Retardance

Retardance is the amount by which a birefringent element delays one polarization component relative to the orthogonal one. Light polarized along the slow axis travels with index $n_s$ and light along the fast axis with $n_f$; after a thickness $d$ the optical path difference is $\Gamma = \Delta n\, d$ with $\Delta n = n_s - n_f$, and the phase difference is

$$\delta = \frac{2\pi\,\Delta n\, d}{\lambda}.$$

Retardance is quoted as the path difference $\Gamma$ in nanometres, as a fraction of a wave $\Gamma/\lambda$, or as the phase $\delta$ in degrees or radians. A quarter-wave plate for 589 nm has $\Gamma = 147$ nm, $\Gamma/\lambda = 0.25$ and $\delta = 90°$. At 633 nm, 1 nm of retardance corresponds to 0.57° of phase. The property arises from birefringence, whether intrinsic to a crystal, induced by stress, or produced by the geometry of a waveguide, and it is the quantity that specifies waveplates, liquid-crystal cells, and the residual stress in optical glass. The older term retardation is used with the same meaning.

Worked values for quartz

Crystalline quartz at 589 nm has $n_o = 1.5443$ and $n_e = 1.5534$, so $\Delta n = 0.0091$. A zero-order quarter-wave plate needs

$$d = \frac{\lambda}{4\,\Delta n} = \frac{589\ \text{nm}}{4 \times 0.0091} = 16.2\ \mu\text{m},$$

too thin to handle as a free plate, which is why zero-order quartz plates are made as two cemented or air-spaced plates with crossed axes whose thicknesses differ by this amount. A single 1 mm quartz plate has $\Gamma = 9.1$ µm, or 15.45 waves at 589 nm. Multi-order quarter-wave plates are ground to a thickness of this order that gives an integer plus 0.25 waves; they act correctly at the design wavelength, but their retardance changes rapidly with wavelength, temperature and angle of incidence.

With $\Gamma$ fixed and dispersion of $\Delta n$ neglected, the retardance in waves scales as $1/\lambda$. A plate that is a quarter wave at 633 nm gives 0.297 waves at 532 nm, which leaves visibly elliptical light where circular was intended. Achromatic retarders combine two birefringent materials, or use total internal reflection in a Fresnel rhomb, to reduce this dependence.

Measurement

The simplest measurement places the sample between crossed polarizers with its axes at 45° to them. The transmitted fraction is

$$T = \sin^2\!\left(\frac{\delta}{2}\right) = \sin^2\!\left(\frac{\pi\,\Gamma}{\lambda}\right).$$

A sample with 100 nm retardance at 550 nm transmits 29.2 %. The measurement determines $\delta$ only modulo $2\pi$ and does not give its sign, so the order and the fast-axis direction need separate information. Compensators resolve this: a Babinet-Soleil compensator adds a calibrated variable retardance until the transmission returns to zero, and the Sénarmont method, with a quarter-wave plate and a rotating analyzer, converts small retardances into an analyzer rotation angle. Automated instruments use a rotating-waveplate or photoelastic-modulator polarimeter to measure the output Stokes parameters or the full Mueller matrix, which separates retardance from dichroism and depolarization.

Viewed in white light between crossed polarizers, a slowly varying retardance produces interference colours, charted on the Michel-Lévy chart. A full-wave plate of about 550 nm retardance, the first-order red or tint plate, turns small retardances into strong colour shifts and is standard in polarized-light microscopy of minerals, crystals and biological fibres.

Stress birefringence

Mechanical stress makes isotropic glass birefringent, with $\Delta n = K\sigma$, where $K$ is the stress-optic coefficient. For N-BK7, $K$ is about $2.77 \times 10^{-12}$ Pa⁻¹, so a uniaxial stress of 1 MPa over a 10 mm path produces about 27.7 nm of retardance. Glass suppliers specify residual stress birefringence in nm/cm for this reason, and lens mounts, adhesives and thermal gradients can add comparable amounts. In polarization-sensitive systems, such as interferometers, polarimeters and lithography optics, stray retardance of a few nanometres converts linear polarization to elliptical and lowers extinction.

In fibers and waveguides

In a polarization-maintaining fiber the retardance grows linearly with length, and the length over which it reaches one wave is the beat length, $L_B = \lambda/\Delta n$. In ordinary single-mode fiber, bends and twists create small, randomly varying retardance that scrambles the output polarization; fiber polarization controllers exploit bend-induced retardance deliberately, coiling fiber into loops that act as quarter- and half-wave plates.

Pitfalls

Retardance quoted in nanometres is independent of wavelength only if $\Delta n$ is, which is approximately true over narrow bands. Tilting a retarder changes both the path length and the effective $\Delta n$, and multi-order plates are especially sensitive. Oblique reflection from metal mirrors and beamsplitter coatings also introduces retardance between s and p polarizations, often several degrees or more, which is easily overlooked when a polarization state is prepared upstream of a folding mirror. In the Jones calculus a linear retarder is diagonal in its own axes, with elements differing by the phase factor $e^{i\delta}$. In practice, errors in axis orientation often exceed errors in $\delta$.

Common questions

What is the difference between retardance and birefringence?

Birefringence $\Delta n$ is a material property. Retardance is the accumulated effect in a particular element, $\Delta n$ multiplied by thickness, and has units of length or phase.

How is retardance converted between nanometres and degrees?

Multiply the path difference by $360°/\lambda$: 147 nm at 589 nm is 90°.

What is a zero-order waveplate?

A retarder whose total retardance equals the nominal fraction of a wave, such as 0.25 waves, with no additional integer number of waves. It changes less with wavelength, temperature and angle than a multi-order plate.

The Jones calculus calculator shows the output polarization of a retarder of any retardance and axis angle.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 8; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 15; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 6; R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).