Confusion about boundary conditions for reflection of light

The reflection at a dielectric interface was analysed in Griffiths introduction to electrodynamics using the following diagram.

I do not understand why the direction of $$\vec E_r$$ and $$\vec B_r$$ are as given. Why can't it be as drawn in a different colour?

I calculated with the new $$\vec E_r$$ and $$\vec B_r$$ and got a different result - with $$\alpha$$ and $$\beta$$ interchanged, implying that $$180^o$$ phase shift happens in reflection from rarer medium.

I need to know why at the interface, the reflected $$\vec E_r$$ and $$\vec B_r$$ should be as drawn by griffith.

Here are my calculation :

The equations of griffiths were:

$$E_{0i} + E_{0r} =\alpha E_{0t}$$

$$E_{0i} - E_{0r} =\beta E_{0t}$$

This is perhaps more obvious to see at normal incidence, where the reflection coefficients are either $$r = \pm \frac{n_2-n_1}{n_1+n_2}$$ depending on whether you started by defining the reflected electric field in the opposite or the same direction as the incident electric field.
• But both results are different right... with $+$ sign, it means that reflected wave is out of phase with incident wave when $n_2 < n_1$. ie : reflection at rarer medium. But with $-$ sign, this happens for $n_2>n_1$. ie: reflection at denser medium. We know that only one is true. ( with $-$ sign) Oct 28, 2020 at 8:09
• @RishabNavaneet No. If you have $n_2>n_1$ and start by drawing the incident and reflected electric fields in the same direction, then $r$ will be negative. If you start by drawing them in opposite directions then $r$ will be positive. Only in the first case can the sign of $r$ be directly interpreted as a phase change. If you do the derivation with a transmitted E drawn in the opposite direction to incident E you would get a negative transmission coefficient, but this is not a phase change... Oct 28, 2020 at 8:17
• Yeahh... I got it now. The electric Field which I drew is already out of phase with incident wave. The reflection of a reflection should give us back the same thing. If I reflect the reflected wave about xy plane again, I see that my arrow points opposite to incident $\vec E$. So it was just the issue with positive direction. Thankyou.. Oct 28, 2020 at 8:30