Waveplate (retarder)

A waveplate, or retarder, is a plate of birefringent material cut so that light polarized along its slow axis travels more slowly than light polarized along its fast axis, and emerges delayed by a set fraction of a wavelength, the retardance. A half-wave plate (retardance λ/2, or π) rotates linear polarization by twice the angle between the input polarization and the plate's fast axis; a quarter-wave plate (λ/4, or π/2) turns linear polarization at 45° to its axes into circular polarization, and circular back into linear. The retardance of a plate of thickness $L$ with birefringence $\mathrm{\Delta }n$ is

$$\mathrm{\Gamma }=\frac{2\pi \mathrm{\Delta }nL}{\lambda },$$

so a true zero-order quartz half-wave plate for 633 nm is only 35 µm thick, which is why most waveplates are thicker multi-order or compound designs.

Retardance

Retardance is quoted in three equivalent ways: as a phase ($\pi \mathrm{/}2$ for a quarter-wave plate), as a fraction of the design wavelength ($\lambda \mathrm{/}4$), or as a path difference $\mathrm{\Delta }nL$ in nanometers (158 nm for a quarter-wave plate at 633 nm). The path difference in nanometers is the natural unit for retarder films and for the stress birefringence of glass, which is quoted as retardance per unit length, in nm/cm; the same path difference is a different fraction of a wave at each wavelength.

Light polarized at an angle to the axes splits into a fast and a slow component, and the retardance sets their relative phase at the exit, which fixes the output polarization state:

Half-wave and quarter-wave plates in use

Rotating a half-wave plate by an angle $\theta$ rotates linear polarization by $2\theta$, which makes it the standard way to set the polarization of a laser beam. Placed before a polarizer or a polarizing beam splitter, it forms a variable attenuator or splitter whose transmission follows Malus's law in the rotated angle.

A quarter-wave plate at 45° after a linear polarizer makes circular polarization. The same pair blocks back-reflections: light reflected from a surface passes the quarter-wave plate twice, which acts as a half-wave plate at 45° and rotates the returning linear polarization by 90°, so the polarizer rejects it. This simple arrangement is often used instead of an optical isolator when the reflection keeps its polarization. A quarter-wave plate followed by a half-wave plate turns any fully polarized state into linear polarization at any chosen angle, and the sequence quarter, half, quarter can realize any retarder and so reach any output state.

In Jones notation, a retarder of retardance $\delta$ with its fast axis at angle $\theta$ is

$$W(\delta ,\theta )=R(-\theta )\left(\begin{array}{cc}1& 0\\ 0& e^{i\delta }\end{array}\right)R(\theta ),$$

where $R(\theta )$ is the rotation matrix; a quarter-wave plate has $\delta =\pi \mathrm{/}2$ and a half-wave plate $\delta =\pi$. The Jones calculus calculator shows what half-wave and quarter-wave plates at any angle do to a chosen input polarization.

Zero-order, multi-order and achromatic plates

The sensitivity of a multi-order plate follows from the formula for $\mathrm{\Gamma }$: a quartz plate with 10.25 waves of retardance at 633 nm, 0.72 mm thick, changes its retardance by about 5.8° for each nanometer of wavelength change, compared with 0.14° per nanometer for a true zero-order plate. Temperature changes scale the same way, by the order of the plate.

Common materials, with birefringence at 1550 nm:

Retarders in fiber and on chips

Fiber paddle polarization controllers use loops of single-mode fiber as retarders: bending the fiber induces stress birefringence that scales as the inverse square of the bend radius, so a fixed number of loops on a paddle gives a fixed retardance, and tilting the paddles turns their axes. Three paddles, usually a quarter-wave, a half-wave and a quarter-wave, convert any input polarization to any output. Electronic polarization controllers do the same with piezoelectric fiber squeezers or liquid-crystal cells, and polarization-maintaining fiber is a retarder whose retardance grows with length. On photonic chips, where waveguides guide TE and TM modes with different effective indices, polarization is usually converted by rotators built into the waveguides, and silica circuits sometimes insert a thin polyimide half-wave plate into a groove across the waveguides.

Common questions

What is the difference between a half-wave plate and a quarter-wave plate?

A half-wave plate delays one polarization by half a wavelength and rotates linear polarization, by twice the angle between the input and its fast axis. A quarter-wave plate delays it by a quarter wavelength and converts linear polarization at 45° into circular polarization, and back.

What is retardance?

The phase delay a retarder introduces between light polarized along its slow and fast axes, $\mathrm{\Gamma }=2\pi \mathrm{\Delta }nL\mathrm{/}\lambda$. It is quoted as a phase, a fraction of a wave, or a path difference $\mathrm{\Delta }nL$ in nanometers.

Can a waveplate be used at a different wavelength?

Only with a different retardance: a plate made for one wavelength has, to first order, a retardance scaled by ${\lambda }_{\text{design}}\mathrm{/}\lambda$ at another, less for a longer wavelength, so a quarter-wave plate for 633 nm is roughly a 0.19-wave plate at 830 nm. Multi-order plates drift much faster, and some wavelengths turn them into full-wave plates with no effect at all.

How do you find the fast axis of a waveplate?

Place the plate between crossed polarizers and rotate it: the transmission falls to zero when either of its axes lines up with a polarizer. To tell the fast axis from the slow one, compare it with a plate whose axes are known, or tilt the plate about one axis and watch the retardance change, which differs for the two axes.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017). M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999). R. Ulrich, S. C. Rashleigh and W. Eickhoff, "Bending-induced birefringence in single-mode fibers," Optics Letters 5, 273 (1980).