Waveplate (retarder)

A waveplate is a slab of birefringent material whose fast and slow axes are oriented perpendicular to the propagation direction. Light polarized along each axis experiences a different phase delay; the relative phase shift after passage through the waveplate is:

$$\Delta\phi \;=\; \frac{2\pi}{\lambda_0} \, \Delta n \, L,$$

where $\Delta n$ is the birefringence and $L$ is the plate thickness.

Standard waveplate types:

Plate	Phase shift	Effect on linear input
Quarter-wave plate (λ/4)	$\Delta\phi = \pi/2$	Linear at 45° to fast axis → circular polarization
Half-wave plate (λ/2)	$\Delta\phi = \pi$	Rotates linear polarization by twice the angle between input and fast axis
Full-wave plate (λ)	$\Delta\phi = 2\pi$	No effect (returns to input state at design wavelength)

The required physical thickness:

$$L \;=\; \frac{\Delta\phi \cdot \lambda_0}{2\pi \, \Delta n}.$$

For quartz at 1550 nm ($\Delta n = 0.009$): $L_{\lambda/2} = 86$ μm. Such a thin "zero-order" plate is mechanically fragile, so commercial waveplates often use either:

• Stacked multi-order plates — two thick plates of crystal axes crossed by 90°, with thicknesses differing by the design retardance. Compensates plate-to-plate thickness errors and is mechanically robust.
• Polymer waveplates — birefringent polymer films laminated between glass substrates. Lower performance but inexpensive and broadband.
• Achromatic compound waveplates — combinations of crystals (quartz + MgF$_2$) chosen to maintain target retardance over a wide wavelength range.

Common materials and applications:

Material	$\Delta n$ at 1550 nm	Use case
Quartz	0.009	Standard zero-order waveplates
MgF$_2$	0.012	Broader wavelength range than quartz
YVO$_4$	0.204	Compact (thinner) for given retardance
Calcite (CaCO$_3$)	$-0.158$	Polarizing prism elements (not typical waveplates)
Liquid crystal	tunable	Variable-retardance electronically-controlled

Mueller-Stokes / Jones matrix representation. A waveplate at retardance $\delta$ with fast axis at angle $\theta$ has Jones matrix:

$$W(\delta, \theta) \;=\; R(-\theta) \begin{pmatrix} 1 & 0 \\ 0 & e^{i\delta} \end{pmatrix} R(\theta),$$

where $R(\theta)$ is the rotation matrix. Combinations of half-wave and quarter-wave plates implement arbitrary polarization transformations — three waveplates suffice to convert any input polarization to any output polarization (Mueller decomposition).

Wavelength dependence. A zero-order waveplate's retardance is exact only at one wavelength. Off-design wavelengths produce different retardance:

$$\Delta\phi(\lambda) \;=\; \Delta\phi_\text{design} \cdot \frac{\lambda_\text{design}}{\lambda}.$$

The deviation produces polarization errors that grow with wavelength offset. Achromatic waveplates mitigate this by combining materials with different dispersion ratios.

Polarization control in fiber. Fiber-based "paddle" polarization controllers use loops of single-mode fiber as variable-retardance waveplates — bending the fiber around a paddle induces stress birefringence proportional to bend radius, and adjusting paddle angles controls polarization. Two or three paddles in series provide arbitrary polarization transformation. Electronic polarization controllers do the same with active piezo elements or Liquid-crystal cells.