  <br>
+  where $`\lambda_z=\lambda / \sin\theta`$ and $`\lambda_y=\lambda / \cos\theta`$, or again
+  <br>
+  $`\dfrac{1}{\lambda_z}=\sqrt{\big(\dfrac{1}{\lambda}\big)^2-\big(\dfrac{n}{\2b}\big)^2}`$   
+  <br>
+from which it is easy to confirm (see definition of $`\omega_c`$) that the
+cut-off wavelength for a $`TE_{0,n}`$ mode is $`\lambda_c=\frac{2b}{n}`$
 
--   We can rewrite the dispersion relation for a TE~0*,n*~ mode using
-    wave- lengths. Eq [4.5](#_bookmark83) becomes:
+![](variation-phase-group-velocity-wersus-omega-guided-mode_temp_L1200.jpg)    
+_Fig. 4.6: The phase and group velocity variations vs angular frequency for a guided mode._
 
-1 = 1 2
+!!!!! *Exercice 4.1 : Refractive index*   
+!!!!! Calculate and plot the refractive index vs the angular frequency for a $`TE`$ wave.
 
-1 2 (4.8)
 
-*λ~z~* V *λ*
 
-− *λy*
->
-where *λ~z~* = *λ/* sin *θ* and *λ~y~* = *λ/* cos *θ*, or again
+#### 4. Power flow
+
+The power density (power per unit surface, units $`[W:m^2]`$)
+traversing the waveguide can be evaluated from the time-averaged
+Poynting vector :
 
-1 = 1 2
 
-*n* 2 (4.9)
->
-from which it is easy to confirm (see definition of *ω~c~*) that the
-cut-off wavelength for a TE~0*,n*~ mode is *λ~c~* = [2*b*]{.underline}
-.
 
-![](media/image232.png)
 
-![](media/image237.png){width="0.12802602799650042in"
-height="9.562445319335083e-2in"}Figure 4.6: The phase and group
-velocity variations vs angular frequency for a guided mode.
 
-chap2 Power flow
 
-The power density (power per unit surface, units \[*W/m*^2^\])
-traversing the waveguide can be evaluated from the time-averaged
-Poynting vector
 >
 (**P**) = ( [$`\overrightarrow{E}`$ × $`\overrightarrow{B}`$]{.underline} )*.* (4.10)
 >