Half-silvered rear surface phase shift

[Alfred Cann]

I have read that the rear surface reflection of a half-silvered mirror undergoes no phase shift. But they caution that this is true only for a dielectric mirror. What is the answer for a metallized mirror? It seems to me that the answer must either be 180 degr., or pass continuously to 180 degr. as we increase the metallization to 100%.

 

[Alfred Cann]

I think I can partly answer my own question.
The amplitudes of the transmitted and reflected waves must both be 1/√2. To satisfy the boundary conditions, their phase shifts must be + and - π/4, so that they add up to match the input wave, 1.0 at zero phase.
Thus, the mirror acts like a 90° hybrid junction, a device well-known in the microwave world.

I would be comfortable with this explanation except for one thing. I can't see how the answer would be any different for a front surface mirror and a dielectric rear surface mirror, yet the literature sez it is. What am I doing wrong?