Are the Fresnel coefficients equal for E and B fields?

[Gabriel Maia]

From the Maxwell Equations we know that there are four boundary conditions for an electromagnetic wave crossing an interface between two dielectric media. For the TE polarisation state, these conditions give us that

$E_{i} + E_{r} = E_{t}$

$B_{i}\,\cos\theta_{i} - B_{r}\,\cos\theta_{r} = B_{t}\,\cos\theta_{t}$

where $E$ and $B$ are the components parallel to the interface of the electric and the magnetic fields, respectively. The indices i, r and t are associated with the incoming, the reflected and the transmitted waves, being then $\theta_{i}$ the angle of incidence, $\theta_{r}=\theta_{i}$ the angle of reflection and $\theta_{t}$ the angle of transmission.

Now, knowing that $B=n\,E/c$ and that $E_{r}/E_{i} = R$ and $E_{t}/E_{i} = T$ we have the set of equations

$1 + R = T$

$1 - R = \frac{\displaystyle n_{2}\,\cos\theta_{t}}{\displaystyle n_{1}\,\cos\theta_{i}}\,T$

Solving these equations, we will arrive at the known Fresnel coefficients for the TE polarisation:

$T = \frac{\displaystyle 2\,n_{1} \cos\theta_{i}}{\displaystyle n_{1}\,\cos\theta_{i}+n_{2}\,\cos\theta_{t}}$

$R = \frac{\displaystyle n_{1}\,\cos\theta_{i}-n_{2}\,\cos\theta_{t}}{\displaystyle n_{1}\,\cos\theta_{i}+n_{2}\,\cos\theta_{t}}$

The problem is that, if I write the equations in terms of the magnetic field, that is, if I use that $E=c\,B/n$, the transmission coefficient becomes different:

$T = \frac{\displaystyle 2\,n_{2} \cos\theta_{i}}{\displaystyle n_{1}\,\cos\theta_{i}+n_{2}\,\cos\theta_{t}}$Is this correct? I could not find a mistake in my derivations so I am inclined to believe it is indeed correct, but if it is so, why do we always talk about the Fresnel coefficients of the Electric field?

Thank you very much.

 

[DrDu]

$E_{t}/E_{i} = T$

Are you sure about this?
I would calculate the transmission comparing the Poynting vector of the incident and the transmitted light. Assuming $\mu=1$,
$E_tB_T/(E_iB_i)=T$ or $n_2E_t^2/n_1E_i^2$ and a similar equation for R.

 

[Gabriel Maia]

Are you sure about this?
I would calculate the transmission comparing the Poynting vector of the incident and the transmitted light. Assuming $\mu=1$,
$E_tB_T/(E_iB_i)=T$ or $n_2E_t^2/n_1E_i^2$ and a similar equation for R.

Doing some math I found out that, in order for

$B_{t} = n\,E_{t}/c$

we must have

$E_{t} = T_{E}\,E_{i}$

$B_{t} = T_{B}\,B_{i}$

So I believe it makes sense for the Fresnel coefficients to be different for both fields.

 

[DrDu]

So I believe it makes sense for the Fresnel coefficients to be different for both fields.

Probably yes. It is a bit confusing that you uses "T" and "R" for the coefficients of the field amplitudes. Most texts I know use "t" and "r" for the amplitudes and use the capital letters for the intensities.