30 Jul 2025

Retardation Plates, the Babinet Compensator, and Polarization States

Quarter- and half-wave plates, continuously variable retardation, and linear, circular, and elliptical polarization states.

waves-and-optics polarization-ellipse quarter-wave-plate half-wave-plate babinet-compensator

At a fixed point in a monochromatic beam travelling along $z$, resolve the electric field along two perpendicular transverse axes:

\[E_x=a\cos\psi, \qquad E_y=b\cos(\psi-\delta), \qquad \psi=\omega t-kz.\]

The phase difference $\delta$ determines the path traced by the tip of $\mathbf E$.

Derivation of the polarization ellipse

Define

\[X=\frac{E_x}{a}=\cos\psi, \qquad Y=\frac{E_y}{b}=\cos\psi\cos\delta+\sin\psi\sin\delta.\]

Then

\[Y-X\cos\delta=\sin\psi\sin\delta.\]

Squaring this equation and using $\sin^2\psi=1-X^2$,

\[(Y-X\cos\delta)^2=(1-X^2)\sin^2\delta.\]

Expanding and collecting terms gives

\[X^2+Y^2-2XY\cos\delta=\sin^2\delta,\]

or

\[\boxed{ \frac{E_x^2}{a^2}+\frac{E_y^2}{b^2} -\frac{2E_xE_y}{ab}\cos\delta =\sin^2\delta}.\]

This equation classifies the polarization:

• Linear polarization: If $\delta=m\pi$, then $E_y/b=(-1)^mE_x/a$, so the tip moves on a straight line. A field along either coordinate axis is also linear.
• Circular polarization: If $a=b$ and $\delta=(2m+1)\pi/2$, then $E_x^2+E_y^2=a^2$. The field magnitude is constant while its direction rotates.
• Elliptical polarization: In the general case the tip traces an ellipse. The sign of $\delta$ fixes the sense of rotation; a handedness label is meaningful only after specifying whether the observer looks with or against the direction of propagation.

Linear and circular polarization are therefore limiting cases of elliptical polarization.

Each trajectory is generated parametrically from $E_x=a\cos\psi$ and $E_y=b\cos(\psi-\delta)$.

Retardation by a birefringent plate

Let the plate’s fast and slow axes have refractive indices $n_f$ and $n_s$, with $n_s>n_f$. A vacuum wavelength $\lambda$ accumulates phases

\[\phi_f=\frac{2\pi n_fd}{\lambda}, \qquad \phi_s=\frac{2\pi n_sd}{\lambda}\]

while crossing thickness $d$. The slow component therefore lags the fast component by

\[\boxed{\delta=\phi_s-\phi_f =\frac{2\pi}{\lambda}(n_s-n_f)d}.\]

The plate changes relative phase, not the component amplitudes, when absorption and reflection losses are neglected.

Quarter-wave plate

A quarter-wave plate produces an odd multiple of $\pi/2$ retardation:

\[\delta=\frac{(2m+1)\pi}{2}, \qquad \boxed{d=\frac{(2m+1)\lambda}{4(n_s-n_f)}}.\]

If incident linear polarization makes $45^\circ$ with the plate axes, its components are equal. The plate makes them differ in phase by $\pi/2$, so the output is circular. At any other non-zero angle to both axes, the component amplitudes are unequal and the output is elliptical. Conversely, a suitable quarter-wave plate converts circular or elliptical light into linear light.

Half-wave plate

A half-wave plate produces an odd multiple of $\pi$ retardation:

\[\delta=(2m+1)\pi, \qquad \boxed{d=\frac{(2m+1)\lambda}{2(n_s-n_f)}}.\]

Let the incident linear field make angle $\alpha$ with the fast axis. Before the plate its components are proportional to

\[\begin{pmatrix}\cos\alpha\\ \sin\alpha\end{pmatrix}.\]

The half-wave retardation changes the relative sign, giving

\[\begin{pmatrix}\cos\alpha\\ -\sin\alpha\end{pmatrix}.\]

Thus the output makes angle $-\alpha$ with the fast axis. If the fast axis is at angle $\phi$ and the incident azimuth is $\theta$, then $\alpha=\theta-\phi$ and

\[\boxed{\theta_{\rm out}=2\phi-\theta}.\]

The half-wave plate therefore rotates the plane of polarization through twice the angle between the incident vibration and the plate axis, with the sign set by their relative orientation.

Babinet compensator

A Babinet compensator uses two birefringent wedges with mutually perpendicular fast axes. Let the local thicknesses traversed in the two wedges be $t_1$ and $t_2$. Because their axes are interchanged, their retardations have opposite signs. The net retardation is

\[\boxed{\delta_B=\frac{2\pi(n_s-n_f)}{\lambda}(t_1-t_2)}.\]

Sliding one wedge changes $t_1-t_2$ continuously. At the position where $t_1=t_2$, the two retardations cancel. The compensator can therefore supply a continuously adjustable retardation of either sign, unlike a fixed quarter- or half-wave plate. An unknown retardation is measured by adjusting the compensator until an analyzer shows the chosen compensation condition; the calibrated wedge displacement then gives $t_1-t_2$.

Fixed plates supply selected phase delays; crossed wedges make the net delay proportional to their local thickness difference.

The ellipse elimination, wave-plate special cases, and half-wave rotation are checked with zero residuals in the Unit III polarization Maxima worksheet.