# Prism Polarizers

This is a continuation from the previous tutorial - ** reflecting and catadioptric afocal lenses**.

## Double Refraction in Calcite

Although many minerals, specifically those which do not have a cubic crystal structure, are doubly refracting, nearly all polarizing prisms used in the visible, near-ultraviolet, and near-infrared regions of the spectrum are made from optical calcite, which exhibits strong birefringence over a wide wavelength range. Polarizing prisms made from other birefringent crystals are used primarily in the ultraviolet and infrared at wavelengths for which calcite is opaque.

Next to quartz, calcite is the most widely distributed of all minerals and usually occurs in an impure polycrystalline form as marble, limestone, or chalk. Optical calcite, or Iceland spar, which is quite rare, originally came from a large deposit on the east coast of Iceland. This source is now exhausted, and optical calcite now comes principally from Mexico, Africa, and Siberia. It has been grown artificially by a hybrid gel-solution method, but maximum edge lengths are only 3 to 4 mm.

Although calcite is much softer than glass, with care it can be worked to an excellent polish. Surfaces flat to one-fifth fringe, or even, with care, one-tenth fringe, which are free from surface defects or perceptible turned edges can be produced using more or less conventional pitch-polishing techniques. Such techniques fail only for surfaces normal to the optic axis, in which case pitch polishing tends to cleave out small tetrahedra. Such surfaces can be polished to a lower surface quality using cloth polishers.

Crystals of calcite are negative uniaxial and display a prominent double refraction. The material can easily be cleaved along three distinct planes, making it possible to produce rhombs of the form shown in Fig. 1.

At points \(B\) and \(H\), a given face makes an angle of \(101°55'\) with each of the other two. At all the other points , two of the angles are \(78°5'\) and one is \(101°55'\).

The optic axis \(HI\), the direction in the crystal along which the two sets of refracted waves travel at the same velocity, makes equal angles with all three faces at point \(H\).

Any plane, such as \(DBFH\), which contains the optic axis and is perpendicular to the two opposite faces of the rhomb \(ABCD\) and \(EFGH\) is called a principal section.

A side view of the principal section \(DBFH\) is shown in Fig. 2.

If light is incident on the rhomb so that the plane of incidence coincides with a principal section, the light is broken up into two components polarized at right angles to each other.

One of these, the ordinary ray \(o\), obeys Snell’s law and has its plane of vibration (of the electric vector) perpendicular to the principal section.

The second, the extraordinary ray \(e\), has its plane of vibration parallel to the principal section.

The refraction of the extraordinary ray in some cases violates Snell’s law, at least in its simple form. The anomalous deflection of the ray is caused by the wavefront becoming ellipsoidal, so that the direction of propagation of the light is not along the wave normal.

This ellipticity causes the velocity of the light in the crystal, and hence its refractive index, to be a function of angle. If light is incident on rhomb face \(EFGH\) parallel to edge \(BF\) of the rhomb, the \(o\) and \(e\) rays, both of which lie in a principal section, are as shown in Fig. 2.

As the angle of incidence is changed in Fig. 2 so that the direction taken by the \(o\) ray approaches that of the optic axis \(HI\), the separation between the \(e\) and \(o\) rays decreases.

If the rhomb is rotated about an axis parallel to \(HD\), the \(e\) ray will precess about the \(o\) ray. However, unlike the \(o\) ray, it will not remain in the plane of incidence unless this plane coincides with the principal section.

The plane containing the \(o\) ray and the optic axis is defined as the principal plane of the \(o\) ray, and that containing the \(e\) ray and the optic axis as the principal plane of the \(e\) ray.

In the case discussed above, the two principal planes and the principal section coincide. In the general case, they may all be different. However, in all cases, the \(o\) ray is polarized with its plane of vibration perpendicular to its principal plane and the \(e\) ray with its plane of vibration in its principal plane (see Fig. 2). In all cases, the vibration direction of the \(e\) ray remains perpendicular to that of the \(o\) ray.

The value of the index of refraction of the \(e\) ray which differs most from that of the \(o\) ray, i. e., the index when the \(e\) ray vibrations are parallel to the optic axis, is called the principal index for the extraordinary ray \(n_e\). Snell’s law can be used to calculate the path of the \(e\) ray through a prism for this case. Snell’s law can always be used to calculate the direction of propagation of the ordinary ray.

Table 1 lists values of \(n_o\) and \(n_e\) for calcite, along with the two absorption coefficients \(a_o\) and \(a_e\), all as a function of wavelength. Since \(n_e\lt{n_0}\) in the ultraviolet, visible, and infrared regions, calcite is a negative uniaxial crystal. However, at wavelengths shorter than 1520 Å in the vacuum ultraviolet, the birefringence \(n_e-n_o\) becomes positive, in agreement with theoretical predictions.

For additional data in the 0.17 to 0.19 μm region, see Uzan et al. The range of transparency of calcite is approximately from 0.214 to 3.3 μm for the extraordinary ray but only from about 0.23 to 2.2 μm for the ordinary ray.

If the principal plane of the \(e\) ray and the principal section coincide (Fig. 2), the wave normal (but not the \(e\) ray) obeys Snell’s law, except that the index of refraction \(n_\phi\) of this wave is given by

\[\tag{1}\frac{1}{n_\phi^2}=\frac{\sin^2\phi}{n_e^2}+\frac{\cos^2\phi}{n_o^2}\]

where \(\phi\) is the angle between the direction of the wave normal and the optic axis (\(\phi\le90°\)) . When \(\phi=0°\), \(n_\phi=n_o\), and when \(\phi=90°\), \(n_\phi=n_e\). The angle of refraction for the wave normal is \(\phi-\beta\), where \(\beta\) is the angle the normal to the surface makes with the optic axis. Snell’s law for the extraordinary-ray wave normal then becomes

\[\tag{2}n\sin{i}=\frac{n_en_o\sin(\phi-\beta)}{(n_o^2\sin^2\phi+n_e^2\cos^2\phi)^{1/2}}\]

where \(i\) is the angle of incidence of light in a medium of refractive index \(n\). Since all other quantities in this equation are known, \(\phi\) is uniquely determined but often must be solved for by iteration. Once \(\phi\) is known, the angle of refraction \(r\) for the extraordinary ray can be determined as follows. If \(\alpha\) is the angle the ray makes with the optic axis (\(\phi\le90°\)), then \(r=\alpha-\beta\) and

\[\tag{3}\tan\alpha=\frac{n_o^2}{n_e^2}\tan\phi\]

Although the angle of refraction of the extraordinary ray determines the path of the light beam through the prism, one must use the angle of refraction of the wave normal, \(\phi-\beta\), in Fresnel’s equation when calculating the reflection loss of the \(e\) ray at the surface of the prism.

For the special case in which the optic axis is parallel to the surface as well as in the plane of incidence, \(\alpha\) and \(\phi\) are the complements of the angles of refraction of the ray and wave normal, respectively. If the light is normally incident on the surface, \(\phi\) and \(\alpha\) are both 90° and the extraordinary ray is undeviated and has its minimum refractive index \(n_e\). In other cases for which the optic axis is not parallel to the surface, the extraordinary ray is refracted even for normal incidence.

If the plane of incidence is neither in a principal section nor perpendicular to the optic axis, it is more difficult to determine the angle of refraction of the extraordinary ray. In such cases, Huygens’ construction is helpful.

## Types of Polarizing Prisms and Definitions

In order to make a polarizing prism out of calcite, some way must be found to separate the two polarized beams.

In wavelength regions where calcite is absorbing (and hence only a minimum thickness of calcite can be used), this separation has been made simply by using a very thin calcite wedge cut so that the optic axis is parallel to the faces of the wedge to enable the \(e\) and \(o\) rays to be separated by a maximum amount . The incident light beam is restricted to a narrow pencil. Calcite polarizers of this type can be used at wavelengths as short as 1900 Å.

In more favorable wavelength regions, where the amount of calcite through which the light passes is not so critical, more sophisticated designs are usually employed. Such prisms can be divided into two main categories, conventional polarizing prisms and polarizing beam-splitter prisms, and a third category, Feussner prisms.

In conventional polarizing prisms, only light polarized in one direction is transmitted. This is accomplished by cutting and cementing the two halves of the prism together in such a way that the other beam suffers total internal reflection at the cut. It is usually deflected to the side, where it is absorbed by a coating containing a material such as lampblack.

Since the ordinary ray, which has the higher index, is the one usually deflected, the lampblack is often mixed in a matching high-index binder such as resin of aloes (\(n_D=1.634\)) or balsam of Tolu (\(n_D=1.628\)) to minimize reflections. When high-powered lasers are used, the coating is omitted to avoid overheating the prism, and the light is absorbed externally.

Conventional polarizing prisms fall into two general categories: Glan types and Nicol types, which are illustrated in Fig. 3.

Glan types have the optic axis in the plane of the entrance face. If the principal section is parallel to the plane of the cut, the prism is a Glan-Thompson design (sometimes called a Glazebrook design); if perpendicular, a Lippich design; and if 45°, a Frank-Ritter design.

In Nicol-type prisms, which include the various Nicol designs and the Hartnack-Prazmowsky, the principal section is perpendicular to the entrance face, but the optic axis is neither parallel nor perpendicular to the face.

Air-spaced prisms can be used at shorter wavelengths than cemented prisms, and special names have been given to some of them.

An air-spaced Glan-Thompson prism is called a Glan-Foucault, and an air-spaced Lippich prism, a Glan-Taylor. In common practice, either of these may be called a Glan prism.

An air-spaced Nicol prism is called a Foucault prism. Double prisms can also be made, thus increasing the prism aperture without a corresponding increase in length. Most double prisms are referred to as double Frank-Ritter, etc., but a double Glan-Thompson is called an Ahrens prism.

In polarizing beam-splitter prisms, two beams, which are polarized at right angles to each other, emerge but are separated spatially.

The prisms have usually been used in applications for which both beams are needed, e. g., in interference experiments, but they can also be used when only one beam is desired.

These prisms are also of two general types, illustrated in Fig. 10; those having the optic axis in the two sections of the prism perpendicular and those having them parallel. Prisms of the first type include the Rochon, Senarmont, Wollaston, double Rochon, and double Senarmont.

Prisms of the second type are similar to the conventional polarizing prisms but usually have their shape modified so that the two beams emerge in special directions. Examples are the Foster, the beam-splitting Glan-Thompson, and the beam-splitting Ahrens.

The Feussner-type prisms, shown in Fig. 12, are made of isotropic material, and the film separating them is birefringent. For negative uniaxial materials the ordinary ray rather than the extraordinary ray is transmitted.

These prisms have the advantage that much less birefringent material is required than for the other types of polarizing prisms, but they have a more limited wavelength range when calcite or sodium nitrate is used because, for these materials, the extraordinary ray is transmitted over a wider wavelength range than the ordinary ray.

The amount of flux which can be transmitted through a prism or other optical element depends on both its angular aperture and its cross-sectional area. The greater the amount of flux which can be transmitted, the better the throughput or light-gathering power (sometimes called etendue or luminosity) of the system.

If a pupil or object is magnified, the convergence angle of the light beam is reduced in direct ratio to the increase in size of the image. The maximum throughput of a prism is thus proportional to the product of the prism’s solid angle of acceptance and its cross-sectional area perpendicular to the prism axis.

Hence, a large Glan-Taylor prism having an 8° field angle may, if suitable magnification is used, have a throughput comparable to a small Glan-Thompson prism with a 26° field angle.

In general, to maximize prism throughput in an optical system, both the angular aperture and clear aperture (diameter of the largest circle perpendicular to the prism axis which can be included by the prism) should be as large as possible.

The quantities normally specified for a prism are its clear aperture, field angle, and length-to-aperture (L/A) ratio. The semi-field angle is defined is the maximum angle to the prism axis at which a ray can strike the prism and still be completely polarized when the prism is rotated about its axis.

The field angle is properly twice the semi-field angle. (Some manufacturers quote a ‘‘field angle’’ for their polarizing prisms which is not symmetric about the prism axis and is thus in most cases unusable.)

The length-to-aperture (L/A) ratio is the ratio of the length of the prism base (parallel to the prism axis) to the minimum dimension of the prism measured perpendicular to the prism base. For a square-ended prism, the L/A ratio is thus the ratio of prism length to width.

In determining the maximum angular spread a light beam can have and still be passed by the prism, both the field angle and the L/A ratio must be considered, as illustrated in Fig. 4.

If the image of a point source were focused at the center of the prism, as in Fig. 4a, the limiting angular divergence of the beam would be determined by the field angle \(2i\) of the prism. However, if an extended source were focused there (Fig. 4b), the limiting angular divergence would be determined by the L/A ratio, not the field angle.

The field angle of a polarizing prism is strongly wavelength-dependent. For example, a Glan prism having an 8° field angle at 0.4 μm has only a 2° field angle at 2 μm. In designing optical systems in which polarizing prisms are to be used, the designer must allow for this variation in field angle. If he does not, serious systematic errors may occur in measurements made with the system.

The next tutorial introduces ** Glan-type prisms**.