Boundary conditions for TM and TE waves

[LCSphysicist]

TL;DR Summary

I am reading about transverse polarization of magnetic and electric field, but i can not understand the boundary conditions it was imposed by the author

Theta in the incident angle
Phi is the refraction angle
'' denotes everything that propagates to the other medium, that is, everything related to refraction
' denotes the reflection in the original medium

I am rather confused, would appreciate any help.
I see the second equation of TE is basically the same as the third equation of it.
Too, i see that the first equation of TM is equal to the second equation of it.

 

[vanhees71]

If this is the standard derivation of the Fresnel formulae for reflection and refraction of a monochromatic em. wave then indeed you only need to deal with the magnetic field, because the magnetic field is given by the electric through $\vec{B}=\frac{1}{c_j} \vec{n} \times \vec{E}$, where $c_j$ ($j \in \{1,2\}$) is the phase velocity of light $c_j=c/n_j$ in medium $j$ and $\vec{n}$ the direction of the wave vector of the corresponding plane waves describing the incoming, reflected, and refracted wave respectively.

 

[LCSphysicist]

If this is the standard derivation of the Fresnel formulae for reflection and refraction of a monochromatic em. wave then indeed you only need to deal with the magnetic field, because the magnetic field is given by the electric through $\vec{B}=\frac{1}{c_j} \vec{n} \times \vec{E}$, where $c_j$ ($j \in \{1,2\}$) is the phase velocity of light $c_j=c/n_j$ in medium $j$ and $\vec{n}$ the direction of the wave vector of the corresponding plane waves describing the incoming, reflected, and refracted wave respectively.

Hello. Thx for the reply. I understand your point, what i don't understand is how to get this equations. That is, what is the math behind it?

If it was give to me, i would probably say that E and H need to be continuous in the separation(1) with the derivative being continuous too(2).

(1) for E agree with the image, but what about B? Why the minus sign there? (H-H' = H'')
(2)It would lead us, for example for TE, (-Ek + E'k' = E''k''), but in the image there is a cos with this equation, why?

 

[vanhees71]

The math behind it are of course Maxwell's equations for all fields having time dependence $\exp(-\mathrm{i} \omega t)$. You find the derivation in any good textbook on electrodynamics and/or optics.