= Phase retardation
{title2=$\delta=2\pi\Delta n\,d/\lambda_0$}
For the overlapping <monochromatic wave> components, let $d$ be the thickness and $\lambda_0$ the vacuum <wavelength>. Define $\Delta n=n_s-n_f>0$ using the slow and fast <refractive indices>. With <complex amplitude> convention $e^{i(kz-\omega t)}$, the accumulated slow-minus-fast <phase difference> is
$$
\delta=(k_s-k_f)d=\frac{2\pi}{\lambda_0}(n_s-n_f)d.
$$
This is the <optical path difference> $\Delta n\,d$ expressed as a <wave phase>. It is a signed accumulation convention; with the opposite time convention the sign reverses. The outgoing <polarization of an electromagnetic wave> depends on $\delta$ modulo $2\pi$. Here the positive slow-minus-fast $\Delta n$ differs from the signed principal-index $n_e-n_o$ used in the parent <birefringence> entry when $n_e<n_o$.
For an original worked example, stipulate $\Delta n=0.0100$ at $\lambda_0=600\,\mathrm{nm}$ and a uniform thickness $d=15.0\,\mathrm{\mu m}$. These are design assumptions, not measured material data. The <optical path difference> is $150\,\mathrm{nm}$, so
$$
\delta=2\pi\frac{150\,\mathrm{nm}}{600\,\mathrm{nm}}=\frac{\pi}{2}.
$$
It is a <quarter-wave plate>. At the same <wavelength> and <refractive indices>, doubling $d$ to $30.0\,\mathrm{\mu m}$ gives $\delta=\pi$, a <half-wave plate>. A $1.0\,\mathrm{\mu m}$ thickness increase changes $\delta$ by $\pi/30$, or $6^\circ$; retaining the same <birefringence> at $500\,\mathrm{nm}$ would instead give $\delta=3\pi/5$ for the $15.0\,\mathrm{\mu m}$ plate. Real <refractive indices> also vary with <wavelength>, so this last calculation explicitly neglects <optical dispersion>.
\Image[/birefringent-waveplate-phase-thickness.png]
{title=<Phase retardation> grows with thickness while overlapping components change <polarization of an electromagnetic wave>}
{description=The stipulated $\Delta n=0.0100$, $\lambda_0=600\,\mathrm{nm}$ model gives quarter-wave and half-wave thicknesses of $15$ and $30\,\mathrm{\mu m}$. Insets trace the normalized <electric field> at a fixed point for equal fast/slow <wave amplitudes>, with $E_f=\cos\tau$ and $E_s=\cos(\tau+\delta)$. The intermediate <ellipse>, circle and rotated line require no spatial beam separation.}
For these equal transmitted <wave amplitudes>, <orthogonal projection> onto a <polarizer> crossed with the initial $45^\circ$ <linear polarization> gives a simple independent readout. After removing a common <wave phase>, the component <complex amplitudes> are proportional to $(1,e^{i\delta})/\sqrt2$; <orthogonal projection> onto $(1,-1)/\sqrt2$ gives $(1-e^{i\delta})/2$. Squaring its <modulus> yields $I_\perp/I_{\mathrm{in}}=\sin^2(\delta/2)$: zero at $d=0$, one half at $15\,\mathrm{\mu m}$, and one at $30\,\mathrm{\mu m}$. This assumes ideal <polarizers>, negligible <specular reflection> and <wave absorption>, overlapping components and stable relative <wave phase>. It measures a periodic function of <phase retardation>, so one brightness measurement cannot determine the thickness uniquely.
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