Why Does Reflection Phase Shift Not Always Apply to Both Polarizations?

[ShayanJ]

Its always said that a reflected light ray acquires a phase shift equal to $\pi$ if $n_1 < n_2$. But considering the Fresnel coefficients, its revealed that its only for the s-polarization reflection coefficient that $n_1 < n_2$ causes the coefficient become negative. The p-polarization reflection coefficient becomes negative only when $\sin^2 \theta_1 > \frac{1}{1+(\frac{n_1}{n_2})^2}$. So why the first sentence doesn't distinguish different polarizations?
Thanks

 

[blue_leaf77]

I guess the author implicitly assumes normal incidence.

 

[ShayanJ]

I just found* that the p reflection coefficient becomes negative when $n_2 < n_1$, exactly the opposite condition for s reflection coefficient!

*$n_2 \cos\theta_1<n_1 \cos\theta_2 \Rightarrow \frac{n_2}{n_1} \cos\theta_1 < \cos\theta_2 \Rightarrow \sin\theta_1\cos\theta_1 < \sin\theta_2 \cos\theta_2 \Rightarrow \sin{2\theta_1} < \sin{2 \theta_2} \Rightarrow \\ \theta_1 < \theta_2 \Rightarrow n_2 < n_1$

 

[blue_leaf77]

What does it have to do with the original problem?

 

[ShayanJ]

It seems we have two conditions that result in $r_p < 0$ but only one for $r_s<0$. So we have the following situations:
1) $n_1 < n_2$ and $\sin^2{\theta_1}<\frac{1}{1+(\frac{n_1}{n_2})^2}$: Only the s polarization shifts phase upon reflection.
2) $n_1 < n_2$ and $\sin^2{\theta_1}>\frac{1}{1+(\frac{n_1}{n_2})^2}$: Both polarizations shift phase upon reflection.
3) $n_1 > n_2$: Only p polarization shifts phase upon reflection.
Well, at least now I have a clearer view. I'm beginning to think that the optics textbooks implicitly assume the light ray to have s polarization. Am I overestimating the number of textbooks that claim $n_1 < n_2$ means there is a phase shift upon reflection? Actually someone asked me this question and I remember in my own optics course that the professor kept repeating that there is phase shift when $n_1 < n_2$. This is also abundant on the internet(this, this, this and this). But I don't remember whether textbooks claim as such or not. It seems to me that textbooks get it right but it became a misunderstanding among people.