 Solution

Adding a known phase $\pi /2$ in the same arm changes the observed phase to $\varphi +\pi /2$. Combining measurements with and without that extra plate determines both quadratures. Explicitly, $p_1-p_2=\cos \varphi$ and $p_1^{(\pi /2)}-p_2^{(\pi /2)}=\cos (\varphi +\pi /2)=-\sin \varphi$. Hence the ordered pair determines $\varphi$ modulo $2\pi$, removing the sign ambiguity of a cosine alone.

It still leaves the integer fringe order undetermined:Thus the thickness remains periodic with period $\lambda /(n-1)$. A known thickness interval shorter than this period, a suitable multiple-wavelength measurement, or continuous phase tracking supplies the missing information. If $n$ is unknown, a single wavelength only determines the excess optical thickness $(n-1)d$ modulo $\lambda$.