Amplitude reflection coefficient for high to low refractive index

[Eruditio]

For a wave traveling from a medium with refractive index n₁ to n₂ with n₂ > n₁, the amplitude reflection coefficient is given by:

r = (n₂ - n₁)/(n₂ + n₁)

Consider a beam of light passing through a single pane of glass. There is a reflection at the front surface of the pane, with the reflected amplitude obtainable through the above equation. If using the above equation for the second interface (the back of the pane, traveling from glass to air), r is negative but equal in magnitude to the first interface (travelling from air to glass). Does this make any difference? If calculating the percentage amplitude reduction, would one have to consider both interfaces, or just the first?

 

[Born2bwire]

Yes, the reflection coefficient going from a high to low index of refraction is negative. The change in sign means that the reflected wave has a phase shift of 180 degrees.

As for the total transmission coefficient, it becomes more complicated because the total reflected and transmitted through a slab is going to be a result of multiple reflections within the slab. For example, you have a contribution from a beam that transmits through directly, t_1*t_2, then you have a contribution from the first order reflection off of the second interface, t_1*r_2*r_2*t_2, and so on and so on. This is a bit simplified because I haven't accounted for interference between these various orders due to the phase differences. It is possible to calculate the result. For example, the refletion coefficient for normal incidence is
$$\Gamma_{in} = \frac{\Gamma_{12}+\Gamma_{23}e^{i2k_2d}}{1+\Gamma_{12}\Gamma_{23}e^{i2k_2d}}$$
Where
$$\Gamma_{ij} = \frac{\eta_j-\eta_i}{\eta_j+\eta_i}$$
and k_2 is the wavenumber in the slab, d is the thickness of the slab, and \eta is the impedance of the medium which is $\sqrt{\frac{\mu}{\epsilon}}$ while the index of refraction is $\sqrt{\mu_r\epsilon_r}$. If the permeability of the slab case is homogeneous then we can see that \Gamma_{ij} gives us the same result if we use the equation I gave using wave impedance or the equation using index of refraction.