# Should the reflection coefficient be negative?

According to Fresnel's equations for the magnetic field parallel to the surface of a dielectric, the reflection coefficient is this: $$r_{s}=\frac{\cos\theta^{i}-\frac{n_2}{n_1}\cos\theta^{t}}{\cos\theta^{i}+\frac{n_2}{n_1}\cos\theta^{t}}\tag{R}$$

In all the examples I have found, the magnetic field's direction "looks"(I'm not sure what is the correct term) at the reader (like this) However, when the magnetic field "looks down", like this

the reflection coefficient I have found is the negative of Fresnel's. $$r_{s}=\frac{-\cos\theta^{i}+\frac{n_2}{n_1}\cos\theta^{t}}{\cos\theta^{i}+\frac{n_2}{n_1}\cos\theta^{t}}\tag{R}$$ Is this correct or are my calculations wrong? Here the user concludes that this is caused by a phase shift, is that the case here too?

• -1 Not useful. You seem to be asking us to check your calculation, or to confirm something which is already stated and explained in another question. – sammy gerbil Feb 10 '18 at 20:14
• I am asking whther or not that result makes sense for the given data, or if soeone has seen similar results for them – Charor Feb 10 '18 at 20:21
• or, can fresnel's equations change based on the magnetic fields's direction? – Charor Feb 10 '18 at 20:27

If you've changed the direction of the incoming magnetic field $\vec{H}_{\pi 0}$ but kept $\hat{u}_\pi$ pointing in the same direction, then you must have implicitly changed the direction of the incoming electric field $\vec{E}_{\pi0}$ (since $\vec{E} \times \vec{H} \propto \vec{k}$.) Since the Fresnel coefficient is defined as the ratio of $\vec{E}_{\alpha0}$ to $\vec{E}_{\pi0}$, it makes sense that flipping the sign of the magnetic (and electric) field flips the sign of the Fresnel coefficient; it's merely a matter of the definitions you choose.