Polarizing Beam-Splitter Prisms
This is a continuation from the previous tutorial - Nicol-type prisms.
The three classic polarizing beam-splitter prisms are the Rochon, Sénarmont, and Wollaston, shown in perspective in Fig. 10a to c and in side view in Fig. 11a to c.
In addition, any polarizing prism can be used as a polarizing beam splitter by changing the shape of one side and removing the absorbing coating from its surface.
Two examples of such prisms are the Foster prism, in which the ordinary and extraordinary rays emerge at right angles to each other, and the beam-splitting Glan-Thompson prism, in which the ordinary ray emerges normal to one side (Figs. 10d and e and 11d and e).
Another prism of this type, the beam-splitting Ahrens prism, is a double beam-splitting Glan-Thompson prism.
In polarizing prisms, the optic axes are always parallel to each other in the two halves of the prism. By contrast, the optic axes in the two halves of the Rochon, Sénarmont, and Wollaston polarizing beam-splitter prisms are at right angles to each other.
Crystal quartz is often used to make these beam splitters, and such prisms can be used down to the vacuum ultraviolet. In applications not requiring such short wavelengths, calcite is preferable because it gives a greater angular separation of the beams (typically \(10^\circ\) as compared to \(0.5^\circ\) for quartz) and does not produce optical rotation.
Rochon Prism
The Rochon prism, invented in 1783, is the most common type of polarizing beam splitter. It is often used in photometric applications in which both beams are utilized. It is also used as a polarizing prism in the ultraviolet, in which case one of the beams must be eliminated, e. g., by imaging the source beyond the prism and blocking off the deviated image.
The paths of the two beams through the prism are shown in Fig. 11a. A ray normally incident on the entrance face travels along the optic axis in the first half of the prism, so that both ordinary and extraordinary rays are undeviated and have the same refractive index \(n_0\).
The second half of the prism has its optic axis at right angles to that in the first half, but the ordinary ray is undeviated since its refractive index is the same in both halves.
The extraordinary ray, however, has its minimum index in the second half, so that it is refracted at the cut according to Snell’s law. Since the deviation angle depends on the ratio \(n_e/n_o\), it is a function of wavelength.
If the angle of the cut is \(S\), to a good approximation the beam deviation \(\delta\) of the extraordinary ray depends on the cut angle in the following manner, according to Steinmetz et al.,
\[\tag{6}\tan{S}=\frac{n_o-n_e}{\sin\delta}+\frac{\sin\delta}{2n_e}\]
This relation holds for light normally incident on the prism face. The semifield angle \(i_\text{max}\) is given by
\[\tag{7}\tan{i_\text{max}}=\frac{1}{2}(n_e-n_o)\cot{S}\]
If the prism is to be used as a polarizer, the light should be incident as shown. Rochon prisms also act as polarizing beam splitters when used backward, but the deviation of the two beams is then slightly less.
When a Rochon prism is used backward, both the dispersion and the optical activity (for quartz) will adversely affect the polarization. Thus, one generally uses a Rochon in the normal manner.
However, an exception occurs when a quartz Rochon is to be used as an analyzer. In this case it is best to reverse the prism and use a detector that is insensitive to polarization to monitor the relative intensities of the two transmitted beams.
A Rochon prism is achromatic for the ordinary ray but chromatic for the extraordinary ray. Since total internal reflection does not occur for either beam, the type of cement used between the two halves of the prism is less critical than that used for conventional polarizing prisms.
Canada balsam is generally used, although the two halves are sometimes optically contacted for high-power laser applications or for use in the ultraviolet at wavelengths shorter than 3500 Å.
Optically contacted crystalline-quartz Rochon prisms can be used to wavelengths as short as 1700 Å, and a double Rochon of MgF2 has been used to 1300 Å in the vacuum ultraviolet.
Optically contacted single Rochon prisms of MgF2 have also been constructed, and the transmission of one has been measured from 1400 Å to 7 μm. Ultraviolet-transmitting cements such as gedamine can be used to extend the short-wavelength limit of calcite prisms to about 2500 Å.
Defects
Quartz and calcite Rochon prisms suffer from several defects. Quartz exhibits optical activity when light is transmitted through it parallel to the optic axis, and
although two mutually perpendicular, polarized beams will emerge from a quartz Rochon prism used in the conventional direction, their spectral composition will not faithfully reproduce the spectral compositions of the horizontal and vertical components of the input.
If such a prism is used backward, different wavelengths emerge from the prism vibrating in different planes. Hence the output consists of many different polarizations instead of the desired two.
Calcite Rochon prisms do not exhibit optical activity but are difficult to make, since when calcite surfaces are cut normal to the optic axis, small tetrahedra tend to cleave out from the surface during pitch polishing.
These tetrahedra may also cleave out during attempts to clean the prisms, and occasionally glass plates are cemented to such surfaces to prevent damage.
Some image distortion will occur in calcite prisms; if nonnormally incident rays pass through the prism, both beams will be distorted along their directions of vibration; i.e., the undeviated beam (\(o\) ray), which vibrates in a vertical plane, will be distorted vertically, and the deviated beam (\(e\) ray), which vibrates in a horizontal plane, will be distorted horizontally.
Glass-Calcite Rochons
Some of the difficulties can be minimized or eliminated by making the entrance half of the Rochon prism out of glass of matching index instead of quartz or calcite.
Both \(o\) and \(e\) rays travel along the same path and have the same reflective index in this half of the prism, so that the birefringent qualities of the quartz or calcite are not being used and an isotropic medium would serve just as well.
By properly choosing the index of the glass, either the ordinary or the extraordinary ray can be deviated, and glasses are available for matching either index of calcite reasonably well over much of the visible region.
The extraordinary ray always suffers some distortion in its direction of vibration, but the distortion of the ordinary ray can be eliminated in the glass-calcite construction.
By properly choosing the refractive index of the glass, we can determine whether the \(e\) ray will be the deviated or the undeviated beam. (Some distortion also arises for deviated beams in the direction of the deviation because of Snell’s law and cannot be corrected in this way.)
Another method of obtaining an undeviated beam was used by Hardy; unable to find a glass with refractive index and dispersion matching those of calcite, he selected a glass with the correct dispersive power and then compensated for the difference in refractive index by putting a slight wedge angle on the calcite surface.
Now a wider selection of glasses is available, but glass-calcite prisms cannot be made strictly achromatic over an extended wavelength range, and thermally induced strains caused by the difference in expansion coefficients in the two parts of the prism may be expected unless the cement yields readily.
Total Internal Reflection in Rochons
When normal Rochon prisms are used as polarizers, one of the beams must be screened off and eliminated. This restriction might be removed by making the cut between halves of the prism at a sufficiently small angle for the extraordinary ray to be totally reflected. Calculations indicate that this approach should be feasible, but it has apparently not been followed.
Sénarmont Prism
The Sénarmont polarizing beam splitter, shown in Figs. 10b and 11b, is similar to the Rochon prism except that the optic axis in the exit half of the prism is coplanar with the optic axis in the entrance half, i.e., at right angles to the Rochon configuration.
As a result, light whose plane of vibration is initially vertical is deviated in the Sénarmont prism, while in the Rochon prism the deviated beam has its plane of vibration horizontal (assuming no optical activity in either case) (compare Fig. 11a and b).
The amount of the deviation in the Sénarmont prism is slightly less than in the Rochon because the extraordinary ray does not have its minimum refractive index.
An alternate form of Sénarmont prism, the right-angle Sénarmont or Cotton polarizer, consists of only the first half of the Sénarmont prism.
Unpolarized light normally incident on the prism face is totally internally reflected at the hypotenuse and is then resolved into two planes of vibration, one parallel to the optic axis and the other perpendicular to it. Double refraction will then occur just as in a normal Sénarmont prism. Such a prism has a transmission equivalent to that of an optically contacted Sénarmont or Rochon but is much less expensive.
Wollaston Prism
The Wollaston prism (Figs. 10c and 11c ) is a polarizing beam splitter, also used as a polarizing prism in the vacuum ultraviolet, that deviates both transmitted beams.
The deviations, indicated in Fig. 11c, are nearly symmetrical about the incident direction, so that the Wollaston has about twice the angular separation of a Rochon or Sénarmont prism.
A normally incident beam is undeviated upon entering the prism, but the \(o\) ray, vibrating perpendicular to the optic axis, has a refractive index \(n_o\) while the \(e\) ray, vibration parallel to the optic axis has its minimum (or principal) index \(n_e\).
At the interface the \(e\) ray becomes the \(o\) ray and vice versa because the direction of the optic axis in the second half is perpendicular to its direction in the first half.
Thus the original \(o\) ray enters a medium of lower refractive index and is refracted away from the normal at the cut, while the original \(e\) ray passes into a medium of higher refractive index and is refracted toward the normal.
On leaving the second half of the prism, both rays are refracted away from the normal, so that their divergence increases. The deviation of each beam is chromatic in Wollaston prisms, which are most commonly used to determine the relative intensities of two plane-polarized components.
Since the light never travels along the optic axis, optical activity does not occur and the relative intensities of the two beams are always proportional to the intensities of the horizontal and vertical polarization components in the incident beam.
For an L/A ratio of 1.0, the angular separation between beams is about \(1^\circ\) for a crystalline-quartz Wollaston prism; it can be as high as \(3^\circ30'\) for an L/A ratio of 4.0.
With a calcite prism, the beams would have an angular separation of about \(19^\circ\) for an L/A ratio of 1.0, but severe image distortion and lateral chromatism results when such large angular separations are used.
These effects can be minimized or the angular separation can be increased for a given L/A ratio by using a three-element Wollaston prism, a modification, apparently suggested by Karl Lambrecht. Divergences as large as \(30^\circ\) can be obtained.
The ellipticity in the emergent polarized beams has been measured by King and Talim. For calcite Wollaston prisms, the ellipticities were in the \(0.004\) to \(0.025^\circ\) range, comparable to those of Glan-Thompson prisms.
Larger values, between \(0.12\) and \(0.16^\circ\), were measured for crystalline-quartz Wollaston prisms. The major contribution, which was from the combined optical activity and birefringence in the quartz rather than from defects within the crystal, cannot be avoided in quartz polarizers.
Foster Prism
This prism, shown in a three-dimensional view in Fig. 10d and in cross section in Fig. 11d, can be used to form two plane-polarized beams separated by \(90^\circ\) from each other.
Its construction is similar to that of a Glan-Thompson prism except that one side is cut at an angle and silvered to reflect the ordinary ray out the other side.
The Foster prism is often used backward as a polarizing microscope illuminator for observing reflecting specimens. For this application, the light source is at \(e\) in Fig. 11d, and unpolarized light enters the right-hand face of the prism.
The ordinary ray (not shown) is reflected at the cut and absorbed in the blackened side of the prism, while the extraordinary ray is transmitted undeviated out the left face of the prism. It then passes through the microscope objective and is reflected by the specimen, returning on its same path to the prism.
Light that is unchanged in polarization will be transmitted undeviated by the prism along the path to the light source. If, however, the plane of vibration has been rotated so that it is at right angles to the optic axis (in the plane of the figure), the light will be reflected into the eyepiece.
The prism thus acts like a crossed polarizer-analyzer combination. If a correctly oriented quarter-wave plate is inserted in the beam between the prism and the microscope objective, the light striking the sample will be circularly polarized, and, after being reflected back through the quarter-wave plate, it will be linearly polarized again but with the plane of vibration rotated by \(90^\circ\).
This light is vibrating perpendicular to the optic axis and will be reflected into the eyepiece, giving bright-field illumination. Foster prisms used in this manner introduce no astigmatism since the light forming the image enters and leaves the prism normal to the prism faces and is reflected only by plane surfaces.
Beam-splitting Glan-Thompson Prism
If a prism design similar to the Foster is used but the side of the prism is cut at an angle so that the ordinary ray, which is deflected, passes out normal to the surface of the prism rather than being reflected, the prism is called a beam-splitting Glan-Thompson prism (Figs. 10e and 11e).
Since no refraction occurs for either beam, the prism is achromatic and nearly free from distortion.
The angle between the two emerging beams is determined by the angle of the cut between the two halves of the prism and hence depends on the L/A ratio of the prism.
For an L/A ratio of 2.414, the angle is \(45^\circ\). The field angle around each beam is calculated for different L/A ratios just as for a conventional Glan-Thompson prism.
By making the prism double, i.e., a beam-splitting Ahrens prism, the incident beam can be divided into three parts, one deflected to the left, one to the right, and one undeviated.
Feussner Prisms
The polarizing prisms discussed so far require large pieces of birefringent material, and the extraordinary ray is the one usually transmitted.
Feussner suggested an alternate prism design in which only thin plates of birefringent material are required and the ordinary ray rather than the extraordinary ray is transmitted for negative uniaxial materials. A similar suggestion was apparently made by Sang in 1837, although he did not publish it until 1891.
In essence, Feussner’s idea was to make the prisms isotropic and the film separating them birefringent, as shown in Fig. 12.
The isotropic prisms should have the same refractive index as the higher index of the birefringent material so that for negative uniaxial materials, e.g., calcite or sodium nitrate, the ordinary ray is transmitted and the extraordinary ray totally internally reflected.
Advantages of this design are
- Since the ordinary ray is transmitted, the refractive index does not vary with angle of incidence and hence the image is anastigmatic.
- Large field angles or prisms of compact size can be obtained.
- The birefringent material is used economically.
Furthermore, because the path length of the ray through the birefringent material is short, a lower-quality material can be used.
Disadvantages are
- For both calcite and sodium nitrate, the extraordinary ray is transmitted over a larger wavelength range than the ordinary ray so that Feussner prisms do not transmit over as as large a wavelength range as conventional prisms.
- The thermal-expansion coefficients of the isotropic and birefringent materials are different, making thermally induced strains likely.
Solutions to the second problem are to use a thixotropic cement, which flows more readily with increasing stress, or to enclose the system in a metal sleeve and use oil instead of cement.
If the ordinary index is matched by the oil, the birefringent material does not even need to be polished very well. Even a cleavage section of calcite can be used, with only a tolerable loss in angular field.
Feussner suggested orienting the optic axis of the birefringent slab perpendicular to the cut, as indicated in Fig. 12a. Since the thermal expansion of the slab is the same in all directions perpendicular to the optic axis, thermally induced strains are minimized in this way.
Shortly after Feussner’s article was published, Bertrand pointed out that the optic axis of the birefringent slab should be parallel to the entrance face of the prism to give the maximum difference between the refractive indices of the ordinary and extraordinary rays. A prism made in this way, sometimes called a Bertrand-type Feussner prism, is shown in Fig. 12b.
Since sodium nitrate is easily obtainable and has a birefringence even larger than that of calcite, attempts have been made to produce polarizing prisms of this material by Wulff, Stober, Tzekhovitzer, West, Huot de Longchamp, and Yamaguti.
However, it is not only deliquescent but also very soft, so that although large single crystals can be obtained, they are difficult to work.
They can be crystallized in the desired orientation from a melt using a technique discovered by West. When sodium nitrate crystallizes from a melt on a mica cleavage surface, one of its basal planes is oriented parallel to the mica cleavage and hence its optic axis is perpendicular to the mica surface. West reports growing single crystals as large as 38x19x2 cm using this technique.
Yamaguti has produced polarizing prisms of sodium nitrate by placing thin, closely spaced glass plates on edge on a mica sheet and then immersing the assembly in a melt of sodium nitrate.
The thin single crystal thus formed was annealed and cemented between glass prisms to form a Bertrand-type Feussner prism. Conceivably, the sodium nitrate could have been grown directly between the glass prisms themselves, but when such thick pieces of glass are used, it is difficult to avoid setting up strains in the crystal and consequently reducing the polarization ratio.
Yamaguti used SK5 glass prisms (\(n_D=1.5889\)) cut at an angle of \(23^\circ\) to form his polarizing prism and reports a field of view of \(31^\circ\), symmetric about the normal to the entrance face.
Another possible birefringent material suitable for a Feussner prism is muscovite mica, and such prisms have actually been constructed and tested. A \(6^\circ\) field angle can be obtained, which is adequate for many optical systems illuminated by lasers.
Noncalcite Polarizing Prisms
Polarizing prisms made of materials other than calcite have been used primarily in the ultraviolet region at wavelengths for which calcite is opaque.
Prism materials used successfully in this region include crystalline quartz, magnesium fluoride, sodium nitrate, and ammonium dihydrogen phosphate. Rutile polarizing prisms have been used beyond the calcite cutoff in the infrared.
A new prism material, yttrium orthovanadate, has been used to make high-transmission polarizers for the visible and near-infrared spectral regions.
Rochon or Wollaston prisms are sometimes made of crystalline quartz for use in the far ultraviolet. The short-wavelength cutoff of the quartz is variable, depending on the impurities present, but can be as low as 1600 Å.
By utilizing magnesium fluoride instead of quartz for the polarizing prisms, the short-wavelength limit can be extended to 1300 Å. Magnesium fluoride transmits to about 1125 Å, but below 1300 Å its birefringence decreases rapidly and changes sign at 1194 Å.
Although it is the most birefringent material available in this region, MgF2 has a much smaller birefringence than that of calcite; hence, a small cut angle and large L/A ratio for the prism are unavoidable. Since absorption does occur, it is desirable to minimize the length of the prism.
Johnson solved this problem by constructing a MgF2 Wollaston prism which requires only half the path length necessary for a Rochon prism. However, both beams are deviated, creating instrumental difficulties.
Steinmetz et al. constructed a double Rochon prism of MgF2 which has the same L/A ratio as the Wollaston prism but does not deviate the desired beam.
Problems with the prism included fluorescence, scattered light, and nonparallelism of the optic axes.
In principle, however, a MgF2 double Rochon polarizing prism should be an efficient, high-extinction-ratio, on-axis polarizer for the 1300- to 3000- Å wavelength range and should also be useful at longer wavelengths. Morris and Abramson reported on the characteristics of optically contacted MgF2 single Rochon prisms.
A different type of polarizer suggested by Chandrasekharan and Damany to take the place of a Rochon or Wollaston prism in the vacuum ultraviolet consisted of a combination of two MgF2 lenses, one planoconcave and the other planoconvex of the same radius of curvature, combined so that their optic axes were crossed.
The combination acted as a convergent lens for one polarization and as a divergent lens for the other. It had the advantage that the polarized beam remained on axis and was focused. A measured degree of polarization of 98.5 percent was obtained at 1608 Å, in good agreement with the calculated value.
Prism polarizers can also be constructed for use in the infrared at wavelengths longer than those transmitted by calcite. Rutile, TiO2, a positive uniaxial mineral with a large birefringence and good transmittance to 5 μm in the infrared, has been used by Landais to make a Glan-Foucault-type crystal polarizer.
Since rutile has a positive birefringence (in contrast to the negative birefringence of calcite), the ordinary ray is transmitted undeviated and the extraordinary ray is reflected out one side.
The next tutorial discusses in detail about specialty single-mode fibers.