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# Waveplates

Author: the photonics expert

Definition: transparent plates with a defined amount of birefringence, used for modifying the polarization of light

Alternative terms: retarder plates, retarders

Optical waveplates (also called wave plates or retarder plates) are transparent plates with a carefully chosen amount of birefringence. They are mostly used for manipulating the polarization state of light beams. A waveplate has a slow axis and a fast axis, both being perpendicular to the beam direction, and also to each other. The phase velocity of light is slightly higher for polarization along the fast axis. The designed value of optical retardance (difference in phase delay for the two polarization directions) is achieved only in a limited wavelength range (see below) and in a limited range of incidence angles.

## Common Types and Applications

The most common types of waveplates are quarter-wave plates (λ/4 plates) and half-wave plates (λ/2 plates), where the difference of phase delays between the two linear polarization directions is <$\pi /2$> or <$\pi$>, respectively, corresponding to propagation phase shifts over a distance of <$\lambda / 4$> or <$\lambda / 2$>, respectively.

Some important cases are:

• When a light beam is linearly polarized, and the polarization direction is along one of the axes of the waveplate, the polarization remains unchanged.
• When the incident polarization does not coincide with one of the axes, and the plate is a half-wave plate, then the polarization stays linear, but the polarization direction is rotated. For example, for an angle of 45° to the axes, the polarization direction is rotated by 90°.
• When the incident polarization is at an angle of 45° to the axes, a quarter-wave plate generates a state of circular polarization. (Other input polarizations lead to elliptical polarization states.) Conversely, circularly polarized light is converted into linearly polarized light.

Within a laser resonator, two quarter-wave plates around the gain medium are sometimes used for obtaining single-frequency operation (→ twisted-mode technique). Inserting a half-wave plate between a laser crystal and a resonator end mirror can help to reduce depolarization loss. The combination of a half-wave plate and a polarizer allows one to realize an output coupler with adjustable transmission.

Many waveplates are made of quartz (crystalline SiO2), as this optical material exhibits a wide wavelength range with very high transparency, and can be prepared with high optical quality. Other possible materials (to be used e.g. in other wavelength regions) are calcite (CaCO3), magnesium fluoride (MgF2), sapphire (Al2O3), mica (a silicate material), and some birefringent polymers. The choice of material can depend on many requirements, e.g. concerning wavelength range, open aperture, absorption losses and high-power capability.

## Zero-order and Multiple-order Plates

There are different kinds of waveplates:

• True zero-order waveplates are so thin that the relative optical phase delay between the two polarization directions is just <$\pi$>, for example, for a half-wave plate. The key advantage of true zero-order waveplates is that they can be used in a substantial wavelength range and range of incidence angles. In addition, their temperature sensitivity is weak.
• While zero-order waveplates offer best performance, the device thickness can be inconveniently small, particularly for strongly birefringent materials such as calcite, so that the optical fabrication becomes difficult and the handling delicate. The latter problem can be eliminated by cementing (or bonding) a zero-order plate to a thicker glass plate, which is not birefringent but provides mechanical stabilization. The resulting interface, however, leads to a lower damage threshold such as 10 MW/cm2.
• Multiple-order waveplates (= multi-order plates) are made so that the relative phase change is larger than the required value by an integer multiple of 2<$\pi$>. Although the performance at the design wavelength can be essentially the same, the optical bandwidth in which the plate has roughly the correct relative phase change is quite limited, e.g. to a few nanometers. Similarly, the allowed range of incidence angles can be quite small. Also, the retardance has a higher temperature sensitivity.
• Low-order waveplates are multiple-order plates with a relatively small order, keeping the mentioned detrimental effects low.
• Effective zero-order waveplates (or net zero-order waveplates) can be made from two multiple-order plates with slightly different thicknesses, which are cemented or optically contacted, or air-spaced for application with higher optical power levels. The slow axis of one plate is aligned with the fast axis of the other plate, so that the birefringence of the two plates is nearly canceled. The difference in thickness must be adjusted to obtain the required net phase change. Such devices can work in a broad wavelength range, since wavelength-induced changes of retardance in one plate are compensated by the other plate. However, the problem of a small usable angular range and temperature range remains.

It is also possible to make achromatic waveplates, combining materials with different chromatic dispersion (e.g. quartz and MgF2), which can have a nearly constant retardance over a very wide spectral range (hundreds of nanometers). Also, there are dual-wavelength waveplates, which have well-defined retardance values at some discrete very different wavelengths. Such features are sometimes required in the context of nonlinear frequency conversion, such as frequency tripling.

## Various Issues

In addition to the fundamental optical performance, various other issues can be relevant:

• The materials should have a high transparency in the wavelength region of interest.
• The sensitivity of the phase retardance to temperature and angle depends on the design.
• High-quality waveplates exhibit a uniform retardance over the full open aperture, or a large portion of it. Where that is not the case, depolarization (more precisely, spatially dependent polarization states) can result.
• Surface reflections can be disturbing, and may be reduced with anti-reflection coatings.
• The optical damage threshold can be important for applications in pulsed laser systems.
• Some waveplates are provided with a housing which makes it easy to rotate the plate by a well-defined angle, and/or to see the orientation of the fast and slow axis.

For some applications, it can be difficult to obtain a waveplate fulfilling all the requirements. Therefore, one sometimes has to use alternative methods, for example for turning the polarization direction of a laser beam. For example, a 90° rotation of polarization (and also the beam profile) can be obtained with a combination of three 45° mirrors, subsequently deflecting the beam to the right, then upwards and finally into the original direction.

## Depolarization Compensators

There are waveplates with spatially varying polarization properties, which are used within a laser resonator to reduce depolarization loss [9].

## Diffractive Waveplates

Waveplates can also be realized based on an entirely different principle of operation: with photonic metasurfaces containing sub-wavelength gratings. These devices are also called diffractive waveplates [6]. They can have remarkable properties, e.g. very wideband operation.

There are optical devices which are somewhat related to waveplates:

• Fresnel rhomb retarders and other types of prism retarders have the same basic function as waveplates, but exploit polarization-dependent phase changes during total internal reflection. This principle allows broadband (achromatic) operation.
• A Babinet–Soleil compensator, containing a pair of birefringent wedges, can be used as a waveplate with a variable degree of retardation. Other versions of tunable waveplates such as the Berek compensator are based on variable tilting of birefringent plates.
• There are various types of fiber polarization controllers with similar functions.
• Electro-optic crystals and liquid crystal devices can be used to make electrically controllable waveplates, also called active retarders.

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### Bibliography

 [1] A. M. Title, “Improvement of birefringent filters. 2: Achromatic waveplates”, Appl. Opt. 14 (1), 229 (1975); https://doi.org/10.1364/AO.14.000229 [2] P. D. Hale and G. W. Day, “Stability of birefringent linear retarders (waveplates)”, Appl. Opt. 27 (24), 5146 (1988); https://doi.org/10.1364/AO.27.005146 [3] S. Nersisyan et al., “Fabrication of liquid crystal polymer axial waveplates for UV-IR wavelengths”, Opt. Express 17 (14), 11926 (2009); https://doi.org/10.1364/OE.17.011926 [4] Z. Qi et al., “Achromatic waveplates for liquid crystal displays”, J. Display Technol. 9 (7), 586 (2013) [5] S. V. Serak et al., “Diffractive waveplate arrays”, J. Opt. Soc. Am. B 34 (5), B56 (2017); https://doi.org/10.1364/JOSAB.34.000B56 [6] N. Tabiryan, G. Cipparronne and T. J. Bunning, “Diffractive waveplates: introduction”, J. Opt. Soc. Am. B 36 (5), DW1 (2019); https://doi.org/10.1364/JOSAB.36.000DW1 [7] D. Rohrbach, B. J. Kang and T. Feurer, “3D-printed THz wave- and phaseplates”, Opt. Express 29 (17), 27160 (2021); https://doi.org/10.1364/OE.433881 [8] X. Chen et al., “Solution-processed inorganic perovskite crystals as achromatic quarter-wave plates”, Nature Photonics 15, 813 (2021); https://doi.org/10.1038/s41566-021-00865-0 [9] L. Veselis et al., “Depolarization compensation with a spatially variable wave plate in a 116 W, 441 fs, 1 MHz Yb:YAG double-pass laser amplifier”, Appl. Opt. 60 (24), 7164 (2021); https://doi.org/10.1364/AO.432573